Numerical simulation of an adobe wall under in-plane loading
Tarque Nicola∗1,
Camata Guido2a, Varum Humberto3b, Spacone Enrico2c and Blondet Marcial1d
1Department of Engineering, Division of Civil Engineering, Pontificia Universidad Católica del Perú, Av.
Universitaria 1801, Lima 32, Peru
2Department of Engineering and Geology, University ‘G. D’Annunzio’ Chieti-Pescara. Viale Pindaro 42, 65127
Pescara, Italy.
3Department of Civil Engineering, University of Aveiro. 3810-193, Aveiro, Portugal
Abstract. Adobe is one of the oldest construction materials that is still used in many seismic countries, and different
construction techniques are found around the world. The adobe material is characterized as a brittle material; it has
acceptable compression strength but it has poor performance under tensile and shear loading conditions. Numerical
modelling is an alternative approach for studying the nonlinear behaviour of masonry structures such as adobe. The
lack of a comprehensive experimental database on the adobe material properties motivated the study developed here.
A set of a reference material parameters for the adobe were obtained from a calibration of numerical models based on
a quasi-static cyclic in-plane test on full-scale adobe wall representative of the typical Peruvian adobe constructions.
The numerical modelling, within the micro and macro modelling approach, lead to a good prediction of the in-plane
seismic capacity and of the damage evolution in the adobe wall considered.
Keywords: adobe masonry; material properties; in-plane behaviour; seismic capacity; numerical modelling
1. Introduction
In many developing countries earthen dwellings are traditional residential solutions because soil
is abundant, readily available and free. Unfortunately these countries are also regions of high
seismicity (Fig. 1). In the majority of the cases, these adobe buildings are built by the owners during
work campaigns in their neighbourhood without taking into account specific seismic reinforcements.
Adobe dwellings are built with sun dried mud blocks and mud mortar. To form the wall, the adobe
bricks run horizontally with the greater dimension parallel to the wall surface (stretcher way) or with
the greater dimension perpendicular to the wall surface (header way) with mud mortar between them,
∗Corresponding author, Associate Professor, sntarque@pucp.edu.pe
a Assistant Professor, g.camata@unich.it
b Associate Professor, hvarum@ua.pt
c Professor, espacone@unich.it
d Professor, mblondet@pucp.pe
mailto:sntarque@pucp.edu.pe
mailto:g.camata@unich.it
mailto:hvarum@ua.pt
mailto:espacone@unich.it
mailto:mblondet@pucp.pe
forming the bed and head joints. The adobe dwellings have excellent acoustic and thermal
characteristics. Due the adobe thermal mass, these buildings are warm during winter and are fresh
and cool during summer. However, their seismic performance is extremely poor due to a perverse
combination of its mechanical properties: relatively high density, extremely low tensile strength, and
brittle failure mode. Every time a strong earthquake occurs, there is widespread damage, economic
losses and casualties due to collapse of earthen houses.
(a) (b)
Fig. 1 Earthen constructions around the world, (a) distribution of earthen constructions (from De Sensi
2003), (b) distribution of earthquake epicenters (from Lowman and Montgomery 1998).
Understanding the seismic behaviour and capacity of earthen structures is a first step toward
reducing their seismic vulnerability. Experimental tests are the primary source of information;
however, they are costly and not necessarily available due to limited laboratory capacities,
specifically in developing countries. Numerical modelling is a valid alternative for evaluating the
seismic behaviour of masonry buildings (e.g. Gambarotta and Lagomarsino 1997; Lourenço 1996;
Magenes and Della Fontana 1998; Stavridis and Shing 2010; Roca et al. 2010; Pelà et al. 2013).
Adobe walls have a very low tensile strength, thus cracks typically initiate in zones subjected to
higher tensile stresses, such as corners of doors and windows. Usually, vertical cracks start at the
connection of perpendicular walls due to the high stress concentration and lack of confinement
elements. Furthermore, horizontal cracks may form close to the façade base allowing it to overturn
due to out-of-plane demands. The typical crack pattern due to in-plane shear forces is X-diagonal
shaped, as shown in Fig. 2. Tarque et al. (2012) shows some limit states and displacement capacities
of adobe walls subjected to in-plane and out-of-plane loads, which are useful to study the seismic
vulnerability of adobe buildings based on a mechanics-based procedure. According to Webster
(2008), the cracking due to in-plane forces are not particularly serious unless the relative
displacement across them becomes large, thus initiating the out-of-plane overturning of the small
wall blocks formed by the cracks.
For adobe structures, the brick and mortar joints are made of similar materials, mainly soil.
Therefore, as a first approximation and without loss of accuracy, it seems reasonable to treat the
adobe masonry as a homogeneous material. The state-of-the-art for the numerical modelling of
unreinforced masonry point to two main approaches within the finite element method: discrete
modelling and smeared crack modelling. Another approach consists in idealizing the structure
through an equivalent frame where each wall is discretized by a set of masonry panels (piers and
spandrels) in which the non-linear response is concentrated (Lagomarsino et al. 2013; Calderini and
Lagomarsino 2008; Magenes and Della Fontana 1998). The piers and spandrels are connected by
rigid joint connections. The smeared crack modelling and the equivalent frame modelling are macro-
modelling approaches; while the discrete approach is considered as part of the micro-modelling. For
adobe masonry, the lack of information and experimental data concerning some of the material
properties, particularly in the inelastic range, makes numerical modelling more uncertain. Therefore,
this work focuses on the calibration of the material properties of an adobe wall cyclically tested until
its collapse in order to numerically reproduce its structural behaviour.
Fig. 2 Typical in-plane damage in adobe walls.
This paper presents the results obtained with three different numerical models developed to
represent experimental tests carried out at the Pontifical Catholic University of Peru (PUCP). The
models are developed in two finite element programme following a simplified discrete and a
smeared cracking approach.
2. Experimental test on an adobe wall
Blondet et al. (2005) carried out a displacement controlled cyclic test (push-pull) on a typical
adobe wall at the PUCP. The test intended to analyse the wall cyclic response and the damage pattern
evolution due to in-plane forces. The wall had an H-shape configuration (Fig. 3a), where the main
longitudinal wall (with a central window opening) was 3.06 m long, 1.93 m high and 0.30 m thick.
The wall had two 2.48 m long transverse walls that were intended to: a) simulate the influence of
the connection with the transversal walls found in typical buildings; b) avoid rocking due to in-plane
actions. The brick composition for the adobe was soil, coarse sand and straw in proportion 5/1/1 in
volume, and for the mud mortar, 3/1/1. The soil was basically obtained from a farm field, the coarse
sand diameter was from 0.5 to 1 mm and the straw was dry grass.
The specimen was built over a reinforced concrete continuous foundation beam. A reinforced
concrete ring beam was built at the top of the adobe wall simulating the gravity loads applied by
traditional Peruvian roof made of wooden beams, canes, straw, mud and corrugated zinc sheets. The
weight of this ring beam was around 16 kN, allowing to have a distributed load of 2.07 kN/m. Neither
additional of vertical loads nor control of the variation was imposed to the system during the test.
The ring beam also ensured a more uniform distribution of the horizontal displacements applied to
the wall. The window lintel was made of wood.
The horizontal displacement load was applied in a series of loading cycles with increasing peak
displacements at the top concrete beam through a servo-hydraulic actuator fixed to a rigid steel
reaction frame (see left top side of Fig. 3a). The top peak displacements were 0.5, 1, 2, 5, 10 and 20
mm; however, the last peak value was not considered for the numerical analyses because it was
associated to unstable sliding wall behaviour. The displacements were applied slowly in order to
avoid dynamic effects, the velocity values were 0.5 mm/min, 1 mm/min; 2 mm/min; 5 mm/min; 10
mm/min and 20 mm/min corresponding to each peak displacement. A total of 17 Linear Variable
Displacement Transducers (LVDT) were placed in the opposite side of the front wall (Fig. 3b).
(a) (b)
(c) (d)
(e) (f)
Fig. 3 Adobe wall tested at PUCP. (a) front side of the adobe wall, (b) distribution of LVDT, rear side of
the adobe wall, (c) damage pattern evolution during the cyclic test, (d) vertical displacements LVDT
9 and 8, (e) horizontal displacement at the wall top LVDT 1, (f) horizontal displacement at the
middle (LVDT 3) and bottom (LVDT 4) part of the window (Blondet et al. 2005).
During the test the cracks started between 1 and 2 mm top displacement at the windows corners
and evolved diagonally up to the top and to the base of the wall. The maximum wall strength was
reached around 2 mm top displacement, after that a clear strength reduction was registered. During
reversal loads, the cracks generated the typical X-shape cracks due to in-plane forces; however,
some unsymmetrical cracking is observed in Fig. 3c which were generated due to the position of the
load application (left top wall) and sequence of degradation in the adobe wall for cyclic loads (first
cracking start from top left to right bottom parts). At 5 mm top displacement large horizontal fissures
appeared at the transversal walls and vertical fissures at the intersection of longitudinal and
transversal walls, with increment of diagonal cracking in the main wall (Fig. 3c).
For 10 mm top displacement cycle a notable loss of lateral strength in the wall was observed
with an increment of crack width and tensile cracking in the adobe bricks. The diagonal cracking in
both directions continue growing in thickness. Horizontal cracks appeared at the base of the
transversal walls, allowing a major sliding mechanism of the walls. At this stage, some rigid blocks
were identified.
Fig. 3d shows the vertical displacement measured by the LVDT 8 and 9 at the centreline of the
two end wall. The positive vertical displacement indicates crack opening measured by LVDT 9 or 8
in the first three adobe rows; while negative values indicates crack closing, crushing and some
sliding in the material. Positive horizontal displacements in Fig. 3d and 3e indicate that the top
displacement goes from left to right; while negative horizontal displacements indicates the contrary,
in both cases the horizontal displacement is referred to LVDT 1. In Fig. 3d is seen that for pushing
loads a complete opening of cracking occurs after 2 mm top displacement at the left wall side with
some sliding in the right wall side due to horizontal fissures. The difference in the vertical values of
LVDT 8 and 9 indicates a non-symmetrical vertical response at the two end walls due to some
rotation at the top concrete beam during the load application and the separation of the complete wall
into a number of rigid parts.
A comparison of Fig. 3e and 3f shows that the maximum deformation is seen at the windows
level, where the displacement exceeds the 10 mm top displacement. In particular for Fig. 3f (LVDT
3 and 4 placed at the right wall side) can be understood from the almost null negative displacements
that the wall is divided into blocks and its movement is controlled by the opening, closing and sliding
phenomenon of cracking.
3. Masonry models within the Finite Element Method
Previous research results have shown that the response of masonry structures up to failure can
be successfully modelled using techniques applied to concrete mechanics (Lotfi and Shing 1994).
According to Lourenço (1996), the numerical modelling of masonry walls within the finite element
method can follow either the micro-modelling of each of its components (discontinuous or discrete
approach) or the macro-modelling of the wall (continuum approach), thus assuming that the masonry
wall is homogeneous. More specifically, the following approaches, illustrated in Fig. 4, can be
described as:
• detailed-micro modelling. Bricks and mortar joints are discretized using continuum elements,
with the brick-mortar interface represented by discontinuous elements;
• simplified micro-modelling. The bricks are modelled as continuum elements, while the
behaviour of the mortar joints and of the brick-mortar interface are lumped in discontinuous
elements;
• macro-modelling. Bricks, mortar and brick-mortar interface are smeared out and the masonry is
treated as a continuum.
The first two modelling techniques are considered part of the discontinuous/discrete approach,
where the failure zones are placed in pre-defined weak paths, such as the mortar joints or brick. The
detailed and simplified micro-modelling approaches are computationally expensive for the analysis
of large masonry structures; however, they represent important research tools that can be used as an
alternative to costly and often time-consuming laboratory experiments (Giordano et al. 2002),
provided adequate material data and precise and reliable constitutive models are available. The third
modelling technique performs well in cases where the damage zones are spread over the wall, and
not limited to few bricks and mortar joints. In the present paper, both the discrete and continuum
approaches are used to calibrate the material properties and to reproduce the response of the adobe
wall described in the previous section and tested at the PUCP.
(a) (b)
(c) (d)
Fig. 4 Modelling strategies for masonry structures, (a) masonry sample, (b) detailed micro-modelling, (c)
simplified micro-modelling, (d) masonry sample (modified from Lourenço 1996).
4. Finite element approaches for crack modelling
4.1 Discontinuous approach
The discontinuous approach, such as the discrete crack model introduced by Ngo and Scordelis
(1967) was first proposed to model the concrete. It assumes discontinuous elements interacting with
material cracks represented as boundaries with zero thickness.
The simplified micro-model of Fig. 4c is considered a discontinuous model, where the
inelasticity in the mortar and in the brick-mortar interface is lumped in a discrete brick-brick
interface. Lourenço (1996) developed a model where brick elements are elastic and are connected
by an inelastic composite interface model. This interface model is capable of describing different
failure modes, such as cracking of the mortar, sliding along the bed or head joints at low values of
normal stress, and crushing at the brick-mortar joints (Fig. 5).
The composite interface model of Fig. 5, also referred to as combined cracking-shearing-
crushing model, is based on the plasticity theory. The main input data for this model are the
constitutive laws for the tension, shear and compression behaviour of the composite interface model
Unit Joint
(Fig. 6). In these laws the primary variable is the fracture energy, which is the area under the
monotonic stress-displacement curve after the peak.
Fig. 5 Constitutive model proposed by Lourenço (1996) based on plasticity concepts.
4.2 Continuum approach
4.2.1 Smeared crack model
The smeared crack model uses continuum elements where the concrete/masonry cracks are
assumed smeared and distributed over the elements. The fracture process is initiated when the
maximum principal stress at an integration point exceeds the material strength. The crack
propagation is mainly controlled by the shape of the softening diagram and the material fracture
energy (Cruz et al. 2004). The tensile and compressive constitutive laws are the main input data for
the smeared crack model, as well as the fracture energy for each of them (see Figs. 6 and 7).
Two approaches are typically followed for the formulation of the smeared crack model: the
decomposed-strain model and the total-strain model. The decomposed-strain model splits the total
strain into the sum of the material strain plus the crack strain. The material strain accounts for the
elastic strain, the plastic strain, the creep strain, the thermal strain, etc. The crack strain describes
the deformations due to crack opening only. The total strain model makes use of the total tensile and
compressive hardening/softening curve of the masonry in terms of stress versus strain.
(a) (b)
Fig. 6 Stress displacement diagrams for quasi brittle materials, (a) tensile behaviour for mortar joints, (b)
shear behaviour for mortar joints, (c) compressive behaviour of brick and mortar joints.
(c)
Fig. 6 (Continuation) Stress displacement diagrams for quasi brittle materials, (a) tensile behaviour for
mortar joints, (b) shear behaviour for mortar joints, (c) compressive behaviour of brick and mortar
joints.
4.2.2 Plastic-damage based model
The plastic-damage model is based on the work developed by Lubliner et al. (1989) and later
extended by Lee and Fenves (1998). This model is a continuum, plastic-based, damage model for
concrete, where the two main failure mechanisms are tensile cracking and compressive crushing of
the material. This model assumes that failure of concrete (or unreinforced masonry in this work) can
be effectively modelled using its uniaxial tensile, uniaxial compressive and plasticity characteristics.
Unlike the smeared crack models, the cracking in the damage-plastic based model is represented by
the damage factors (dt, dc) that reduce the modulus of elasticity in tension and compression for
reversal loading (Fig. 7). In contrast to the classical theory of plasticity, the damaged plasticity model
uses a set of variables that alters the elastic and plastic behaviour.
(a) (b)
Fig. 7 Response of concrete under tensile and compressive loads implemented in Abaqus for the concrete
damaged plasticity model, (a) tensile behaviour, (b) compressive behaviour (modified from
Wawrzynek and Cincio 2005).
5. Finite element models of the adobe wall: analysis and comparison
5.1 Monotonic seismic response
The adobe wall tested by Blondet et al. (2005), Fig. 3, is modelled using three finite element
approaches described before, i.e.: the discrete simplified micro-model, the continuum approach
within the total-strain model and the continuum approach within the concrete damaged plasticity
model. The first two models are analysed with Midas FEA, the former using solid elements and the
second using shell elements. The last model is analysed with Abaqus/Standard using shell elements.
In all cases the geometric nonlinearity is taken into account. The configurations of the numerical
models are shown in Fig. 8.
All models include the top and base reinforced concrete beams, the adobe walls and the timber
lintel. The base was fully fixed. Since the experimental test was displacement-controlled, a
monotonic top displacement was applied to the numerical model at one vertical edge of the top
concrete beam up to a maximum displacement of 10 mm. In this first part, just a monotonic
displacement is considered for the calibration of material parameters. At higher displacement levels,
wall instability started during the experimental test and though the test was not interrupted, the
results beyond 10 mm were not considered reliable (Blondet et al. 2005).
The loading sequence in the numerical monotonic analysis is as follows: the gravity loads are
applied first, followed by imposed horizontal displacements. As in the tests, the top concrete beam
is free to have vertical displacements due to geometric non linearity and deformation of the adobe
material. In Midas FEA, the lateral displacement is imposed incrementally following the arc-length
iterative procedure in combination with the initial stiffness. The convergence criterion is controlled
through a displacement and energy norm ratios of 0.005 and 0.01, respectively. In Abaqus/Standard,
the top-displacement is imposed following a full Newton-Raphson iterative procedure. An automatic
stabilization is selected for the convergence criterion, with a specified dissipated energy fraction of
0.001 and an adaptive stabilization with maximum ratio of stabilization to strain energy of 0.01. In
both software platforms the nonlinear geometric effects are considered.
(a) (b)
Fig. 8 Numerical models of the tested adobe wall, (a) model in Midas FEA (solid elements), (b) model
in Midas FEA and Abaqus/Standard (shell elements).
As previously mentioned, a complete database of material properties for adobe bricks and walls
used in Peru is not available. The scarce available data refers to compressive strength and elastic
properties only (e.g. modulus of elasticity). The lack of data for defining the inelastic properties of
adobe, such as the fracture energy in compression and tension, should be obtained through an
appropriate experimental campaign. However, in this study an approximation of these inelastic
parameters is obtained through a correlation study between the global experimental and numerical
results of the adobe wall tested at the PUCP. These numerical material values should be considered
as a mere reference for other analyses since additional experimental tests are needed to obtain the
mechanical properties of different adobe materials (whose strength changes, for example, depending
on the soil type) and building techniques.
5.1.1 Discrete approach: composite interface model
The adobe bricks, the concrete beams and the lintel are modelled using 8-node hexahedron (solid)
elements with elastic and isotropic material properties. The elastic properties of the different
materials are shown in Table 1. E is the modulus of elasticity and was calibrated based on Blondet
and Vargas (1978) to match the initial stiffness measured in the experimental test; υ is the Poisson’s
ratio, γm is the weight density. The crack propagation (inelasticity) follows the mortar joints, which
are modelled using the three dimensional combined interface model with 4 integration points. This
interface model is based on the model proposed by Lourenço (1996).
Table 1 Elastic material properties.
Adobe blocks Concrete Timber
E (MPa) υ γm (N/mm3) E (MPa) υ γm (N/mm3) E (MPa) υ γm (N/mm3)
230 0.2 2e-05 22000 0.25 2.4e-05 10000 0.15 6.87e-06
Table 2 shows the calibrated material parameters used for the combined interface model. kn is
the normal stiffness modulus, kt is the shear stiffness modulus, c is the cohesion, φo is the frictional
angle, ψ is the dilatancy angle, φr is the residual friction angle, ft is the tensile strength, GI
f is the
fracture energy for Mode I (related to the tensile softening), a and b are factors to evaluate the
fracture energy for Mode II (computed as GII
f= a(σ+b) and related to the shear behaviour). In
MIDAS FEA the compression cap is not implemented when dealing 3D interface model, so in this
model the compression for the mortar is elastic; however, as it is explained later the in-plane
behaviour of the adobe wall is less influence by its compressive strength.
Table 2 Material properties for the interface model (mortar joints).
Structural Mode I Mode II
kn
(N/mm3)*
kt
(N/mm3)*
c
(N/mm2)
φo
(deg)
Ψ
(deg)
φr
(deg)
ft
(N/mm2)*
GI
f
(N/mm)*
a
(mm)*
b
(N/mm)*
8 3.2 0.05 30 0 30 0.01 0.0008 0 0.01
The earth mortar properties, marked with * in Table 2, were calibrated based on the experimental
results and observations, namely the pushover envelope curve and the observed damages and failure
pattern of the tested adobe wall. The softening curve in tension (exponential) follows the same
shapes specified by Lourenço (1996) for clay masonry, with lower strength characteristics typically
for adobe masonry.
The damage pattern and displaced configuration at the end of the analysis are shown in Fig. 9.
The crack pattern follows the experimental results: the cracks go from the top left (where the
displacement is applied) to the bottom right of the wall (Fig. 3). The horizontal cracks in the
transversal walls were also observed during the experimental tests. Since the applied load is
monotonic, the FE model cannot capture the X-shape crack pattern. The presence of the two
transversal walls prevents wall rocking, correctly reproducing the tested wall response. The
maximum displacement reached at the top of the wall is around 6.2 mm, thereafter the program
stopped due to convergence problems associated to large element distortions.
Fig. 9 Deformation pattern of the adobe wall considering the combined interface model for a top
displacement of 6.2 mm, Midas FEA.
5.1.2 Continuum approach
Total-strain model
First order shell elements are used in the finite element model created in Midas FEA. The model
uses rectangular 4-node shell elements. The element size is about 100 x 100 mm; which ends on a
characteristic element length of 141 mm (diagonal of the element size). The concrete beams and the
lintel are represented by elastic and isotropic materials. The adobe masonry (bricks plus mortar),
which is defined here as an isotropic material, makes use of the tensile and compressive constitutive
laws for representing material inelasticity within the total-strain approach. An exponential function
is defined for the tensile behaviour and a parabolic one, similar to the function given by Lourenço
(1996), is given for the compressive response. Since experimental data for the softening behaviour
in tension and compression is not available, the inelastic strain values are assumed based on the ones
for clay masonry and calibrated through the experimental pushover curve. The obtained values are
reported in Table 3, where εp is the plastic strain related to the maximum compressive strength and
h is the characteristic element length.
Table 3 Adobe masonry material properties used for the continuum approach.
Elastic Tension Compression
E
(N/mm2)
υ γm
(N/mm3)
h
(mm)
ft
(N/mm2)
GI
f
(N/mm)
fc
(N/mm2)
cG
(N/mm)
εp
(mm/mm)
200 0.2 2e-05 141 0.04 0.01 0.45 0.155 0.002
The modulus of elasticity E of the composite material (adobe plus mortar) was calibrated to
match the initial stiffness of the pushover envelope curve obtained from the experimental test. A
compressive strength fc close to 0.70 MPa is specified in the literature (test on adobe piles, Blondet
and Vargas 1978), but the complete curve of the uniaxial behaviour law is unavailable. The authors
believe that tests on adobe piles don’t represent the actual adobe composite behaviour because they
consider a portion not representative of the wall. In this work a fc= 0.45 MPa is specified. Tensile
strength and fracture energy are calibrated in this model using the experimental test results, since no
data is found in the literature for this type of constructions (adobe). Since a fixed crack model is
selected, a reduction of the shear stiffness β= 0.05 was specified for the analysis. The elastic material
properties for the concrete and timber are the same as those used in the discrete model presented in
Table 1.
Fig. 10 shows the wall strains and crack pattern of the model (e.g. closed, partially open, fully
open, etc.) at the end of the analysis. The strain pattern is in good agreement with the failure pattern
observed in the test (Fig. 3d) if just a monotonic load is assumed. The target maximum displacement
was 10 mm. As in the previous model, cracks start at the openings corners and progresses diagonally
to the wall edges. Fig. 10a shows that the maximum strains are reached at the contact zone of the
wall with the timber lintel and at the corner openings, due to stress concentration in these zones.
(a) (b)
Fig. 10 (a) Strain (mm/mm) and (b) crack status at the middle surface of the adobe wall considering the
total-strain model for a top displacement of 9.38 mm (P: partially open; O: fully open; C: closed).
Concrete damaged plasticity model
The concrete damaged plasticity model, which accounts for both tensile cracking and
compressive crushing, can be effectively applied to masonry too. In this model the effect of material
cracking (especially for unloading as for cyclic loadings) is represented through damage factors in
tension and compression. As in the previous case, this model is built with rectangular 4 nodes-shell
elements. The size of each element is about 100 x 100 mm. Similarly to the total strain model, the
concrete damaged plasticity model defines material inelasticity separately in tension and
compression. In Abaqus/Standard, input data is defined based on the material properties presented
in Table 3. Fig. 11 shows the damage pattern associated to tensile stresses at the top displacement
of 10 mm. The diagonal cracks control the wall behaviour and extend from the opening corners to
the wall corners. The largest strains are observed at the contact points between the timber lintel and
the wall and at the opening corners (Fig. 11). The horizontal crack pattern at the transversal walls is
in agreement with the actual failure pattern observed during the tests.
Fig. 11 Maximum plastic strain distribution considering the concrete damaged plasticity model (middle
surface) for a top displacement of 10 mm (units of strain in mm/mm), Abaqus/Standard.
5.1.3 Comparison of the analytical and experimental results in terms of force
displacement curves
The three numerical models reproduced fairly well the monotonic global response of the tested
adobe wall, as shown in Fig. 12, as well as the stress distribution and the crack pattern, even though
not all analyses reached the maximum imposed displacement of 10 mm due to convergence
problems. The material properties (parameters) adopted in these models were obtained from a
parametric study changing the parameters one by one (e.g. tensile strength, compression strength,
fracture energy, shape of the constitutive laws, etc). The best values were determined by comparison
between the numerical with the experimental pushover envelop curve. A correct parametric study
should vary all parameters at the same time to look for a non-linear optimization. However, in this
work it was not possible due to the great lack of information about the material properties.
When the response of the wall starts being inelastic (approximately at 1 mm top displacement),
small differences between the numerical and experimental results are observed. However, as the
crack patterns stabilize, all the numerical curves match well the experimental response (after a top
displacement of 2 mm, see Fig. 12).
Fig. 12 Force-displacement curves: experimental results and numerical modelling.
The model with the combined interface law (discrete approach) tends to be more unstable, due
to local large deformations in the cracks region, which however describes better the physical
discontinuity between the cracks’ faces (Fig. 9). Continuum models are more stable because the
inelastic properties are smeared in the wall and not concentrated just at the mortar joints, as in the
discrete modelling approaches. The first elements that reach the tensile strength are located at the
opening and lintel corners, where stress concentration occurs. Also, horizontal cracks appear at the
transversal walls.
As it is well known, the response of quasi brittle materials is more sensitive to the variation of
the tensile strength than to the variation of the compressive strength. To study the influence of the
strength, additional analysis were done considering values of fc varying from 0.30 to 0.70 MPa, with
a fracture energy proportional to the corresponding for fc= 0.45 MPa (i.e. Gc/fc= 0.344 mm). This
comparative analysis was performed with the plastic-damage model and the results are shown in
Fig. 13. It is seen that the compressive strength influences the maximum lateral strength of the adobe
wall, though it does not affect the post peak behaviour and failure pattern; which is controlled in
part by the fracture energy. The main differences in terms of lateral strength are seen between 2 and
4 mm of top displacement. A lower difference is seen for the pushover curves computed with fc
=0.45 to 0.70 MPa.
Another sensitivity study was carried out varying the tensile strength (between 0.03 and 0.06
MPa), keeping the tensile fracture energy at 0.01 N/mm and the compressive strength at 0.45 MPa.
The analyses were carried out with the plastic-damage model, and the results are shown in Fig. 14,
proving the pronounced influence of the tensile strength on the wall response. Increasing the material
tensile strength induces an increase in the lateral strength and a concentration of the damage in the
adobe wall diagonal. On the other hand, for lower values of tensile strength the damage is more
distributed in the adobe wall, representing better the damage pattern observed in the test (Fig. 3d).
Fig. 13 Sensitivity of the adobe wall response to the
variation of fc (considering in all analyses ft=
0.04 MPa, GI
f= 0.01 N/mm & Gc/fc= 0.344
mm).
Fig. 14 Sensitivity of the adobe wall response to the
variation of ft (considering in all analyses fc=
0.45 MPa, Gc/fc= 0.344 mm & GI
f= 0.01
N/mm).
The last sensitivity analysis intended to assess the influence of the compressive and tensional
fracture energy in the response of the adobe wall. The results presented in Fig. 15 indicates the low
influence of Gc on the global response even increasing this value up to 3 times. This means that more
important than the compressive fracture energy is the initial and maximum compressive strength.
Instead, the results presented in Fig. 16 show a significant wall strength variation in contrast to the
measured in the experimental test, pointing out the importance of a correct definition of the tensile
fracture energy to properly represent the behaviour of adobe masonry walls. A lower value of GI
f
induces a more brittle behaviour of the adobe masonry without any energy dissipation capacity.
Fig. 15 Sensitivity of the adobe wall response to the
variation of Gc (considering in all analyses
fc= 0.45 MPa, ft= 0.04 MPa & GI
f =0.01
N/mm).
Fig. 16 Sensitivity of the adobe wall response to the
variation of GI
f (considering in all analyses
fc= 0.45 MPa, Gc/fc= 0.344 mm & ft= 0.04
MPa).
5.2 Cyclic seismic response
In this section, the finite element model created with the plastic-damage model was upgraded
for the representation of the cyclic behaviour through the calibration of the damage factors in tension
dt and in compression dc, and the stiffness recovery values wt and wc, for tension and compression,
respectively (see Fig. 7). The damage factors control the cracking closing and the unloading
stiffness. The values of the tensile and compressive strengths, and its respective fracture energies,
are specified in Table 3.
In the concrete damaged plasticity model, supported by experimental tests, it is considered that
after concrete cracking the compressive stiffness is recovered upon crack closure when the load
changes from tension to compression. On the other hand, in the implemented model the tensile
stiffness is not recovered when the load changes from compression to tension once crushing micro-
cracks have developed in compression. These modelling assumptions are taken for the adobe
masonry simulation.
The finite element model created for this analysis is similar to the one showed in Fig. 8b (4-
node shell elements). Unlike the previous model, the displacement load is applied at both vertical
edges of the concrete beam (top of the wall) for simulating the cyclic loading. The history of imposed
lateral displacements consisted of one cycle of ±1, ±2, ±5 and ±10 mm. For convergence control just
an automatic stabilization of 0.0002 is selected.
5.2.1 Calibration of damage factors (dt and dc) and stiffness recovery values (wt and
wc)
Since no experimental data exist concerning the mechanical characterization of adobe masonry
under reversal loads in tension and compression, stiffness recovery values (wc and wt) and damage
factors (dt and dc, see Fig. 7) are proposed to consider its influence in the adobe structures response
(Tarque 2011). The idea is to represent numerically the effect of the cracks closing during the
transition from tension to compression stress with calibrated damage factors, which varies between
0 and 1. With the best adopted values for these parameters (Table 4), the force-displacement curves
obtained with the numerical models are compared with the results of the cyclic test.
By default, Abaqus/Standard assumes wc= 1 and wt= 0, which represents full stiffness recovery
when in the integration point is installed compression stresses, and no stiffness recovery when tensile
stress is installed. For adobe masonry, wc between 0.5 and 0.6 give good representation of the
behaviour under reversal loads, here a wc= 0.5 is used. Major attention has been put to the tensile
behaviour, in contrast to the compressive, since the tension stress controls the global in-plane
response of adobe walls. The calibrated tensile damage factors (dt) are showed in Table 4.
As graphically represented in Table 4, it is not reached zero crack displacement when the load
passes from tension to compression, even with large values of tensile damage factors. So, a residual
crack aperture appears for reversal loadings and this effect will be reflected in the numerical cyclic
curve response (see Fig. 18). In Table 4, the crack displacement values presented are equal to the
crack tensile strain times the characteristic element length (h= 141 mm).
An alternative way for the identification and interpretation of the tensile damage occurred in the
adobe masonry at the end of the analysis is evaluating the tensile damage factor distribution (Fig.
17). The tensile damage factor is a non-decreasing quantity associated with the tensile failure of the
material. In Fig. 17, the grey zones indicate the regions which already are in the softening branch of
the tensile constitutive law, that and can be considered as damaged zones.
Table 4 Proposed tensile damage factor.
Damage factor dt Plastic disp. (mm)
0.00 0.000
0.85 0.125
0.90 0.250
0.95 0.500
Graphical representation of the degradation stiffness
Fig. 17 Damaged zones in tension at the end of the cyclic loading demand (middle shell surface).
Here, the numerical formation of X-diagonal cracks is evident, as typical for adobe masonry
walls cyclically loaded in-plane. However, due to the geometric non linearity specified in the model
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 0.2 0.4 0.6 0.8 1
Crack displacement (mm)
Te
ns
ile
s
tre
ng
th
(M
Pa
)
Tensile curve
Degradated stiffness
for unloading
and the degradation of the adobe material -even at early steps- the failure pattern is not symmetric
at all for cyclic loading, similar results were seen in the test (Fig. 3c). So, the numerical results
obtained here reproduced quite well the global failure pattern observed in the experimental test (Fig.
3d).
5.2.2 Global force-displacement response
The material properties given in Table 3 and the tensile damage factors proposed in Table 4 are
result of the iterative calibration process, matching quite well the numerical results with the
experimental ones in terms of strength, lateral capacity and crack pattern. These proposed material
properties for numerical models is an important contribution of this research, since there is not
available a complete experimental database for the adobe material properties. Here the compressive
strength value should be interpreted as a lower bound, leaving the possibility to use higher values.
Comparing the numerical and experimental pushover curves, it is observed an acceptable agreement
for the loading branches and acceptable agreement is achieved for the cyclic reversal demands (Fig.
18). When the load changes sign, from positive to negative and vice versa, the numerical results
does not follow the experimental curve at the first load steps, showing larger deformations due to
the incapacity of representing the complete closing of the tensile cracks. This fact results in an
overestimation of the energy dissipation and could be solved modifying the damage plasticity law
specified in continuum models to take into account the pinching effect.
(a)
(b) (c)
Fig. 18 Comparison between the experimental and numerical force-displacement response for the cyclic
demand: (a) wall top LVDT 1, (b) middle part of the window (LVDT 3), (c) bottom part of the
window (LVDT 4).
-50
-40
-30
-20
-10
0
10
20
30
40
50
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
H
or
iz
on
ta
l f
or
ce
(k
N
)
Horizontal displacement (mm)
LVDT 1
Numerical
-50
-40
-30
-20
-10
0
10
20
30
40
50
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
H
or
iz
on
ta
l f
or
ce
(k
N
)
Horizontal displacement (mm)
LVDT 3
Numerical
-50
-40
-30
-20
-10
0
10
20
30
40
50
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
H
or
iz
on
ta
l f
or
ce
(k
N
)
Horizontal displacement (mm)
LVDT 4
Numerical
6. Conclusions
This paper focuses on the numerical analysis of the response of an adobe wall, previously tested
under cyclic loads. Due to the lack of a full experimental database for the characterization of the
adobe material properties (brick and mortar), the main objective of this work was the calibration of
material properties (i.e. tension, compression and shear) in order to reproduce numerically the in-
plane seismic response of the tested adobe wall.
For the monotonic analysis, three numerical models were created. The first considered the
nonlinearity at the mortar joints (discrete crack model, simplified micro-modelling), and the other
two considered the nonlinearity smeared over the FE mesh (total-strain and concrete damaged
models, macro-modelling). All three models were able to reproduce with good accuracy the in-plane
response of the tested adobe wall, including the crack initiation and propagation. In what regards
convergence issues, the distributed (smeared) model used in this study showed a more stable
numerical behaviour.
Using the smeared crack approach, a parametric study was performed in order to demonstrate
the limited influence of the compressive strength and the major influence of the tensile strength on
the global behaviour of adobe walls loaded in-plane. More specifically, for the more stable
continuum model analysis, the compressive strength for the adobe masonry was taken as 0.45 MPa
with a ratio Gc/fc= 0.344 mm, and the tensile strength, ft, as 0.04 MPa with Gf= 0.01 N/mm.
Furthermore, the cyclic response of the adobe wall was reproduced and for this the damage factors
for tensile behaviour were calibrated to match quiet well the global behaviour (damage pattern) and
the experimental force-displacement curve (capacity).
The lack of experimental data for the evaluation of the hardening/softening behaviour of the
adobe composite material (brick and mortar) introduces uncertainties in the numerical modelling of
the adobe structures’ response. Even though, the material properties calibrated and used in this work
represent well the experimental behaviour of the tested adobe wall, there is a strong need extensive
experimental campaigns aimed at characterizing the mechanical properties of different adobe
materials and construction techniques. Provided additional material test data is available, numerical
analyses such as those presented in this paper may be used to gain insight on the seismic behaviour
of different adobe structures.
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5. Finite element models of the adobe wall: analysis and comparison