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Abstract
An integrated methodology for the characterization of the response of rubble
masonry is presented. The behaviour at collapse of a wall belonging to the
Guimarães castle (Portugal) is investigated through a rigid-plastic homogenization
procedure, accounting both for the actual disposition of the blocks constituting the
walls and texture irregularity, given by the variability of block dimensions.
A detailed survey is conducted by means of a photogrammetric technique, allowing
for a precise characterization of blocks dimensions and disposition. After a
geometric simplification assuming mortar joints reduced to interfaces, homogenized
masonry in- and out-of-plane strength domains are evaluated on a number of
different representing elements of volume (RVEs) having different sizes and
sampled on the walls of the castle. Strength domains are obtained using a finite
element (FE) limit analysis with a heterogeneous discretization by triangles and
interfaces.
Finally, a series of limit analyses are carried out on the façade for the safety
assessment under seismic loads by means of two numerical models, the first one
being a heterogeneous model and the second a homogenized approach. The
reliability of the results, in terms of limit load and failure mechanism, provided by
the homogenized model, when compared to the heterogeneous one is satisfactory.
Keywords: homogenization; quasi-periodic masonry; geometrical survey; limit
analysis; structural safety, sampled RVEs.
1 Introduction
Masonry constituted by the assemblage of blocks with variable dimensions is very
common in both existing and historical buildings in many countries. However, the
complexity of the problem, and the number of variables required by accurate
1
Paper 0123456789
Limit Analysis of Loaded Out-of-Plane Rubble
Masonry : A Case Study in Portugal
G. Milani1, Y.W. Esquivel2, P.B. Lourenço2, B. Riveiro3 and D.V. Oliveira2
1Department of Architecture, Built Environment & Construction Engineering
Technical University of Milan, Italy
2Department of Civil Engineering, University of Minho, Guimarães, Portugal
3Department of Materials Engineering, Applied Mechanics & Construction
University of Vigo, Spain
Civil-Comp Press, 2013
Proceedings of the Fourteenth International Conference on
Civil, Structural and Environmental Engineering Computing,
B.H.V. Topping and P. Iványi, (Editors),
Civil-Comp Press, Stirlingshire, Scotland
numerical heterogeneous FE analyses [1]-[3], usually preclude the study of these
structures in the inelastic range through commercial software. As a consequence, it
can be stated that, at present, the numerical analysis of masonry structures randomly
assembled, remains a very challenging problem, despite the efforts recently
expended by many authors to tackle the problem using stochastic homogenization
schemes. In recent years, the interest in the conservation of historical buildings and
in finding efficient numerical models, has led to a significant number of numerical
model for historical masonry buildings, from very simple to complex ones [5]-[7],
which are able to simulate the behaviour of the material under different type of
loads. The choice depends on the degree of accuracy, sought in the analysis for each
particular case.
The present paper deals with the characterization of the response of quasi-periodic
masonry by means of a geometrical study and a statistical analysis of stone units,
homogenization of masonry and structural implementation. For this purpose, it was
decided: (a) to carry out the geometrical investigation of stones units from an
existing case study (Guimarães castle), to obtain statistical parameters and
distribution of the height and length of the stones units, with the aim of determining
the adequate size of the representative volume elements; (b) to perform a
homogenized limit analysis on a number of representative volume elements (RVEs)
to obtain their in-plane and out-of-plane failure surfaces; (c) to carry out a series of
structural limit analyses under static horizontal loads up to collapse of one of the
walls (Alcaçova) and to compare limit loads and failure mechanisms provided by the
homogenized model with those obtained by means of a detailed heterogeneous
model, in order to check the reliability of the approach proposed and its applicability
for practical purposes.
2 Masonry homogenization: a brief state-of-the-art
Homogenization theory may represent a powerful tool in problems involving
periodic arrangements of heterogeneous materials, where the study of the whole
structure within a heterogeneous approach is impractical or impossible, due to the
computational effort. A non-rigorous homogenization may be also attempted for
random composites, provided that a suitable portion of the whole structure (test
window) is extracted.
The basic idea of homogenization [4] [6] consists in introducing averaged quantities
representing the macroscopic strain and stress tensors (respectively
Σ
and
E
) as
follows:
dY
AY
∫
>==< )(
1uεεE
dY
AY
∫
>==< σσΣ 1
( 1 )
where A stands for the area of the elementary cell,
ε
and
σ
stand for the local
quantities (stresses and strains respectively) and <*> is the averaging operator.
Periodicity conditions are imposed on the stress field
σ
and the displacement field
u, given by:
2
∂
∂
+
=
Y
Y
onperiodic-anti
on
perper
σn
uuEyu
( 2 )
where
u
is the total displacement field and
per
u
stands for a periodic displacement.
Here, it is worth noting that, in equation ( 2 ), the periodicity of the displacement
fluctuation
per
u
forces corresponding boundary segments to exhibit the same shape
in the deformed configuration.
Let
m
S
,
b
S
and
hom
S
denote respectively the strength domains of mortar, blocks
and homogenised macroscopic material. It has been shown that a kinematic upper
bound approximation of
hom
S
can be derived through the support function
( )
D
hom
π
as follows (see also [6] for further details):
( )
( ) ( ) ( )
() ( ) ( )
+
=
⊗+⊗
Γ
=
=
∈∀≤∑
∑=
∫ ∫
∫Γ=∂
Y S
Y
dSdY
P
dSP
R
S
nvd
v
vnnvDvD
DDD
v
]];[[
2
1
|inf
:
|
hom
6hom
hom
ππ
π
π
( 3 )
where:
-
per
vDyv+=
is the microscopic velocity field;
-
per
v
is a periodic velocity field;
- d and D are respectively the microscopic and macroscopic strain rate fields;
- S is any discontinuity surface of v in Y, n is the normal to S;
-
( ) ( )
]][[]]
[[2/
1]];[[ vnnvnv ⊗+⊗=
π
;
-
( ) ( ){ }
yσd:σd
σ
S∈= ;max
π
.
The above considerations hold on a Representative Element of Volume (REV) that
should generate the entire wall by repetition. The REV is defined as the smallest
volume that contains all the essential information about the microstructure. On the
boundaries of the REV, periodicity and anti-periodicity conditions should be
assigned in agreement with equation ( 2 ).
However, the identification of a REV for old masonry is not always an easy task, as
is the case here analysed. The most straightforward approach is represented by large
sampling of several REVs with different dimensions in different positions of the
wall under study and in the average evaluation of their ultimate behaviour, both for
in-plane loads under different directions of the load with respect to the bed joint [8]
and their flexural response, as detailed in [9].
3 Guimarães castle: geometry of the masonry units
The origin of the castle dates back to the 10th century and the fights against the
Moors in the Iberian Peninsula. In the 11th century, the first King of Portugal was
born there. Later, between the 12th and 14th centuries, the castle was enlarged and the
defence capacity was improved. At a certain stage, the castle was abandoned and
3
suffered damage caused by time, and by the subsequent changes of use. In the 20th
century, important restoration works have been carried out. The current condition is
shown in Figure 1, where the pentagonal plan view of the castle is reported. The
castle is surrounded by eight square towers, which delimit the main square, with a
main tower (“Torre de Menagem” in Portuguese and “Keep” in English) in the
centre. The main wall under study in this paper is the so-called “Alcáçova” Wall,
which is originally the highest and most protected part of an Iberian medieval castle,
with a defence function and where the civil or ecclesiastical authorities lived. The
word was later used to define the part of the castle where the governor lived.
(a) (b)
Figure 1: Castle of Guimarães: a) Plan view and (b) “Alcáçova” wall
The masonry of the castle is made using granite stone ashlars in the external leaves.
The masonry features horizontal courses and is relatively regular, despite the fact
that the height of the courses is not constant and that the length of the units is rather
variable. In order to represent this feature, a statistical description considering mean,
standard deviation, coefficient of variation and probability distribution of the size of
the stone units from four walls was made: Wall1, Wall2, Tower wall and Alcaçova
wall, see Figure 2 and Table 1. The walls were analysed separately and together as a
single group. The objective was to characterize the length l and height h of stone
units and the results are shown in Table 1. The procedure was to identify the stone
units in a first step and then to define the best fit probabilistic distribution, which is a
lognormal (skew) distribution for both variables.
The following aspects from the geometric data are relevant: (a) there is a large
variation between the mean value of the stone length and height in the four walls
selected for sampling (0.60 to 0.85 m in length and 0.34 to 0.46 m in height). The
ratio between the maximum and minimum averages in the different samples is
similar in length and height (about 75%); (b) the stone geometrical ratio is rather
important for the quality of the masonry bond. The value of h/l for the average
geometrical dimensions is about 56% (1:1.8). Only in Wall2, a slightly different h/l
ratio is found, equal to 63% (1:1.6); (c) the scatter found in the length is always
4
much larger than the scatter found in the height, being the scatter in the full sample
not so much different from the scatter in the individual samples; (d) wall 2 is the
sample with the lowest scatter and the Alcaçova is the sample with the largest
scatter, despite the fact that the Alcaçova sample is three times larger than Wall 2;
(e) the difference between averaging the total sample weighted by the number of
samples or weighted equally is only moderate, with about 5% change in the
dimensions; (f) the probabilistic distribution for the length is clearly skewed,
requiring a lognormal distribution. The probabilistic distribution for the height is
symmetric, meaning that a normal distribution can be used.
Results
Wall
Number of
units
Ratio
Height
/
Length
h / l
Length Height
Average
[m]
(CoV)
Typical [m]
(Frequency) Range
[m] Height [m]
(CoV)
Typical [m]
(Frequency
)
Range
[m]
Wall
W1 110 0.51 0.76
(34%) 0.70
(27%) 0.40-
1.70 0.39
(20%) 0.40
(38%) 0.20-
0.40
Wall
W2 110 0.63 0.70
(27%) 0.65
(27%) 0.45-
1.35 0.44
(19%) 0.40
(25%) 0.25-
0.60
Tower 110 0.54 0.85
(37%) 0.65
(34%) 0.50-
1.80 0.46
(17%) 0.50
(35%) 0.15-
0.60
Alcaçov
a 308 0.56 0.60
(44%) 0.45
(25%) 0.25-
2.10 0.34
(23%) 0.40
(35%) 0.15-
0.60
Full
Sample Weighted
average 0.56 0.69
(40%) 0.55 / 0.65
(21%) / (20%) 0.25-
2.10 0.38
(24%) 0.40
(35%) 0.15-
0.60
Table 1: Geometric Data Measured
4 Homogenized limit analysis of RVEs
Next, a study on different representative volume element (RVE) samples from the
Alcaçova wall is presented. The RVEs are firstly analysed under in-plane load in
order to obtain membrane failure surfaces at different orientations of a load with
respect to the bed joint, considering masonry with weak and strong mortar joints,
aiming at representing a possible injection intervention. The RVEs are also analysed
under out-of-plane load in order to obtain the out-of-plane surface failure at
increasing compressive loads. The result allows subsequent implementation of the
obtained failure surfaces in the study of the full masonry wall. In-plane failure
surfaces are described by horizontal strength (σh) and vertical strength (σv). Out-of-
plane failure surfaces are described by horizontal bending moment (M11), vertical
bending moment (M22) and torsional moment, or torsion (M12).
5
(a)
210
180
150
12090
60
30
140
120
100
80
60
40
20
0
l (cm)
Frequency
1
0
2
33
1
2
44
10
24
2727
55
69
125
133
90
50
8
64564840
322416
90
80
70
60
50
40
30
20
10
0
h (cm)
Frequency
1
4
33
4
11
51
34
33
44
49
60
80
28
32
26
39
27
54
18
9
88
3
9
(b)
Figure 2: Geometric Survey of the Units: (a) Identification in the Alcaçova Wall; (b)
Distribution of Length (l) and Height (h) in the Entire Sample
The Alcáçova wall is built using two external leaves with an average thickness of
400mm, separated by an infill. It was decided to consider three RVEs of different
size: the first size, called 3x3, has dimensions three times the mean width and the
mean height of stone; the second size, called 4x4, is four times the mean width and
the mean height of stone; and the third size, called 5x5, is fifth times the mean width
and the mean height of stone. For each size of RVE, three different samples located
randomly on the wall are taken into account, as schematically represented in Figure
3. In addition, three artificial RVEs were built using mean size stones and periodic
arrangement in order to compare the failure surfaces between the RVEs with quasi
periodic arrangement and the RVEs with periodic arrangement using average
geometry. A linearized Lourenço and Rots [5] failure criterion is adopted for joints
reduced to interfaces and a classic Mohr-Coulomb failure criterion is used for brick
interfaces, as in [1][9].
6
Figure 3: Location of 3x3 Representative Volume Elements (RVEs)
The in-plane homogenized failure surfaces (σv – σh) are obtained keeping a ϑ angle
fixed. This angle measures the rotation of the principal stresses with respect to the
material axes. Three different ϑ angles are considered ϑ=0°, ϑ=22.5° and ϑ=45°
(Figure 4-a) in analogy to [8][9]. For each RVE and in each orientation ϑ, 32 values
with steps of half of 22.5º have been calculated. The 32 points were then connected
to draw failure surfaces. The optimization problem arising in order to obtain the
failure surface is solved by using an algorithm code developed in [6][9].
Different hypotheses on the mechanical properties of the mortar joints are
investigated, simulating a scarcely resistant masonry (actual situation) and a strong
material (hypothesis of rehabilitation with injection of mortar having good
mechanical properties).
For masonry with weak mortar, the compressive strength of masonry is assumed
equal to 12 MPa and the ultimate tensile strength of joints is assumed equal to 0.05
MPa. The compressive strength of stones is assumed equal to 89.5 MPa and their
ultimate tensile strength is equal to 0.93 MPa [10][11]. For masonry with strong
mortar, only the ultimate tensile strength of joints is changed, assuming a value
equal to 0.3 MPa.
A full discussion and presentation of the results is provided in [12]. Here, for the
sake of conciseness, only a small sample of the huge amount of experimental results
obtained by the authors is reported.
Figure 4b shows typical in-plane homogenized failure surfaces for RVEs of masonry
with weak mortar at different orientations of the load with respect to the bed joint.
The usual anisotropic behaviour of masonry is found. Figure 4c shows a comparison
between in-plane homogenized failure surfaces obtained from RVEs of the same
size and artificial RVEs with periodic arrangement for masonry at a given
orientation. Finally, Figure 4d shows a comparison between the mean values of in-
plane homogenized failure surfaces at a given orientation for all sizes of the RVEs,
where it is shown that small difference are found. These results seem to indicate that
the average of 3 masonry samples, with minimum size of 3x3, provides a reasonable
estimate of the true failure surface. Further details on these results can be found in
[11].
7
(a)
(b)
(c)
(d)
Figure 4: Homogenization: (a) ϑ angle orientations of the external load with respect
to the bed joint; (b) example of a result with different orientations; (c) example of a
result for different cells of the same size; (d) example of a result for cells of different
size.
8
Failure modes obtained from representative volume elements are depicted in Figure
5, where a qualitative comparison with experimental results [13] is also shown. A
staircase crack in the 4x4 representative volume element is found independently of
the quality of the mortar. It is noted that dilatancy is present in the numerical model,
even if it is believed that the influence in the global behaviour is very low (the upper
boundary is allowed to move up, meaning that an artificial confining stress built up
does not occur).
(a)
(b)
Figure 5: Qualitative comparison of mode failure between a masonry RVE with load
orientations equal to ϑ=0º and 45º: (a) Numerical; (b) Experimental.
5 Out-of-plane homogenized failure surfaces
Out-of-plane loads are responsible for the majority of failures experienced in
masonry structures, and especially in historical buildings, whose façades are usually
characterized by a relative small thickness in comparison with height and length and
a box behaviour is hardily present due to deformable floors. For this reason it is
paramount to evaluate out-of-plane homogenized failure surfaces (M11-M22 and M11-
M12), which are obtained, similarly to the in-plane case, from a combination of
homogenization techniques and limit analysis. Again, plasticity and associated flow
rule for the constituent materials are assumed. The RVE is subdivided into 12 layers
along the thickness (a conservative thickness is considered, assuming only the
Alcaçova wall external leaf, with h = 400 mm). For each layer, the out-of-plane
components σi3 (i=1, 2, 3) of the micro-stress tensor σ are set to zero, meaning that
9
only the in-plane components σij (i,j=1,2 ) are considered active and constant in the
thickness.
The out-of-plane homogenized failure surfaces in sections in the space of bending
moment (M22) and horizontal bending moment (M11) are generated from the
integration of in-plane homogenized stress for which the algorithm requires the
following data: RVE thickness, hereafter assumed equal to 0.40 m; the number of
layers in which the thickness of the RVE will be divided, selected as twelve layers;
the compressive vertical load, which is considered at three different levels N22=0
(top), N22=self-weight/2 (mid-height), N22= self-weight (bottom) of the Alcaçova
wall; and the values of the in-plane failure surfaces. On the other hand ,for obtaining
the out-of-plane homogenized failure surfaces in sections in the space of torsion
(M12) and horizontal bending moment (M11), the algorithm requests the geometry of
the mesh, number of elements and the properties of the masonry, using a process
similar to the in-plane case.
Figure 6a shows out-of-plane homogenized failure surfaces (M11-M22) for a sampled
RVE at the three increasing vertical compressive loads previously discussed. As it
can be seen, the vertical compression applied increases not only the horizontal
bending moment (M11) but also the vertical bending (M22) and torsion (M12). This
means that bed joints, in general, contribute to masonry vertical and torsion ultimate
moment due to the friction effect of interlocking units. In some cases, due to
insufficient staggering of the stones in the RVE with strong mortar, M11 does not
increase as a straight vertical crack is obtained. Figure 6b shows out-of-plane
homogenized failure surfaces (M11-M12) for RVEs of masonry with increasing
vertical compressive loads. Again, the vertical compression load applied usually
increases not only the horizontal bending moment but also the vertical bending
moment (M11) and torsion (M12). Finally, Figure 6c shows a comparison between the
mean values of out-of-plane homogenized failure surfaces of RVEs of the same size
when the compressive load is maximum and equal to N22=133 kN/m. As it can be
observed, the vertical bending moment (M22) exhibits similar values for the different
cell sizes (as well as the torsion M12). The horizontal bending moment (M11) exhibits
some scatter for the different average results, as it is more sensitive to the
compressive loads. However, the scatter may be considered as moderate for
engineering purposes.
6 Limit analysis of the alcaçova wall of Guimarães castle
The present study is focused in the part of the Alcaçova wall located above a much
thicker panel, composed by two external leaves of stone masonry and an infill
material in the middle, see Figure 4. The wall is connected to secondary small
buildings and rooms. The analysis of the whole wall would require full 3D
computations, which are outside the scope of the present paper. Therefore, attention
is devoted exclusively to the Alcaçova wall.
10
(a)
(b)
(c)
Figure 6: Results of out-of-plane homogenization: example of failure surfaces in (a)
(M11-M22) plane and in (b) (M11-M12) plane for increasing vertical compression; (c)
comparison between the mean values for different RVEs sizes.
11
The portion of the wall under consideration may be reasonably assumed as
supported on three edges, one corresponding to the lower edge, the remaining two
corresponding to vertical boundaries. The dimensions of the wall are 14.25 m in
length and 6.85 m in height. The thickness of the external leaves is 0.40 m. The wall
has seven openings, labelled from O-1 to O-7 for the sake of clearness in Figure 4.
The openings represent approximately 20% of the area of the entire wall. Above
each opening a lintel is present.
Two numerical models are utilized and the results obtained critically compared: the
first is a heterogeneous model, where the actual disposition of the blocks is
considered in detail, whereas the second is a homogenized model where strength
domain obtained in the previous section are utilized.
Limit analyses are performed using the out-of-plane model utilized for e.g. in [9] ,
where plastic dissipation is allowed only at the interfaces between adjoining
elements.
Results obtained comprise limit loads and the possible collapse mechanisms,
whereas no information on displacements is provided. External seismic load
depending on the load multiplier is applied to the model as out-of-plane pressure on
single elements following a first mode distribution.
Collapse mechanism obtained by means of both the heterogeneous and the
homogenized model assuming joints with weak mortar are depicted in Figure 7. The
actual thickness of the plate elements is visualized for the sake of clearness, as well
as all the cracks forming the failure mechanism are labelled.
Cracks a4, a5, a13, a14, a17 belong to two types of vertical cracks located in the
middle third of the wall. These cracks are caused by the vertical bending moment.
Crack a4 is a vertical line and forms for the inexistent staggering of the blocks in
eleven masonry courses in this area. Crack a13 propagates following head and bed
joints, whereas Cracks a8 and a18 follow the horizontal alignment of the blocks.
Cracks a1, a2, a3, a6, a7, a9, a10, a11, a12, a15, a16 appear on corners and
surrounding different portions of the openings, and are mainly diagonal, propagating
for a combined effect of horizontal and vertical bending moment. This combination
of effects is caused by the edge constrains applied to the wall, representing the
connections of the Alcaçova wall with orthogonal walls and with the base horizontal
base. It is important to remark that these types of cracks develop on areas close to
the lateral edges. Finally, it can be stated that globally the failure mechanism is
similar to a local overturning, as usually occurs for historical masonry buildings.
When dealing with the homogeneous model, it should be taken in to account that
cracks can only follow the mesh lines. As can be seen from Figure 10, where the
failure mechanism obtained by means of the homogenized model is represented,
such model appears in term of cracks, in satisfactory agreement with the
heterogeneous one. In particular, crack b1 is similar to a1, b2 to a2, b3 to a3. b4
appears slightly different and propagates on the right side, but maintaining the same
pattern (vertical crack). b5 is again similar to a5, a6 to b6, b7 to a9, b8 to a11, b14 to
a19, b15 to a14, b16 to b15, b17 to a16, b18 to a18 and b19 to a17. Cracks b10, b11,
b12 and b20 exhibit some discrepancies. However, from a global point of view, the
agreement seems quite satisfactory, both models providing comparable failure
mechanisms.
12
(a)
(b)
Figure 7: Typical Wall Collapse: (a) Heterogeneous and (b) Homogenous Model.
Form simulations results, it is found that the limit analysis load of the heterogeneous
model with weak mortar equals 8.9% of the self-weight and the limit load of the
homogeneous model is equal to 6.9% of the self-weight, which are again in
reasonable agreement (20% difference). The limit load of the heterogeneous model
with strong mortar is equal to 34% of the self-weight, whereas the limit load of the
homogeneous model is equal to 32% of the self-weight, again in reasonable
agreement (10% difference). It is also interesting to observe that the introduction of
strong mortar significantly increases the limit load of the structure (almost four
times). A synopsis of the failure loads found is summarized in Table 2.
Joints
Heterogeneous model
(% of self-weight)
Homogenous model
(% of self-weight)
Accuracy of
homogeneous model (%)
Weak
mortar
8.7
6.9
80
Strong
mortar
34.4
31.6
90
Table 2. Comparison between heterogeneous and homogenized limit loads.
13
Finally, it is noted that the analysis was performed on a standard PC Intel Pentium
Dual 2.12 GHz equipped with 3GB RAM. A comparison in terms of processing
time, only for computing, indicates that the homogeneous model saves about 95%
calculation time (30 vs. 600 seconds) and mesh preparation times (three vs. sixteen
hours).
7 Conclusions
Non-linear tools often imply expensive computational costs, a good knowledge
about non-linear processes and a large time to build the model and perform the
analysis. This problem was addressed here by means of a geometrical investigation
and homogenization of masonry. In particular, the behaviour at collapse of one
perimeter wall belonging to the Guimarães castle in Portugal was investigated by
means of a rigid-plastic homogenization procedure, accounting for the actual
disposition of the blocks constituting the wall and the texture irregularity given by
the variability of dimensions in the blocks.
After a simplification of the geometry and assuming mortar joints reduced to
interfaces, homogenized masonry in- and out-of-plane strength domains were
evaluated on a number of different representing elements of volume (RVEs) having
different sizes and sampled on the walls of the castle.
By means of such strength domains, a homogenized limit analysis was carried out
on a wall of the castle (Alcaçova) and results were compared with those provided by
a standard heterogeneous discretization of the domain. The comparison, in presence
of both good and weak mortar joints, have proved a satisfactory reliability of the
homogenization proposed performed on different REVs with blocks having variable
dimension.
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14
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