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Engineering Structures 32 (2010) 1466–1478
Contents lists available at ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Numerical models for the seismic assessment of an old masonry tower
Fernando Peña a,∗ , Paulo B. Lourenço b , Nuno Mendes b , Daniel V. Oliveira b
a
Instituto de Ingeniería, Universidad Nacional Autónoma de México, Edificio 2 – 401, Circuito Escolar, Ciudad Universitaria, 04510 Mexico city, Mexico
b
University of Minho, Department of Civil Engineering, ISISE, Campus Azurem, 4800-058 Guimarães, Portugal
article
info
Article history:
Received 21 August 2008
Received in revised form
20 April 2009
Accepted 25 January 2010
Available online 19 February 2010
Keywords:
Dynamics
Non-linear analysis
Pushover
Tower
Rigid elements
abstract
The present paper describes the seismic assessment of the Qutb Minar in Delhi, India. Three models
with different levels of complexity and simplification were developed. The use of these models allows
one to overcome the complexity of the study of the seismic behavior of ancient masonry structures; by
combining the results of the different models it is possible to obtain a better and more comprehensive
interpretation of the seismic behavior. The models were used for non-linear static (pushover) and nonlinear dynamic analyses. The static and dynamic analyses give different behaviors, indicating that the
push-over analysis should be used carefully in the seismic assessment of masonry structures. For the
static analysis, the base of the tower is the most vulnerable part; while according to the dynamic analysis,
it is the upper part of the tower. This last behavior is according to the historical damage suffered by the
tower due to earthquakes. The different behaviors can be explained by the influence of the higher modes
of vibration.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
The Qutb Minar is the highest monument in India and one of the
tallest stone masonry towers in the world. It is considered one of
the most important monuments of Delhi. The minaret was built as
a part of the Quwwatul-Islam Mosque to call the prayers and as a
sign to glorify the victory of Islam against idolatry. Its construction
started around 1200 and was finished by 1368 [1].
The seismic history of the city is a testimony of the risk that
historical constructions are subjected to. Delhi suffers near and far
seismic activity. According to the Indian Seismic code [2], Delhi
is in the Indian seismic zone IV, which is considered to have a
severe seismic intensity. In general, earthquakes with magnitude
of 5 to 6 are common, while a few earthquakes of magnitude 6 to
7 and occasionally of 7.5 to 8 have been also registered [3]. Thus,
restoration and conservation works have been carried out in the
Qutb Minar since its construction.
In the framework of the EU-India Economic Cross Cultural Programme ‘‘Improving the Seismic Resistance of Cultural Heritage
Buildings’’, aimed at the preservation of ancient masonry structures with regard to the seismic risk, the seismic assessment of the
Qutb Minar was carried out.
The evaluation of the seismic behavior of ancient masonry
structures requires specific procedures, since their response to
∗
Corresponding author. Tel.: +52 55 56233600x8404.
E-mail addresses: [email protected] (F. Peña), [email protected].pt
(P.B. Lourenço), [email protected] (N. Mendes),
[email protected] (D.V. Oliveira).
0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2010.01.027
dynamical loads often differs substantially from those of ordinary
buildings. In order to obtain a reliable estimate of the seismic risk,
it is desirable to perform full dynamical analyses that describe the
effective transmission and dissipation of the energy coming from
the ground motion into the structure [4]. However, sometimes the
available analytical tools require a great amount of computational
resources that are not commonly available. Therefore, procedures
are necessary to overcome the complexity of the study of the
seismic behavior of ancient masonry structures.
In general, modeling the non-linear mechanical behavior
of ancient masonry structures by means of three-dimensional
models is not possible because it requires a great amount of
computational resources. However, the use of simplified models
in combination with very refined models allows one to overcome
those restrictions. The results obtained from the complex models
can be used as the basis for a better conception of the simplified
models, which can resort to different analysis tools. Moreover, by
combining the results of different models, it is usually possible
to obtain a better and more comprehensive interpretation of the
seismic behavior [4].
The seismic assessment of the Qutb Minar has been performed
by using a stepped strategy, conveniently divided into:
• Background, which means gathering the previous information
that it is possible to collect, as for example: the historical
information (behavior, damages, repairs, etc); the materials and
geometry description; and preliminary studies (survey of the
structure, monitoring, field research and laboratory test, etc).
• Numerical analysis tools, which is related to the selection
of the different analysis tools. Here the following aspects
F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
1467
should be considered [5]: the relation between the analysis tool
and the information sought; the cost and time requirements;
and the use of an analysis tool that can be validated and
assessed.
• Process of tuning or the model’s calibration, where the data
obtained in the first step, as well as the results of other models,
can be used for the calibration.
• Types of analysis, which depending on the analysis tools
chosen and the computer resources available can be: elastic,
non-linear, static, dynamic, etc. In general, the types of analysis
can also be divided in an analysis for validation of the numerical
tools and an analysis for the seismic assessment.
• Parametric analyses, where, if the data collected or the tuning
process are not enough to define all necessary variables,
some values are proposed using as reference other similar
structures or materials. Then, parametric analyses are required
to overcome the uncertainty associated with the use of these
‘‘estimated values’’.
As a consequence of the above, the paper is based on a case
study but also provides a very relevant discussion on an issue
that has received insufficient attention: the seismic analysis of
masonry structures without a box behavior. The seismic analysis of
masonry structures with a box behavior received much attention
in the past and is relatively well consolidated in the research
environment, e.g. [6–9]. On the contrary, the seismic analysis
of masonry structures without a box behavior remains complex,
with some apparent consensus in the use of a limit analysis of
block mechanisms, e.g. [10–12]. The use of a limit analysis is not
trivial for special structures, which constitute most monumental
structures. Thus, this paper also presents a significantly wider
contribution to the field, by demonstrating the possible inadequacy
of push-over analysis to tackle the seismic assessment of masonry
structures without a box behavior.
2. Background
2.1. Historical information
The Qutb Minar is one of the tallest stone masonry towers in the
world. The tower has five balconies (Fig. 1). The construction began
during the reign of Qutb-ud-din around 1202, but the erection
stopped at the first storey. The next ruler, Iltutmish, added the next
three storeys. The tower was damaged by lightning in 1326 and
again in 1368. In 1503 Sikandar Lodi carried out some restoration
and enlargement of the upper storeys [1].
The minaret has five storeys. The topmost of the original four
storeys was replaced by two storeys in 1368. Balconies are placed
at the end of each storey. The minaret had originally a cupola,
which fell during an earthquake in 1803 and was replaced by a
new cupola in late Mughal style in 1829. However, it was removed
in 1848. In 1920 some facing stones needed some repairs and
major structural work was carried out in 1944. The damaged stones
were replaced using lime mortar and stainless-steel dowels. The
foundation was strengthened by grouting in 1971, and further
work was carried out in 1989 to replace damaged facing blocks and
to strengthen the inner core [1].
2.2. Geometrical and structural description
The Minar is a typical example of the classical Indo Islamic
architecture. It directly rests on a 1.7 m deep square ashlar masonry
platform with sides of approximately 16.5 m, which in turn overlies
a 7.6 m deep lime mortar rubble masonry layer, also square, with
sides of approximately 18.6 m. The bedrock is located around
50–65 m below the ground level. The Minar cross-section is
Fig. 1. Qutb Minar in Delhi, India.
circular/polilobed, the base diameter being equal to 14.07 m and
tapering off to a diameter of 3.13 m at the top, over a height of
72.45 m (Fig. 2(a),b). The tower is composed of an external shell
corresponding to a three leaf masonry wall and a cylindrical central
core [13].
The core and the external shell are connected by a helicoidal
staircase and by 27 ‘‘bracings’’ composed of stone lintels with an
average cross section of 0.40 × 0.40 m2 . The staircase is spiral,
disposed around the central masonry shaft, and it is made of Delhi
quartzite stone. Each storey has a balcony and the uppermost
storey finishes with a platform. Each storey has a different pattern
in construction and ornamentation (Fig. 2(b)): (a) the first storey
has angular flanges alternating with rounded flutes; (b) the second
storey has circular projections; (c) the third storey has star shaped
projections; and (d) the fourth and fifth storey projections are
round.
In plan the minaret can be considered as approximately circular,
with a base of 14.07 m of diameter and tapering to a diameter
of 3 m at the top, with a total height of 72.45 m. The Minar is
also provided with diffuse ventilation openings that can be divided
in some smaller openings on three levels and larger openings as
windows and doors (Fig. 2(c),(d)). In correspondence with the
second and third levels of the smaller openings the cross section
of the tower decreases almost to 50% of the total [13].
2.3. Materials
The Minar outer shell is built of three leaf masonry (Fig. 2(a)). In
the first three storeys the external veneering is of ashlars of red and
buff colored sandstone, whereas the internal veneering is of ashlars
in Delhi quartzite stones. In the two upper stories the external
veneer is made of white marble stones and the internal veneering
is of red sandstone. The infill is composed of rubble stone masonry,
mainly with stones taken from the temples destroyed during
the Islamic dominion. These three layers are held up with iron
dowels incorporated frequently in between the masonry layers.
The central shaft is of rubble masonry with a Quartzite stone facing.
The mortar used is of lime with brick powder as an aggregate. The
thickness of the outer shell tapers from 3 m at the base to 0.6 m at
the top [1,13].
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41.71 m
c
a
b
d
Fig. 2. Geometrical survey: (a) vertical section of the minaret; (b) cross sections at different levels; (c) small size ventilation openings; (d) tapered windows.
Fig. 3. Mode shapes and the associated frequencies in Hz for the first 10 modes [13].
2.4. Survey of the structure
Ambient vibration tests were performed by the University of
Minho with the aim of defining the modal parameters (natural
frequencies, mode shapes and damping coefficients) of the Minar,
as well as to evaluate the degree of connection between the
central shaft of the tower and the external shell. For details on
the experimental testing program, please refer to [13]. These tests
were carried out considering several test positions, at different
heights, in order to proceed with the modal identification of the
tower.
Several natural frequencies and the corresponding modes were
defined. The Unweigth Principal Component (UPC) technique was
used to define the dynamic parameters. It operates in the time
domain and is based on the Stochastic Subspace Identification (SSI)
method, which allows a modal parameters estimation with high
frequency resolution [13]. Fig. 3 depicts the modal shape and the
associated frequency for the 10 measured modes. Eight bending,
one torsional and one axial mode shapes were estimated. It is
stressed that the two first bending modes were not clearly defined
at the top, especially at the fourth balcony. The bending modes
directions are almost perpendicular for the closely spaced pairs of
frequencies. This is due to the axisymmetric cross section of the
tower.
3. Numerical analysis tools
Three different numerical models were considered to evaluate
the structural behavior of the minaret. Two models use the well
known Finite Element Method, both are three-dimensional models
but one uses 3-D solid elements (Solid model) while the other
one was performed with 3-D composite beams (Beam model). The
third model uses 2D in-plane elements based on the Rigid Element
Method (Rigid model).
3.1. Three-dimensional Solid model
The Solid model was implemented by using the Finite Element
(FE) software DIANA [14]. The external shell was modeled as a
three layered wall. The central shaft was modeled using a single
type of masonry. The foundation, the doors and windows of the
minaret were also modeled (Fig. 4).
The three levels of openings situated below the first, second
and fifth balconies were represented in the FE model with different
materials, with the mechanical properties defined as a percentage
of the materials in the surrounding areas, namely the outer layer
of the outer shell. In order to include the openings, the material’s
properties of the first, second and third openings were thus
considered as 70%, 50% and 60% of the material in the outer layer
near each opening. It should be noted that these layers cross all
three vertical layers.
The helicoidal staircase was modeled using flat shell elements,
forming horizontal slabs, where its thickness and elastic modulus
were optimized, in order to minimize the differences between
the experimental results and the FE model, as explained below in
Section 4. The stairs have an important role, because they make the
connection between the inner shaft and the outer wall.
The full model involves 65,912 elements with 57,350 nodes,
resulting in about 172,000 degrees of freedom (DOF). The base
of the foundation was considered as fully restrained, given the
existence of the foundation block.
F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
a
1469
c
b
y
z
x
Fig. 4. Solid model: (a) general view; (b) staircases modeled as horizontal slabs; (c) detail of the FE model.
a
b
Fig. 5. Material constitutive axial laws: (a) Solid and Beam element [14]; (b) Rigid element [15].
3.2. Three-dimensional Beam model
The Beam model was performed using the DIANA code [14]
based on the Finite Element Method. Three dimensional tapered
beam elements of three nodes based on the Mindlin–Reissner [14]
theory were used. Composite beams were adopted in order to take
into account the different layers of materials, as well as the center
core of the minaret, in a cross section. Each composite cross section
was defined with four different materials according to the layer.
The balconies were considered as additional localized masses. The
model has 41 nodes (120 DOF) and 20 elements.
It is noted that, in this model, it is not possible to model the
influence of the staircase, thus a perfect connection between the
shaft and the core is considered. In addition, the openings were
neglected. These simplifications were made after the analysis of
the results obtained by the Solid model, which shown that they
have minor influence in the global behavior of the structure.
3.3. In-plane Rigid model
A simplified in-plane model of the minaret based on the Rigid
Element Method was developed. The Rigid Element Method idealizes the masonry structure as a mechanism made of rigid elements
and springs. For further details about the method please refer
to [15]. The rigid elements used can be defined only with a rectangular cross section. For this reason an equivalent square cross section and equivalent isotropic material were considered (Table 1).
In total, the numerical model has 39 elements and 117 DOF.
3.4. Material model
Isotropic materials have been considered for all models because
the masonry is not made by blocks and layers of mortar, where
an orthotropic behavior can be expected [16]. The masonry
found is instead a rubble material made with irregular stones
embedded in thick lime mortar, where no weak planes can be
detected. Therefore, the mechanical behavior of this composite can
be approached as an isotropic material, similar to concrete [17].
Orthotropic behavior can be expected for the external veneer
since it is made of stone blocks and layers of mortar. However,
the global behavior of the minaret seems to be not influenced
very much by the external veneer, as the rigid element model
shows, and bending along the vertical direction will be the
predominant behavior. Therefore, this layer can also be considered
as a homogenized isotropic material in order to reduce the
computational effort.
The models consider that the materials of levels 1 to 3 are
different from the materials of levels 4 and 5. This hypothesis was
adopted because the last two levels were constructed in a later
period, the adopted stone is different and the experimental modal
shapes show that the deformation is concentrated on the last two
levels, see Fig. 3.
Finally, the elastic properties of the material were obtained by
calibration of the models with the frequencies measured in the
tower (see next Section). The strength properties were taken from
the literature [1,18–20] and parametric analyses, varying these
parameters, were performed in order to evaluate the influence of
these properties in the global behavior of the Minaret.
The constitutive model used to simulate the non-linear
properties of the Solid and Beam model was a traditional smeared
cracking model, with an exponential tension softening and a
constant shear retention of 0.01 [21]. A parabolic compressive
behavior was considered for all materials (Fig. 5(a)), see [14]
for details. The compressive strength (fc) was different for each
material, while for the tensile strength (ft) a very low value was
assigned (50 kPa). The tensile fracture energy (Gf ) was set equal
to 20 Nm/m2 . The objective is to simulate a very weak material in
tension.
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F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
Table 1
Equivalent square cross section and the mesh considered in the Rigid model.
Area (m2 )
Inertia (m4 )
Level
Equivalent thickness (m)
Real
Equivalent
Real
Equivalent
Error(%)
0
11.67
127.36
136.25
6.98
1762.80
1547.10
−12.24
1
7.75
54.69
60.08
9.86
360.07
300.85
−16.45
2
5.91
31.62
34.97
10.58
123.43
101.91
−17.43
3
4.45
18.08
19.81
9.54
38.94
32.71
−16.00
4
3.45
10.80
11.91
10.26
14.25
11.82
−17.00
5
2.41
5.18
5.85
12.94
3.58
2.85
−20.51
Error (%)
In the case of the Rigid model, the constitutive law for axial
springs is parabolic in compression and bi-linear in tension with
softening (Fig. 5(b)). A Mohr–Coulomb law is considered for the
shear springs in order to relate the shear stresses with the axial
stresses [15]. For this model only compressive (f 0 c), tensile (ft) and
shear (fs) strength are necessary, as well as the friction angle (ϕ ).
The compressive strength was different for each material, while
the tensile and shear strength as well as the friction angle were
considered equal for both materials, as was done in the smeared
cracking model.
The damping is considered as a viscous damping C , depending
on the mass M- and stiffness K by means of the Rayleigh
formulation C = aM + bK , where a is the mass proportional
damping constant and b is the stiffness proportional damping
constant. The value of these two parameters was optimized in
order to obtain similar values as the experimental damping for
each frequency derived from the environment vibration tests,
being: a = 0.2256, b = 0.00057.
4. Process of tuning the model
Model updating was performed in order to match the natural
frequencies arising from the experimental investigation. Each
model was updated independently and the elastic modulus was
the parameter considered for the calibration of the different sets
of materials, ranging between reasonable values for the different
types of masonry.
Considering that the models present an axial symmetry, the
numerical pairs of corresponding bending mode shapes and
frequencies are identical (and orthogonal). Thus the averaged
experimental values were considered for the model calibration.
The modal updating was performed by using the methodology
proposed by Douglas and Reid [22], in which the frequencies of the
structure can be estimated by means of:
fiD (X1 , X2 , . . . , XN ) = Ci +
N
X
Aik Xk + Bik XkN
(1)
i=1
where f D is the frequency estimated, Xk are the variables to
calibrate, A, B and C are constants, and N is the number of
frequencies used in the calibration. Thus, the variables are obtained
by an optimization of the function J:
J =
m
X
wi εi2
(2)
i=1
εi = fiEMA − fiD (X1 , X2 , . . . , XN )
(3)
where f
is the experimental value of the frequencies and w is a
weight factor.
The optimization process took place in the GAMS software [23]
that determines the optimal values by a comparison of the
EMA
optimization variables in limit cases with the experimental
values, and then determines the values that minimize the error
between the numerical and experimental values. Matching the
frequencies does not guarantee that the mode shapes are also
correctly matched. It is also straightforward to incorrectly match
one frequency but it is unexpected to incorrectly match several
frequencies, as done here. The mode shapes obtained with the
analyses will be only qualitatively compared with the measured
modal shapes in order to corroborate the results. This is due to the
low quality of some of the mode shapes and the low number of
stations used in the dynamic identification procedure [13].
As the Solid model has a very refined description of the
geometry of the structure, ten variables were optimized. They were
the elastic modulus of the different layers of the shaft and core,
as well as the thickness and the elastic modulus of the staircase
(Table 2). Five variables were considered in the Beam model that
corresponds with the different materials of the minaret (Table 3).
Finally, due to the simplifications made in the Rigid Element model,
only two isotropic homogeneous materials were considered, one
for levels 1 to 3 and the other for levels 4 and 5 (Table 4).
5. Types of analyses
The different types of analysis performed can be divided into
two main groups: (a) validation of the analysis tools; and (b)
evaluation of the seismic behavior of the tower. In the first group
we have the eigenvalue and self-weight analyses. The calculus of
the frequencies and modal shapes allow one to validate the models
in the elastic field; while the self-weight analysis allow one to
verify the correct geometrical description of the structure and the
properties of the materials. In this context, the 3D Solid model was
used to validate the hypothesis in which the other two models are
based, mainly: the stairs are sufficiently stiff to consider that the
inner shaft works together with the external shell; the openings
can be neglected, since they do not influence the global behavior
of the minaret.
In the second group, the non-linear static (pushover) and
dynamic analyses are considered. It is worth noting that the
Solid model was not used for a dynamic analysis because the
great amount of computational resources and time required.
The dynamic analyses were carried out using five synthetic
accelerograms compatible with the design spectrum of the Indian
Seismic code for Delhi considering the experimental damping for
the first mode of 2.5%. Fig. 6 shows the design spectra for the
different values of damping. The vertical lines correspond to the
periods of the structure. For the minaret the design spectrum
corresponds to a damping of 2.5%, which is the damping emerged
from the environment tests. For this spectrum, the maximum
ground acceleration corresponds to 0.25 g, while the constant
branch of the spectrum has a spectral acceleration of 0.63 g. It can
be seen that the first period is in the lower part of the spectrum
F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
1471
Table 2
Optimized values of the materials used in the Solid model.
Material
Elastic modulus (GPa)
Specific mass (Kg/m3 )
Poisson’s coefficient
f 0 c (kPa)
ft (kPa)
Shaft 1–3
Shaft 4–5
External shell inner layer 1–3
External shell external layer 1–3
External shell core layer 1–3
External shell core layer 4–5
External shell inner layer 4–5
External shell external layer 4–5
Stairs
1.545
0.300
6.171
2.000
0.785
0.300
6.602
2.000
3.689
1800
1800
2600
2300
1800
1800
2600
2600
2000
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
1000
600
5200
2500
1000
600
3000
5200
elastic
50
50
50
50
50
50
50
50
Table 3
Optimized values of the materials used in the Beam model.
Material
Elastic modulus (GPa)
Specific mass (Kg/m3 )
Poisson’s coefficient
f 0 c (kPa)
ft (kPa)
Masonry infill 1–3
Masonry infill 4–5
Sandstone
Quartzite
Marble
1.00
0.60
2.50
5.21
3.00
1800
1800
2300
2600
2600
0.2
0.2
0.2
0.2
0.2
1000
600
2500
5200
3000
50
50
50
50
50
Table 4
Optimized values of the materials used in the Rigid model.
Material
Elastic modulus (GPa)
Specific mass (Kg/m3 )
Poisson’s coefficient
f 0 c (kPa)
ft (kPa)
fs (kPa)
ϕ (o )
Masonry 1–3
Masonry 4–5
3.18
0.57
1900
1900
0.2
0.2
3000
1500
35
35
45
45
15
15
6. Analyses for validation
model presents the higher errors for the frequency associated to
the torsion (Table 5). This model calculated the fifth mode as
torsional, while the ambient vibration presented this mode as the
seventh. The reason for this difference is possibly the staircase.
The Solid model calculated the ‘‘correct sequence’’ of the modes,
with an error associated to this torsional mode of about 8%. On
the other hand, the model that presents in general the smaller
errors in the calculated frequencies (less than 5%) is the simplified
Rigid Element model; except for the vertical mode that has an error
up to 10%. Here, it is noted that the vertical mode was difficult
to calibrate. Solid and Beam models consider this mode as the
8th, while in the experimental test this is the 10th mode. Note
that the true structure and the Solid model are not axisymmetric,
meaning that a clear orientation of the modes in two essentially
orthogonal directions x and z have been found. On the contrary,
the Beam model is fully axisymmetric, meaning that the directions
x and z lose their meaning. As any set of orthogonal directions
would provide two equal eigenvectors, the frequencies related to
the consecutive bending modes in two orthogonal directions have
been lumped to a single value in Table 5. Finally, the Rigid model
is two-dimensional, meaning that only one frequency is found
for each pair of consecutive bending modes and a torsional mode
cannot be found.
The Solid model considers the different openings (windows,
doors and the small ventilation openings) of the minaret. These
openings should give some kind of asymmetry to the structure.
However, the differences between the pairs of bending modes
obtained are very low and showing that the differences in the pairs
of bending modes obtained experimentally are not only related
with the openings. These differences can be related to other aspects
of the structure as differences in the quality and properties of the
materials, differences in the geometry of the minaret, some cracks,
etc.
6.1. Eigenvalue analysis
6.2. Self-weight analysis
In general, all the models have a good match between the
experimental and calculated frequency values (Fig. 7). The Beam
A self-weight analysis was performed in order to evaluate
the stress pattern. All three models give similar results. The
Fig. 6. Design spectrum of the Indian Seismic Code for Delhi for different values of
damping.
(0.20 g), while the third to tenth periods are located in the constant
branch of the spectrum (0.63 g).
It is noted that the response spectrum can have some influence
on the result and several codes define a near- and far-fault
earthquake, which must be associated with a macro-zonation
map. It is believed that the use of historical earthquake records,
particularly if scaled to a different PGA, raise also a lot of doubts
on representativeness. For the purpose of discussing the safety
of the particular structure and discussing the difficulty of using a
push-over analysis for the monumental structures without a box
behavior, it is believed that the choice of using the code response
spectrum is adequate. The procedure adopted is aligned with the
proposal of recent codes, e.g. Eurocode 8 [24], which clearly states
that the code’s elastic response spectra should be used for a nonlinear time history analysis.
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F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
Fig. 7. Modal shapes for the models, Solid (left), Beam (center) and Rigid (right).
Table 5
Comparison among experimental frequencies and the frequencies obtained with the three different models.
Mode shape
1
2
3
4
5
6
7
8
9
10
Comment
Frequencies (Hz)
1st Bending, z
1st Bending, x
2nd Bending, x
2nd Bending, z
3rd Bending, x
3rd Bending, z
1st Torsion
4th Bending, x
4th Bending, z
1st Vertical
Experimental
Solid
Error (%)
Beam
Error (%)
Rigid
Error (%)
0.789
0.814
1.954
2.010
3.741
3.862
4.442
5.986
6.109
6.282
0.71
0.71
2.07
2.09
3.55
3.59
4.80
5.98
6.02
5.35
−10.01
−12.77
0.734
−8.42
0.778
−3.17
2.257
13.87
1.886
−4.84
4.129
8.62
3.582
−5.77
—
—
5.93
3.98
−5.10
−7.04
8.06
−0.10
−1.45
−14.83
3.656
−17.69
6.665
10.21
6.419
5.65
6.098
−2.93
7.061
12.41
Table 6
Compressive stresses at the bottom of each material layer (middle plane) of the Beam model and the map of the vertical stresses in the Rigid model.
Material
Maximum axial stress σmax (KPa)
Compressive strength f 0 c (kPa)
σmax /f 0 c (%)
Masonry infill 1–3
494
2000
24.70
Masonry infill 4–5
102
600
17.00
Sandstone
617
2500
24.68
Quartzite
1290
5208
24.77
127
3000
4.23
Marble
total weight of the minaret is around 75,000 kN and the
average compressive stress at the base is 0.6 MPa. The maximum
compressive stress is 1.29 MPa at the bottom of the external
layer of the shell (the reference adopted in the middle plane of
each layer). The materials remain in the elastic range and the
maximum stresses at the base of the minaret are around 24% of
the compressive strength (Table 6).
6.3. Performance of the numerical models
The objective in making different models was to combine the
results of different models in order to obtain a better and more
comprehensive interpretation of the seismic behavior. Thus the
3D Solid model was, mainly, used to validate the hypothesis
on which the other two models are based. In this context, the
Rigid model (Pa)
computational efficiency of the simplified models (Beam and Rigid
models) compared with the 3D Solid model was exploited in order
to perform a full non-linear dynamic analysis. This type of analysis
is precluded in the Solid model due to the level of refinement of
the mesh and the high computational cost (CPU time and space
in hard disk), see Table 7. The Solid model takes 24 more times
to run than the Beam model and 12,000 more times to run than
the Rigid model, making the non-linear time integration analysis
unwieldy. It is worth noting that the Beam model requires 480
more times to run than the Rigid model; although both models
have almost the same number of degrees of freedom. This is mainly
due to the number of integration points of each model. The Rigid
model has 195 integration points, while the Beam model has
14,000.
The Rigid model is able to perform fast non-linear dynamic
analysis with few computational resources. For example, the Beam
F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
1473
Table 7
Comparison between the degrees of freedom (DOF) and the computational cost for a non-linear static analysis using the three different models.
Model
Beam (3D)
Rigid (2D)
Solid (3D)
1
2
DOF
120
117
172,000
DOF ratio
CPU time (min)1
CPU ratio2
Space in hard disk (MBytes)
Space ratio2
1
1
1400
120
0.25
2880
1.0
0.002
24
92.3
0.7
6958
1.0
0.008
75.4
Using a one-core Pentium IV Processor (3 Ghz), 1 Gb in RAM and for non-linear static analysis.
Using the Beam model as reference.
a
b
c
Fig. 8. Pushover analysis, capacity curves.
model (in the scope of general purpose software) requires 23 h to
perform a non-linear dynamic analysis, while the Rigid model (in
the scope of specific, stand alone software) needs only 20 min for
the same analysis.
7. Analyses for the evaluation of the seismic behavior
7.1. Non-linear static analysis
A non-linear static pushover analysis was carried out considering a uniform acceleration distribution. The load was applied with
an increasing acceleration in the horizontal direction and a control point at the top of the tower was considered. Fig. 8 shows the
capacity curve lateral displacement–load factor (shear base/selfweight). Rather similar behaviors were found for the models. It can
be seen that the average load factor is 0.21 for the Beam and Rigid
models, while the load factor for the Solid model is 0.18. It is worth
noting that the materials do not fail by compressive stresses and
the tower collapses by overturning at the base.
7.2. Non-linear dynamic analysis
A non-linear dynamic analysis was performed with the Rigid
and Beam model, considering a Rayleigh model for damping with
the damping values emerged from the investigation campaign
a
Fig. 10. Results of the Rigid model for record 1 at time 18.3 s: (a) deformed shape;
(b) axial stresses [Pa]; (c) axial strains.
(approximately equal to 2.5% for the first frequency). The dynamic
analyses were carried out using five synthetic accelerograms
compatible with the design spectrum of the Seismic Indian
code.
Fig. 9 shows the load factor (shear force/own weight) and the
absolute maximum displacements for each level obtained with the
Rigid model. It is worth noting that these values are the maximum
and do not necessary occur at the same time. The average load
factor at the base is 0.20 and remains almost constant for the first
level, while for the second balcony the average load factor is 0.25.
The average load factors of the third and fourth balconies are of
0.75 and 1.10 respectively. This means that the amplification of the
seismic loads is concentrated at the last two levels. Displacements
of levels 1, 2 and 3 increase practically in a linear way, while the
displacements of level 5 are in general almost a double of the
displacements of level 4.
Fig. 10 shows the deformed shape and the map of axial stresses
and strains at the instant 18.3 s of record 1 of the Rigid model.
As can be seen, only levels 4 and 5 present a lateral deformation
(Fig. 10(a)). The axial stresses in levels 1 and 2 are practically
uniform (Fig. 10(b)), since the bending moment is not present.
Axial compressive stresses on levels 4 and 5 are concentrated on
one side due to the bending moment and therefore the axial strains
are concentrated on the fourth balcony (Fig. 10(c)).
b
Fig. 9. Maximum absolute results along the height of the minaret for dynamic analyses with the Rigid model: (a) load factor; and (b) lateral displacements.
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F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
a
b
Fig. 11. Maximum absolute results along the height of the minaret for dynamic analyses with the Beam model: (a) load factor; and (b) lateral displacements.
a
b
Fig. 12. Comparison between the Rigid and Beam models for average results along the height of the minaret for dynamic analyses: (a) load factor; and (b) lateral
displacements.
These results show that levels 4 and 5 are the most vulnerable.
Especially level 5 presents the highest drift (3.0%). Analyses show
that the higher modes of vibration have a great influence on the
seismic response of the minaret.
Fig. 11 shows the maximum load factor and displacements for
each level with the Beam model. The average load factor at the base
is 0.16 and increases to 0.18 for the first level. The second balcony
has an average load factor of 0.28, while the third and fourth
balconies have an average load factor of 0.47 and 0.9 respectively.
The maximum displacements at the top vary from 0.35 to 0.65 m,
having a maximum drift of 6%. The results show that for all the five
records levels 4 and 5 are the most vulnerable too.
On the other hand, Fig. 12 shows the comparison between
the Rigid and the Beam models for the dynamic analysis. The
compared results are the average of the maximum load factors and
displacements for each storey. Both models have similar behavior:
the two top storeys are the most vulnerable part of the minaret.
The load factors calculated by the Rigid model are, in average, 20%
greater than the Beam model; while the displacements are 30%
lower. This means that Rigid model is slightly more rigid than the
Beam model. This is mainly due to the simplifications considered
in the Rigid model (see Section 3.3).
7.3. Remarks about the analyses performed
The different models present a similar behavior under different
types of load. However, the results obtained from the non-linear
static and dynamic analyses indicate a quite different response of
the structure to earthquakes.
The non-linear static analysis shows that the lowest part of
the structure exhibited a diffuse cracking and a base overturning
mechanism could be detected. On the other hand, the non-linear
dynamic analyses carried out indicated that the part of the Qutb
Minar more susceptible to seismic damage coincides with the
two upper levels, where the highest accelerations and drifts were
found.
The differences in the results between the static and dynamic
analyses can be explained by the high influence of the higher
modes in the seismic behavior of the tower. In fact, the
standard non-linear static analyses do not take into account the
participation of the different modes. The results of the non-linear
dynamic analyses are more representative of the real seismic
behavior of the tower, since the damage by previous earthquakes
has been concentrated in the last two levels. In this context, it
is possible to conclude that the most vulnerable part of the Qutb
Minar is the two top storeys.
8. Parametric analyses
For the study of the seismic assessment of ancient masonry
structures, parametric studies should be made when the real
properties of the material or the loads that the structure could
suffer are not well known or defined. In this particular case, as the
strength properties of the materials were taken from the literature
and differences found in the behavior between the static and
dynamic analyses, parametric analyses were carried out in order
to evaluate their influence.
It is worth noting that the parametric analyses were only
performed with the Beam model in order to have a better
description of the geometry and the material’s distribution (cross
section and different material layers), as well as to take into
account the influence of the torsional mode.
F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
Fig. 15. Damping curves used in the parametric analyses.
Fig. 13. Pushover analyses, capacity curves with different distribution of lateral
loads.
8.1. Non-linear static analysis
In order to study the influence of the distribution of the lateral
force into the pushover analysis, a second non-linear static analysis
was performed. Three different configurations of lateral loads
were considered: the forces proportional to the mass (uniform
acceleration); the linear distribution of the displacement along the
height as proposed by the Seismic Indian Code [2]; and the forces
proportional to the first modal shape.
Fig. 13 shows the three computed capacity curves. The
maximum load factor that the structure can resist depends very
much on the distribution of the forces: 0.205, 0.108 and 0.072 for
the proportional mass, the linear displacement and the first mode’s
load distribution, respectively. The load factor proportional to the
first mode is only 35% of the load factor proportional to the mass,
while the load factor proportional to the linear distribution is 53%.
It is worth noting that the collapse section changes too. In the case
of the analysis considering the forces proportional to the mass, the
section of the collapse is located at the base, while for the other
two analyses the Minaret collapses at the first balcony.
A modal pushover analysis (MPA) was performed in order to
take into account the higher modes [25]. The first mode has an
effective mass equal to 37.9% of the total mass. Thus, in order to
take into account at least the 90% of the mass, it is required to
consider seven modes for each direction, see Table 8. Fig. 14 shows
the displacement and drift of each storey obtained with the MPA
and they are compared with the average results obtained with the
dynamic analyses. As it can be observed, the obtained behavior
is similar for the first three levels, where the displacements and
drifts are similar to the non-linear dynamic analyses. However, the
behavior of levels four and five are quite different. The MPA cannot
reproduce satisfactory the failure mechanism of the last two levels.
a
1475
This difference is due to the change of the dynamic properties of
the tower during the damage process (see Table 8).
The change of the dynamic properties means that the hypotheses of the modal pushover analysis cannot be fulfilled, namely:
(a) the influence (participation factors and effective mass) of each
mode cannot be considered constant during the analysis; (b) the
peak acceleration for each mode will be different since the frequency changes; (c) the number of the required modes to have at
least the 90% of the effective mass also changes, in this particular case changing from 7 to 19; (d) the lateral force distribution for
each mode will change; (e) the modal damping will vary as the frequencies change; for the first frequencies it will be greater, while
for the higher modes it will decrease (see Fig. 15).
The change of the dynamic properties can be possibly taken
into account considering an adaptative modal pushover analysis.
However, this type of analysis becomes impractical. In this case,
it is possibly better to perform a non-linear dynamic analysis, in
which the full dynamic behavior is directly taken into account.
8.2. Non-linear dynamic analysis
In order to evaluate the influence of the most relevant
parameters in the seismic behavior of the Minar, non-linear
dynamic analyses were performed varying the tensile strength,
the tensile fracture energy and the damping. In total 75 analyses
were carried out, since three different values for each variable and
non-linear dynamic analyses using the previous five compatible
accelerograms with the Indian Seismic code were considered.
The values of the tensile strength were considered as the 2.5,
5 and 10% of the compressive strength of the masonry, while the
values of the fracture energy were considered proportional to the
values of the tensile strength, according to [18–20]. Table 9 shows
the values taken into account in these analyses.
b
Fig. 14. Comparison between the modal pushover analysis and the non-linear dynamic analyses; (a) displacements; and (b) drifts.
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F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
Table 8
Periods and effective mass of the modes considered during the modal pushover analysis (original) and calculated after the non-linear dynamic analysis for the seismic record
1 (damaged).
Mode
2
3
5
8
14
17
21
Period (s)
Effective mass (%)
Original
Damaged
Difference (%)
Original
Damaged
Difference
1.36
0.44
0.24
0.15
0.10
0.08
0.06
1.67
0.85
0.51
0.36
0.26
0.22
0.18
22
92
112
138
149
183
183
37.9
20.3
11.9
8.8
5.6
3.3
2.5
25.8
17.3
10.1
4.5
11.5
4.0
1.5
−32
−15
−15
−49
90.3
74.7
−17
Total
105
20
−39
Table 9
Variables considered in the parametric analyses.
Parameter
Reference value
Mid value
Maximum value
Tensile Strength (kPa)
Tensile Fracture Energy (Nm/m2 )
Damping for the first frequency (%)
Mass proportional damping a
Stiffness proportional damping b
50
20
2.5
0.2256
0.00057
100
50
5.0
0.4361
0.00119
200
100
8.0
0.7194
0.0017
The initial analyses were performed considering the damping
values emerged from the investigation campaign. However, these
damping ratios were obtained through low amplitude vibrations
(ambient vibration), which are sometimes not representative of
the damping that the structure can present during a seismic event.
Historical masonry constructions subjected to seismic events can
present damping ratios around 8%–10% [26]. Therefore, three
different damping ratios were used: the damping obtained from
the ambient vibration and a proportional damping to ambient
vibration but considering 5% (a = 0.4361, b = 0.00119) and 8%
(a = 0.7194, b = 0.00220) for the first frequency (Fig. 15).
Fig. 16 shows the average results for the five synthetic records
with damping of 2.5% (experimental) varying the tensile strength
(left) and the fracture energy (right). It can see that the fracture
energy has practically no influence on the average behavior
of the minaret in terms of the forces and displacements. The
tensile strength has more influence in the global behavior. This
influence can be noticed in the top storey, in which the lateral
displacement and the drifts have differences around 10% between
extreme values. The shear forces and load factors increase with this
parameter.
On the other hand, the parameter that has more influence on the
global behavior is damping. Fig. 17 shows the average maximum
displacements and drifts for different values of damping. As can
be seen, the global behavior of the tower is similar to previous
analyses, being independent of the damping ratio considered. The
last two storeys of the minaret are the most vulnerable part of
the tower. However, considerable differences in the values of the
displacements and the drift were found. In general, the first 2
storeys have a similar behavior independently of the damping and
strength characteristics. But in the last three storeys the drifts and
displacements increase as the damping decreases.
9. Conclusions
A simple strategy of analysis for the seismic assessment of
the Qutb Minar in Delhi, India was presented. Three different
models with different levels of complexity and simplifications
were developed. The use of these three models allows one to
overcome the complexity on the study of the seismic behavior
of ancient masonry structures; since combining the results of
the different models it is possible to obtain a better and more
comprehensive interpretation of the seismic behavior. It is clear
that this strategy can be used for other similar structures. The
models present a similar behavior under the same loads and types
of analysis. However, the results obtained from the non-linear
static and dynamic analyses indicate quite a different response of
the structure to earthquakes.
The non-linear static analysis shows that the lowest part of
the structure exhibited a diffuse cracking and a base overturning mechanism could be detected. On the other hand, the nonlinear dynamic analyses carried out indicated that the part of the
Qutb Minar more susceptible to seismic damage coincides with the
two upper levels, where the highest accelerations and drifts were
found. The differences in the results between the static and dynamic analyses are due to the high influence of the higher modes
in the seismic behavior of the tower. In fact, the non-linear static
analyses do not take into account the participation of the different
modes. The modal pushover analysis, which considers the influence of the higher modes, cannot reproduce the appendix like behavior of the last two levels satisfactorily. This is due to the change
of the dynamic properties of the tower during the damage process.
The results of the non-linear dynamic analyses can be
considered more representative of the real seismic behavior of
the tower, since the historical damage by earthquakes has been
concentrated in the last levels. In this context, it is possible to
conclude that the most vulnerable part of the Qutb Minar is the
two top storeys.
The results from the different approaches allow one to conclude
that: (a) the distribution of the lateral forces has a large influence
on the pushover analyses. The section at the minaret fails and the
load factor depends significantly on the lateral force distribution;
(b) it is not recommended to perform pushover analyses to study
the seismic behavior of masonry towers, since it is not possible to
neglect the influence of the higher modes; (c) in this particular
case, the modal pushover analysis cannot reproduce the results
obtained with the non-linear dynamic analysis; (d) damping has a
large influence on the dynamic time integration results in terms of
displacements, whereas the effect of the tensile strength and the
fracture energy is only minor; (e) it is recommended to perform
a full non-linear dynamic analysis instead of a static non-linear
analysis for the seismic assessment of the tower.
Acknowledgement
This research was part of the activities of the EU-India Economic
Cross Cultural Programme ‘‘Improving the Seismic Resistance of
Cultural Heritage Buildings’’, Contract ALA-95-23-2003-077-122.
F. Peña et al. / Engineering Structures 32 (2010) 1466–1478
1477
(a) Displacements.
(b) Load factors.
Fig. 16. Average results along the height of the minaret for the parametric dynamic analyses varying the tensile strength (left) and the fracture energy (right).
a
b
Fig. 17. Average results along the height of the minaret for a Strength of 50 kPa and a Fracture Energy of 20 Nm/m2 varying the damping; (a) displacements; and (b) drifts.
F. Peña acknowledges funding from the FCT grant contract
SFRH/BPD/17449/2004, as well as the partial support of the project
PAPIIT IN105409.
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