ADVANCED MASTERS IN STRUCTURAL ANALYSYS OF MONUMENTS AND HISTORICAL CONSTRUCTION Master’s Thesis Sarah Francisca Dynamic characterisation of the bell tower of Sant Cugat Monastery. University of Minho Spain | 2020 Dynamic characterisation of the bell tower of Sant Cugat Monastery. Sarah Francisca Spain | 2020
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ADVANCED MASTERS IN STRUCTURAL ANALYSYS OF MONUMENTS AND HISTORICAL CONSTRUCTION
Master’s Thesis
Sarah Francisca
Dynamic characterisation of the bell tower of Sant Cugat Monastery.
University of Minho
Spain | 2020
Dynam
ic c
hara
cteri
sati
on o
f th
e b
ell t
ow
er
of
Sant
Cuga
t M
onast
ery
. Sa
rah
Fran
cisca
Sp
ain
| 2
020
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ACKNOWLEDGEMENTS
I would like to express my most sincere appreciation to all those who supported and encouraged me
throughout this rewarding experience with the SAHC master’s program. I would like to express gratitude
to all the professors who taught the coursework at the University of Minho, as each one increased both
my knowledge base and passion for conservation engineering. In addition to the wonderful professors,
I would like to thank the master’s consortium for the scholarship that I was awarded as it greatly
influenced my decision to pursue this opportunity.
I would like to thank my dissertation supervisors, Professors Climent Molins and Nirvan Makoond, for
providing me with the opportunity to work on such an interesting topic and develop my knowledge in the
field. I would also like to thank them for their constant support and encouragement despite the tough
times with COVID-19. Without them, this thesis would not have been possible.
I would like to recognize the influence that my former University Professor and conservation engineering
mentor, Jack Vandenberg, had on my decision to pursue this master’s. Without his encouragement and
mentorship, I do not think that I would be where I am today. I would also like to thank former SAHC
students and colleagues, Carol Kung and Sandryne Lefebrve, for sharing their SAHC experience with
me and encouraging the endeavour.
Lastly, I would like to give immense gratitude to my family for always pushing me to achieve my dreams
and supporting me with everything that I do. I would also like to acknowledge the amazing people that
became my family during my time in Portugal. I am forever grateful for their kindness, friendship, and
support. SAHC 2020 will never be forgotten.
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ABSTRACT
Recent investigations have revealed that the bell tower of the Monastery of Sant Cugat could be
continuing to experience lateral displacement linked to an active deterioration mechanism. As such, it
was envisioned that obtaining key dynamic characteristics of the bell tower through full-scale ambient
vibration testing (AVT) may help calibrate a numerical finite element (FE) model to better understand
the deterioration mechanisms affecting the tower, if any. Therefore, it was the objective of this
dissertation to develop a robust procedure for the dynamic identification of the bell tower including a
preliminary state-of-the-art literature review, the creation of a suitable numerical model to obtain
expected modal properties, and a detailed dynamic testing procedure to be conducted in the future.
Both a simplified and more detailed FE model were constructed in DIANA FEA to obtain expected modal
parameters. The modal parameters were compared, and it was concluded that the simplified model
validifies the full solid model. However, the solid model should be utilized for the dynamic
characterisation as it is more accurate and is able to produce three-dimensional global and local mode
shapes. Various restraint scenarios were analysed in the FE models since the supports and connections
of the bell tower are unknown. This included modelling the tower fixed as a cantilever and with lateral
restraints. The possibility of poor soil-structure interaction was also considered through the
implementation of boundary springs at the base of the tower. An iterative sensitivity analysis was
conducted to obtain ranges of stiffness for the boundary surfaces. Following the analyses, it was
observed that eigenfrequencies tend to decrease with reduced stiffness at the boundary surfaces.
Therefore, when analysing the results from the dynamic testing, lower eigenfrequencies likely indicate
loss of stiffness at one or more connection surfaces. The aim of the iterative model updating procedure
is to identify the source of the observed flexibility.
The information obtained from the literature review and the FE models was utilized to design a dynamic
testing campaign using AVT to obtain experimental modal parameters. Accelerometer locations were
suggested according to the modal parameters obtained in the preliminary FE models and the testing will
require 2 triaxial accelerometers, 4 uniaxial accelerometers, 10 cables and 10 channels to be connected
to the centralized data acquisition system. Two acquisitions were recommended to capture the modal
properties of both the bell tower and the bells.
Once the dynamic testing has been conducted, modal analysis software may be used to identify the
modal parameters of the bell tower through operational modal analysis identification techniques and
calibration of the numerical model can be achieved through iterative modification of the defined updating
parameters. Following model calibration, the cause of the lateral displacement of the bell tower may
become apparent. In addition, the calibrated model may be used to analyse the dynamic interaction
between the bells and the supporting structure.
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ABSTRACTE
Investigacions recents han revelat que el campanar del monestir de Sant Cugat podria continuar
l'experiència de desplaçament lateral lligat a un mecanisme de deteriorament actiu. Com a tal, es va
preveure que l'obtenció de característiques dinàmiques clau del campanar a través de proves de
vibració ambiental a gran escala (AVT) pot ajudar a calibrar un model numèric d'elements finits (FE) per
comprendre millor els mecanismes de deteriorament que afecten la torre, si escau. Per tant, va ser
l'objectiu d'aquesta dissertació per desenvolupar un procediment robust per a la identificació dinàmica
del campanar incloent una revisió preliminar de la literatura d'última generació, la creació d'un model
numèric adequat per obtenir propietats modals que s'esperava, i un detallat procediment de proves
dinàmiques que es durà a terme en el futur.
Es van construir models de FE simplificats i complets a Diana FEA to per obtenir paràmetres
modalsque s'esperava. S'han comparat els paràmetres modals i es va concloure que el model
simplificat validifica tot el model sòlid, però, el model sòlid s'ha d'utilitzar per a la caracterització
dinàmica, ja que és més precís i és capaç de produir formes de mode tridimensional globals i locals.
En els models fe es van analitzar diversos escenaris de contenció, ja que els suports i les connexions
del campanar són desconeguts. Això incloïa la modelització de la torre fixada com a voladís i amb
restriccions laterals. La possibilitat de la mala interacció de l'estructura del sòl també es va plantejar
mitjançant l'aplicació de fonts de frontera a la base de l'estructura. Es va realitzar una anàlisi de
sensibilitat iterativa per obtenir rangs de rigidesa per a les superfícies de la frontera. Després de les
anàlisis, es va observar que les eigenfreqüències tendeixen a disminuir amb la rigidesa reduïda a la
superfície de la frontera. Per tant, en analitzar els resultats de les proves dinàmiques, els
eigenfreqüències inferiors indiquen la pèrdua de rigidesa en una o més superfícies de connexió.
L'objectiu del procediment d'actualització del model iteratiu és identificar l'origen de la flexibilitat
observada.
La informació obtinguda a partir de la revisió bibliogràfica i dels models FE es va utilitzar per dissenyar
una campanya de proves dinàmiques amb AVT per obtenir paràmetres modals experimentals. Els
acceleròmetres van ser col·locats segons els paràmetres modals obtinguts en els models de FE
preliminars. Es van recomanar dues adquisicions per capturar les propietats modals tant del campanar
com de les campanes. Un cop realitzades les proves dinàmiques, es pot emprar un programari d'anàlisi
modal per identificar els paràmetres modals del campanar a través de tècniques d'identificació d'anàlisi
modal operacional i el calibratge del model numèric que es pot aconseguir mitjançant la modificació
iterativadels paràmetres d'actualització definits. Després del calibratge del model, pot arribar a ser
evident la causa del desplaçament lateral del campanar. A més, el model calibrat es pot utilitzar per
analitzar la interacció dinàmica entre les campanes i l'estructura de suport.
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2.1.2 Material Properties ................................................................................................................................ 7
2.3.1 Signal Pre-Processing ........................................................................................................................... 13
2.6 Dynamic Action of the Bells ................................................................................................................25
2.6.1 Bell Systems in Europe ......................................................................................................................... 25
3.1.2 Construction Chronology ..................................................................................................................... 33
3.1.3 The Bells ............................................................................................................................................... 37
3.2.3 Outward Tilt of SE façade ..................................................................................................................... 44
4.4.1 Simplified Beam Model ........................................................................................................................ 66
4.4.2 Full 3D Solid Model .............................................................................................................................. 68
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4.5 FE Model for Dynamic Calibration ......................................................................................................71
5. DYNAMIC TESTING PLAN ............................................................................................... 75
5.1 Data Acquisition .................................................................................................................................75
The results obtained from the various methods should be compared using the Modal Assurance
Criterion (MAC) which correlates two sets of modal vectors as follows [4]:
𝑀𝐴𝐶 =(∅𝐴,𝑘
𝑇 ∙∅𝐵,𝑗)2
(∅𝐴,𝑘𝑇 ∙∅𝐴,𝑘)∙(∅𝐵,𝑗
𝑇 ∙∅𝐵,𝑗) (4)
Where ∅𝑨,𝒌 is the k-th mode of data set A, and ∅𝑩,𝒋 is the j-th mode of data set B. A MAC value of 1
implies perfect correlation of the two mode shape vectors, while a value of 0 indicates uncorrelated,
orthogonal vectors [4]. In general, a MAC greater than 0.80 is considered good and one less than 0.40
is considered poor [4]. If the MAC is low and the standard deviation is high between two mode shapes
identified through different techniques, this indicates that insufficient data was collected or an error was
incurred during the testing procedure [8].
2.4 FE Model Calibration
The final step in dynamic characterisation is the calibration of the hypothetical numerical model. This
procedure is illustrated in Figure 8 and involves the examination of the differences in main mode shapes
and frequencies between the hypothetical (FE) and experimental (OMA) results. If the difference is found
to be negligible, the model may be considered accurate, whereas, if a difference in values exists, the
updating parameters must be iteratively changed until convergence is achieved.
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Figure 8 – Example of a dynamic-based assessment procedure for bell towers [6]
2.4.1 Manual Tuning
The updating parameters should first be assessed through a rough comparison between results of the
FE model and experimental OMA. This can be achieved through iteratively modifying the updating
parameters in the FE model until the differences in natural frequencies between the FE model and OMA
are minimized. This procedure is known as manual tuning and typically, the experimental value of the
first bending mode is used as a reference [13] [15].
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This procedure is conducted by varying one updating parameter at a time until a satisfactory agreement
between results is achieved for the main global mode shapes (less than 5% error) [3] [4] [7] [8] [11] [12]
[13] [15]. To improve the efficiency of this method, the range of dynamic properties to be assessed may
be limited by examining only the following four ratios [13]:
• Ratio between the first flexural frequency in X and the first torsional frequencies,
• Ratio between the first flexural frequency in X and first flexural frequencies in Y,
• Ratio between first and second flexural frequencies in X and,
• Ratio between first and second flexural frequencies in Y.
The model can be further validated using the modal assurance criterion (MAC) defined as follows [13]:
𝑀𝐴𝐶 =(∅𝐴𝑉𝑇
𝑇 ∙∅𝐹𝐸𝑀)2
(∅𝐴𝑉𝑇𝑇 ∙∅𝐴𝑉𝑇)∙(∅𝐹𝐸𝑀
𝑇 ∙∅𝐹𝐸𝑀) (5)
Where Where ∅𝑨𝑽𝑻 is the modal displacement from the experimental data set and ∅𝑭𝑬𝑴 is the
corresponding modal displacement from the FE model. However, before computing the MAC, the
experimental results must be converted to real valued ones since mode shapes cannot be scaled in an
absolute way using OMA [11]. This is achieved by scaling the experimental mode shapes so that the
mode shape vector component of one of the channels is equal to 1, and then transforming the predicted
mode shapes at the approximate points of the accelerometers in the FE model to the simplified
coordinate system [11]. A MAC value of 1 indicates a perfect correlation of the two mode shape vectors,
while a value close to 0 indicates completely uncorrelated, orthogonal vectors. Typically, a MAC value
of 0.80–0.85 is considered acceptable [13].
Aside from manual tuning, which is largely based on trial and error, other system identification
techniques exist that can help refine the linear elastic model such as: the Inverse Eigen-sensitivity (IE)
method, the Douglas-Reid (DR) method [4], the Genetic Algorithm technique (GA) [14], and Sensitivity
Analysis (SA) [13], all of which are summarized below. Note that these methods are typically time
consuming and complex and therefore, may not be the most efficient or effective approach to calibrate
the FE model for the Sant Cugat Bell Tower.
2.4.2 The Inverse Eigen-Sensitivity (IE) Method
The IE method analyses the functional relationship between the measured responses “R” and the
structural updating parameters “X” of the model expressed in terms of a Taylor series expansion as
follows [4]:
𝑹𝒆 = 𝑹(𝑿𝟎) + 𝑺(𝑿 − 𝑿𝟎) (6)
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Where Re is the vector associated with the reference experimental response data, R(X0) is a vector
containing the responses from the model corresponding to the starting choice X0 of the updating
parameters, and S is the sensitivity matrix [4]. This Equation is then rearranged to derive the following
iteration scheme to evaluate X [4]:
𝑿𝒏+𝟏 = 𝑿𝒏 + 𝑯[𝑹𝒆 − 𝑹(𝑿𝒏)] (7)
Where the gain matrix H is computed using either the Moore-Penrose pseudo-inverse or following the
Bayesian estimation theory [4].
2.4.3 The Douglas-Reid (DR) Method
The DR Method analyses the relationship between any modal response of the FE model where the
structural updating parameters Xk (k=1,2,…N) of the model are approximated based on the current
values of Xk through the following Equation [4]:
𝑅𝑖∗ = (𝑋1, 𝑋2, … , 𝑋𝑁) = ∑ [𝐴𝑖𝑘𝑋𝑘 + 𝐵𝑖𝑘𝑋𝑘
2]𝑁𝑘=1 + 𝐶𝑖 (8)
where 𝑹𝒊∗ represents the approximation of the i-th response of the FE model [3].
Table 5 outlines an example of this procedure, comparing the preliminary manual tuning scheme (lower
value, base value and upper value) and the numerical schemes (DR and IE) that were used to refine
the updating parameters and fully calibrate the FE model [3].
Table 5 – Updated structural parameters of a case study [3]
2.4.4 The Genetic Algorithm Technique (GA)
The GA method is a stochastic algorithm used for solving optimization problems based on a natural
selection process that mimics biological evolution [14]. A population of “chromosomes” with a uniform
random distribution must first be selected and several parameters must be defined [14]. Following the
manual tuning scheme, the relative errors between experimental and numerical modal frequencies may
be used as fitness functions for the first natural frequencies to obtain the possible ranges for the defined
updating parameters [14].
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Following the manual tuning procedure, a parametric analysis may be performed by changing any
parameters with a high variation coefficient ranging from the lower bound of its range to the optimal
value as determined through the GA analysis until good correlation is achieved [14]. The following Table
outlines an example of this procedure showing the updated elastic properties, the initial range value and
the standard deviations of all parameters obtained by means of the GA after 3 runs [14].
Table 6 – Updated structural parameters of a case study through GA optimization [13]
2.4.5 Sensitivity Analysis (SA)
SA approaches are based on “element-level sensitivity equations” which relate the mode shapes of the
structure to the changes of the chosen updating parameters based on functions derived by changes in
stiffness [13]. Following preliminary manual tuning, the SA method can be used to further refine the
results by considering only the most sensitive updating parameters through a sensitivity analysis using
the following sensitivity index [13]:
𝑆𝑖,𝑗 =ln(𝜀𝑗)−𝑙𝑛(𝜀𝑗0)
ln(𝑥𝑖)−𝑙𝑛(𝑥𝑖0) (9)
where xi0 and xi are the initial nominal value of the i-th updating parameter and its value incremented
at a given percentage, ej0 is the error of the j-th output of the model corresponding to the nominal model,
and ej is the error corresponding to setting all parameters to their nominal value while setting the i-th
parameter to xi [13]. The SA technique calculates the sensitivity coefficient (Si,j) as the rate of change
of the j-th output of the model with respect to a change in the i-th input xi. An example of the results of
one case study are depicted in Figure 9 [13]. The computed sensitivity index allows for the identification
of the parameters most sensitive to change. In the example depicted below, parameters E2, k1, k2, and
α showed to be most sensitive to change [13].
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Figure 9 – Local sensitivity index of a case study [13]
Once the updating parameters are chosen based on the sensitivity index, the model updating can
continue using a procedure where a parameter vector is defined for both the measured and FE
computed modal quantities and encloses the properties to be updated. This procedure is based on the
computation of the objective function over a multi-dimensional grid [13]. The structural parameters for
the case study mentioned above were computed using the SA technique as shown in Table 7.
Table 7 – Updating parameters for structural identification [13].
2.5 Soil-Structure Interaction
The soil–structure interaction is an important factor when attempting to identify seismic vulnerability or
sources of damage in masonry towers as differential soil settlement often causes structural issues. In
addition, it can be non-conservative to assume the soil to be rigid and perfectly fixed to the foundation
of the bell tower [13]. Therefore, should the subsurface conditions be unknown and of interest, it may
be appropriate to utilize experimental dynamic testing such as AVT to assess the stiffness of the soil-
structure interaction [13]. This can be achieved by adopting the Winkler model for soil and introducing
uniform linear elastic constraints (springs) as model updating parameters on the bottom surface of the
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FE model in attempt to simulate the deformability of the ground [10] [11] [13]. See Figure 10 depicting
an example of the Winkler model where q is a distributed load (the masonry tower) applied to a set of
springs representing the subsurface soil.
Figure 10 – Example of Winkler model [21]
To obtain a range of expected spring stiffnesses, the FE model may be analysed comparing rigid
constraints versus elastic restraints [10]. In the report by F. Lorenzoni et al. (2017), a range of vertical
spring constants was obtained by merging literature data with the sensitivity analysis of the bell tower
under study resulting in an expected spring constant between 5.108 N/mm3 and 109 N/mm3 [10]. It is
recommended to assume that the horizontal springs have a value equal to 1/10th of the vertical stiffness
to account for the limited connection with the soil in this direction and the impact on shear behaviour of
the foundation [10]. An example of a soil-structure sensitivity analysis is depicted in Figure 11.
Figure 11 – a) Sensitivity analysis showing variations in numerical frequency with the change in spring stiffness; b) variation of MAC index with and without elastic foundations [10]
Once the experimental results have been recorded, the model may be calibrated by iteratively changing
the spring constants in the FE model until the modal parameters are in alignment with the experimental
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data. This procedure is typically achieved by keeping the Young’s Modulus constant and equal to the
value that was obtained by model calibration assuming a rigid soil-structure interaction [10].
Ultimately, the calibrated models with foundation springs and without (i.e. rigid foundation) should be
compared. If the model with springs is more in alignment with the experimental results, this indicates
that there is likely a poor connection between the soil and the foundation and subsurface conditions may
be influencing the behaviour of the structure [10]. If this is the case, subsurface conditions should be
further examined through a geotechnical investigation.
2.6 Dynamic Action of the Bells
Oscillation of the bells is one of the strongest forces that bell towers are subjected to as the induced
forces to the structure are often amplified when the bells are swinging [6] [22]. The impact of the vertical
forces are often neglected as the axial stiffness of masonry towers is higher than their bending stiffness
and therefore, no resonance problems are expected [6] [14]. However, the horizontal forces are of
concern as historic masonry structures were not designed to resist large lateral forces.
There are various techniques provided in literature to analyse the interaction between ringing bells and
the supporting masonry structure. Currently, the primary technique for assessing this interaction is
through the comparison of the natural frequencies of the bell tower with those of the bells’ oscillation [6].
This work can be combined with numerical models to simulate the interaction between frequencies [6].
2.6.1 Bell Systems in Europe
Forces induced by the bells to their supporting structures vary with time and depend on the
characteristics of the bells and the way in which they are rung [6] [14]. Within Europe, the bells can be
classified into three main categories: Central European, Spanish, and English. Each system presents
certain characteristics of frequency, oscillation, unbalance and turn rate which results in a different
structural impact to the supporting structure [2] [14] [22]. See Figure 12 depicting the various bell ringing
systems.
In the Spanish system, a counterweight provides a high level of balance and the bells rotate continuously
in the same direction [12] [14]. In the Central European system, the bells tilt on their axis at swing angles
between 55 and 160̊ with no counterweight, often causing a highly unbalanced system which exerts
considerably more horizontal dynamic load on the supporting structure [12] [14]. In the English system,
the bells rotate in a complete circle, changing the direction of the swing in each cycle [6] [14].
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Figure 12 – Different bell ringing systems (from left to right): Central European, Spanish, English [2]
The difference between the dynamic forces caused by bells swinging according to the Spanish vs.
Central European systems is illustrated in Figure 13.
Figure 13 – Typical dynamic horizontal forces induced by bells swinging according to the Spanish system (left) and Central European system (right) [2]
As can be observed, the Spanish system presents substantially lower levels of unbalance compared to
the Central European system [2] [14]. Subsequently, the horizontal forces induced by Spanish bells are
the smallest, approximately 0.15 times the weight of the bell assemble, and largest in the Central
European and English systems, approximately 2 (CE) to 4 (English) times the weight of the bell
ensemble [1].
2.6.2 Static Analysis
Depending on the angular velocity and balance of the bells, the forces that they induce to the supporting
structure can be considerable [14] [23]. Therefore, these forces should be evaluated to enable decisions
regarding structural strengthening and/or changing of the bells’ operating system to be made [6]. To
determine the quantity of these forces for a certain bell, a static analysis can be conducted considering
the characteristics of the bells including their dimensions, weight, centre of gravity, unbalance, initial
angular velocity, moment of inertia, swing angle, and maximum nondimensional horizontal and vertical
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forces [3] [12]. With this information, analytical models can be used to estimate the dynamic forces
induced by a specific bell using the following Equations [6] [12]:
Where a is the distance of G1 from C1 (G1 being the bells centre of gravity and C1 being the axis of
rotation), φ is the angle of the bell from the downward vertical of C1, g is acceleration due to gravity, t
is time, and M is the mass of bell and yoke [6]. The geometrical quantities are described in Figure 14.
Figure 14 – Simplified geometrical quantities of a bell [6]
Note that the total force transmitted to the supports in each direction is equal to the sum of the horizontal
forces of all the bells that turn in that direction at any given time [6]. Therefore, to reduce the horizontal
force, the arrangement of the bells is usually carefully considered [6].
2.6.3 Dynamic Analysis
It is impossible to calculate the sum of the horizontal forces of all the bells that turn in a particular
direction at any given time using the static method described above. In addition, the dynamic nature of
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the bells often induces an amplification effect to the forces that are transferred to the supporting
structure. Therefore, to evaluate the effect of the bell swinging on the modal parameters of the
supporting tower, a Fast Fourier Transform (FFT) analysis of these forces must be conducted when the
bells are ringing [14]. This type of analysis can be achieved through AVT during the ringing of the bells
and identification of bell harmonics through modal identification procedures in the frequency domain
(see Section 2.3) [9].
If one of the predominant harmonics of the bells interacts with a natural frequency of the supporting bell
tower, a large dynamic amplification factor (DAF) may be induced which could impact the stability of the
structure [3]. In general, the predominant harmonic in the Central European system is the second
horizontal force, in the English system the third horizontal force, and the Spanish system the first
horizontal force [6]. This procedure is depicted in Figure 15 for a Spanish system.
Figure 15 – Example of a frequency analysis of a Spanish bell system [1]
As expected for a Spanish system, the first harmonic is strongly predominant however, is distant from
the bell towers’ typical natural frequencies, therefore, the bells will have a negligible influence on the
DAF [1]. However, should the first harmonic have been closer to the natural frequency of bell towers, or
the second and third frequencies been greater in amplitude, there likely would have been a considerable
DAF.
Once the dynamic properties of both the bell tower and the bells have been identified, the DAF can be
calculated through a parametric analysis considering the bell’s swing velocity, damping factor ( 𝝃),
harmonic component ( 𝛀𝒊), and the vibration frequencies ( 𝝎𝒋), as follows [14]:
𝐷𝐴𝐹𝑖𝑗 =1
√(1−(Ω𝑖𝜔𝑗
)
2
)
2
+(2𝜉(Ω𝑖𝜔𝑗
))
2 (12)
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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 29
If the DAF is found to be less than 1 for all bell ringing schemes, no dynamic interaction is considered
between the bell tower and the bells [24].
An additional check may be conducted according to DIN 4078 through the comparison of the first three
modal frequencies of the tower subjected to the swinging bells and the first three natural frequencies of
the structure itself. If the corresponding modes are separated by more than 20%, the dynamic interaction
between the bells and the supporting structure may be considered negligible [24]. In addition, one case
study showed that when the first to third bell harmonics have a frequency between the first and second
tower frequencies, the horizontal displacement at the tower’s highest level can induce damage to the
structure, regardless of swing angle [14].
An alternate method to compute the DAF is through introducing the maximum static vertical and
horizontal forces caused by the bell swinging to the model at the height where the bells are situated,
running a modal analysis and comparing results between the tower with and without bell ringing [3] [12].
An example of the DAFs computed using this method for a bell tower subjected to various bells is
provided in Table 8.
Table 8 – Example of DAF evaluation of a historic bell tower for a given mode shape [3]
This Table shows that the San José and Minerva bells produced a DAF of approximately 5, indicating
that when those bells are rung, they can impact the dynamic characteristics of the bell tower by a factor
of 5, which is considerable.
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Dynamic characterisation of the bell tower of Sant Cugat Monastery
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3. THE CASE STUDY: BELL TOWER OF SANT CUGAT MONASTERY
The Sant Cugat Monastery is a 9th century Benedictine Abbey located in Sant Cugat del Vallès,
Catalonia, Spain, 20 km NW of Barcelona as shown in Figure 16. The Monastery is composed of two
main structures: the basilica church and the cloister; however, this report is focused on the bell tower
located within the body of the basilica church as depicted in Figure 17.
Figure 16 – Main façade of the Sant Cugat Monastery showing the rose window and bell tower (left); map showing the location of the Monastery (right) [25]
Figure 17 – Plan view of the Sant Cugat Monastery showing the bell tower within the main body of the basilica and the cloister [25]
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The bell tower of the Sant Cugat Monastery is the subject of this case study as an outward tilt has been
observed in the tower since the early 19th century, causing great concern regarding the stability of the
structure. It is the objective of this dissertation to develop a robust procedure for the dynamic
identification of the bell tower based on the information obtained in the literature review and the
development of a representative numerical model. Once the dynamic testing has been conducted
according to the procedure presented in this report, it will be possible to identify the hypothetical cause(s)
of the displacement through the calibration of a finite element model using the experimental results. This
information will help inform the owner of the Monastery as to the level of concern associated with the
leaning of the bell tower. However, prior to the development of the dynamic characterisation plan, an
historic and geometric survey were conducted to obtain a better understanding of the structural history
and existing condition of the Monastery.
3.1 Historic Survey
As with any conservation project, it is essential to understand the building of study through an historic
survey prior to the development of any additional work. This is due to the unique nature of historic
buildings, each having their own construction history and historic values. The historic survey will
examine the initial construction of the Monastery and later interventions to ensure a proper
understanding of the existing condition of the structure. This is of particular importance for designated
heritage buildings, such as the Sant Cugat Monastery, as they have numerous character-defining
elements that should be protected. The following subsections will discuss the historic significance of
the Monastery along with a detailed chronology of historical events regarding the construction of and
later interventions made to the Monastery of Sant Cugat, with a focus on the bell tower.
3.1.1 Historic Values
The Monastery of Sant Cugat was designated as a National Monument in Spain in 1931 due to its
significant historic and architectural values. The Monastery is of particular historic significance as it was
one of the first monasteries to be built in Catalunya and sits upon the remains of an Ancient Roman
castrum [26]. In addition, the Monastery has withstood attacks by the Muslims in 985 AD and the War
of the Spanish Succession in the early 18th century [27].
Architecturally, the church is composed of three naves without a transept, and three apses of semi-
circular plan on the interior and polygonal plan on the exterior [28]. The Monastery is also an excellent
example of the architectural transition from the Romanesque period to the Gothic period. This is depicted
through the Romanesque sobriety of the church displayed in the lack of decoration and light, the gothic
grandeur of the 13th century dome, the large gothic rose window on the main façade, the Romanesque
vaults in the apse, the gothic vaults crossings the rest of the church, and the three Gothic lateral chapels
on the south side [26].
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With respect to the bell tower, it was built in several periods and was not completed until the 18th century
[29]. It too has foundations on the ancient Roman fortress and displays decorative motifs characteristic
of the Lombard style, as well as two stone arches of Islamic Influence [29].
3.1.2 Construction Chronology
The Monastery was constructed between the 9th and 14th centuries and features a classic basilica plan,
a bell tower integrated into the SE façade, and a Romanesque cloister located adjacent to the NW
façade of the basilica [26]. The basilica and the bell tower were constructed upon the remains of a
Roman castrum, however, the rest of the church was founded directly on soil with a shallow foundation
[26] [30]. The masonry is comprised of stone blocks sourced from the local quarry “Pedrer de
Campanya” and is composed of detrital carbonate rocks, likely limestone [30].
The bell tower was built in several periods and was not completed until the 18th century [1]. The tower
began construction in 1062 AD as an external element founded upon one of the tower ruins of the former
Roman fortress [1]. The original structure extended up to a floor level that was used to install the first
bells as can be seen in the 16th century painting of the Martyrdom of Sant Cugat by Germanic artist
Ayne Bru shown in Figure 18 [1].
Figure 18 - Painting of the Martiri de Sant Cugat by Ayne Bru (1502-1507) showing the state of the Monastery at the time; note the construction of the bell tower and the arch connecting the bell tower to the dome [23]
In 1760, the construction of the bell tower advanced upwards under the initiative of Abbot Gayolà [1].
Under his campaign, the bell tower was finished through the construction of a new upper level housing
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the liturgical bells and two smaller superimposed structures sitting atop the main tower to host the clock
bells as shown in Figure 19 [1].
Today, the bell tower is divided into several floors, some of which can be visited. On the ground floor, is
the Chapel of Mercy, which in the 16th century was divided into two levels to accommodate the
Renaissance organ. On the proceeding level, it is possible to see the early Romanesque barrel vaults,
above which, an internal staircase leads to a metal passageway which extends above the church’s
baroque chapels and beneath the old Gothic vaults that house them, as shown in Figure 19 [1]. The
level above is occupied by a clock from the late seventeenth century which has been restored and is
still responsible for the ringing the clock bells today, followed by the liturgical bell house which hosts the
four liturgical bells and finalizes the main body of the bell tower [1].
Figure 19 – SE façade of the bell tower today (left); metal passageway above the churches Baroque chapels (right) [30]
A summarized chronology of the construction and interventions made to the Sant Cugat Monastery is
provided in the following list accompanied by some historic photos as shown in Figure 21 to Figure 24:
• 9th century AD – the construction of the Monastery was founded and dedicated to unite the 5th
century church housing the remains of Sant Cugat [27]
• 10th century AD – expansion of the monastery [27]
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• 985 AD – damage to Monastery during an attack of Muslim troops led by al-Mansur Ibn Abi
Aamir, who then repaired it and added the minaret [27]
• 1062 AD: initial construction of the Bell Tower upon the foundations of the former Roman
fortress [1]
• Mid-12th c. - 1337: construction of the new monastery and cloister adjacent to the church [27]
• 1350 - 1383: construction of the fortification walls and towers [27]
• 1502-1507: painting of the Martiri de Sant Cugat by Ayne Bru, now located at the Museu
Nacional de Catalunya [27]
• 16th century: separation of the main level of the bell tower into two floors to accommodate the
Renaissance organ [27]
• 1760: dismantling of the arch joining the octagonal dome with the bell tower and initiation of the
construction of the upper portion of the bell tower [1]
• 18th century: completion of the bell tower [27]
• 1701-1714: war of the Spanish Succession causing damage to the structure [27]
• 1782: construction of a new Sacristy in the eastern corner of the church, between the bell tower
and the apse [30]
• 1789: restoration work completed on damaged portions of the structure [27]
• 1835 – 1950: abandonment of the monastery [27]
• 1851: restoration works under the supervision of architect Elies Rogent due to the observation
of several large cracks and outward displacement in the bell tower, dome, SE wall and arches
of the SE lateral nave. This work included the following [29] [30]:
o extraction of the earth and rubble infill located above the vaults next to the dome,
o reinforcement of two arches in the SE lateral nave between the dome and the bell tower
with 60 cm wide brick walls intended to allow both the dome and bell tower to stand
independently,
o installation of two tie rods at the base of the dome to reduce lateral thrust, and
o repair works on the pendentives under the dome
• 1931: declaration of the Monastery as a National Monument [27]
• 1992: geotechnical study conducted by the Architectural Heritage Service of the Generalitat de
Catalunya which concluded the following [30]:
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o the monastery lies over a 2 m deep layer of quaternary materials (soil, sand and rubble)
followed by a lower layer varying between 1 and 8 m in depth composed of tertiary
materials (silts, clays and sandy silts),
o the water table is at a depth of 10 m,
o the tertirary level is very near the surface beneath the Church (being only centimetres
from the surface at the SE wall). However, it is located much deeper as you approach
the cloister, and
o the depth of the foundation was found to be 2.3 m on the SE wall, 3.1 m on the main
SW façade, and 1.7 m on the NW wall between the church and the cloister.
• 1995 – 1996: restoration works by the Architectural Heritage Service of the Generalitat de
Catalunya conducted to stabilize the SE wall of the basilica. This was achieved primarily through
strengthening of selected arches, consolidation of select gothic vaults, insertion of steel ties in
the buttresses of the SE wall, and stitching of the wall between the Sacristy and the Bell Tower
to the main body of the church with steel tie bars as depicted in in Figure 20 [30].
Figure 20 – Drawings of the intervention performed in the 1990s: left – stitching of the wall between the sacristy
and the bell tower to the main body of the church; right – reinforcement of the buttresses supporting the
southern wall with steel ties [26]
• 2000: Geometric survey conducted by the municipality of Sant Cugat del Vallès as part of the
Master Plan for the rehabilitation of the Monastery [30]
• 2006: Structural stabilization project for the central and northern nave of the church [29]
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Figure 21 – Painting of the Monastery of Sant Cugat in 1842 showing the NE façade of the
Church and bell tower [31]
Figure 22 – Picture of the Monastery of Sant Cugat in 1915 showing the SE and NE façades of the Church and
bell tower [32]
Figure 23 – Photograph of the Monastery of Sant Cugat in 1890 showing the NE and SE façades of the Church and bell tower [33]
Figure 24 – Photograph of the Monastery of Sant Cugat in 1920 showing the main SW façade of the Church and bell
tower [34]
3.1.3 The Bells
Within the bell tower, there are four liturgical bells (c. 1940s) located between levels 6 and 7, and two
clock bells (c. 1623) located within the two superimposed bell structures on top of the main body [35].
The liturgical bells were previously hung upon timber bridges in the windows to be rung using the
traditional Catalan yoke system. However, following the Spanish Civil War in 1936, the six liturgical bells
of Sant Cugat were destroyed, with only the clock bells (marking the hour and quarter hour) remaining
in the upper bodies [35]. Since the war, four new liturgical bells have been installed (Severa, Juliana,
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Semproniana and Gugada) in a non-elastic metal structure, fixed to the wall, in the Central European
way and without yoke [35]. This "modern" installation does not allow the manual ringing of the bells, nor
does it allow for the reproduction of the traditional rhythm of the bells [35]. In addition, this system may
cause damage to the bell tower due to the lack of counterweight. See Figure 25 and Figure 26 depicting
the various bells.
Figure 25 – Picture of traditional Spanish Yoke bells with counterweights c. 1920s (left) [35]; Picture of the clock bell in level 7-8 today (right) [36]
Figure 26 – “Modern” church bell structure today [36]
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3.2 Geometrical Survey
Following an historic survey, it is critical to understand the state of the existing structure prior to the
development of any additional work. Therefore, a thorough review of existing documentation regarding
the current structure of the bell tower was conducted prior to the development of the numerical model.
3.2.1 Structure
The geometry and structure of the bell tower of Sant Cugat Monastery were based primarily on an as-
found drawing set created by the Departament de Cultura of the Generalitat de Catalunya in 1991 [37]
and pictures of the Monastery found online. The drawing set was imported into AutoCAD, scaled to size,
and measured to obtain the necessary dimensions to create the 3-dimensional model. See Appendix A
for the amalgamation of drawings pertinent to the bell tower and for key dimensions utilized for the
creation of the model.
The bell tower of Sant Cugat Monastery measures approximately 42 m in height and has a floor plan
measuring approximately 7.2 m by 7.2 m. The floor plan is assumed to remain constant until the point
at which two smaller bell structures rest atop the main structure. However, the NW wall has an additional
thickness of 60 cm up to Level 5 from the 1851 restoration campaign. The structure of the bell tower is
composed primarily of limestone masonry walls, floors, and cross vaults. Based on the drawings, it was
assumed that the masonry walls measure approximately 1.35 m in thickness and that the masonry
cross-vaults measure approximately 0.25 m in thickness and are covered with a less dense infill beneath
floor levels. The structure is separated into nine levels as shown in Figure 27 and has cross vaults and
floor structures at levels 2, 3, 5, 6 and 7.
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Figure 27 – Basic dimensions from existing drawing set [37]
Locations and measurements of openings were obtained from the drawing set and were included in the
3D model. Between levels 1 and 6 there are large openings on the interior NW façade and multiple
smaller openings (windows) on the exterior SE façade. The NE façade had no apparent openings and
the SW façade had door openings at the first, second and third floor levels, after which point, an external
staircase proceeds into the tower and continues upwards to the top of level 7. Between levels 6 and 7,
there are 2 large semicircular windows on each façade approximately 3.7 m in height and 1.3 m in width.
Between levels 7 and 8, there is one large semi-circular window on each façade measuring
approximately 3.3 m in height and 1.2 m in width, hosting one clock bell. Lastly, between levels 8 and
9, there is also one large semi-circular window on each façade measuring approximately 2.5 m in height
and 0.75 m in width, hosting a second clock bell. See Figure 28 and Figure 29 depicting the geometry
of the bell tower as created in AutoCAD 3D.
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Figure 28 – Elevations of all four facades of the 3D model of the bell tower (from left to right: NW, NE, SE, SW)
Figure 29 – Sections of the 3D model of the bell tower showing openings and cross-vaults (from left to right: NE-SW section looking towards SE façade, NW-SE section looking towards NE façade ,NE-SW section looking
towards NW façade)
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The cross-vaults were modelled by intersecting two barrel vaults with thicknesses of 0.25 m within the
external masonry walls where the load is transferred. The vault geometry is depicted in Figure 30
showing how the masonry infill (blue) is located atop the vaults in order to model the different mechanical
properties.
Figure 30 – Construction of the barrel vaults: left – masonry infill; middle – masonry infill located with the vault structure; right – 3D solid model of the vault, infill is hidden
Following vault construction, the entire model was built by extruding the main floor plan to each level,
attaching the vault-floor structure, extruding the remaining thicknesses of the floors and subtracting
openings from the exterior walls as shown in Figure 31.
Figure 31 – Finalized 3D solid geometry as constructed in AutoCAD to be imported into DIANA FEA software
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3.2.2 Connections
The bell tower is surrounded by the main body of the Church on the NE, SW and NW façades and
therefore, the connection between the bell tower and church must be modelled to allow for some load
transfer and movement at intersection points. Based on the drawings, it appears that the SW and NW
walls of the tower are connected to the main body of the Church through large cross-vaults intersecting
the tower between levels 4 and 5. However, on the NE façade, the tower appears to be connected to a
small, rectangular masonry substructure up to a height of 9.1 m and then by a smaller rectangular
substructure up another 5.8 m. Therefore, the connection points were modeled as solid bodies
integrated into the walls of the tower for implementation in the FEM software.
A brief assessment of the lateral connections between the church and the bell tower was conducted by
Professor Climent Molins of UPC on July 14th, 2020. This was achieved through the inspection of the
intrados’ of the vaults within the bell tower at levels 4 and 5. As depicted in Figure 32 and Figure 33,
cracks were observed at the intersections between the bell tower and the SW wall of the church at both
levels. Cracking indicates stress concentrations within the masonry, however, also indicates some form
of restraint introduced by the connecting vaults. Therefore, the impact of the lateral connections must
be accounted for in the FE models.
Figure 32 – Observed crack pattern at the intersection between the vault intrados and the SW wall of the church at Level 4 [36]
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Figure 33 – Observed repaired cracks at intersection between the vault intrados and the SW wall at Level 5 [36]
Lastly, the foundation was assumed to be fixed where all surfaces meet the ground due to the lack of
information regarding subsurface conditions. However, due to the observed outward tilt of the tower,
flexible foundations were also considered in the modal analyses.
3.2.3 Outward Tilt of SE façade
An outward tilt of the bell tower of Sant Cugat Monastery has been observed and of concern as early as
the 19th century. In June 2019, the outward tilt of the bell tower was measured by the Department of
Urban Planning and Projects as shown in Table 9.
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Table 9 – Results from June 2019 bell tower survey [38]
Position measured
from Edge Height (m) Inclination (%)
Displacement
(m)
Lower tower
North 15.4 0.63 0.10
East 31.40 1.34 0.42
South 31.37 1.50 0.47
West 16.2 0.46 0.07
Intermediate tower
North 5.85 0.45 0.03
East 5.85 1.00 0.06
South 5.85 0.50 0.03
West 5.85 1.10 0.06
Upper tower
North 3.19 1.00 0.03
East 3.19 1.46 0.05
South 3.19 1.00 0.03
West 3.19 0.63 0.02
From these results, it was concluded that the edges of the bell tower have inclinations varying between
0.50% and 1.50%, predominantly in the South and Southeast directions, resulting in a maximum
displacement of 42 to 47 cm with respect to the vertical at the highest point of the tower edges [38]. A
laser scan was also conducted on the SE façade of the church in July 2019 to verify the measured
displacements of the bell tower and to identify any displacement in the SE façade of the surrounding
church, the results of which are depicted in Figure 34 [39].
Figure 34 – Results from July 2019 laser scan of the SE façade of the Sant Cugat Basilica [39]
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The laser scan confirmed the outward displacement of the tower, measuring a maximum inclination of
1.79% and lateral displacement of 52 cm, similar to the maximum displacement observed in the June
2019 survey. The laser scan confirmed that the SE church façade is characterized by a significant
inclination as well, varying between 0.34 % (0.04 m) and 2.50 % (0.30 m) along its length [39].
Both the survey and the laser scan conducted in 2019 confirm the outward displacement of the tower,
however, it is uncertain if the displacement of the bell tower is caused by poor connections with the
surrounding church structure, differential subsurface conditions, or a combination of both. Therefore, it
is recommended that dynamic testing be conducted to help identify the source of the displacement so
that the problem can be remediated, and the cultural asset may be conserved for future generations.
3.2.4 Limitations & Assumptions
Due to COVID-19, access to the Monastery was not possible and therefore all geometrical and structural
information was based solely on a review of existing documentation and therefore, may not be indicative
of the actual structure. For example, additional deadloads may exist within the bell tower that could not
be identified, nor could the identification of any damages or interventions be conducted. Therefore, it
was assumed that the drawings are accurate and that the structure is in relatively good condition
throughout.
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4. EXPECTED MODAL PARAMETERS
Following the construction of the geometry of the bell tower, numerical modelling was conducted to
obtain its hypothetical dynamic properties and to be used for model calibration following dynamic testing.
Both a simplified beam and full solid model were created with various restraint scenarios to be compared
and to ensure accuracy of the more detailed solid model. The following subsections outline the
construction of the finite element models along with an analysis of their results.
4.1 Simplified Beam Element Estimation
Preliminary modal parameters of the bell tower at Sant Cugat Monastery were obtained through the
creation of a representative simplified beam model in finite element software DIANA FEA [17]. This was
achieved through the following procedure.
4.1.1 Geometry
Class I 3D beam elements were drawn in DIANA FEA to represent the bell tower by splitting the structure
into 18 critical cross-sections including hollow sections, solid floors and large openings as described in
Table 10 and Table 11.
Table 10 – Main sections of bell tower
Section Z (m) Height (m) Description
1 0 3.5 3.5 Opening at level 1
2 3.5 5.7 2.2 Above opening at level 1
3 5.7 6.7 1 Floor at level 2
4 6.7 9.1 2.4 Opening at level 2
5 9.1 11.3 2.2 Above opening at level 2
6 11.3 11.65 0.35 Floor at level 3
7 11.65 12.2 0.55 Connection on NE façade
8 12.2 13.83 1.63 Connection on NW façade
9 13.83 18.65 4.82 Below level 5
10 18.65 19.33 0.68 Floor at level 5
11 19.33 25.38 6.05 Between levels 5-6
12 25.38 25.84 0.46 Floor at level 6
13 25.84 32.338 6.498 Between levels 6-7
14 32.338 33.008 0.67 Top of main tower
15 33.008 37.9704 4.9624 Between levels 7-8
16 37.9704 38.659 0.6886 Top of level 8
17 38.659 41.3936 2.7346 Between levels 8-9
18 41.3936 42 0.6064 Top of bell tower
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Table 11 – Section properties
Section Plan view Area (m2) Ixx (m4) Iyy (m4) Ixy (m4) It (m4/rad)
1
26.8 149.6 179.7 0.8956 14.41
2, 5, 7, 8
30.3 185.6 184.9 -0.506 277.3
3, 6
49.9 222.2 213.5 -0.6727 368.1
4
27.4 156.4 181.9 1.443 14.41
9, 11, 13
26.5 159.7 159.6 -0.9807 209.7
10, 12, 14
48.5 200.6 200.0 -0.5496 311.9
15
7.11 10.21 10.81 -0.0269 15.10
16
12.4 12.49 13.30 0 21.65
17
2.38 0.7481 0.7228 -0.0014 0.9780
18
3.05 0.7891 0.7582 0 1.3
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For each section, the moment of inertia about X (Ix) and Y (Iy), the product moment of inertia (Ixy), the
torsional moment of inertia (It), and any eccentricities were inputted into the software, an example of
which is depicted in Figure 35.
Figure 35 – Section properties for Section 1
The area and moment of inertias were computed using AutoCAD command MASSPROP whereas the
torsional moment of inertias were computed by simplifying sections as rectangles, hollow rectangles or
C-channels and using a moment of inertia calculator [40]. Note that an eccentricity of -0.3 m in the
global Y direction was applied to sections 1 to 8 to align the structure as it is. The total height of the
beam model measures 42 m, identical to that of the 3D solid model.
4.1.2 Materials
The material properties were chosen as indicated in Table 12 and are based on the values obtained
from the literature review.
Table 12 – Material properties used in FE model
Material Class Material
Model
Young’s
Modulus (MPa)
Poisson’s
Ratio
Specific Weight
(kN/m3)
Masonry Concrete and
Masonry
Linear elastic
isotropic 2500 0.15 22
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4.1.3 Loads
The only load considered was the self weight of the structure. This was computed by DIANA based on
the inputted mass densities and was imposed to the structure as a global load. The total load for analysis
was computed as indicated in Table 13.
Table 13 – Applied loads due to self weight as calculated in AutoCAD or DIANA FEA
Self Weight (kg) Applied Load (kN)
2 302 000 22 583
4.1.4 Boundary Conditions
The tower was assumed to be supported on the bottom surface and at the three lateral connections
between the bell tower and church. The base of the tower was assumed to be fully fixed and therefore
was modelled as such with the bottom face of the tower restrained from translation and rotation in all
directions. The connections between the three walls of the tower and the main church were more
complicated as they cannot be assumed to be fully fixed or fixed at all. In particular, it is hypothesized
that the SW and NE walls have little-to-no connection to the main body of the church due to the observed
outward tilt of the tower separate from the body of the main church. Therefore, the tower was modelled
with three different support variations:
A ) Cantilever fixed at the base: this restraint scenario represents the worst case, where there
is no lateral restraint provided by the adjacent NE, NW or SW walls. This is likely not the
case, however, was considered to obtain a Lower Limit for the dynamic properties.
B ) Fixed at the base and supported laterally on the interior connection surface: this
restraint scenario represents a situation where there is a rigid connection between the tower
and the intersecting church vault on the NW façade, but no additional lateral restraint
provided by the adjacent NE or SW walls. This is a more probabilistic scenario as it has
been hypothesized that the NE and SW walls are improperly connected to the tower. In
addition, this scenario will provide modal results that lie within the lower and upper limits.
C ) Fixed at the base and supported laterally on all three connection surfaces: this
restraint scenario represents the most retrain, where the tower is fully laterally supported by
the intersecting church bodies on the adjacent NE, NW and SW walls. This is almost
certainly not the case, however, will be considered to obtain an Upper Limit for the dynamic
properties.
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Note that due to the two-dimensional nature of the simplified model, the restraints could not be applied
to surfaces and could only be applied to lines or nodes. The restraint scenarios that were analysed are
depicted in Figure 36.
Figure 36 – Support conditions from left to right: A: fixed at the base, B: fixed at the base and laterally restrained in global Y direction, C: fixed at base and restrained in global X and Y directions
4.1.5 Mesh
The mesh properties were assigned as depicted in Figure 37 using a 0.2 m mesh (209 elements), default
mesher type and linear interpolation. A smaller mesh size was implemented for comparison; however,
it was found that due to the simple nature of the model, the element size had a negligible impact on the
results (<0.5% difference between eigen frequencies). Therefore, the model with 0.2 m element size
was used for analysis of the results, similar to that of the 3D model.
Figure 37 - Mesh as defined in the FE model: top view (left), full view (right)
A B C
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4.1.6 Linear Self Weight Analysis
A linear elastic analysis was conducted first to ensure the reaction forces were as expected for the self
weight of the structure. This was achieved using the “structural linear static” analysis tool in DIANA FEA
in which the Parallel Direct Sparse solution method was utilized with a convergence tolerance of 1e-08.
As seen in Table 14, the model was in equilibrium and in the correct order of magnitude.
Table 14 – Linear elastic self weight analysis results
Applied Load (kN) Sum of Reaction Forces (kN) % Difference
22 583 22 583 0
4.1.7 Linear Modal Response Analysis
The modal parameters (mode shapes, eigenfrequencies, global participation factors and mass
participation percentages) were identified using the “Structural Modal Response” analysis function in
DIANA FEA. This was achieved using the eigenvalue analysis parameters as indicated in Table 15.
Table 15 – Eigenvalue analysis parameters
Parameter Input
Stiffness Matrix linear elastic
Mass Matrix Type Consistent
Solver Method Implicitly restarted Arnoldi method
Solver Type Parallel direct
Number of Eigenfrequencies 30
Maximum # of Iterations 30
Convergence Criterion Tolerance 1e-06
4.2 Full Solid Element Estimation
To obtain a more accurate representation of the modal parameters of the bell tower of Sant Cugat
Monastery, a representative 3-dimensional solid finite element model was constructed in DIANA FEA.
This was achieved through the following procedure.
4.2.1 Geometry
The solid 3D geometry was imported from AutoCAD as a .iges file. This allowed for the solid elements
created in AutoCAD to be imported as 3D solid structural elements in DIANA FEA. Any changes that
needed to be made throughout the iterative procedure were conducted in AutoCAD and re-imported into
DIANA.
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4.2.2 Materials
The material properties were chosen as indicated in Table 16 based on the findings from the literature
review.
Table 16 – Material properties used in FE model
Material Class Material Model
Young’s
Modulus
(MPa)
Poisson’s
Ratio
Specific
Weight
(kN/m3)
3D Surface
Interface Springs
(kN/m3)
Masonry Concrete and
Masonry
Linear elastic
isotropic 2500 0.15 22
250 – 1250000
Vault Infill Concrete and
Masonry
Linear elastic
isotropic 2500 0.15 18
4.2.3 Loads
The only load considered for both the analyses was the self weight of the structure. This was computed
by DIANA based on the inputted specific weights and was imposed to the structure as a global load.
The total load for analysis was computed as indicated in Table 17.
Table 17 – Applied loads due to self weight as calculated in AutoCAD or DIANA FEA
Self Weight (kg) Applied Load (kN)
2 288 854 22 446
4.2.4 Supports & Connections
The tower was supported on the bottom surface and at the three lateral connection surfaces between
the tower and the church. The base of the tower was assumed to be fully fixed and therefore was
modelled as such with the bottom face of the tower restrained from translation and rotation in all
directions. The connections between the three walls of the tower and the main church were more
complicated as they cannot be assumed to be fully fixed or fixed at all. In particular, it is hypothesized
that the SW and NE walls have little-to-no connection with the main body of the church due to the
observed outward tilt of the tower separate from the body of the main church. Therefore, the tower was
modelled with five different support scenarios to capture the range of possible dynamic properties of the
bell tower:
A ) Cantilever fixed at the base: this restraint scenario represents the worst case, where there
is no lateral restraint provided by the adjacent NE, NW or SW walls. This is likely not the
case, however, was considered to obtain a Lower Limit for the dynamic parameters.
B ) Fixed at the base and supported laterally on the interior connection surface: this
restraint scenario represents a situation where there is a rigid connection between the tower
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and the intersecting church vault on the NW façade, but no additional lateral restraint
provided by the adjacent NE or SW walls. This is a more realistic scenario as it has been
hypothesized that the NE and SW walls are improperly connected to the tower. In addition,
this scenario will provide modal results that lie within the lower and upper limits.
C ) Fixed at the base and supported laterally on all three connection surfaces: this
restraint scenario represents the most restrained case, where the tower is fully laterally
supported by the intersecting church bodies on the adjacent NE, NW and SW walls. This is
likely not the case, however, was considered to obtain an Upper Limit for the dynamic
parameters.
D ) Fixed at the base, fixed laterally on the interior connection surface and supported
laterally with springs on the SW and NE connection surfaces: similar to scenario B, this
restraint scenario represents a situation where there is a rigid connection between the tower
and the intersecting church vault on the NW façade, however there is some lateral restraint
provided by the adjacent NE and SW walls. The additional lateral restraints are modeled
with boundary surfaces with reduced stiffnesses compared to the restraint provided by a
rigid support. This is a more realistic scenario compared to scenario B, as even if there is a
poor connection between the tower and the adjacent NE and SW walls, there is likely some
connection which can be modeled with the boundary surface.
E ) Fixed at the base and supported laterally with springs on all connection surfaces:
This restraint scenario is hypothesized to be the most likely as it is very unlikely that the
connections between the tower and the church, if any, are completely rigid. The reduced
rigidity was modeled by introducing boundary surfaces with reduced stiffness on the
adjacent NW, NE and SW walls.
The restraint scenarios that were analysed are depicted in Figure 38 to Figure 41.
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Figure 38 - Support conditions from left to right: A: fixed at the base, B: fixed at the base and laterally restrained on the interior connection surface, C: fixed at base and laterally restrained on all three connection surfaces
Figure 39 – Plan view of support conditions from left to right: A, B and C
Figure 40 - Support conditions for Scenario D: fixed at the base, fixed laterally on the interior connection surface and supported laterally with springs on the SW and NE connection surfaces
A B C C
A B C
D D
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Figure 41 - Support conditions for Scenario E: fixed at the base, supported laterally with springs on all connection surfaces
Scenario A can be considered the most flexible scenario, where there are no lateral restraints supporting
the tower. On the contrary, Scenario C is the most rigid case where there are fixed lateral restraints at
all connection surfaces. In reality, the dynamic properties will lie within this range, therefore, Scenarios
D and E were created with springs to simulate the real situation. The boundary springs are comprised
of structural plane interface elements with an associated stiffness applied perpendicular to the boundary
plane in kN/m3.
The spring stiffness along with the elastic modulus of the masonry are parameters that will need to be
updated using an iterative procedure following the dynamic testing. However, an expected range for the
spring stiffness was obtained by conducting an iterative sensitivity analysis of the first three global
modes (bending in X, bending in Y and torsion) and comparing the eigen frequencies to those obtained
from the first three restraint scenarios (A-C). Through this analysis, a range of spring constants was
obtained as shown in Table 18.
E E
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Table 18 – Spring stiffness range for lateral restraints
Range
Description
Spring Stiffness
(kN/m3)
Mode Shape
Description
Scenario D – Eigen
frequency (Hz)
Scenario E – Eigen
frequency (Hz)
Low end – close
to Scenario A 500
Global bending about
X 2.19 1.08
Global bending about
Y 1.11 1.10
Global torsion 6.18 3.51
Average -
between Scenario
A and Scenario
B/C
50 000
Global bending about
X 2.19 1.59
Global bending about
Y 1.66 1.65
Global torsion 6.30 5.19
High end – close
to Scenario B/C 1 000 000
Global bending about
X 2.19 2.13
Global bending about
Y 2.08 2.16
Global torsion 6.58 6.61
Therefore when calibrating the model, spring stiffnesses ranging from 500 to 1 000 000 kN/m3 may be
considered until modal parameters match those obtained in the dynamic testing campaign. However,
for the sake of the analysis of the hypothetical model, the spring stiffness which produced results
between Scenarios A and B/C was considered (50 000 kN/m3).
4.2.5 Mesh
Two meshes were considered for increased accuracy and comparison of results. The mesh properties
were assigned as indicated in Table 19 and are depicted in Figure 42.
Table 19 – Mesh properties as defined in the FE model
Mesh Element Size [m] Mesher Type Mid-side node location # of elements
Coarse 0.3 Tetra/Triangle Linear interpolation 320 630
Fine 0.2 Tetra/Triangle Linear interpolation 978 067
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Figure 42 – Model with coarse mesh (left) vs. fine mesh (right)
It was found that the modal frequencies between coarse and fine meshes were consistent, varying by
an average of only 2.1%. Therefore, the model was considered to be accurate and the fine mesh was
utilized for further analyses.
4.2.6 Linear Self Weight Analysis
A linear elastic analysis was conducted first to ensure the reaction forces were as expected for the self
weight of the structure. This was achieved using the “structural linear static” analysis tool in DIANA FEA
in which the Parallel Direct Sparse solution method was utilized with a convergence tolerance of 1e-08.
As can be seen in Table 20, the model was in equilibrium and in the correct order of magnitude.
Table 20 – Linear elastic reaction forces in the global Z direction
Applied Load (kN) Sum of Rection Forces (kN) % Difference
22 446 22 454 0.0375
4.2.7 Linear Modal Response Analysis
The modal parameters (mode shapes, eigenfrequencies, global participation factors and mass
participation percentages) were identified, with both fine and coarse meshes, using the “Structural Modal
Response” analysis function in DIANA FEA. This was achieved using the eigenvalue analysis
parameters as indicated in Table 21.
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Table 21 – Eigenvalue analysis parameters
Parameter Input
Stiffness Matrix linear elastic
Mass Matrix Type consistent
Solver Method Implicitly restarted Arnoldi method
Solver Type Parallel direct
Number of Eigenfrequencies 30-70
Maximum # of Iterations 30
Convergence Criterion Tolerance 1e-06
4.3 Results & Analysis
The following section outlines the main results from both the static and dynamic analyses including the
comparison of the main modal parameters (mode shapes and frequencies) for the beam and solid
models and for each restraint scenario.
4.3.1 Linear Self Weight Analysis
First, a comparison of the applied load was conducted between the simplified beam model and full 3D
solid model, the results of which are provided in Table 22.
Table 22 – Linear elastic self weight analysis results
Model Applied Load (kN) Sum of Reaction
Forces (kN)
Simplified Beam 22 583 22 583
3D Solid 22 446 22 454
% Difference 0.6
From this comparison, it was found that the solid model had only 0.6 % less mass compared to the
simplified beam model. Therefore, the simplified model proved to be an accurate representation of the
bell tower.
4.3.2 Linear Modal Response Analysis
Further comparison was conducted between the first three corresponding global mode shapes of the
simplified beam model and the full 3D solid model as shown in Table 23 to Table 25. As can be observed,
the modal frequencies vary by less than 8% and therefore, the simplified model can be considered to
validate the 3D solid model for further analysis and for the dynamic characterisation of the Sant Cugat
bell tower.
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In addition, the first ten modal parameters between the simplified and full models were compared as
indicated in Table 26. See Appendix B for a depiction of each mode shape listed in Table 26.
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Table 26 – Modal Frequency comparison between beam and solid models (brackets indicate the mode number)
Mode Shape Simplified Beam Model 3D Solid Model
Description Line Diagram 3D Diagram f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. C)
[Hz]
f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. D)
[Hz]
f (Sc. E)
[Hz]
f (Sc. C)
[Hz]
1st global
bending mode
about X axis
1.1256
(1)
2.2957
(3)
2.2958
(3)
1.0756
(1)
2.1624
(2)
2.1633
(2)
1.5796
(1)
2.1658
(1)
1st global
bending mode
about Y axis
1.1707
(2)
1.1707
(1)
2.1502
(2)
1.0886
(2)
1.0975
(1)
1.6526
(1)
1.6383
(2)
2.2109
(2)
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Mode Shape Simplified Beam Model 3D Solid Model
Description Line Diagram 3D Diagram f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. C)
[Hz]
f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. D)
[Hz]
f (Sc. E)
[Hz]
f (Sc. C)
[Hz]
Global torsion
mode
6.1573
(5)
6.1566
(4) - -
6.0721
(4)
6.1927
(4)
5.1347
(3)
6.6020
(3)
2nd global
bending mode
about X axis
6.0309
(4) - -
4.4418
(4)
6.6314
(5)
6.6335
(5)
5.3498
(4)
6.6705
(4)
3rd global
bending mode
about Y axis,
global torsion
-
- - - 4.4574
(5) - -
5.7751
(5)
6.8807
(5)
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Mode Shape Simplified Beam Model 3D Solid Model
Description Line Diagram 3D Diagram f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. C)
[Hz]
f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. D)
[Hz]
f (Sc. E)
[Hz]
f (Sc. C)
[Hz]
Global
elongation in Z
axis
7.5631
(6)
7.5592
(5)
7.5592
(4)
7.0420
(6)
7.0455
(6)
7.0496
(6)
7.0465
(6)
7.0926
(6)
Local bending
mode above
level 6 about Y
axis
- - 9.7533
(6)
7.2213
(7)
7.2493
(7)
7.9923
(7)
7.9812
(8) -
3rd global
bending mode
about X axis
- 10.178
(7)
10.179
(7) -
9.4400
(8)
9.4675
(8)
7.6198
(7)
9.4625
(7)
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Mode Shape Simplified Beam Model 3D Solid Model
Description Line Diagram 3D Diagram f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. C)
[Hz]
f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. D)
[Hz]
f (Sc. E)
[Hz]
f (Sc. C)
[Hz]
Local bending
mode above
level 7 about X
axis
12.127
(9) - -
7.2799
(8) -
13.623
(11)
13.612
(12)
13.620
(10)
4th global
bending mode
about Y axis,
local torsion
above level 6
12.086
(8)
12.087
(8)
12.088
(8)
8.8469
(9)
9.6053
(9)
11.533
(9)
11.509
(11)
9.7927
(8)
Local bending
mode above
level 6 about Y
axis
12.337
(10)
12.376
(9) - - -
13.880
(12)
13.874
(13)
13.962
(11)
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Mode Shape Simplified Beam Model 3D Solid Model
Description Line Diagram 3D Diagram f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. C)
[Hz]
f (Sc. A)
[Hz]
f (Sc. B)
[Hz]
f (Sc. D)
[Hz]
f (Sc. E)
[Hz]
f (Sc. C)
[Hz]
Global torsion
mode -
- - - 9.9383
(10)
11.533
(10)
12.131
(10)
10.309
(9) -
4th global
bending mode
about X axis
-
- - - 10.330
(11) - -
10.982
(10) -
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From this comparison, it can be concluded that the simplified beam model validifies the 3D solid model
as it has similar global mode shapes with corresponding frequencies varying by less than 8%. The
discrepancies between corresponding mode shape frequencies could be caused by several factors such
as the linear nature of the simplified model, the lack of openings, vaults and connection surfaces in the
simplified model, and the lack of surface restraints in the simplified model. Therefore, although the
simplified beam model validifies the 3D solid model, the 3D solid model should be utilized for the
dynamic characterisation of Sant Cugat Monastery as it is more accurate and is able to produce three-
dimensional global and local mode shapes.
In addition, it can be observed that eigenfrequencies tend to decrease with reduced stiffness at the
connection surfaces (Scenarios B-E). Therefore, when analysing the results from the dynamic testing,
lower eigenfrequencies likely indicate loss of stiffness at one or more of the connection surfaces. The
aim of the iterative model updating procedure is to identify the source of the observed flexibility.
4.4 Soil-Structure Interaction
In the previous subsections, the tower was assumed to be fully supported on the bottom surface with a
perfectly rigid connection between soil and structure. However, a rigid foundation is not necessarily the
case, especially since the outward tilt of the tower may be caused by subsurface soil settlement.
Therefore, this section examines the impact of a flexible foundation on the hypothetical dynamic
properties of the Sant Cugat bell tower. This was achieved through the implementation of reduced
stiffness at the base of both models and an iterative sensitivity analysis to obtain the expected range of
stiffness.
4.4.1 Simplified Beam Model
For the sensitivity analysis of the simplified beam model, the tower was modelled with two different
support scenarios to identify an appropriate range of stiffness:
A ) Cantilever fixed at the base: this restraint scenario represents the best case, where the
base node is fixed from translation and rotation in all directions. This is likely not the case,
however, was considered to obtain an Upper Limit for the dynamic parameters.
B ) Cantilever with reduced stiffness at the base: this restraint scenario represents a soil-
structure interface with reduced stiffness at the base of the structure implemented through
two discrete rotational boundary springs (one acting in the X axis and one in the Y axis) with
assigned stiffness. In this case, an additional restraint was implemented at the base node
to restrict translation and torsion.
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Note that due to the two-dimensional nature of the simplified model, the restraints could not be applied
to surfaces and could only be applied to lines or nodes.
The dynamic properties from Scenario A can be considered the best-case scenario, where the soil-
structure interaction is perfectly fixed. However, it is probable that there is some flexibility/movement in
the foundation, therefore, Scenario B was created with springs to simulate this situation. The boundary
springs are comprised of discrete rotational boundary spring elements (SP1RO) located at the base
node with an associated stiffness applied in the X and Y directions in kNm/rad.
A range for the spring stiffness was obtained by conducting an iterative sensitivity analysis of the first
three global modes (bending in X, bending in Y and second global bending in Y) and comparing the
eigen frequencies to those obtained from Scenario A. Through this analysis, a range of spring constants
was obtained as shown in Table 27.
Table 27 – Spring stiffness range for soil-structure interaction
Range
Description
Spring Stiffness
(kNm/rad)
Mode Shape
Description
Scenario B – Eigen
frequency (Hz)
Low end – close
to fully flexible 1.75e+06
Global bending about
X 0.21160
Global bending about
Y 0.21186
2nd global bending
mode about Y 1.7205
Mid-range 1e+08
Global bending about
X 0.92925
Global bending about
Y 0.95466
2nd global bending
mode about Y 1.7221
High end – close
to Scenario A
(fully rigid)
1e+12
Global bending about
X 1.1255
Global bending about
Y 1.1707
2nd global bending
mode about Y 1.7236
Therefore, when calibrating the model, spring stiffnesses ranging from 1.75e+06 to 1e+12 kNm/rad may
be considered until modal parameters match those obtained in the dynamic testing campaign. However,
for the sake of the analysis of the hypothetical model, the spring stiffness which produced results in the
mid-range was considered (1e+08 kNm/rad). See Table 28 for a comparison of the first global
frequencies between Scenarios A and B.
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Table 28 - Modal Frequency comparison between restraint scenarios for mid-range spring stiffness
Mode Description Sc. A: rigid
f [Hz]
Sc. B: flexible
f [Hz]
1st global bending
mode about X axis
1.1256
(1)
0.92925
(1)
1st global bending
mode about Y axis
1.1707
(2)
0.95466
(2)
2nd global bending
mode about Y
axis, global
torsion mode
1.7236
(3)
1.7221
(3)
As can be observed, the eigenfrequencies decrease with increased flexibility at the foundation.
Therefore, when analysing the results from the dynamic testing, lower eigenfrequencies may indicate
loss of stiffness at the soil-structure interaction, however, may also be indicative of loss of stiffness at
lateral connections surfaces. The aim of the iterative model updating procedure is to identify the source
of the observed flexibility.
4.4.2 Full 3D Solid Model
For the sensitivity analysis of the 3D solid model, the tower was modelled with three different support
scenarios to identify an appropriate range of stiffness:
A ) Cantilever fixed at the base: this restraint scenario represents the best case, where the
soil-structure interface is perfectly fixed. This is likely not the case, however, was considered
to obtain an Upper Limit for the dynamic parameters.
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B ) Cantilever with reduced stiffness at the base: this restraint scenario represents a soil-
structure interface with uniform reduced stiffness across the base of the structure
implemented through a fixed boundary interface with reduced stiffness to represent
subsurface instability.
C ) Cantilever with reduced stiffness at the base and fixed hinges along the NW edge of
the foundation: this restraint scenario represents a situation where the northern edge of
the foundation is fully rigid however the remainder of the surface is subjected to reduced
stiffness. In this case, the northern edge is restrained from translation in all directions,
however, is free to rotate, whereas the base of the tower is modeled with a fixed boundary
interface with reduced stiffness to represent subsurface instability. This scenario was
examined as the tower is exhibiting outward lateral displacement in the SE direction,
indicating that soil settlement may be occurring past the NW edge of the tower’s foundation.
The restraint scenarios that were analysed are depicted in Figure 43 and Figure 44.
Figure 43 - Support conditions from left to right: A: cantilever fixed at the base, B: cantilever with reduced stiffness at the base, C: cantilever with reduced stiffness at the base and fixed hinges along the NW edge of the
foundation
A B C
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Figure 44 – Plan view of support conditions from left to right: A, B and C
The boundary springs in Scenarios B and C are comprised of structural plane interface elements (T18IF)
with an associated stiffness applied perpendicular to the boundary plane in kN/m3, with a shear stiffness
assumed to be infinite (1+E28 kN/m3) for simplification. A range for the spring stiffness of the soil-
structure interaction was obtained by conducting an iterative sensitivity analysis of the first three global
modes (bending in X, bending in Y and torsion) and comparing the eigen freuquncies to those obtained
in Scenario A. Through this analysis, a range of spring constants was obtained as shown in Table 29.
Table 29 – Spring stiffness range of soil-structure interaction
Range
Description
Spring
Stiffness
(kN/m3)
Mode Shape Description Scenario B – Eigen
frequency (Hz)
Scenario C – Eigen
frequency (Hz)
Low end – close
to fully flexible 100
Global bending about X 0.020601 0.04467
Global bending about Y 0.022583 0.43707
2nd global bending mode
about Y and global torsion 2.8208 1.7827
Mid-range 100 000
Global bending about X 0.55904 0.73449
Global bending about Y 0.59260 0.65628
2nd global bending mode
about Y and global torsion 2.9032 1.9057
High end – close
to Scenario A
(fully rigid)
15 000 000
Global bending about X 1.0691 1.0700
Global bending about Y 1.0836 1.0839
2nd global bending mode
about Y and global torsion 3.4395 3.4423
Therefore, when calibrating the model, spring stiffnesses ranging from 100 to 15 000 000 kN/m3 may be
considered until modal parameters match those obtained in the dynamic testing campaign. However,
for the sake of the analysis of the hypothetical model, the spring stiffness which produced mid-range
results was considered (100 000 kN/m3). See Table 30 for a comparison of the first global frequencies
between Scenario A, B and C.
A B C
Dynamic characterisation of the bell tower of Sant Cugat Monastery
ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 71
Table 30 – Modal Frequency comparison between restraint scenarios for mid-range spring stiffness
Mode
Description
Sc. A: rigid
f [Hz]
Sc. B: flexible
f [Hz]
Sc. C: supported on NW edge
f [Hz]
1st global bending
mode about X axis
1.0756
(1)
0.55904
(1)
0.73449
(2)
1st global bending
mode about Y axis
1.0886
(2)
0.59260
(2)
0.65628
(1)
2nd global bending
mode about Y axis,
global torsion mode
3.4293
(3)
2.9032
(3)
1.9057
(3)
Similar to the simplified beam model, the eigenfrequencies decrease with increased flexibility at the
foundation. Therefore, when analysing the results from the dynamic testing, lower eigenfrequencies may
indicate loss of stiffness at the soil-structure interaction, however, may also be indicative of loss of
stiffness at lateral connection surfaces. The aim of the iterative model updating procedure is to identify
the source of the observed flexibility.
4.5 FE Model for Dynamic Calibration
Following dynamic testing, it is recommended that the final numerical model for calibration has boundary
conditions and updating parameters as indicated in Table 31. Spring connections were chosen for all
connection surfaces to allow for the analysis of the rigidity of these connections.
Dynamic characterisation of the bell tower of Sant Cugat Monastery
ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 72