advanced masters in structural analysys of monuments …

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

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Dynamic characterisation of the bell tower of Sant Cugat Monastery

ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL

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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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Table of Contents

1. INTRODUCTION ............................................................................................................... 1

2. LITERATURE REVIEW: DYNAMIC IDENTIFICATION OF MASONRY TOWERS ........................ 3

2.1 Numerical Modelling Techniques ........................................................................................................ 6

2.1.1 Geometry ............................................................................................................................................... 7

2.1.2 Material Properties ................................................................................................................................ 7

2.1.3 Boundary Conditions.............................................................................................................................. 8

2.1.4 Updating Parameters ............................................................................................................................. 8

2.2 Experimental Dynamic Testing ............................................................................................................ 9

2.2.1 Testing Procedure ................................................................................................................................ 10

2.2.2 Typical Modal Parameters ................................................................................................................... 11

2.3 Modal Identification ...........................................................................................................................12

2.3.1 Signal Pre-Processing ........................................................................................................................... 13

2.3.2 Modal Identification Techniques ......................................................................................................... 13

2.3.3 Cross Validation - Modal Assurance Criterion (MAC) .......................................................................... 18

2.4 FE Model Calibration ..........................................................................................................................18

2.4.1 Manual Tuning ..................................................................................................................................... 19

2.4.2 The Inverse Eigen-Sensitivity (IE) Method ........................................................................................... 20

2.4.3 The Douglas-Reid (DR) Method ........................................................................................................... 21

2.4.4 The Genetic Algorithm Technique (GA) ............................................................................................... 21

2.4.5 Sensitivity Analysis (SA)........................................................................................................................ 22

2.5 Soil-Structure Interaction ...................................................................................................................23

2.6 Dynamic Action of the Bells ................................................................................................................25

2.6.1 Bell Systems in Europe ......................................................................................................................... 25

2.6.2 Static Analysis ...................................................................................................................................... 26

2.6.3 Dynamic Analysis ................................................................................................................................. 27

3. THE CASE STUDY: BELL TOWER OF SANT CUGAT MONASTERY ....................................... 31

3.1 Historic Survey ...................................................................................................................................32



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3.1.1 Historic Values ..................................................................................................................................... 32

3.1.2 Construction Chronology ..................................................................................................................... 33

3.1.3 The Bells ............................................................................................................................................... 37

3.2 Geometrical Survey ............................................................................................................................39

3.2.1 Structure .............................................................................................................................................. 39

3.2.2 Connections ......................................................................................................................................... 43

3.2.3 Outward Tilt of SE façade ..................................................................................................................... 44

3.2.4 Limitations & Assumptions .................................................................................................................. 46

4. EXPECTED MODAL PARAMETERS ................................................................................... 47

4.1 Simplified Beam Element Estimation ..................................................................................................47

4.1.1 Geometry ............................................................................................................................................. 47

4.1.2 Materials .............................................................................................................................................. 49

4.1.3 Loads .................................................................................................................................................... 50

4.1.4 Boundary Conditions............................................................................................................................ 50

4.1.5 Mesh .................................................................................................................................................... 51

4.1.6 Linear Self Weight Analysis .................................................................................................................. 52

4.1.7 Linear Modal Response Analysis .......................................................................................................... 52

4.2 Full Solid Element Estimation .............................................................................................................52

4.2.1 Geometry ............................................................................................................................................. 52

4.2.2 Materials .............................................................................................................................................. 53

4.2.3 Loads .................................................................................................................................................... 53

4.2.4 Supports & Connections ...................................................................................................................... 53

4.2.5 Mesh .................................................................................................................................................... 57



4.3 Results & Analysis ..............................................................................................................................59



4.4 Soil-Structure Interaction ...................................................................................................................66

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

5.1.1 Dynamic Testing: AVT .......................................................................................................................... 75

5.1.2 Required Equipment ............................................................................................................................ 75

5.1.3 Location of Accelerometers ................................................................................................................. 76

5.1.4 Testing Procedure ................................................................................................................................ 80

5.2 Modal Parameter Identification .........................................................................................................81

5.3 Calibration of the Finite Element Model .............................................................................................81

5.4 Computation of DAF ...........................................................................................................................83

6. CONCLUSIONS ............................................................................................................... 85

6.1 Expected Modal Parameters ..............................................................................................................85

6.2 Dynamic Testing Plan .........................................................................................................................86

6.3 Future studies ....................................................................................................................................87

REFERENCES ......................................................................................................................... 89

APPENDIX A – AS-FOUND DRAWING SET .............................................................................. 93

APPENDIX B – MODE SHAPE COMPARISON......................................................................... 103


ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 1

1. INTRODUCTION

The Monastery of Sant Cugat is located in Sant Cugat del Vallès, 20 km NW of Barcelona, and consists

of various components built predominantly between the 12th to 15th centuries. Recent investigations

have revealed that its bell tower, constructed between the 12th and 18th centuries, may be continuing to

experience lateral displacement linked to an active deterioration mechanism. As such, it is envisioned

that obtaining key dynamic characteristics of the bell tower through full-scale ambient vibration testing

(AVT) will help calibrate a numerical finite element model to better understand the deterioration

mechanisms affecting the tower, if any. See Figure 1 depicting the southern portion of the Monastery

including the Basilica and Bell Tower.

Figure 1 – The Sant Cugat Monastery [1]

The key modal parameters that can be extracted from AVT include the natural frequencies of the bell

tower, their associated mode shapes, and the damping ratios of the mode shapes caused by the

excitation induced by the ringing of the bells. This information can then be used to calibrate the numerical

model with the aim to identify weaknesses in the structure.

Therefore, the objective of this dissertation is to develop a robust procedure for the dynamic identification

of the bell tower of Sant Cugat Monastery. This includes a preliminary state-of-the-art literature review

regarding dynamic identification of masonry bell towers, the creation of a suitable numerical model to

obtain expected modal properties, and lastly, a detailed dynamic testing procedure to be conducted in

the future. The dynamic testing procedure will outline the logistics of the in-situ testing and the

recommended procedure for model calibration.



2. LITERATURE REVIEW: DYNAMIC IDENTIFICATION OF MASONRY

TOWERS

A comprehensive literature review of existing documentation related to the dynamic characterisation of

historic masonry bell towers was conducted to obtain a thorough understanding of the subject matter.

The literature review focuses on topics such as dynamic testing and modal identification, numerical

modelling and calibration, and the dynamic impact of swinging bells. The aim of the review is to analyse

and compare findings from different authors to generate a thorough and efficient dynamic testing plan

for the bell tower of Sant Cugat Monastery. The literature review is comprised of over 20 published

documents, the most influential of which are summarized below.

• Salvador Ivorra, Maria José Palomo, Gumersindo Verdu and Alberto Zasso (2005) [2]

researched the dynamic forces produced by swinging bells in historic bell towers including the

calculation of the maximum forces that they may induce to their supporting structure. This was

achieved through a comparison of the three most common bell ringing systems in Europe:

Central European, English and Spanish. The study found that the forces transmitted to the

supporting structure are significantly lower in the Spanish system as opposed to the others, and

emphasized that the dynamic amplification caused by the interaction between the supporting

structure and the bells involves the measurement of the dynamic characteristics of both the

bells and the tower.

• Salvador Ivorra and Francisco J. Parallarés (2006) [3] conducted dynamic investigations on

a masonry bell tower in Valencia, Spain. In this study, different numerical models were

calibrated based on ambient vibration tests to determine the modal characteristics of the tower.

This work was complimented by an analysis of the inertia forces caused by the bell swinging

and the computation of possible dynamic amplification factors. Due to the good balance of the

Spanish bells, it was found that the horizontal forces developed during their swinging is quite

low and may be considered negligible.

• C. Gentile and A. Saisi (2007) [4] conducted ambient vibration testing (AVT) of an historic

masonry tower in Monza, Italy, to characterize its dynamic properties and to conduct a damage

assessment due to concerning through-wall cracks. This study involved full scale AVT of the

tower, modal identification through operational modal analysis, and finite element modelling and

updating. A good correlation was observed between experimental and theoretical modal

behaviour and therefore the updated models were deemed to be adequate to provide reliable

predictions in the safety assessment of the tower.

• Alemdar Bayraktarm Temel Turker, Baris Sevim, Ahmet Can Altunisik and Faruk Yildirim

(2009) [5] obtained the modal parameters of the bell tower of Hagia Sophia through ambient



vibration testing. Through this analysis, it was found that the theoretical and experimental mode

shapes were in good correlation, however, the theoretical frequencies were greater than the

experimental frequencies. This study highlights common difficulties present in the dynamic

characterisation of masonry bell towers including the complex and often unknown mechanical

properties of masonry walls, the difficulties involved with accurately modelling structural

damage, and the unknown boundary conditions often associated with historical constructions.

• Salvador Ivorra, Francisco J. Pallarés and Jose M. Adam (2011) [6] created a guideline for

the dynamic considerations of masonry bell towers as the oscillations of the bells often interact

with the tower’s natural frequencies causing dynamic amplification. The paper emphasizes how

the arrangement of the bells within the bell frame can considerably impact the dynamic

interaction between the bells and the supporting structure. Therefore, modifying the excitation

frequency of the bells or adding counterweights proves to be much less expensive than

modifying the natural frequencies of the tower through structural strengthening.

• Dora Foti, Salvador Ivorra Chorro and Maria Francesca Sabba (2012) [7] conducted a

dynamic investigation of an ancient masonry bell tower in Mola di Bari, Italy. The aim of this

research was to calibrate a numerical model through operational modal analysis to better

understand the dynamic behaviour of the structure.

• Mariella Diaferio, Dora Foti and Nicola Ivan Giannocaro (2014) [8] conducted the dynamic

characterisation of an old masonry tower in Bari, Italy through ambient vibration testing

accompanied with ground penetration radar (GPR) technology. Operational modal analysis was

utilized to estimate modal parameters of the tower whereas, GPR was utilized to estimate the

wall composition at each level to characterize the masonry in the finite element (FE) model. The

results allowed for the calibration of a complete FE model which was then utilized to characterize

the tower and to evaluate the dynamic interaction between the tower and the cathedral walls.

• N. Nisticò, S. Gambarelli, A. Fascetti and G. Quaranta (2016) [9] conducted the experimental

dynamic testing and numerical modelling of an historic belfry in Rome. Although this study is

not concerning a bell tower, it provided good insight regarding the impact of swinging bells on

supporting masonry structures. Ambient vibration testing was conducted under different

dynamic loading scenarios which confirmed that severe vibrations are induced when the bells

swing, and thus a slight reduction of the swing angle was recommended.

• F. Lorenzoni, M.R. Valluzzi, M. Salvalaggio, A. Minello and C. Modena (2017) [10]

performed a modal analysis to characterize four ancient water towers located in Pompeii, Italy.

This paper reports the outcomes of ambient vibration tests and the extraction of the towers



modal properties using various operational modal analysis techniques. The obtained data was

then used to study the soil-structure interaction of the towers through model updating.

• Massimiliano Ferraioli, Lorenzo Miccoli, Donato Abruzzese and Alberto Mandara (2017)

[11] conducted a dynamic characterisation and seismic assessment of two medieval masonry

towers located in Italy. This paper describes the full-scale ambient vibration testing and modal

identification of the towers followed by the calibration of their associated finite element models.

The challenges associated with model updating were highlighted due to the many uncertain

geometrical and mechanical properties of historic masonry structures. This uncertainty causes

difficulties in conducting accurate structural assessments and highlights the importance of

independent numerical modelling and simulation of the mechanical behaviour of the masonry

itself. This paper also conducted a seismic vulnerability analysis, emphasizing that the seismic

performance of masonry towers is greatly influenced by their geometry and slenderness, the

thickness of their perimeter walls, the percentage of openings, and their boundary conditions.

• Climent Molins and Nirvan Makoond (2017) [12] analysed the dynamic behaviour of an

historic bell tower in Lleida, Spain. This investigation aimed to characterise the dynamic

behaviour of the bell tower through an analysis of the dynamic impact of different bell ringing

systems on the supporting structure. The characterisation involved full scale ambient vibration

testing, extraction of modal parameters using various operational modal analysis techniques,

and the construction and calibration of a finite element model.

• Massimiliano Ferraioli, Lorenzo Miccoli and Donato Abruzzese (2018) [13] conducted a

dynamic characterisation of an historic bell tower in Vico, Italy, using a sensitivity-based

technique for model tuning. This study involved full-scale ambient vibration testing, operational

modal analysis and dynamic-based finite element (FE) modelling. Uncertain parameters of the

FE model were adjusted to match the experimental mode shapes, then, the most sensitive

parameters were chosen as updating parameters for the sensitivity-based model tuning

technique. It was found that the soil–structure interaction was very sensitive to adjustment and

therefore, it was recommended that a geotechnical study be conducted to verify the subsurface

conditions.

• David Bru, Salvador Ivorra, Michele Betti, Jose M. Adam and Gianni Bartoli (2019) [14]

conducted a parametric dynamic interaction assessment between bells and their supporting

masonry tower structures. This included the analysis of a case study through experimental

operational modal analysis and numerical evaluation of the dynamic interaction. This paper

suggested a method to analyse the dynamic properties of masonry bell towers as a significant

dynamic interaction between the harmonic bell forces and the fundamental tower modes may

exist. The numerical analysis was performed on a calibrated finite element model.



• Milorad Pavlovic, Sebastiano Trevisani and Antonella Cecchi (2019) [15] presented a

procedure for the structural identification of masonry towers as the determination of their

structural behaviour is often complex due to material inhomogeneity, manufacturing

imperfections, geometric configuration and/or structural interaction between the layers in multi-

leaf masonry structures. In this research, an efficient procedure was proposed based on

experimental measurements and numerical modelling with the aim to estimate average

mechanical characteristics under service loads. In summary, the procedure involved the

acquisition of the fundamental frequency of the tower using digital tromographs followed by finite

element model calibration based on experimental data.

An analysis of the findings from the literature review is provided below, divided into the following

subsections: Numerical Modelling Techniques, Experimental Dynamic Testing, Modal Identification,

Finite Element (FE) Model Calibration, Soil-Structure Interaction and the Dynamic Action of the Bells.

2.1 Numerical Modelling Techniques

Dynamic characterisation requires the combination of experimental test data and analytical methods

such as finite element (FE) modelling to determine the global dynamic properties of a structure [16].

Therefore, often before dynamic testing is conducted, a preliminary FE model is created to obtain the

expected modal characteristics of the structure which helps identify where the sensors should be placed

[16]. When constructing the FE model, various parameters must be defined and/or assumed such as:

geometry, material properties, boundary conditions and updating parameters. See Figure 2 for

examples of numerically modeled bell towers from existing literature.

Figure 2 – Examples of FE models from existing literature [4] [5] [8] [13]



2.1.1 Geometry

The geometry of the numerical model is often developed in three-dimensional FE software (such as

DIANA FEA [17]) and is typically based on a geometric survey of the structure and existing structural

and architectural drawings [4] [11] [13]. In general, it was observed that 3D solid elements were utilized

to represent the load-bearing masonry walls to allow for a uniform mass distribution and the accurate

representation of openings and floors [4] [7] [11] [13]. Following the construction of the geometry, a FE

mesh was implemented to create discretized finite elements and in most cases, the mesh was refined

until the change in frequency between considered mode shapes was negligible [11] [13]. The following

commonalities between analysed FE models were observed:

• FE element type: 8-node solid (most common), 4-node tetrahedral solid, 6-node solid

• Number of nodes: 4 000 – 54 000

• Number of solid elements: 2 500 – 180 000

As can be observed, the number of nodes and elements varies considerably depending of the

refinement of the mesh. In general, the larger the number of elements, the better the mass distribution

and resulting distribution of forces.

Oftentimes, in addition to the full 3D solid model, a simplified model was also created to obtain a

preliminary idea of the expected mode shapes and to validify the 3D model. However, due to the many

assumptions required to idealise the behaviour of the tower in the simplified models, the more detailed

3D models were always chosen for dynamic characterisation [4] [7] [12].

2.1.2 Material Properties

Due to the variable, inhomogeneous and nonlinear nature of masonry as a building material, it is difficult

to assume homogeneous, linear elastic properties for the hypothetical numerical model. However, the

purpose of dynamic testing is to obtain global dynamic properties of the entire structure. Therefore, for

the purposes of this type of analysis, linear elastic behaviour is considered acceptable. The following

common assumptions regarding material properties were observed from the literature review:

• Linear elastic isotropic material behaviour [7] [10] [11] [13]

• Constant self-weight per unit volume of masonry [4] [7] [11] [13] [15]

• Constant Poisson’s ratio [4] [11] [13] [15]

A range of linear elastic material properties was defined based on the literature review as shown in

Table 1.



Table 1 – Reference material properties from different case studies

Article E [MPa] v p (kg/m3)

[4] 985 – 1380 0.07 – 0.2 -

[3] - 0.15 -

[7] - 0.15 1700

[12] 11 200 0.2 2300

[6] - - 1220 - 1835

[13] 1800 - 2600 - -

[11] 1800 - 5000 0.15 – 0.2 1600-1900

[9] 2000 - 3600 0.3 1800

[15] 2200 0.2 1800

[5] 1300 0.15 1600

It was observed that Young’s Modulus (E) varies from 1 000 to 11 000 MPa (average of 3078 MPa),

Poisson’s ratio (v) varies from 0.07 to 0.3 (average of 0.177), and the specific weight varies from 1200

to 2300 kg/m3 (average of 1750 kg/m3). These values can be utilized as a reference for masonry tower

FE model construction, however, if possible, in situ material testing should be conducted to identify a

more accurate representation of the material properties.

2.1.3 Boundary Conditions

In general, most FE models were modelled with a fixed base due to the uncertainty of subsurface

conditions [4] [7] [8] [9] [12] [14] [15]. However, the soil-structure interaction should not be ignored,

particularly if there is structural concern regarding seismic resistance or lateral displacement. Some

papers assessed the soil-structure interaction through the implementation of a uniform distribution of

discrete linear elastic springs of constant stiffness at the base; this procedure is known as the Winkler

soil model [10] [11] [13]. With the implementation of these springs, it is possible to identify the stiffness

of the soil-structure interaction through the model calibration procedure.

Moreover, lateral restraints along exterior facades were also considered in many numerical models as

bell towers are often confined by a surrounding structure (i.e. the main body of the church). Similar to

the soil-structure interaction, these connections were modelled with a uniform distribution of linear elastic

springs in many case studies [4] [7] [8] [12] [13]. One of the case studies modelled the constraining walls

as 3D solids, however, concluded that introducing lateral springs would have been more efficient [14].

2.1.4 Updating Parameters

To calibrate the FE model, it is necessary to identify “updating parameters” that can be iteratively

changed, assessed, and compared against the experimental results until a high level of precision is

attained. In every explored case study, the Young’s Modulus of the masonry (E) was utilized as an

updating parameter as changes in E will considerably change the dynamic properties. In some case



studies, the masonry was modeled in different sections to identify areas of hypothesized low quality

masonry (lower E) versus high quality masonry (higher E) to be confirmed with the dynamic testing [4]

[13] [14].

In addition to Young’s Modulus, the spring constants (K) located at the foundation and connections

between the bell tower and its surrounding structure were chosen as updating parameters [4] [8] [10]

[11] [13]. These parameters were chosen to help identify the presence of poor connections and/or

subsurface conditions when compared with the experimental data. In some cases, the structural density

of the masonry was also chosen as an updating parameter [8] [14].

2.2 Experimental Dynamic Testing

Following the construction of the preliminary FE model, experimental dynamic testing is typically

conducted. Using dynamic testing, it is possible to evaluate the dynamic characteristics of a structure

(i.e. natural frequencies, mode shapes and damping ratios), develop realistic numerical models, identify

structural issues due to dynamic actions (i.e. interaction between vibrations induced by bell ringing and

the supporting structure), identify “weak points” within the structure, and evaluate structural health,

among many other uses [16]. In addition, experimental dynamic testing is the only non-destructive

technique used to measure parameters associated with global structural behaviour and therefore, it is a

useful tool when other experimental techniques are not viable [5] [16]. This is especially useful for

historic constructions as their structural details and mechanical properties are often unknown and

destructive testing is often prohibited.

Dynamic testing methods are classified in two main groups: Experimental Modal Analysis (EMA) and

Operational Modal Analysis (OMA) [5]. In EMA, structures are excited by measured forces and the

corresponding response is recorded. In contrast, OMA does not require any excitation and requires the

measurement of only the output signals [5].

In all reviewed articles that conducted experimental dynamic testing, Ambient Vibration Testing (AVT)

was chosen to extract modal parameters. AVT is based on OMA principles and is centred on the

assumption that the ambient excitation is a stationary Gaussian white noise stochastic process in the

frequency range of interest which requires only environmental excitations (i.e. wind, traffic, humans) [5]

[15] [16]. As OMA does not require any artificial excitement, AVT is a non-destructive test and therefore

has minimal interference with the normal use of the structure and its cultural/historic value [4] [5] [7] [9]

[15]. In addition, AVT has been proven to be especially suitable for flexible systems (such as tall, slender

towers) as even vibrations of low intensity generally produce very clear signals [4] [7].



2.2.1 Testing Procedure

The dynamic response of a structure can be measured using any type of device able to convert physical

quantities (i.e. displacements, accelerations, velocities, strains) into proportional electrical signals to be

processed by a data acquisition system (DAQ) [16]. For AVT, it is common to utilize piezoelectric

accelerometers due to their relatively low cost, high sensitivity, good signal-to-noise ratio, and no need

for an external power source [16]. The response signals are converted into discrete digital signals and

recorded as acceleration time-histories which can then be processed and analysed by modal

identification software. Due to the low signals recorded by the sensors under ambient excitation, it is

crucial to amplify the signals and filter them to obtain the desired output [13].

In the analysed case studies, it was found that piezoelectric accelerometers (uniaxial, biaxial and triaxial)

were installed at various locations within the bell towers depending on the desired outcome. For

example, in the case studies that were assessing global damage, accelerometers were installed at all

levels and in all directions to be compared with future tests; this involved the installation of several

accelerometers (20+) [4] [8]. However, much of the research that was examined was regarding dynamic

characterisation. In these cases, a fewer number of accelerometers were installed in strategic positions

related to the hypothetical mode shapes obtained from the preliminary FE models. This typically involved

2 to 10 accelerometers with one or more acquisitions. See Figure 3 for examples of accelerometer

placement.

Figure 3 – Examples of accelerometer placement for dynamic characterisation [3] [7] [11]



Accelerometers are typically installed to the surfaces of the bell tower through screws or putty. In

general, the piezoelectric accelerometers discussed in the reviewed articles had a working range of

0.025 to 2000 Hz and a sensitivity of 1 V/g. The sampling frequency was set between 100 and 2000 Hz

(the upper bound was utilized to capture the harmonics of the bells) and the acquisition time varied

between 5 and 40 minutes. An empirical rule for sampling time is equal to 2000 times the highest natural

period of interest from the hypothetical FE model [16].

With respect to the number of acquisitions required for dynamic testing, typically only one acquisition is

conducted in addition to a short preliminary test to ensure the DAQ is functioning properly [12]. However,

in the case of bell towers, it is also useful to measure the structural response subjected to the ringing of

the bells as this helps identify any interaction between the bells and the supporting structure. Therefore,

in the papers examining the dynamic impact of the bells, multiple acquisitions were conducted under

different bell ringing schemes [3] [7] [12] [14].

2.2.2 Typical Modal Parameters

Based on existing research and documentation, it is expected that the main frequencies of historic

masonry bell towers are between 0.9 and 2 Hz, with the first two modes being global bending in

orthogonal directions and the third mode being global torsion [3]. Table 2 outlines this phenomenon

through the analysis of several case studies as presented in one of the reviewed papers [3]. The same

study was conducted with the cases that were analyzed as apart of this research as indicated in Table

3. In both cases, the results were found to be generally consistent with the expected modal frequencies

of historic masonry bell towers.

Table 2 – Comparative study of some real cases [3]



Table 3 – Comparative study of experimental frequencies

Ref. Tower height (m) Natural frequencies (Hz)

1 2 3 4

Ivorra [3] 41 1.294a 1.489a 3.979a 4.321b

Diaferio [8] 60 2.04a 2.26a 7.03a 7.60a

Ferraioli [13] 40 1.31a 1.44a 2.91b 4.42a

Bru [14] 40 0.86a 0.98a 3.55b 3.99a

Ferraioli [11] 45.5 1.05a 1.37a 4.81b 4.89a

Ferraioli [11] 41 1.26a 1.29a 3.10b 6.15a

Turker [5] 23 2.56a 2.66a 6.22b 8.10a

a Bending mode; b Torsion mode.

Once the natural frequencies have been identified, it is simple to produce large structural deformations

in the experimental data with small energy input. This was achieved in several papers through the

oscillation of the bells and allowing for the free vibration of the tower while recording the vibration time

history. This type of test allows for the calculation of the logarithmic viscous damping coefficient [3]. A

typical damping coefficient for masonry buildings is equal to 0.015, however, is often lower in masonry

bell towers due to their flexible and slender nature [14]. Table 4 outlines typical damping coefficients of

masonry bell towers from the analysis of several case studies.

Table 4 – Comparative study of experimental damping coefficients

Ref. Tower height (m) Damping Coefficient Damping percentage [%]

Ivorra [3] 41 0.0159 1.59

Foti [7] 35 0.01745 1.75

UPC [12] 53 0.0061 0.61

Lund [18] various 0.014 – 0.078 1.4 – 7.8

Ivorra [6] various 0.0061 – 0.025 0.61 – 2.5

Bru [14] 40 0.0137 – 0.0166 1.37 – 1.66

Turker [5] 23 0.01932 -0.03616 1.93 – 3.62

Low damping ratios are characteristic of masonry bell towers and may cause a high dynamic

amplification factor when the excitation frequencies of the swinging bells come close to one of the

tower’s natural frequencies [12]. When the dynamic parameters of the tower subjected to bell ringing

are obtained, the dynamic amplification factor may be computed and the possible interaction between

excitation and natural frequencies can be analysed [6].

2.3 Modal Identification

Once the dynamic testing has been conducted, modal analysis software (such as ARTeMIS [19]) may

be used to identify the experimental dynamic parameters of the structure through OMA identification

techniques [4] [20]. The estimation is conducted through various system identification methods, in either

the frequency domain or the time domain, that are related to the equations of motion [20]. In general,



modal parameter identification consists of signal pre-processing, modal identification and parameter

extraction as discussed in the following subsections [12].

2.3.1 Signal Pre-Processing

Oftentimes, raw signals measured by the accelerometers lack certain characteristics required for further

processing such as amplitude, power level and bandwidth [12]. The signals may also be masked by

superimposed interference or excessive noise levels. Therefore, before conducting any system

identification and subsequent modal analysis, preliminary processing of the acceleration time-histories

must be carried out [12]. Examples of pre-processing include the following and are depicted in Figure 4

[12] [16]:

• Averaging of experimental signals to ensure data can be used with confidence,

• Decimation of the sampling frequency to a lower sampling rate to account for sampling

frequencies that are too high with respect to the bandwidth of interest. Another advantage of

decimation is that it reduces the processing time as it reduces the number of values in the time

histories, and,

• High or low pass filters to remove disturbances at very high or low frequencies.

Figure 4 – Signal pre-processing techniques: top – averaging, left – decimation, right - filtering [16]

2.3.2 Modal Identification Techniques

Once the raw data is pre-processed and comprehensible for the modal identification software, many

modal identification techniques exist in literature that may be used to extract the experimental modal

parameters. These techniques are based on complex mathematical processes, hence why modal



analysis software is typically used for modal estimation. However, the principles behind the most

common modal identification techniques will be discussed below and a more in-depth explanation of the

mathematics can be found in existing literature such as System identification methods for (operational)

modal analysis: review and comparison by Edwin Reynders (2012) [20].

Due to the inability to scale the measured mode shapes and the uncertainties often apparent in the

frequency content of the ambient excitation, it is important to use different techniques for modal

parameter extraction to gain confidence in the data [7] [12]. At a minimum, it is recommended that OMA

be conducted with one technique in both the frequency and time domains to ensure that the data is

accurate [16].

Frequency Domain Techniques

Frequency domain techniques are nonparametric methods which compute dynamic properties through

the use of the Fourier transformation allowing any harmonic function to be expressed as a summation

of periodic terms [16]. The Fourier transformation represents the distribution of frequencies of all

infinitesimal harmonic components in which a recorded signal can be decomposed [16]. The most

common OMA techniques in the frequency domain are the classical “Peak Picking” technique and the

more recent “Frequency Domain Decomposition” technique, both based on the evaluation of the spectral

eigenvalue matrix G(f) [4] [5]:

𝑮(𝒇) = 𝑬[𝑨(𝒇)𝑨𝑯(𝒇)] (1)

Where the vector A(f) collects the acceleration responses in the frequency domain, superscript H

denotes a complex conjugate transpose matrix, and E denotes the expected value. The diagonal terms

of the matrix G(f) are the real valued auto-spectral densities (ASD) while the other terms are the complex

cross-spectral densities (CSD) [4]. The ASD’s and CSD’s are estimated from the recorded time-histories

by averaging, dividing and filtering the data into several points [4].

Peak Picking (PP)

Peak Picking involves identifying natural frequencies from resonant peaks in the power spectral density

(PSD) plots which are obtained by converting the measured data to the frequency domain through the

Fast Fourier Transform (FFT) [11] [13]. This method is based on the evaluation of the G(f) matrix

discussed above, where the natural frequencies are identified from resonant peaks in the ASD’s and in

the amplitude of the CSD’s [4] [13]. The mode shapes are then obtained from the amplitude of square-

root ASD curves and the CSD phases are used to determine directions of relative motion [4]. Peak

Picking leads to reliable results provided that the basic assumptions of low damping and well-separated

modes are satisfied [4] [11] [13].



Frequency Domain Decomposition (FDD)

Frequency Domain Decomposition is an approximate decomposition of the response time histories into

a set of independent, single degree of freedom systems for each mode [5]. The FDD procedure identifies

modal frequencies through the location of the peaks of the first singular value of the spectral matrix, G(f)

[4]. This process involves the following main steps [4] [5]:

1. Estimation of the spectral density matrix from raw time series data,

2. Singular Value Decomposition (SVD) of the spectral density matrix at each frequency,

3. Averaging of singular values of all data sets if multiple data sets exist, and,

4. Peak picking of the highest average SVD to identify resonant frequencies and estimate

corresponding mode shapes.

Note that for well-separated modes, only the highest singular value should be considered whereas for

closely spaced modes, other singular values should be considered as well [12].

The FDD method is an improvement from the PP method as it effectively separates signal space from

noise space, the evaluation of mode shapes is automatic, and in the case of closely spaced modes,

every singular vector corresponding to a non-negligible singular value represents a mode shape

estimate [4]. Therefore, it is recommended to conduct FDD in opposition or addition to the PP method

when conducting OMA. See Figure 5 depicting a comparison between the PP and FDD interface for a

given data set [12].

Figure 5 – Example of modal analysis data using the PP method (left) and FDD method (right) [12]



Enhanced Frequency Domain Decomposition (EFDD)

The FDD method is limited by the frequency resolution of the spectral density estimates which can lead

to heavily biased modal estimates [15]. To increase the frequency resolution, the Enhanced Frequency

Domain Decomposition method can be used as it estimates modal parameters through computing the

inverse Fast Fourier Transform of each spectral density function for each mode shape in the time domain

[16]. Figure 6 shows a typical diagram obtained by the EFDD method where the red lines identify the

first modal frequencies [8].

Figure 6 – Example of an EFDD diagram and the identified frequencies [5]

Time Domain Techniques

Time domain techniques identify modal parameters through parametric methods which fit the response

of each measurement point to a mathematical model representative of the dynamic behaviour of the

structure [16]. The most common OMA technique in the time domain is “Stochastic Subspace

Identification” which typically uses data-driven system identification techniques to extract modal

parameters [8] [5] [12]. Time domain techniques are robust and allow for accurate modal parameter

estimation, however, are not as efficient or simple as the frequency domain techniques as more

processing time is typically required during parameter estimation [15].

Stochastic Subspace Identification (SSI)

Stochastic Subspace Identification works directly with the raw time history output data, without the need

to convert the data to correlations or spectra [5] [12]. The SSI algorithm identifies the state space

matrices based on the experimental measurements by using robust numerical techniques, however,



once the mathematical description of the structure is found, it is straightforward to determine the modal

parameters [5].

The numerical technique is based on first order differential equations and fits a model directly to the raw

time series data based on the State-Space Formulation and from the analysis of the response time

series [9] [15]. Under the hypotheses of unknown excitation and a time-invariant linear system, the SSI

is established on the following representation of the equation of motion [8]:

𝒙𝑝+1 = 𝑩𝒙𝒑 + 𝒘𝒑 (2)

𝒚𝒑 = 𝑪𝒙𝒑 + 𝒗𝒑 (3)

Where xp is the discrete-time state vector (which collects displacements and velocities at a point in time),

wp is the process noise (due to disturbances and model inaccuracies), yp is the output vector, vp is the

measurement noise (due to measurement errors), B is the discrete state matrix (which depends on

mass, stiffness and damping properties of the structure), and C is the discrete output matrix (which maps

the state vector into the measured output) [8].

Different SSI-based techniques are utilized to manipulate these equations to extract the modal

parameters and stabilization diagrams are analyzed to differentiate between real modes and noisy

modes [8]. Natural frequencies and damping ratios are obtained from the eigenvalues of B, and mode

shapes are computed from the product of the output matrix C with the eigenvectors of the state matrix

B [8]. Figure 7 shows a typical diagram obtained by SSI where a modal identification software was able

to identify stable frequencies as indicated by the red lines [8].

Figure 7 – Example of an SSI diagram and the identified frequencies [8]



2.3.3 Cross Validation - Modal Assurance Criterion (MAC)

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.



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].



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)



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].



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].



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



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



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].



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



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]:

𝐻(𝑡) = 𝑀 ∙ 𝑎 ∙ [∅̇2 ∙ 𝑠𝑖𝑛∅(𝑡) − ∅̈ ∙ 𝑐𝑜𝑠∅(𝑡)] (10)

𝑉(𝑡) = 𝑀 ∙ 𝑔 − 𝑀 ∙ 𝑎 ∙ [∅̇2 ∙ 𝑐𝑜𝑠∅(𝑡) − ∅̈ ∙ 𝑠𝑖𝑛∅(𝑡)] (11)

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



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)



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.



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]



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].



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



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]



• 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]:



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]



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,



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]



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.



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.



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)



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



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]



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.



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]



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.



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



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



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



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.



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



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.



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


250 – 1250000

Vault Infill Concrete and

Masonry

Linear elastic


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




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


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.



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



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



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


Y 1.11 1.10

Global torsion 6.18 3.51

Average -

between Scenario

A and Scenario

B/C

50 000


X 2.19 1.59


Y 1.66 1.65


High end – close

to Scenario B/C 1 000 000


X 2.19 2.13


Y 2.08 2.16


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



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.


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


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.



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.


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.


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.



Table 23 – Beam vs. solid – Scenario A: corresponding mode shapes

Simplified Beam Model 3D Solid Model %

Difference Mode

Frequency

(Hz) Mode Description Mode

Frequency

(Hz) Mode Description

1 1.13 1st global bending mode

about X axis 1 1.08

1st global bending mode

about X axis 4.5

2 1.17 1st global bending mode Y

axis 2 1.09


about Y axis 7.3

6 7.56 Global elongation and

torsion 6 7.04

Global elongation and

torsion, local bending of top

two tiers about XY axes

7.1

Table 24 – Beam vs. solid – Scenario B: corresponding mode shapes


Difference Mode

Frequency


Frequency



about Y axis 1 1.10


about Y axis 6.5


about X axis 2 2.16


about X axis 6.0


torsion 6 7.05

Global elongation and

torsion, local bending of

top two tiers about Y axis

7.0

Table 25 – Beam vs. solid – Scenario C: corresponding mode shapes


Difference Mode

Frequency


Frequency



about X axis 1 2.17


about both X (more) and

Y axes

5.8


about Y axis 2 2.21


about both X and Y

(more) axes

-2.8


torsion 6 7.09

Global elongation, local

bending of top two tiers

about Y axis

6.4

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.



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)





[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)





[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)





[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)





[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) -



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.



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


X 0.21160


Y 0.21186

2nd global bending

mode about Y 1.7205

Mid-range 1e+08


X 0.92925


Y 0.95466

2nd global bending

mode about Y 1.7221

High end – close

to Scenario A

(fully rigid)

1e+12


X 1.1255


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.



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.



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



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





High end – close

to Scenario A

(fully rigid)

15 000 000





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



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.



Table 31 – Updating Parameters

Updating

Parameter

Estimated

Lower Limit Mid-range

Estimated

Upper Limit Location

Young’s

Modulus (E) 1 000 MPa [6] 2 500 MPa 7 500 MPa [6] All masonry

Soil-Structure

Interface (K1) 100 kN/m3

100 000

kN/m3

15 000 000

kN/m3

NE Lateral

Interface (K2) 10 kN/m3 50 000 kN/m3

5 000 000

kN/m3

NW Lateral


5 000 000

kN/m3

SW Lateral


5 000 000

kN/m3

FE Models for all three scenarios (lower limit, mid-range and upper limit) were created and compared

against one another for use in the model updating procedure. The first three global modal parameters

in both the X and Y directions were compared in Table 32.



Table 32 – Modal Frequency comparison between Upper and Lower limits

Mode

Description

Lower Limit:

f [Hz]

Mid-range:

f [Hz]

Upper Limit:

f [Hz]

1st global

bending mode

about X axis

0.038038

(1)

1.5371

(1)

3.7082

(2)

1st global

bending mode

about Y axis

0.050585

(2)

1.5703

(2)

3.6510

(1)

Global

elongation in

Z-axis

0.17193

(3)

4.4054

(3)

- -

2nd global

bending mode

about Y axis,

global torsion

mode

1.7900

(4)

5.0875

(4)

11.359

(3)



Mode

Description

Lower Limit:

f [Hz]

Mid-range:

f [Hz]

Upper Limit:

f [Hz]

2nd global

bending mode

about X axis

2.3667

(5)

5.1658

(5)

12.144

(6)

3rd global

bending mode

about Y axis,

global torsion

2.5200

(6)

5.8727

(6)

16.832

(8)

Global torsion

mode

6.1899

(10)

10.489

(9)

23.149

(9)

3rd global

bending mode

about X axis

6.2410

(11)

10.511

(10)

16.410

(7)

These models help identify the absolute upper and lower boundaries for the modal properties of the

Sant Cugat bell tower. As discussed previously, the eigenfrequencies decrease with increased flexibility,

therefore when analysing the results from the dynamic testing, lower eigenfrequencies may indicate loss

of stiffness at one or more of the connection surfaces. Therefore, the aim of the iterative model updating

procedure is to identify the source of the observed flexibility and subsequent source(s) of damage.



5. DYNAMIC TESTING PLAN

The information obtained from the state-of-the-art literature review and the analysis of the hypothetical

numerical models were used to design a robust dynamic testing plan for the bell tower of Sant Cugat

Monastery. The dynamic testing plan includes guidelines for data acquisition, modal parameter

identification, calibration of the FE model and computation of the possible dynamic amplification factors

induced by the ringing of the bells.

5.1 Data Acquisition

The following subsection outlines the logistics required for the dynamic testing campaign. This includes

the modal testing technique to be utilized, the identification of equipment to be used, the placement of

sensors and an outlined testing procedure.

5.1.1 Dynamic Testing: AVT

It is recommended that output-only, Ambient Vibration Testing (AVT) be utilized to obtain the

experimental modal parameters of the bell tower of Sant Cugat Monastery. This is recommended as

AVT is based solely on measurements of a structure’s response caused by ambient excitation sources

(i.e. traffic, wind, people) and without the use of artificial excitation. This is excellent for historical

constructions which are typically large, structurally complex and subject to multiple restrictions due to

their historic significance. In addition, OMA testing is typically more cost efficient than EMA which

requires the use of expensive equipment.

5.1.2 Required Equipment

The sensors and data acquisition system (DAQ) recommended to be used for the dynamic testing

campaign are indicated in Table 33 and Table 34. Piezoelectric accelerometers are recommended due

to their relatively low cost, high sensitivity, good signal-to-noise ratio, and no need for an external power

source. The DAQ is required to convert the response signals into discrete digital signals which can then

be processed and analysed by modal identification software.



Table 33 – Data acquisition equipment to be used

Model/ Name cRIO-9064 (Main Controller) NI 9234 NI 9230

Quantity 1 2 1

Max no. of channels depends on chosen modules 4 3

Max sampling rate [kHz] limited by module 51.2 12.8

Input range limited by module ± 5V ± 30V

Lower cut-off frequency (at -3dB) [Hz]

limited by module 0.5 0.1

Description Embedded Real-Time controller with reconfigurable FPGA.

Chassis can hold up to 4 modules.

Sound and Vibration

Input

Sound and Vibration

Input

Table 34 – Accelerometers to be used

Model / Name 356B18 393A03 393C 393B12

Quantity 2 1 1 2

Brand PCB Piezotronics PCB

Piezotronics PCB Piezotronics

PCB Piezotronics

Type Piezoelectric Piezoelectric Piezoelectric Piezoelectric

Uniaxial/ Triaxial Triaxial Uniaxial Uniaxial Uniaxial

Sensitivity [mV/g] 1000 1000 1000 10000

Mass [g] 25 210 885 210

Frequency Range [Hz] 0.5 to 3000 (±5%)

0.3 to 5000 (±10%)

0.5 to 2000 (±5%)

0.3 to 4000 (±10%)

0.025 to 800 (±5%) 0.01 to 1200 (±10%)

0.15 to 1000 (±5%)

0.10 to 2000 (±10%)

Resolution

Broadband Resolution

[g] 0.00005 0.00001 0.0001 0.000008

Spectral Noise (1 Hz)

[g/√Hz] 0.0000114 0.000002 - 0.0000013

Spectral Noise (100

Hz) 0.0000012 0.0000002 - 0.00000013

Effective Resolution

[g] 0.00005 0.00001 0.0001 0.000008

Measurement Range [g] ±5 ±5 ±2.5 ±0.5

Needs individual power source?

No No No No

5.1.3 Location of Accelerometers

To calibrate the FE model, dynamic testing must be conducted so that the modal properties obtained in

the hypothetical models can be compared with those obtained during the dynamic testing. Therefore, it

is critical that the accelerometers be placed in predetermined locations where large deformations are



expected to occur. In this regard, it is typical to look at the first two to three global mode shapes as these

activate the greatest percentage of mass in the structure. As can be observed in Sections 4.3.2, 4.4.1,

4.4.2 and 4.5, mode shapes are depicted with different colours indicating the amount of displacement

experienced by the structure subjected to the given mode shape (blue indicating negligible displacement

whereas red indicates the highest displacement). When choosing locations for the accelerometers, it is

recommended to place them in green to red zones opposed to blue zones as these locations are more

likely to produce a good signal-to-noise ratio.

In all hypothetical mode shapes, it was observed that most of the activated mass occurs at the top of

the bell tower (above level 6). Therefore, as the main body of the bell tower is of greatest concern due

to its large mass, it is recommended that accelerometers be placed at levels 6 and 7 to best record the

dynamic properties of the tower. In addition to the ambient vibrations recorded at this height, level 6 is

the location where the liturgical bells are housed and therefore, dynamic testing will be able to identify

the overall damping of the structure subjected to the reverberation of the bells and the predominant

harmonic frequencies of the bells. With this information, it is possible to calculate the dynamic

amplification factors generated by the interaction between the tower’s modal frequencies and the bell’s

excitation which may influence the tower’s dynamic behaviour.

Secondly, it can be observed that deformations are restricted to above level 4 in restraint scenarios

which assume there is some connection between the bell tower and the surrounding church (Scenarios

B to E). Therefore, it is also recommended that accelerometers be placed at level 4, as close to the

connection surfaces as possible, to help identify if the connections are playing a role in the dynamic

parameters of the bell tower. This information will help identify the rigidity of the connection at these

points, if any, which can then be translated into the spring stiffnesses of the FE model during calibration.

Lastly, the orientation of the accelerometers is of equal importance to their position within the structure

since the direction in which they are placed is the only direction in which they measure vibration.

Therefore, since the global bending of the structure about both the X and Y axes is of greatest

importance for calibrating the FE model, most of the sensors should be oriented in the X and Y

directions. However, global torsion is of interest in addition to global bending, therefore placing two

accelerometers in opposing corners is recommended.

Based on these considerations, it is recommended that accelerometers be placed as indicated in Figure

45, Figure 46 and Table 35. This configuration would require 2 triaxial accelerometers, 4 uniaxial

accelerometers, 10 cables and 10 channels to be connected to the centralized DAQ system. It is

recommended that the DAQ system is located on the sixth floor, at a central position. This would require

an extension cord approximately 40 - 50 m in length, assuming the only power outlet available is within



the main body of the church. Due to accessibility constraints, all accelerometers are to be located on

the interior walls aside from those located above the roof at level 7.

Table 35 – Proposed accelerometer locations

Level Name Location Direction Number / type of

accelerometers

Approximate length

of cable required (m)

3 NS1 Interior SE wall at first stair

landing NW 1 uniaxial 1 x 15 m

4 NS2 Interior NW wall SE 1 uniaxial 1 x 10 – 15 m

4 EW1 Interior SW wall NE 1 uniaxial 1 x 10 – 15 m

6 NS3, EW2 Interior wall in E corner NW, SW 1 triaxial 2 x 3 - 5 m

6 EW3 Interior wall in W corner NE 1 uniaxial 1 x 3 - 5 m

7 NS4, EW4, Z1 Base of superimposed bell

structure on NE wall

SE, NE,

elevation 1 triaxial 3 x 10 – 15 m

Figure 45 – Proposed accelerometer locations – plan view



Figure 46 – Proposed accelerometer locations – section view



5.1.4 Testing Procedure

Following the identification of sensor locations, the date for the dynamic testing can be set. It is

recommended that testing be conducted on a calm day to avoid excessive noise caused by wind, traffic

and/or people as large amounts of noise often result in less accurate data. It is also recommended that

the sampling frequency be set between 100 and 2000 Hz to ensure that the bell harmonics are captured

in the recorded data.

Each test should have a duration equal to approximately 2000 times the highest natural period of interest

(lowest modal frequency). In the case of the Sant Cugat Monastery, the lowest modal frequency is 1.07

Hz from Scenario A, Mode 1. This results in a recommended sampling time of approximately 30 minutes

per acquisition. It is recommended that two acquisitions of 30 minutes be conducted, one during the

ringing of the clock bells and the other during the ringing of the liturgical bells. This way, the modal

properties of the bell tower and the predominant harmonics of the different bell ringing schemes may be

captured. Although it is not required to conduct a 30-minute test to obtain the modal properties of the

bells, it is recommended to ensure the first acquisition properly captured the modal parameters of the

bell tower.

Table 36 outlines the recommended procedure to be followed for the ambient vibration testing at the

bell tower of the Sant Cugat Monastery:

Table 36 – Recommended AVT testing procedure

Recommended

Schedule Item

Required Time

(h)

6:00 – 6:30 /

17:00 – 17:30 Set up data acquisition system (DAQ) and connect computer at Level 6. 0.5

6:30 – 7:30 /

17:30 – 18:30

Install accelerometers by gluing or screwing them to the walls at the identified

locations. Gluing them with a putty that can later be removed is recommended

to minimize the impact to the historic structure.

1

7:30 – 7:45 /

18:30 – 18:45

Obtain preliminary signal measurements to ensure the sensors are working, to

characterize the signal-to-noise ratio, and to obtain an approximate range for

the resonant frequencies involved. This can be achieved through a 5-minute

acquisition.

0.25

7:55 – 8:25 /

18:55 – 19:25

Acquisition #1: Conduct the test, ensuring the ringing of the clock bells (rung at

the hour and quarter hour) is captured within the measured duration. 0.5

8:45 – 9:15 /

19:45 – 20:15

Acquisition #2: Conducted during the ringing of the liturgical bells (before mass)

to compute all possible dynamic amplification factors. Mass is conducted at Sant

Cugat Monastery, Monday – Friday at 9:00 and 20:00.

0.5

9:15 – 9:45 /

20:15 – 20:45

After obtaining all data, conduct a quick modal analysis to ensure the

experimental results are sufficient for modal identification. 0.5

9:45 – 10:45 /

20:45 – 21:45 Clean up and removal of accelerometers. 1

Total Time 4.25



5.2 Modal Parameter Identification

Once the dynamic testing has been conducted, modal analysis software (such as ARTeMIS, [19]) may

be used to identify the main modal parameters through OMA identification techniques. Before

conducting the modal analysis, preliminary processing of the acceleration time-histories should be

carried out to refine the data such as: offset removal, averaging, decimation and high/low pass filters.

Following pre-processing, it is recommended that OMA be conducted with one or more identification

techniques in both the frequency and time domains to ensure that the experimental data is accurate.

For example, obtaining modal parameters using the Peak Picking and Enhanced Frequency Domain

Decomposition methods in the frequency domain will ensure that all modes are identified, even closely

spaced modes, noisy modes, and modes with low resolution. Stochastic Subspace Identification should

also be conducted in the time domain, to ensure precision. For all identification methods, it is

recommended to verify the automatic mode estimation to ensure the maximum number of stable modes

are identified according to the stabilization diagrams provided by the software.

The results obtained from the various methods should be compared using the Modal Assurance

Criterion (MAC) which correlates the two sets of modal vectors as discussed in Section 2.3.3.

5.3 Calibration of the Finite Element Model

The final step in dynamic characterisation is the calibration of the hypothetical numerical model. The

defined updating parameters (Young’s modulus, spring stiffnesses at lateral connections with the

adjacent Church, and soil-structure interaction springs) should be assessed through the comparison of

modal results between the FE model and the experimental OMA. This can be achieved through

iteratively modifying the updating parameters in the FE model until the differences in the main global

natural frequencies are minimized. The updating parameters for the Sant Cugat Bell Tower are indicated

in Table 37.

Table 37 – Updating Parameters

Updating Parameter Reference Value Estimated

Lower Limit

Estimated Upper

Limit

Young’s Modulus (E) 2 500 MPa 1 000 MPa [6] 7 500 MPa [6]

Soil-Structure Interface (K1) 100 000 kN/m3 100 kN/m3 10 000 000 kN/m3

NE Lateral Interface (K2) 50 000 kN/m3 10 kN/m3 5 000 000 kN/m3

NW Lateral Interface (K3) 50 000 kN/m3 10 kN/m3 5 000 000 kN/m3

SW Lateral Interface (K4) 50 000 kN/m3 10 kN/m3 5 000 000 kN/m3

When calibrating the model, it is recommended that the stiffness values be iteratively modified

downwards from the Upper Limit model until the numerical and experimental modal properties are in



alignment. Alternately, the Lower Limit model may be used as a reference. This way, the analyst need

only decrease or increase the updating parameters.

The model calibration can be achieved by using the experimental value of the first bending mode as a

reference and varying one updating parameter at a time until a satisfactory agreement between results

is achieved. This is followed by updating additional parameters to maximize the correlation between the

main mode shapes. Typically, the first parameter to be updated is Young’s modulus (E), followed by

lateral connections and soil-structure interaction.

It is recommended that the main global mode shapes be refined until the percentage error is less than

5% as computed using Eq. 14. At minimum, this should be the case for the first two global bending

modes.

||𝑓𝐹𝐸𝑀−𝑓𝐴𝑉𝑇|

𝑓𝐴𝑉𝑇| ∙ 100% (14)

The model should be further validated through the correlation of measured and predicted modal

displacements using the modal assurance criterion (MAC) as defined in Section 2.4.1. 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]. See Figure 47 and Section 5.1.3 for the approximate locations of the

accelerometers.



Figure 47 – Approximate points to be selected for comparison with experimental displacements

Once the model has been calibrated, the cause of the lateral displacement of the bell tower may become

apparent through the analysis of the calibrated updating parameters. For example, if the modal

parameters are most accurate with weak lateral interface elements, this may indicate that there are poor

connections between the bell tower and the church that need to be strengthened. If the modal

parameters are most accurate with weak soil-structure interface elements, this may indicate that there

are poor subsurface conditions allowing for differential soil settlement of the bell tower and that further

geotechnical investigations should be conducted.

5.4 Computation of DAF

Lastly, it is recommended that the dynamic interaction between the bells and the supporting structure

be examined, especially since the traditional Spanish bells were replaced with Central European style

bells in the 1940s. Due to the high unbalance of the Central European system, the bells may exert

considerably more horizontal dynamic loads on the supporting structure compared to the Spanish bell

system. Therefore, if the dynamic amplification caused by the ringing of the new bells is substantial, the

lateral displacement currently exhibited by the bell tower may be intensified.

Therefore, following model calibration, the dynamic interaction between the bells and the tower should

be examined. This can be achieved through applying the time history of the bell forces as a dynamic

force to the calibrated FE model and analysing the modal results. The time history of the bells may be



extracted from AVT tests in which the bells were ringing or may be obtained in a laboratory setting. Once

the dynamic properties of both the bell tower and the bells have been identified, the dynamic

amplification factor (DAF) comparing the deformed shapes of corresponding mode shapes between

static and dynamic models may be computed for a given mode shape as follows:

𝐷𝐴𝐹 =𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑑𝑦𝑛𝑎𝑚𝑖𝑐 𝑚𝑜𝑑𝑒𝑙

𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑠𝑡𝑎𝑡𝑖𝑐 𝑚𝑜𝑑𝑒𝑙 (12)

Alternately, to evaluate the effect of the bell swinging on the modal parameters of the supporting tower,

the bell harmonics may be identified through modal identification procedures in the frequency domain

(see Section 2.3). If it is found that one of the predominant harmonics of the bells interacts with a natural

frequency of the tower, it is likely that there is a large DAF which could impact the stability of the structure

[3]. Therefore, 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 (13)

If a DAF greater than 1 is found to exist, actions should be implemented to prevent this interaction.

Mitigation measures may include changing the bell ringing scheme so as to not coincide with the natural

frequencies of the tower, reverting the bell system to the historic Spanish system which is proven to

induce less horizontal thrust to the supporting structure, or structural strengthening of the bell tower

itself.



6. CONCLUSIONS

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 the objective of this dissertation to develop a robust procedure for the dynamic identification of the

bell tower with the aim to better understand its deterioration mechanisms. The dissertation included a

comprehensive literature review, the creation of a suitable numerical model, and a detailed dynamic

testing procedure to be conducted in the future.

6.1 Expected Modal Parameters

Following the literature review and historic survey, the geometry of the bell tower was obtained from

existing drawings, however, could not be confirmed on site due to COVID-19 restraints. With the

obtained geometry, numerical modelling was conducted to obtain the hypothetical dynamic properties

of the bell tower and to be used for model calibration following dynamic testing. Both a simplified beam

model and 3D solid model were created in DIANA FEA software to ensure accuracy. Mechanical

properties for the preliminary analyses were assumed for the masonry based on the literature review as

follows: Young’s modulus of 2500 MPa, Poisson’s ratio of 0.15 and mass density of 22 kN/m3. The

models considered the self-weight of the structure as the only load which amounted to approximately

22 500 kN. In addition, boundary surfaces with reduced stiffness were considered at lateral connection

points between the tower and the church, and at the soil-structure interface to ensure accurate

representation of the structure.

Both a coarse and fine mesh were analysed to ensure consistency of results and to validate the model.

For both models, a mesh element size of 0.2 m was chosen for the analysis of the results, resulting in

209 elements for the simplified beam model and 978 067 elements for the 3D solid model. The mesher

type was tetra/triangle as this produced the best results.

Following discretization, linear self-weight analysis was conducted to ensure the reaction forces were

as expected, followed by linear modal response analysis to obtain the modal parameters. Depending

on model complexity (beam vs. solid) and mesh density, the analyses took between 5 and 30 minutes

each to run and required almost 100% capacity of the memory and CPU power on a computer with a

64-bit operating system, i7-6700HQ Intel® Core and 16 GB RAM.

It was found that results between coarse and fine meshes varied by less than 2% for both models and

therefore, the models were considered valid and further analysis was conducted with the results

obtained from the fine mesh models. It was also concluded that the simplified beam model validifies the

3D solid model as it had 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 2D 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 full 3D solid model, the 3D 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.

An iterative sensitivity analysis was conducted on the 3D solid model to obtain ranges of spring stiffness

for the boundary surfaces. Through this analysis, it was found that the lateral connection stiffnesses

may lie between 10 and 50 000 kN/m3 whereas, the stiffness of the soil-structure interface may vary

between 100 and 15 000 000 kN/m3.

From the analysis of the various restraint scenarios in the 3D solid model, 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 of the connection surfaces. The aim of the iterative model updating procedure is to identify

the source of the observed flexibility. The model updating may be achieved following dynamic testing

through the analysis of three finalized 3D solid FE models representative of a fully fixed model, a fully

free (cantilever) model and a model lying within the two extremes. These may be compared against one

another and used to help identify the absolute upper and lower boundaries for the modal properties of

the Sant Cugat bell tower.

6.2 Dynamic Testing Plan

The information obtained from the literature review and the preliminary FE models was utilized to design

a robust dynamic testing campaign for the bell tower using AVT to obtain experimental modal

parameters. In the plan, AVT was chosen as it is excellent for historical constructions due to their

structural complexity and numerous restrictions instilled by their historical significance. Accelerometers

were recommended to be placed near the lateral connection surfaces and at the top of the tower to

measure global bending and torsion modes. Modal analysis software was recommended to identify the

experimental modal parameters through OMA identification techniques in both the frequency and time

domains.

Once the experimental modal parameters have been extracted, calibration of the hypothetical numerical

model can be achieved through the comparison of the hypothetical modal results and the experimental

results. This is followed by iterative modification of the defined updating parameters until convergence

is achieved. Once the model has been calibrated, the cause of the lateral displacement of the bell tower

may become apparent through the analysis of the updating parameters.



6.3 Future studies

Following the calibration of the FE model, additional studies are recommended to compliment and

validate the hypotheses made based on the findings.

Firstly, should a large dynamic amplification factor of the bell-structure interaction be computed, it is

recommended that mitigation measures be investigated to reduce the impact of bell ringing on the

structure. This may include actions such as changing the bell ringing scheme, reverting the bell system

to the historic Spanish system, or structural strengthening of the bell tower itself.

Secondly, a geotechnical survey beneath the bell tower and SE wall of the church is recommended to

better understand the subsurface conditions on which the structure lies. If a poor soil-structure interface

is concluded from the calibrated FE model, a geotechnical investigation may help confirm the

hypothesized poor subsurface conditions. Similarly, an investigation into the lateral connections

between the bell tower and the surrounding church walls should be conducted to identify if the bell tower

was properly integrated into the surrounding church. If weak lateral connection surfaces are found to

best calibrate the model, the poor connections may be validated through this study. Information on the

quality of connections may be obtained through non-destructive testing such as ground penetrating

radar, or destructive testing such as investigative openings.

Lastly, although the mechanical properties obtained from the calibrated FE model represent the global

linear behaviour of the structure, these properties may be used as baseline parameters for nonlinear

analyses that may be conducted to better understand stress distributions and/or seismic behaviour of

the tower. To achieve this, the calibrated FE model must be used in conjunction with additional material

testing since several nonlinear parameters must be obtained or assumed to conduct nonlinear analysis

due to the anisotropic, non-homogeneous nature of masonry.



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[36] C. Molins, Sant Cugat , 2020.

[37] G. d. C. Department de Cultura, “As-found drawing set for the Sant Cugat Monastery,” Generalitat de Catalunya

, 1991.

[38] Tècnic Topograf Municipal, “Informe Tècnic: Monestir de Sant Cugat del Vallès,” Ajuntament de Sant Cugat,

Sant Cugat del Vallès , 2019.

[39] S. d. P. A. Direccio General del Patrimoni Cultural, “Desplom dels contraforts de la façana sud del monesti de

sant cugat,” Generalitat de Catalunya, 2019.

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https://skyciv.com/free-moment-of-inertia-calculator/. [Accessed 01 05 2020].



APPENDIX A – AS-FOUND DRAWING SET



APPENDIX B – MODE SHAPE COMPARISON



3D Solid Model - Restraint Scenarios Simplified Beam Model

A B C D E A B C

Plan View

Mode Shape

1st global bending

mode about X axis

1.0756 Hz (1) 2.1624 Hz (2) 2.1658 Hz (1) 2.1633 Hz (2) 1.5796 Hz (1) 1.1256 Hz (1) 2.2957 Hz (3) 2.2958 Hz (3)

1st global bending

mode about Y axis




Mode Shape 3D Solid Model - Restraint Scenarios Simplified Beam Model

A B C D E A B C

2nd global bending

mode about Y axis,

1st global torsion

mode

-

-

-

3.4293 Hz (3) 4.1201 Hz (3) 5.7341 Hz (3) 1.7236 Hz (3) 1.7236 Hz (2)

2nd global torsion

mode

-

-

6.0721 Hz (4) 6.6020 Hz (3) 6.1927 Hz (4) 5.1347 Hz (3) 6.1573 Hz (5) 6.1566 Hz (4)




A B C D E A B C

2nd global bending

mode about X axis

- -

4.4418 Hz (4) 6.6314 Hz (5) 6.6705 Hz (4) 6.6335 Hz (5) 5.3498 Hz (4) 6.0309 Hz (4)

3rd global bending

mode about Y axis,

global torsion

-

-

- - -

4.4574 Hz (5) 6.8807 Hz (5) 5.7751 Hz (5)




A B C D E A B C

1st global elongation

in Z, local bending of

top two tiers about

XY axes


Local bending mode

(above level 6)

about Y axis

-

- -

7.2214 Hz (7) 7.2493 Hz (7) 7.9923 Hz (7) 7.9812 Hz (8) 9.7533 Hz (6)




A B C D E A B C

3rd global bending

mode about X

-

-

9.4400 Hz (8) 9.4625 Hz (7) 9.4675 Hz (8) 7.6198 Hz (7) 10.178 Hz (7) 10.179 Hz (7)

Local bending mode

(above level 7)

about X axis

-

- -

7.2799 Hz (8) 13.620 Hz (10) 13.623 Hz (11) 13.612 Hz (12) 12.127 Hz (9)




A B C D E A B C

4th global bending

mode about Y axis,

local torsion above

level 6

8.8469 (9) 9.6053 (9) 9.7927 (8) 11.533 Hz (9) 11.509 Hz (11) 12.086 Hz (8) 12.087 Hz (8) 12.088 Hz (8)

Local bending

mode about Y axis

above level 6

- -

-

13.962 (11) 13.880 Hz (12) 13.874 Hz (13) 12.34 Hz (10) 12.376 Hz (9)




A B C D E A B C

3rd global torsion

mode

-

- - -

9.9383 (10) 11.533 (10) 12.131 Hz (10) 10.309 Hz (9)

4th global bending

mode about X axis

- - -

- - -

10.330 (11) 10.982 Hz (10)

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