Development and validation of empirical seismic vulnerability models for historic masonry towers

Universidade do Minho Escola de Engenharia Francesco Testa Development and Validation of Empirical Seismic Vulnerability Models for Historic Masonry Towers March 2025 UMinho | 2025 Francesco Testa Development and Validation of Empirical Seismic Vulnerability Models for Historic Masonry Towers Francesco Testa Development and Validation of Empirical Seismic Vulnerability Models for Historic Masonry Towers Doctoral Thesis Civil Engineering Work conducted under the supervision of Professor Paulo José Brandão Barbosa Lourenço Doctor Alberto Barontini Universidade do Minho Escola de Engenharia March 2025 Copyright and Terms of Use for Third Parties This academic work is made available under the following terms of use, in accordance with internationally recognised standards and best practices regarding copyright and related rights. This work may be freely used, shared, and adapted by third parties under the conditions specified in the license below. If intended use falls outside the scope of this license, prior written permission must be obtained from the author via the University of Minho’s Institutional Repository (RepositóriUM). License https://creativecommons.org/licenses/by/4.0/ Acknowledgements The present thesis, Development and Validation of Empirical Seismic Vulnerability Models for Historic Masonry Towers , has been made as part of my Ph.D. programme from January 2021 to March 2025 at the Institute for Sustainability and Innovation in Structural Engineering (ISISE), Department of Civil Engineering, University of Minho, Portugal. The thesis has been supported by the Portuguese governmental agency, Foundation for Science and Technology (FCT), whose financial contribution is gratefully acknowledged. I would like to express my sincere gratitude to my supervisor, Prof. Paulo B. Lourenço, for giving me this opportunity, guiding me, providing the necessary tools, and sharing knowledge, advice and suggestions throughout my Ph.D. studies. I am deeply grateful to my supervisor, Dr. Alberto Barontini, for providing support, indispensable assistance and careful reading, as well as for engaging in insightful conversations and sharing precious comments on the research. Furthermore, I would like to thank Dr. Nicola Chieffo, from the Department of Civil Engineering at the University of Huddersfield, for the collaboration in the research presented in Chapter 3. I am very thankful to Prof. Giuseppe Brando, Dr. Maria Giovanna Masciotta and Dr. Giorgia Cianchino, from the Department of Civil Engineering at the University Chieti-Pescara, for sharing post-earthquake survey data, instrumental in the research involved in Chapters 4 and 5, and participating in fruitful discussions. Last, but not least, I would like to thank mom and dad for always being there when I needed them most, my sister, Pina, for being my source of inspiration and Luisa for the endless love, patience and encouragement. Guimarães, March 2025 Statement of Integrity I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. Resumo Esta tese tem como objetivo avaliar a vulnerabilidade sísmica de torres históricas em alvenaria e pretende desenvolver métodos empíricos com base em dados recolhidos após sismos. Embora existam modelos empíricos bem estabelecidos na literatura para avaliar a vulnerabilidade sísmica de estruturas, poucos estudos abordam sistematicamente as torres históricas em alvenaria, sendo ainda mais limitados aqueles que se focam na sua validação e na análise comparativa. Assim, o principal objetivo é analisar, validar e melhorar os modelos de vulnerabilidade existentes, com especial foco na avaliação do comportamento distinto e interligado dos vários componentes, o fuste e o campanário, que constituem as torres. São explorados dois conjuntos de dados distintos. O primeiro resulta de uma revisão bibliográfica extensa, reunindo dados de danos causados por sismos anteriores em Itália, compilados numa base de dados em acesso aberto. O segundo aborda algumas limitações do conjunto anterior, menos abrangente, através da recolha e filtragem de informações relativas à sequência sísmica da Itália Central entre 2016-2017, disponível na plataforma Da.D.O., Base de Dados de Danos Observados, disponibilizada pelo Departamento de Proteção Civil Italiano. Este estudo analisa o desempenho de várias medidas de intensidade sísmica, incluindo a intensidade macro-sísmica MercalliCancani-Sieberg (MCS), a Aceleração Máxima do Solo (PGA) e a Aceleração Espectral (SA), de forma a melhorar a precisão das previsões de dano. A estimativa da SA recorre a modelos de movimento do solo existentes, juntamente com novas formulações empíricas desenvolvidas para estimar o período fundamental das torres. Além disso, a investigação conduzida analisa como a vulnerabilidade e o nível de dano são influenciados pelas características da ação sísmica, tais como eventos principais simples ou múltiplos, e fatores relacionados com as torres, tais como geometria, interação com estruturas envolventes, o nível de manutenção e a localização geográfica. A investigação integra metodologias como Matrizes de Probabilidade de Dano (DPMs), funções de vulnerabilidade e de fragilidade, e introduz novos modelos que consideram diferentes classes de torres e cenários sísmicos, com especial ênfase nos efeitos de danos cumulativos. Palavras-chave: Avaliação empírica de vulnerabilidade; Dados pós-desastres; Danos cumulativos; Património construído; Torres em alvenaria. Abstract This thesis aims to assess the seismic vulnerability of historic masonry towers and seeks to advance current empirical methods derived from post-earthquake survey data. Although empirical models for evaluating the seismic vulnerability of structures are well-established in the literature, few studies systematically address historic masonry towers, and even fewer focus on their validation and comparative analysis. Therefore, the main objective of the thesis is to analyse, validate and enhance existing vulnerability models, with a particular focus on assessing the distinct and interconnected behaviour of tower’s shaft and belfry components. Two distinct datasets are exploited. The first is derived from an extensive literature review, gathering damage data from past earthquakes in Italy within a dataset shared in open access. The second addresses some limitations of the previous, less comprehensive, by collecting and filtering information about the 2016-2017 Central Italy seismic sequence, hosted in the Da.D.O., Database of Observed Damage, platform released by the Italian Department of Civil Protection. This study examines the role of various seismic intensity measures, including Mercalli-Cancani-Sieberg (MCS) macroseismic intensity, Peak Ground Acceleration (PGA), and Spectral Acceleration (SA), in improving the accuracy of damage predictions. SA estimation resorts to existing ground motion models alongside newly developed empirical formulations to predict the towers’ fundamental period. Additionally, this research investigates how vulnerability and damage levels are affected by key features of seismic actions, such as the occurrence of single or multiple main shocks, and factors related to the towers, such as geometry, interaction with surrounding structures, maintenance level, and geographical location. To this end, the research integrates methodologies such as Damage Probability Matrices (DPMs), vulnerability and fragility functions and introduce novel models that account for different tower classes and seismic scenarios, with particular emphasis on cumulative damage effects. Keywords: Empirical vulnerability assessment; Post-disaster data; Cumulative damage; Built heritage; Masonry towers. I Contents 1 CHAPTER 1: INTRODUCTION 1.1 MOTIVATIONS 1 1.2 OBJECTIVES 5 1.3 OUTLINE 7 10 CHAPTER 2: SEISMIC RISK ASSESSMENT: STATE OF THE ART 2.1 SEISMIC RISK 11 2.2 ASSESSMENT METHODOLOGIES: AN OVERVIEW 13 2.3 EMPIRICAL METHODOLOGIES 16 2.3.1 Damage Probability Matrices (DPMs) 25 2.3.2 Vulnerability functions 27 2.3.3 Vulnerability index methods 32 2.3.4 Fragility functions 37 2.4 ANALYTICAL METHODOLOGIES 43 CHAPTER 3: VALIDATION AND IMPROVEMENT OF EXISTING VULNERABILITY MODELS THROUGH AN 51 OPEN DATASET 3.1 INTRODUCTION 52 3.2 OPEN DATABASE OF MASONRY TOWERS STRUCK BY EARTHQUAKES 53 3.3 DAMAGE PROBABILITY MATRICES (DPMs) 62 3.3.1 Entire dataset and individual events 62 3.3.2 Typology of tower and individual intensity groups 69 3.4 VULNERABILITY FUNCTIONS 73 3.5 FRAGILITY FUNCTIONS 82 3.6 CONCLUSIONS 84 Contents VIII Figure 4.32: Observed mean damage compared with the vulnerability functions accounting for the tower typology (confined and integrated); (a) mechanism A; (b) mechanism B; (c) overall macroelement. ...................................................................................................................................................... 128 Figure 4.33: Fragility curves of D3 and D4 exceedance fitted to confined (diamond points) and integrated (circle points). ................................................................................................................ 129 Figure 4.34: Observed mean damage compared with the vulnerability functions: (a) regional-based analysis and (b) provincial-based analysis. ....................................................................................... 130 Figure 4.35: Fragility curves of D3 and D4 exceedance fitted to towers in Marche (diamond points) and Umbria (circle points). .................................................................................................................... 131 Figure 4.36: Fragility curves of D3 and D4 exceedance fitted to towers in Macerata (diamond points) and Perugia province (circle points). ............................................................................................... 131 Figure 5.1: Flowchart of the methodology adopted in this Chapter. .................................................. 135 Figure 5.2: Comparison of GMMs predictions with Da.D.O. values for the M6.2 August 24, 2016 seismic shock. ................................................................................................................................ 143 Figure 5.3: Comparison of GMMs predictions with Da.D.O. values for the M5.5 October 26, 2016 seismic shock. ................................................................................................................................ 143 Figure 5.4: Comparison of GMMs predictions with Da.D.O. values for the M6.1 October 26, 2016 seismic shock. ................................................................................................................................ 144 Figure 5.5: Comparison of GMMs predictions with Da.D.O. values for the M6.6 October 30, 2016 seismic shock. ................................................................................................................................ 144 Figure 5.6: Comparison of GMMs predictions with Da.D.O. values for the M5.7 January 21, 2017 seismic shock. ................................................................................................................................ 144 Figure 5.7: Correlations between the highest PGA prior to inspection and the corresponding epicentral distance. Da.D.O. values (dark blue circles) and GMM prediction (light blue circles) using: (a) SP96; (b) ITA08. ............................................................................................................................................ 145 Figure 5.8: Correlation between the highest PGA prior to inspection computed based on Da.D.O. values and the highest PGA predicted using: (a) SP96; (b) ITA08. .............................................................. 146 Figure 5.9: Geometric features: (a) plan view, thickness ( s ), maximum side length ( Lmax ) and minimum side length ( Lmin ); (b) elevation, total height ( Htot ), effective height ( Heff ). ...................................... 149 Figure 5.10: Fundamental frequency, total height and aspect ratio correlations of the training set. ... 151 Figure 5.11: Correlations between first natural frequency and geometric characteristics of the training. ...................................................................................................................................................... 151 Contents IX Figure 5.12: Histograms of the occurrence for the continuous variables of the entire database (white colour) and of the training set (blue colour). .................................................................................... 152 Figure 5.13: Distribution of the samples according to the plan configuration, masonry material and cross-section shape: (a) training set; (b) validation set. .................................................................... 154 Figure 5.14: Prediction of the first natural frequency for the entire training set: (a) models and training samples; (b) models, training and validation samples. ..................................................................... 156 Figure 5.15: Histograms of the occurrence of the fundamental period for the analysed towers. ........ 158 Figure 5.16: Correlations of the tower heights with the fundamental period (a), and with the fundamental frequency (b). ............................................................................................................. 158 Figure 5.17: Spectral accelerations of the analysed towers as function of the fundamental period (a) and epicentral distance (b). ............................................................................................................. 159 Figure 5.18: Spectral accelerations, epicentral distances, and fundamental period correlations. ....... 159 Figure 5.19: Comparison of the spectral accelerations with the response spectrum (source: INGV). . 160 Figure 5.20: Decay of seismic effects with the epicentral distance (GT0): (a) damage grades for Mechanism A; (b) damage grades for Mechanism B; (c) damage grades for overall macroelement; (d) maximum SA estimated at the tower site. Scatter plot (blue points) and mean values for 10 km bins (red diamonds). .............................................................................................................................. 162 Figure 5.21: Decay of maximum SA with epicentral distance: (a) after the first event (GT1), and (b) after the last event (GT2). Scatter plot (blue points) and mean values for 10 km bins (red diamonds). ...... 163 Figure 5.22: DPMs as functions of SA. ............................................................................................ 164 Figure 5.23: DPMs for the entire set using two distinct intensity measures: (a) SA and (b) PGA ........ 165 Figure 5.24: PGA-SA correlations for both individual damage mechanisms (a, b) and overall macroelement (c). Damage grades highlighted through the colour scale. ......................................... 166 Figure 5.25: Calibration using Eq. (5.15), (5.16), (5.17), to fit the data. .......................................... 167 Figure 5.26: Observed mean damage, plotted with circles, compared with the developed vulnerability functions for the entire dataset (GT0) as functions of SA. ................................................................. 168 Figure 5.27: Observed mean damage as functions of the SA: (a) GT1 (b) GT2. ................................ 171 Figure 5.28: Fragility functions for GT0 using SA. ............................................................................ 172 Figure 5.29: Fragility functions for GT0 using PGA. .......................................................................... 173 Figure B.1: DPMs as functions of the PGA: (a) Good; (b) Acceptable; (c) Poor. ................................. 191 Figure B.2: Fragility functions for towers with good, acceptable and poor maintenance states: (a, c, and e) traditional approach; (b, d and f) optimised approach. ................................................................. 192 Contents X Figure B.3: DPMs as functions of the PGA: (a) Htot/Lmin ≤ 4 ; (b) Htot/Lmin > 4 . ......................... 194 Figure B.4: Fragility functions for towers characterised by aspect ratio equal or lower than 4: (a) traditional approach; (b) optimised approach. ................................................................................. 194 Figure B.5: Fragility functions for towers characterised by aspect ratio greater than 4: (a) traditional approach; (b) optimised approach................................................................................................... 195 Figure B.6: DPMs as functions of the PGA: (a) Confined; (b) Integrated. ........................................... 195 Figure B.7: Fragility functions for confined towers: (a) traditional approach; (b) optimised approach. 196 Figure B.8: Fragility functions for integrated towers: (a) traditional approach; (b) optimised approach. ...................................................................................................................................................... 196 Figure B.9: DPMs as functions of the PGA: (a) Marche; (b) Umbria; (c) Macerata; and (d) Perugia. .. 198 Figure B.10: Fragility functions for towers located in Marche and in Umbria: (a and c) traditional approach; (b and d) optimised approach. ........................................................................................ 199 Figure B.11: Fragility functions for towers located in Macerata and in Perugia: (a and c) traditional approach; (b and d) optimised approach. ........................................................................................ 200 Figure C.1: Flowchart methodology for the development and validation of empirical formulation for predicting the first natural frequency of historic masonry towers. ..................................................... 203 Figure C.2: Prediction of the first natural frequency: (a) isolated; (b) bounded towers. ...................... 205 Figure C.3: Prediction of the first natural frequency for brick, stone and mixed historic towers according to their boundary condition: (a) isolated; (b) bounded. ..................................................................... 205 Figure C.4: (a) Comparison between the prediction based on the total height and effective height for bounded towers; (b) Prediction of the first natural frequency for brick, stone and mixed bounded towers. ...................................................................................................................................................... 205 Figure C.5: Prediction of the first natural frequency based on the aspect ratio (a) and on the total height and minimum side length (b) for the entire training set. ................................................................... 207 XI List of Tables Table 2.1: Typical timetable of technical activities adopted after seismic events. ................................ 17 Table 2.2: Correlation between the global damage index and the global damage level. ....................... 24 Table 2.3: Average values of V0 and Q to be used in Eq. (2.6) for distinct building typologies. ............ 28 Table 2.4: Vulnerability modifier parameters. .................................................................................... 30 Table 2.5: Corrective coefficients proposed for the shaft and belfry mechanisms. .............................. 30 Table 2.6: Coefficients of Eq. (2.8)-(2.9) proposed for the shaft and belfry mechanisms. .................... 32 Table 2.7: Existing list of parameters proposed for masonry buildings (Met1) and for historic masonry towers (Met2), together with scores and importance weights. ............................................................ 34 Table 3.1: Earthquake events considered in the present study. .......................................................... 54 Table 3.2: Maximum MCS intensity associated to each earthquake event and range of intensity at the location of the investigated towers. .................................................................................................... 55 Table 3.3: Number of towers included in the database to which a damage score has been assigned, grouped by earthquake event, damage mechanism and macroseismic intensity. ................................ 60 Table 3.4: Number of towers included in the database. ..................................................................... 62 Table 3.5: Comparison of the mean damage found with literature values for masonry towers. ........... 67 Table 3.6: DPMs for towers as entire macroelements (Macroelement damage). ................................. 67 Table 3.7: DPMs for tower damage mechanism (Mechanism A). ....................................................... 68 Table 3.8: DPMs for belfry damage mechanism (Mechanism B). ....................................................... 69 Table 3.9 DPMs for the tower mechanism (Mechanism A), grouped by macroseismic MCS intensity .. 73 Table 3.10 DPMs for the belfry mechanism (Mechanism B), grouped by macroseismic MCS intensity 73 Table 3.11 Vulnerability and ductility indexes for historic masonry towers .......................................... 74 Table 3.12 Constraints adopted in the analysis ................................................................................. 77 Table 4.1: General information about the seismic sequence. ............................................................. 89 Table 4.2: Main statistics of the geometrical data. ............................................................................. 95 Table 4.3: Calibration of vulnerability function coefficients. .............................................................. 112 Contents XII Table 4.4: Recalibration coefficients for the two distinct subsets and for the seismic intensity measures. ...................................................................................................................................................... 114 Table 4.5: Median and standard deviation for the distinct damage grades. ....................................... 116 Table 4.6: Cumulative lognormal distribution parameters for the fragility functions. .......................... 118 Table 4.7: Median and standard deviation for damage states D3 and D4 of the distinct subsets accounting for the different key features considered in this study. .................................................... 122 Table 4.8: Recalibration coefficients for the three distinct subsets based on the state of maintenance. ...................................................................................................................................................... 124 Table 4.9: Statistics of the subsets considering the maintenance states based on type of damage. ... 126 Table 4.10: Statistics of the subsets considering the location based on the type of damage. ............ 132 Table 5.1: SP96 attenuation law coefficients for the horizontal maximum PGA ( g ). ........................... 137 Table 5.2: SP96 attenuation law coefficients for the horizontal maximum 5% damped PSV ( cm/sec ) for different fundamental periods (natural frequencies) of the investigated structures. ........................... 137 Table 5.3: ITA08 attenuation law coefficients for the horizontal maximum PGA ( cm/sec2 ). .............. 138 Table 5.4: ITA08 attenuation law coefficients for the horizontal maximum 5% damped SA ( cm/sec2 ) for different fundamental periods (natural frequencies) of the investigated structures. ........................... 139 Table 5.5: ITA10 attenuation law coefficients for the horizontal maximum PGA ( cm/sec2 ). .............. 141 Table 5.6: ITA10 attenuation law coefficients for the horizontal maximum 5% damped SA) ( cm/sec2 ) ...................................................................................................................................................... 141 Table 5.7: Summary of the characteristics of the GMMs considered in this study. ............................ 142 Table 5.8: Summary of the R-squared values. ................................................................................. 145 Table 5.9: Statistics of the entire masonry towers database, the training set and the validation set. .. 150 Table 5.10: Equations based on the total height: existing and new formulations and their performance. ...................................................................................................................................................... 155 Table 5.11: Seismic stations and ground motion time histories characteristics (source: INGV). ......... 160 Table 5.12: Correlation coefficients between the intensity measures and the damage levels. ............ 167 Table 5.13: Calibration of vulnerability function coefficients and performance metrics results. .......... 167 Table 5.14: Calibration of vulnerability function coefficients. ............................................................ 168 Table 5.15: Performance metrics of the vulnerability models using SA and PGA. .............................. 170 Table 5.16: Recalibration coefficients for the two distinct subsets and for the seismic intensity measures ...................................................................................................................................................... 171 Table 5.17: Median and standard deviation for the distinct damage grades, SA-based functions. ...... 172 Contents XIII Table 5.18: Median and standard deviation for the distinct damage grades, PGA-based functions. ... 173 Table 5.19: AIC values for the fragility functions derived using the SA and the PGA. ......................... 174 Table A.1: A novel open database of structural damage observed in historic masonry towers .......... 184 Table B.1: Median and standard deviation for damage states (from D1 to D5) of the distinct subsets accounting for the state of maintenance. ......................................................................................... 193 Table B.2: Median and standard deviation for damage states (from D1 to D5) of the distinct subsets accounting for the geometry and interactions. ................................................................................. 197 Table B.3: Median and standard deviation for damage states (from D1 to D5) of the distinct subsets accounting for the location (region and province). ............................................................................ 201 Table C.1: Summary of the existing and novel empirical formulations proposed for predicting the frequency of historic masonry towers. ............................................................................................. 203 Table C.2: Summary of single independent parameters formulas, according to Eq. (C.1), Eq. (C.2) and Eq. (C.3). ....................................................................................................................................... 204 Table C.3: Summary of two independent parameters formulas, according to Eq. (C.4) and Eq. (C.5). ...................................................................................................................................................... 206 Table C.4: Summary of three independent parameters formulas, according to Eq. (C.6) and Eq. (C.7). ...................................................................................................................................................... 207 Table C.5: Summary of three independent parameters formulas, according to Eq. (C.8) and Eq. (C.9). ...................................................................................................................................................... 208 Table C.6 Summary of four independent parameter formulas, according to Eq. (C.10) and Eq. (C.11). ...................................................................................................................................................... 209 XIV List of Abbreviations/Acronyms/Symbols GEM Global Earthquake Database NSPP National Seismic Prevention Plan INGV National Institute of Geophysics and Volcanology DPMs Damage Probability Matrices SA Spectral Acceleration MCS Mercalli-Cancani-Sieberg EMS-98 European Macroseismic Scale Da.D.O. Database of Observed Damage PGA Peak Ground Acceleration GMM Ground Motion Model MM Modified Mercalli MSK Medveded-Sponheuer-Karnik JMA Japan Meteorological Agency ITACA Italian Accelerometric Archive GNDT National Group for Earthquake Defence PGV Peak Ground Velocity PGD Peak Ground Displacement SV Spectral Velocity PSV Pseudo Spectral Velocity SD Spectral Displacement BDPF Binomial Density Probability Function LSE Least Square Estimation MLE Maximum Likelihood Estimation ULS Ultimate Limit State DBMI15 Italian Macroseismic Database EC8 Eurocode 8 Contents XV RSS Residual Sum of the Square 𝑅2 Coefficient of Determination MSE Mean Squared Error PCC Pearson Correlation Coefficient AIC Aikake Information Criterion 𝐻 Hazard 𝑉 Vulnerability 𝐸 Exposure 𝑑 Individual Damage Level 𝜌 Importance Coefficient 𝑖𝑑 Global Damage Index 𝐷 Global Damage Level 𝜇 Global Mean Damage Level 𝑉𝐼 Vulnerability Index 𝑄 Ductility Index 𝑉0 Average vulnerability Index 𝑉j Corrective vulnerability Parameters 𝐼𝑣 Global Vulnerability Score 𝑞𝑖 Vulnerability Score Parameters 𝑤𝑖 Weight Parameters 𝑝 Damage Probability 𝜙 Standard Normal Cumulative Distribution Function 𝜃 Median Estimate Parameter 𝛽 Standard Deviation Estimate Parameter 𝑀 Magnitude 𝑅 Site-to-source Distance 𝑀𝑤 Moment Magnitude 𝐻𝑡𝑜𝑡 Total Height 𝐻𝑒𝑓𝑓 Effective Height 𝐿𝑚𝑖𝑛 Minimum Side Length 𝐿𝑚𝑎𝑥 Maximum Side Length Contents XVI 𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛 Aspect Ratio 𝑠 Base Wall Thickness 𝑓1 First Natural Frequency 𝑇1 Fundamental Vibration Period 1 CHAPTER 1 INTRODUCTION 1.1 MOTIVATIONS Earthquakes are natural hazards that can have catastrophic effects on human lives, built environment, infrastructure, economies and ecosystems, hindering the development of the affected regions. Severe events, such as the 2023 Morocco earthquake (Mw=6.9, resulting in 2960 deaths) and the 2023 SyriaTurkey earthquake (Mw=7.7, with over 60000 deaths), serve as recent reminders of the destructive power of seismic activity worldwide. In the European context (Figure 1.1), Italy stands out due to its high seismicity, which is the result of its geographical location along the boundary of the Eurasian and African tectonic plates. This geotectonic setting has led to several significant seismic events throughout its history, including the 1976 Friuli earthquake (Mw=6.5 and 978 fatalities), the 1980 Irpinia earthquake (Mw=6.9 and 4689 fatalities), the 1997 Umbria-Marche earthquake (Mw=6.0 and 14 fatalities), the 2002 Molise earthquake (Mw=5.7 and 32 fatalities), the 2009 L’Aquila earthquake (Mw=6.3 and 295 fatalities), the 2012 Emilia earthquake (Mw=5.9 and 24 fatalities) and the 2016 Central Italy earthquakes (Mw=6.2 and 327 fatalities). Figure 1.1: European fatalities from earthquakes over the last twenty years (Source: The International Disaster Database, EM-DAT (Delforge et al., 2023) - with major processing by Our World in Data). CHAPTER 1. INTRODUCTION 8  Chapter 1 provides an overview of the motivations and objectives of this research, outlining the significance of assessing the seismic vulnerability of historic masonry towers and the goals of the study.  Chapter 2 provides an introduction to seismic risk, including a review of state-of-the-art methodologies for assessing seismic vulnerability, with a specific focus on historic masonry towers. It offers a comprehensive overview of the methodologies commonly employed in largescale assessments, as applied in this research. Additionally, this Chapter discusses the principles outlined in the Italian Guidelines for seismic safety assessment and mitigation of heritage structures, particularly with regard to towers.  Chapter 3 compiles damage data from an extensive literature review on Italian masonry bell towers affected by past seismic events, creating an open-access database. This database is used as a testbed to assess existing territorial scale vulnerability models. The data is initially analysed through DPMs, with a comparison to analogous DPMs available in the literature. The observed mean damage is then computed for various macroseismic intensities, allowing an evaluation of the predictive capabilities of existing vulnerability functions. Development of new vulnerability functions is performed, and cross-validation strategies are employed to assess the effects of sample size on the reliability and the predictive performance of these models. Finally, fragility functions are derived using cumulative probability represented by the binomial distribution, and their validity is assessed by comparing them to observed probabilities.  Chapter 4 analyses damage data from historic masonry towers following the 2016-2017 Central Italy seismic sequence. The data, sourced from the Da.D.O. platform (Dolce et al. 2019), are used to develop vulnerability models, including DPMs, vulnerability functions, and fragility functions, with PGA employed as the seismic intensity measure. The models are initially developed and compared under different scenarios (towers affected by a single main shock or multiple shocks of the sequence) to investigate the impact of cumulative damage. Key tower features, such as location, geometry, configuration, and state of maintenance, are considered to assess their influence on vulnerability. Fragility functions are derived by fitting a lognormal cumulative distribution function to the observed probabilities of damage exceedance using the Maximum Likelihood Estimation (MLE) method.  Chapter 5 proposes novel vulnerability models based on the data analysed in Chapter 4, using SA as an alternative seismic intensity measure to PGA. Since SA is not included in the Da.D.O. database, an existing GMM is applied to estimate it. This requires knowledge of the CHAPTER 1. INTRODUCTION 9 fundamental period of the structures. While several empirical relationships for predicting the fundamental period of historic masonry towers are available in the literature, this Chapter introduces novel empirical laws derived from an open dataset compiled through an extensive literature review. The newly developed vulnerability and fragility functions are compared to those based on PGA, using various performance metrics to highlight the impact of intensity measure selection on the accuracy and effectiveness of these models.  Chapter 6 summarises the main innovative aspects and outcomes of this work, offering conclusions and recommendations for future research in the field of seismic vulnerability assessment of historic masonry towers. 10 CHAPTER 2 SEISMIC RISK ASSESSMENT: STATE OF THE ART ABSTRACT This Chapter presents an overview of seismic risk and systematically reviews the methodologies for assessing the vulnerability of masonry structures, with a specific focus on slender buildings and towers. It highlights current, advanced techniques of increasing complexity and detail, prioritising those most suitable for large-scale territorial applications. While various well-established classifications are presented, these methodologies are primarily organised by distinguishing between empirical and analytical approaches, reflecting the emphasis on the former in the subsequent Chapters. Both solutions are critically evaluated, addressing their advantages and limitations, and discussing relevant illustrative case studies. CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 11 2.1 SEISMIC RISK Seismic risk, 𝑅, may be defined as the characterisation of the potential consequences of an earthquake, commonly represented as an estimate of the expected losses, such as damage, fatalities and economic losses, resulting from the event. Risk assessment is typically concerned with a large asset stock rather than a single building, often categorising specific types (e.g., residential dwellings, public buildings, heritage, infrastructures, critical facilities) and it is a function of how the seismic hazard, 𝐻, combines with the exposure, 𝐸, of the considered assets and their vulnerability, 𝑉, thus being a spatial and temporal convolution of these three specific models, as in Figure 2.1. 𝑅 =𝑓(𝐻,𝐸,𝑉) (2.1) Figure 2.1: Seismic risk components. More recently, the capacity as qualitative parameter has been introduced in the above equation by (UNISDR, 2015) for a more comprehensive definition of the risk. Developing robust models which can accurately describe and predict the characteristics of such entities and their dynamic interactions in order to assess the overall seismic risk, is a challenging multidisciplinary task. It requires the collaboration among various experts, including seismologists, hazard modellers, data analysts and structural engineers. In general terms, the seismic hazard 𝐻 refers to the probability of a seismic event which has the potential to cause fatalities, injuries, building damage, services downtime and socio-economic interruptions within a specific territorial region. The main parameters characterising the seismic hazard are the frequency of occurrence (the return period) and the force (the severity) of the expected ground shaking in each area. Hazard assessment involves evaluating the seismicity, through appropriate and suitable seismic intensity measures. Several intensity measures are today available in the literature to characterise strong ground motions affecting the built environment. Typical seismic intensity measures are the macroseismic intensity, magnitude, Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV), Peak Ground Displacement (PGD), Spectral Acceleration (SA), Spectral Velocity (SV), Spectral Displacement (SD), Arias intensity, among others. Today, hazard maps used for design purposes are mostly given in terms of PGA, presenting the spatial distribution of the expected severity for a given CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 12 return period. Therefore, the seismic hazard can be generally defined as the probability of occurrence of an event exceeding a certain value of PGA within a given time span. Several strategies can be undertaken for assessing the seismic hazard, ranging from deterministic (Silva, 2016) to probabilistic approaches (Dolce et al., 2021). In this field, significant advances have been made and the seismic hazard can be nowadays computed by means of hazard curves at any site in Europe (Pagani et al., 2014). The exposure, 𝐸, refers to the location and qualitative attributes (e.g., occupants, building type, properties, and artefacts, collections, paintings and sculptures, if present) of the assets at risk, describing the possible consequences of a hazardous event in economic, cultural and social terms. Typically, data related to the location, typology classification, construction age, cost of the building stock and occupancy class, with potential number of occupants, are required for the development of robust exposure models. This information, stored in a building inventory when available, of great interest for planning decision and for safeguarding human life, allowing to extrapolate qualitative estimates of the people affected and economic losses caused by the event (Dolce et al., 2021; Lagomarsino & Podestà, 2004b). The vulnerability, 𝑉, refers to the susceptibility of buildings to damage when exposed to a seismic event. The damage depends on the response of buildings to the ground shaking and their capacity. The more a building is vulnerable, the greater is the expected damage. The damage mechanism type and extent can be related primarily to key physical features and characteristics of the investigated building stock and their surrounding that govern their response to the seismic action. These features include but are not limited to the type of the structural system, geometry regularity, materials and soil conditions. Other relevant characteristics, such as construction age and state of maintenance, including existing damage, may influence the susceptibility. In risk assessment, the damage can be modelled in different ways. Suitable discrete damage scales (e.g., the European Macroseismic Scale, EMS-98) are typically adopted in empirical studies, especially in post-earthquake investigations, to conduct the statistical analysis of the event and its consequences and generate the distributions of observed damage. Differently, engineering demand parameters, such as drift or crack width, are mainly used in numerical investigations, to realistically reproduce possible damage scenarios. Among these three components of the seismic risk, existing hazard and exposure can be hardly modified and often evolve slowly over time. Distinct levels of accuracy in their assessment can be ensured with different model refinements, depending on the objectives of the assessment, expertise and economic budget. Current efforts are primarily focused on enhancing vulnerability models for CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 13 seismic risk assessment and mitigation, since reducing the vulnerability of buildings is a more viable approach to decreasing the seismic risk. Therefore, understanding the seismic risk and developing effective models for evaluating it and its components ensure consistent structural condition screening, that is paramount to support the emergency response and disaster planning, safeguard critical buildings and, at the same time, minimise human and economic potential losses (Ferreira et al., 2021). The structural condition screening, in seismic risk assessment, is enabled by the identification and prediction of adequate loss metrics (i.e., damage, casualties, costs) measured in correlation with the increasing levels of one (or more) seismic intensity measures. Existing models designed for this purpose are comprehensively described in the following Sections. 2.2 ASSESSMENT METHODOLOGIES: AN OVERVIEW Although various criteria have been discussed in literature to classify risk analysis and especially vulnerability assessment methods, as for instance in (Calvi et al., 2006; Vicente et al., 2011), hereafter the following taxonomy is used to introduce and discuss existing strategies for masonry structures at territorial scale: (i) analytical models; (ii) empirical models; (iii) heuristic models; (iv) hybrid models. Analytical models are based on the idealisation of the investigated structural typology through numerical simulations, defined to quantitatively estimate the structural response to expected local seismic intensities (Shabani et al., 2021; Zizi et al., 2021). As the level of details of the involved models may vary significantly depending on the scope, these methods can be further subdivided to account for the accuracy, complexity, computational resources needed and input data required, addressing all different scales from large territorial studies to building scale assessments. Empirical models are directly based on experimental or observational data, typically collected after a seismic event, resorting to a datadriven evaluation of the structural performance through statistical methods (Rota et al., 2008). The adoption of real observational data makes them extremely reliable, although, at the same time, limits their applicability to contexts for which statistically relevant data of past events are available, and their scalability and generalizability to other contexts may be questionable (Rossetto & Ioannou, 2018). Heuristic methods rely on expertise and technical judgment, qualitative criteria, and rule-based decisionmaking to estimate the vulnerability of a building typology and/or to identify the main factors affecting its seismic performance (Giuliani et al., 2021; Romão et al., 2016). These methods are particularly useful when quantitative observational data or computational simulations are not readily available nor easy to achieve. Focusing on expert insights and opinions rather than detailed structural analysis, they have been extensively used for rapid territorial scale assessments within simple frameworks. Finally, CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 14 hybrid models result from possible combinations among the models previously described (Cardinali et al., 2022; Kappos, 2016; Kappos et al., 2006; Sandoli et al., 2023). Within this latter class, various combinations of approaches are included as for instance the use of observed data to calibrate numerical models, enhancing their reliability, and then exploit them to expand the investigation to unknown seismic scenarios. Among these methodologies, the empirical models and the analytical models have been extensively adopted for assessing the seismic risk and vulnerability of historic masonry towers and slender structures, in some cases hybridised with expert decision regarding the weight of influential factors. Due to the relevance for the scopes of the present thesis, these methodologies are shortly presented hereafter, and more details are provided in the following Section, alongside with their practical and potential applications. Examples of common tools adopted by empirical models are the Damage Probability Matrices (DPMs), vulnerability functions and fragility functions, as summarised in Figure 2.2. While these models statistically analyse post-earthquake damage data to establish specific relationships between building typology, its features and observed damage relative to seismic intensity, vulnerability index methods, reported in Figure 2.2, often rely on expert judgement to enhance the models. This approach allows for the inclusion of information that may not be readily available from past events. Figure 2.2: Empirical models. CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 15 These models differ from each other depending on how the data are used and which information is needed to extract. DPMs, originally proposed by (Braga et al., 1982; Whitman et al., 1973), are tabular representations, extensively adopted in empirical studies, that show the frequency of occurrence (probability) of different damage states according to a given scale for a building or structure typology, given a specific level of seismic intensity. The vulnerability functions are mathematical or graphical relationships that describe the expected damage level or loss (e.g., cost, downtime) as a function of a seismic intensity measure (Lagomarsino & Giovinazzi, 2006). The vulnerability index methods are procedures that assign a single numerical vulnerability score to buildings based on the main parameters that remarkably affect their seismic behaviour (Benedetti & Petrini, 1984). The definition of these relevant parameters and their weighted contribution to the overall vulnerability often result from expert opinions and heuristic approaches. Finally, the fragility functions are probabilistic models that represent the likelihood of a building or structure reaching or exceeding a specific damage state as a function of seismic intensity measures (Rossetto & Elnashai, 2003; Rota et al., 2008). Irrespective of the tool, numerical data can be used as surrogate or to integrate observational ones, supporting model development for specific building typologies. Further details on empirical methods are provided in Section 2.3. Examples of analytical approaches for historic masonry buildings at various level of detail and complexity have been discussed in the Italian document for the preservation of the cultural heritage, namely the “Guidelines for the assessment and mitigation of the seismic risk of the Cultural Heritage” (DPCM, 2011). This document establishes a framework for structural analysis tailored to the peculiar characteristics of the built heritage. In particular, three analytical levels of evaluation (LV1, LV2 and LV3) with an increasing complexity are suggested for specific structural masonry building typologies, including masonry churches and masonry towers, among others. A summary of the three levels of evaluation proposed specifically for slender masonry structures is sketched in Figure 2.3. Varying the level of analysis, with its corresponding complexity, affects the feasibility of the methods for the two distinct scales of assessment: the territorial and the individual building scale. More specifically, the first level of evaluation (LV1) is suggested for assessing the vulnerability of a large number of buildings over a defined territory, an urban centre or a region, with the main goal of identifying the most critical buildings, establishing the priorities for future interventions and supporting the seismic risk management (Bartoli, Betti, & Monchetti, 2017; Casapulla et al., 2018; Formisano & Marzo, 2017). The remaining two levels of evaluation (LV2 and LV3) are recommended for the vulnerability assessment of individual buildings when more refined analysis are required to better understand the CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 16 building's response and design future preservation strategies in a cost-effective manner (D’Ayala & Speranza, 2003; Lagomarsino & Cattari, 2015; Tanganelli et al., 2024; Torelli et al., 2020). In general, the first level of evaluation can be considered the preliminary step toward more accurate and detailed methods of analysis (DPCM, 2011). Further details on analytical methods are provided in Section 2.4. Figure 2.3: Summary of LV1, LV2 and LV3 levels of evaluation proposed by the Italian Guidelines for slender structures. 2.3 EMPIRICAL METHODOLOGIES The development of empirical models requires damage data, in specific seismic events, for structures associated with specific building typologies, spanning at various territorial scales (i.e., urban, regional, national). The development of these models also requires information on the corresponding earthquake characteristics. Damage and earthquake data are typically collected and assessed following the seismic events, often as part of technical activities aimed at enabling rapid safety and usability verifications and supporting emergency management efforts. A typical timetable of these activities in case of ground shaking is provided in Table 2.1. Generally, the technical activities are coordinated by the Department of Civil Protection or a similar national agency, together with other organisations, authorised private entities, research centres and universities. Earthquake data generally includes the processing of soil and building accelerometric data, as well as the site investigations to define the macroseismic Mercalli-Cancani-Sieberg (MCS) intensity and other intensity measures. To this end, most seismically active countries maintain a strategically deployed CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 17 monitoring network. Without the knowledge of the earthquake data, it is not possible to assess the seismic hazard and consequently the seismic risk. The MCS intensity scale, as other similar intensity scales like the Modified Mercalli (MM), the Medveded-Sponheuer-Karnik (MSK), the European Macroseismic Scale (EMS-98), and the Japan Meteorological Agency (JMA), is a predate modern instrumental method to record and characterise seismic intensity (Dolce et al., 2019, 2021). Therefore, it has been extensively used in the past to characterise the location and the intensity of an earthquake. Table 2.1: Typical timetable of technical activities adopted after seismic events. No. Time Main activities Description 1 from 2 min up to 5-30 min Epicentre and Magnitude evaluation Collecting and processing seismometric network data 2 from 10 min up to 60 min Simulated damage scenarios and data processing of monitoring systems Software simulation of the earthquake impact on constructions; Collecting and processing soil and building accelerometric data. 3 from 6 hours up to 7-14 days Site surveys for macroseismic effects Site evaluation of macroseismic intensity; Geological surveys for landslides, surface faulting, and soil liquefaction. 4 from 6 hours up to 6-12 months Temporary monitoring of soil and structures Installing temporary soil and accelerometric stations and structure monitoring systems. 5 from 24 hours up to 6-12 months Post-earthquake damage and safety assessment Building inspections for damage and usability assessment; Technical evaluation for temporary houses. MCS intensity is based on observed effects of an earthquake and their qualitative description, namely observed damage and human reaction, at a particular location (Musson, 2009). Intensity data can be collected from field surveys, eyewitness reports, and secondary sources, making it useful for regional or national risk assessments, especially in areas with limited instrumentation or to investigate past events prior to the introduction of the monitoring networks. For this reason, many traditional vulnerability assessment models have been originally developed by relying on these intensity measures. Nonetheless, several limitations of the assessment according to the MCS scale are gradually leading to its replacement by other intensity measures. Among these limitations, it is worth mentioning the subjectivity of the evaluation that relies on human observations, its resolution, often limited to coarse spatial scales and the impossibility of calculating it in real-time, supporting emergency response. Additionally, as it reflects the effects of the earthquake, it is not solely related to hazard, as an intensity measure typically should be, but it is also affected by the vulnerability of the local building stock (Musson, 2009). With the recent growth of instrumental monitoring, the use of PGA as seismic intensity measure has become very popular (Dolce et al., 2019, 2021). The PGA is a measure of the maximum acceleration experienced by the ground during an earthquake, as registered by a seismic station. The CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 24 After the identification of each occurring mechanism and the evaluation of its damage level according to the EMS-98 grading system (𝑑𝑖), a global damage index for the whole tower is computed (𝑖𝑑) with a maximum equal to the unit value, based on the following equation (weighted sum criteria): 𝑖𝑑= 15 ∑𝜌𝑖 𝑑𝑖 𝑛𝑖=1 ∑𝜌𝑖 𝑛𝑖=1 (2.2) where 𝜌𝑖 denotes the weight to be assigned to each of the n possible damage mechanisms, based on its importance for the structural behaviour of the whole building. Focusing in this work on the tower macroelement only, an importance weight equal to one is considered for both damage mechanisms (see Chapters 3, 4 and 5), as commonly done for the global assessment of churches, for instance in (De Matteis & Zizi, 2019). Therefore, the expression becomes: 𝑖𝑑= 1 10 ∑ 𝑑𝑖 2 𝑖=1 (2.3) An alternative would be to give more relevance to the shaft mechanism, Mechanism A, as the consequences of failure are likely to be greater. However, this approach was not pursued in this work. The global damage index (𝑖𝑑) is, thus, the average of the damage levels observed in the activated damage mechanisms. This is a continuous index within the 0-1 range. To obtain a discrete final global damage level (𝐷𝑘) for each tower aligned with EMS-98 scale, the correlations provided in (Lagomarsino & Podestà, 2004c) and reported in Table 2.2 are used. Table 2.2: Correlation between the global damage index and the global damage level. Global damage index (𝒊𝒅) Global damage level (𝑫𝒌) Description 𝑖𝑑≤0.05 0 No damage 0.05<𝑖𝑑≤0.25 1 Negligible to moderate damage 0.25<𝑖𝑑≤0.40 2 Moderate damage 0.40<𝑖𝑑≤0.60 3 Substantial to heavy damage 0.60<𝑖𝑑≤0.80 4 Very heavy damage 𝑖𝑑>0.80 5 Collapse With the advent of digital technologies, significant efforts have been made to digitise this essential information about past events. Some of the most well-known databases available include the GEM CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 25 Database, the Cambridge Earthquake Damage and Casualty Database, the CATDAT Database, and the EM-DAT International Disaster Database (Rossetto & Ioannou, 2018). In Italy, the Da.D.O. (Database of Observed Damage) web platform has been conceived by the Italian Civil Protection Department and realised by the European Centre for Training and Research in Earthquake Engineering to allow a long lasting and easy storage and retrieval of information (Di Meo et al., 2023; Dolce et al., 2019). The platform includes the digitisation of the survey forms from the physical inspection carried out during or following eight seismic events of national importance, namely, the 1997 Umbria-Marche, the 2002 Molise, the 2003 Piedmont, the 2004 Salò, the 2009 L’Aquila, The 2012 Emilia, the 2016-2017 Central Italy, and the 2017 Ischia earthquakes. In the cases of churches affected by these events, the damage data, catalogued with reference to the pre-defined damage mechanisms, can be displayed together with the characteristics of the events in terms of MCS intensity and in terms of PGA, elaborated by the National Institute of Geophysics and Volcanology (INGV). The collected data on hazards and losses serve as foundational inputs for statistical analyses, which are used to derive damage distributions, develop vulnerability and risk models, and evaluate their predictive accuracy. For ordinary masonry buildings, ancient churches, and historic towers, commonly adopted methodologies include DPMs, vulnerability functions, vulnerability index methods, and fragility functions, each of which is discussed in detail in the following Sections. 2.3.1 Damage Probability Matrices (DPMs) Among the earliest methods developed for risk analysis at the territorial scale, DPMs play a pivotal role in interpreting the vulnerability of asset typologies based on observed damage data. These matrices form the foundation of probabilistic risk assessment frameworks, offering a structured approach to quantify the occurrence of specific damage levels for various asset classes, given a range of intensity measure. To construct DPMs, observational data is typically aggregated into ranges of intensity measure, with the results visualised as tables or histograms for easier interpretation and application. Emerging in the 1970s, after the 1971 San Fernando earthquake (Whitman et al., 1973), DPMs gained prominence following their application to analyse the damage from major seismic events. In the case of Italian earthquakes, DPMs were extensively adopted since the systematic study of data from the 1980 Irpinia earthquake (Braga et al., 1982). In the case of masonry towers, DPMs are used to describe the discrete probability distribution of the exceedance of specific damage levels, considering the individual collapse mechanisms (Mechanisms A and B) and the entire tower macroelement. In particular, when the DPMs are developed for the individual damage mechanisms, the mean damage 𝜇𝑑 is given by: CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 26 𝜇𝑑= ∑𝑑𝑘,𝑖 𝑛𝑖=1𝑛 𝑘=[0,..,5] (2.4) where 𝑑𝑘,𝑖 is the damage level attributed to either the shaft or the belfry of the i-th tower, depending on the damage mechanism under study, and 𝑛 denotes the number of towers investigated. When the DPMs are developed for the entire macroelement, the mean damage 𝜇𝐷 is given by: 𝜇𝐷= ∑𝐷𝑘,𝑖 𝑛𝑖=1𝑛 𝑘=[0,..,5] (2.5) where 𝐷𝑘,𝑖 is the global damage level obtained for the i-th tower, and 𝑛 denotes the number of towers analysed. As DPMs represent a statistical relationship between the intensity measure of the hazard and the observed damage levels for a specific asset typology, their reliability depends on the consistency of these input data. If the assets within a typology share similar characteristics (e.g., construction materials, age, design standards) and the hazard’s effects are uniform across the area of observation, the damage distributions are expected to exhibit relatively low scatter within each intensity range. When applied to different building typologies, DPMs serve as an effective vulnerability screening tool, allowing for the preliminary identification of the most critical assets. However, DPMs provide a single output (distribution) per asset class, without explicitly accounting for unique features of the investigated building stock in their model definition, except as criteria for selecting homogeneous classes. Consequently, DPMs can be used to assess the seismic vulnerability of buildings not included in the original dataset and to predict their expected damage levels for events with specific intensity measures, only assuming that the buildings under assessment belong to the same typology, share similar characteristics, and will experience analogous hazard conditions. Numerous studies have reported DPMs for different types of masonry structures, including churches subjected to various seismic events, since the 1976 Friuli and the 1997 Umbria-Marche earthquakes (Doglioni et al., 1994; Lagomarsino & Podestà, 2004c). Recent studies have also focused on specific typologies of churches, as for instance, one-nave (Ceroni et al., 2022; De Matteis & Zizi, 2019; Ruggieri et al., 2022) or three-nave (De Matteis et al., 2016; De Matteis, Brando, & Corlito, 2019). However, most of the studies have treated churches as a whole structural system, rarely reporting and discussing the individual behaviour of each single macroelement. This prevents critical insights into the localised vulnerabilities that significantly influence the overall seismic performance of these structures. CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 27 DPMs for distinct macroelements, including masonry towers, are provided by (De Matteis et al., 2016; De Matteis & Zizi, 2019), while (Canuti et al., 2021; Hofer et al., 2018; Ruggieri et al., 2022) also investigated distinct damage mechanisms (i.e., shaft and belfry). Therefore, the contribution of the distinct components and their damage mechanisms on the overall tower behaviour is not fully assessed in the literature. Additionally, in the case of masonry towers, the sources rarely provide a complete set of DPMs for various ranges of intensity measures, with just a few exceptions, e.g., (Canuti et al., 2021). The scarce sources available are extensively discussed in Chapter 3. Finally, most of the existing studies developed DPMs using the MCS scale, which was traditionally one of the first measures adopted for seismic intensity. Nonetheless, recent advancements in territorial scale seismic risk assessment have shifted towards the use of alternative intensity measures that provide more quantifiable and objective metrics. In particular, PGA has been increasingly adopted to develop DPMs for masonry churches in general and the tower macroelements in particular (Ceroni et al., 2022; Sisti et al., 2023). 2.3.2 Vulnerability functions While DPMs provide discrete probability distributions that quantify the frequency of specific damage levels for a given intensity measure, vulnerability functions aggregate this information into a single predictive curve, offering a continuous estimation of the expected damage, through a representative metric expressed as a function of the intensity measure. In this context, DPMs and vulnerability functions are closely related tools, with DPMs serving as a foundation for constructing the curves. Additionally, they are often complementary in a study as the vulnerability functions focus on a single metric of the damage for the class, while the DPMs allow an insight into the variability within the asset class. Commonly, the expected damage metric adopted is the mean damage, 𝜇𝑑 and the intensity measure follows the macroseismic scale, 𝐼𝑀𝐶𝑆. Due to their ease-of-use and interpretation, vulnerability functions are likely the most common methods used in the literature for assessing the seismic vulnerability of ordinary and monumental masonry buildings, enabling a rapid screening of the structural condition across the considered study area. For instance, these functions have been extensively used within the RISK UE Project (Mouroux & Le Brun, 2008), “An advanced approach to earthquake risk scenarios”, to provide the seismic risk scenarios in some European cities, including Barcelona, Catania, Nice, and Thessaloniki. A similar approach has been adopted in Italy to estimate the expected damage to masonry churches and other monuments (DPCM, 2011). CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 28 To build a vulnerability model for a monumental masonry building typology, the function originally proposed by (Lagomarsino & Giovinazzi, 2006; Lagomarsino & Podestà, 2004c) is usually assumed, as follows: 𝜇=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+𝛼𝑉𝐼−𝛽 𝑄)] (2.6) where 𝜇 is the expected mean damage 0≤𝜇≤5 correlated to the macroseismic damage levels defined by the EMS-98 scale, 𝐼𝑀𝐶𝑆 is the MCS intensity varying from 5 up to 12, 𝑄 is the ductility index, 𝑉𝐼 is the vulnerability index and 𝛼 and 𝛽 are coefficients that assume constant values of 6.25 and 13.1, respectively, irrespective of the building typology. Therefore, the expected mean damage 𝜇 is correlated with increasing levels of macroseismic intensity 𝐼𝑀𝐶𝑆 through the definition of two intrinsic parameters of the monumental building typology, namely, the ductility index, Q , and the vulnerability index, 𝑉𝐼. Q accounts for the structural response in nonlinear regime. It controls the rate of increase of the damage with the intensity and is typically defined for the building typology. The vulnerability index, 𝑉𝐼, may be assumed as equal to an average value, 𝑉0, for the entire class of assets, adjusted, if needed, to take into account relevant features of each specific building that influence their structural response. Average values of the vulnerability and the ductility indexes have been provided in literature for different building typologies according to expert judgement or by calibration to the real observed data collected from past seismic events. Different pairs of 𝑄 and 𝑉0 values for monumental masonry structures suggested by (Despotaki et al., 2018; Lagomarsino, 2006; Lagomarsino et al., 2004) are shown in Table 2.3. Table 2.3: Average values of V0 and Q to be used in Eq. (2.6) for distinct building typologies. Monumental building typology Vulnerability index 𝑉0 Ductility index Q Arch bridges 0.296 2.30 Castles 0.456 2.30 Churches 0.890 3.00 Columns 0.456 1.95 Monasteries 0.736 2.30 Mosques 0.730 2.65 Obelisks 0.456 1.95 Palaces 0.616 2.30 Temples 0.500 1.95 Towers 0.776 2.30 Trilithes 0.456 1.95 Triumphal arches 0.456 2.30 CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 29 The vulnerability functions obtained for the monumental masonry building classes with the coefficients in Table 2.3 are illustrated in Figure 2.9. Just by looking at the vulnerability functions, the expected mean damage computed for masonry churches is the greatest over the entire range of the intensity. Historic towers are one of the most vulnerable building typologies, together with mosques, palaces and masonry arch bridges. It is worth noting that bell towers are not included in the masonry tower class discussed by the authors. Figure 2.9: Simplified vulnerability functions for distinct monumental masonry building typologies. In this framework, a list of corrective factors 𝑉𝑗 were originally suggested for masonry churches by (Lagomarsino et al., 2004) and subsequently adopted for other historic structural typologies, including towers, by (Despotaki et al., 2018). These factors contribute to the estimation of the vulnerability index as: 𝑉𝐼=𝑉0+∑𝑉𝑗 (2.7) Table 2.4 presents the parameters and categories considered along with their corresponding modifiers. With respect to the original list of parameters, the plan regularity is neglected for historic towers due to its minimal influence, as suggested by (Sepe et al., 2008). The corrective factors 𝑉𝑗 enable the adjustment of vulnerability functions for an entire asset class by incorporating specific building characteristics. This approach requires minimal knowledge, making it well-suited for large building stock and territorial-scale assessments. However, even the limited information necessary to accurately evaluate these corrective factors is often unavailable in large-scale assessments based on existing data sources. Standardised tools for data collection, such as inspection forms, in fact, frequently lack fields to capture all the required information. In this case, the vulnerability CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 30 index can be assumed equal to the average value for the class, as shown in Table 2.3, producing a less refined vulnerability assessment. Table 2.4: Vulnerability modifier parameters. Parameters Classification Vulnerability score State of preservation Poor Medium Good +0.04 0 -0.04 Damage level Severe Light None +0.04 +0.02 0 Architectural transformation Yes No – +0.02 0 – Recent interventions Yes No – -0.02 +0.02 – Site morphology Ridge Slope Flat ground +0.04 +0.02 0 Additional parameters for churches Position Included Additions Isolated -0.02 +0.02 0 Lateral wall height Low (<6m) Medium (>6 m and < 12m) High (>12m) -0.02 0 +0.04 Plan regularity (nave typology) Central One Three -0.02 0 +0.02 Elevation regularity (e.g., emerging elements) Yes No – +0.04 0 – Besides the development of vulnerability functions for masonry towers as independent structural typology and following the seminal contribution provided by (Lagomarsino et al., 2004; Lagomarsino & Podestà, 2004c), few authors focused on the analysis of the vulnerability of the bell tower macroelements, leveraging a much larger datasets based on churches inspections carried out in the aftermath of the main Italian earthquakes and using the previously discussed inspection form of the civil protection (Canuti et al., 2021; Ceroni et al., 2022; Curti et al., 2008; Sepe et al., 2008). A first noteworthy contribution is the work of (Curti et al., 2008), who exploited the collection of a wide dataset of historic masonry towers affected by different events, namely the 1976 Friuli, the 1997 Umbria-Marche, the 2002 Molise and the 2004 Lombardia earthquakes. This was used to calibrate adhoc vulnerability and ductility indexes for the two distinct set of mechanisms (shaft and belfry) by fitting the vulnerability curve, Eq. (2.6), proposed by (Lagomarsino & Giovinazzi, 2006; Lagomarsino & Podestà, 2004c), to the observed data. The two pairs of vulnerability and ductility indexes for the shaft and belfry mechanisms proposed by (Curti et al. 2008) are summarised in Table 2.5. Table 2.5: Corrective coefficients proposed for the shaft and belfry mechanisms. Monumental Building Typology 𝑉0 Q Source Tower macroelement damage 0.776 2.30 (Lagomarsino et al., 2004) Damage Mechanisms Mechanism A (shaft) 0.890 2.00 (Curti et al., 2008) Mechanism B (belfry) 0.940 1.49 (Curti et al., 2008) CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 31 The formulations suggested by (Curti et al., 2008) are illustrated in Figure 2.10 in comparison with the curve proposed for the tower building typology. From the comparison, the bell tower shafts (Mechanism A) and belfries (Mechanism B) are expected to undergo greater expected mean damage than the tower typology. Among the two components, belfry presents higher susceptibility than shaft. (a) (b) Figure 2.10: Vulnerability functions for the shaft (a) and the belfry (b) mechanisms, compared with the vulnerability function for the tower typology of monuments. A multi-level methodology for assessing the seismic vulnerability of historic masonry towers was proposed in (Sepe et al., 2008), analysing a set of 107 historic towers. In the preliminary stage, the structural vulnerability was roughly estimated using limited information to identify the most critical towers within the investigated stock. Initial insights were gained by correlating the level of conservation of the towers with the MCS intensity, and by comparing the existing vulnerability function proposed by (Curti et al., 2008) with the observational data. Afterwards, the towers that deviated significantly from the expected trend were flagged for more detailed structural analysis. (Canuti et al., 2021) exploited the surveys of 541 masonry churches in Marche region inspected in the aftermath of the 2016-2017 Central Italy seismic sequence to provide an overview on the occurred damage and structural vulnerability. Among the most recurrent mechanisms, bell towers received particular attention. The Authors reported the DPMs for distinct ranges of macroseismic intensity, separating failure mechanisms to the shaft and the belfry. Then, the distributions of the observed damage were compared with those estimated by the existing vulnerability models proposed by (Curti et al., 2008), revealing great consistency between observations and model predictions. Although the use of macroseismic intensity in predictive vulnerability models is widespread, as already discussed for DPMs, several methodologies are now exploring the use of PGA as a ground motion intensity measure. PGA-based approaches are gaining increasing popularity (Ceroni et al., 2022; De Matteis & Zizi, 2019). For masonry towers, (Ceroni et al., 2022) proposed multiple vulnerability curves CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 32 as a function of the MCS intensity and PGA by fitting the Eq. (2.8)-(2.9) to the observed damage for 633 single nave churches struck by the 2016-2017 Central Italy sequence. The proposed calibration coefficients are shown in Table 2.6. Unlike the original formulation of the vulnerability functions, the Authors expressed the damage data in terms of mean damage index rather than mean damage level, thus having a continuous variable in the 0-1 range. Furthermore, the vulnerability and ductility indexes were not included in the novel predictive formulations. 𝜇𝑖𝑑=0.5 [1+𝑡𝑎𝑛ℎ(𝑎 𝐼𝑀𝐶𝑆−𝑏)] MCS (2.8) 𝜇𝑖𝑑=0.5 [1+𝑡𝑎𝑛ℎ(𝑎′ log (𝑃𝐺𝐴)−𝑏′)] PGA (2.9) Table 2.6: Coefficients of Eq. (2.8)-(2.9) proposed for the shaft and belfry mechanisms. MCS PGA Damage Mechanisms 𝒂 𝒃 𝒂′ 𝒃′ Source Mechanism A (shaft) 0.25 2.08 0.89 2.54 (Ceroni et al., 2022) Mechanism B (belfry) 0.25 1.87 0.89 2.22 (Ceroni et al., 2022) Besides these few mentioned cases, validations of existing vulnerability functions for historic masonry towers on post-earthquake damage data are quite rare in the literature. Therefore, in the present thesis, these existing models are extensively tested on empirical observations, evaluating their predictive performance and suggesting recalibrations for specific datasets at hand. 2.3.3 Vulnerability index methods While DPMs and vulnerability functions have been extensively used to describe the probability distribution of damage levels based on empirical observations for classes of assets with expected homogeneous behaviour, a widespread family of methods, the so-called vulnerability index methods, have been developed by addressing the risk assessment problem from a different perspective. Indeed, vulnerability index methods provide a way to estimate the susceptibility of a building based on a set of qualitative or semi-quantitative criteria stressing the features that contribute the most to it. Each criterion is defined by a specific parameter, which is assigned a vulnerability score and an associated weight. These methods calculate the overall numerical vulnerability index, 𝐼𝑣, to an asset aggregating the scores of each parameter, weighted according to its relative importance. Therefore, these methods do not provide a direct estimation of the expected damage, rather highlight assets that are more susceptible to damage. Typical applications are aimed at identifying the critical structures for further detailed evaluation. These critical structures are generally detected as the ones with highest CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 33 vulnerability score obtained from the comparison among all the structures included in the investigated building stock (Sepe et al., 2008). At the same time, the assessment of structural parameters related to the seismic behaviour allows a more refined analysis, capable of capturing more differences within a class of assets. The number of parameters and the type of assessment enable rapid application to large building stocks, with most parameters requiring only a visual inspection supported by basic tools. Nonetheless, when the building stock is extremely large the vulnerability index methods can be computationally and economically expensive. Furthermore, the judgment of the expert teams involved in the evaluation plays a dominant role in the results. Several vulnerability index methods exist in literature. As previously mentioned, their basic functioning consists in the definition of a score 𝐼𝑣 as the weighted sum of a set of relevant features: 𝐼𝑣= ∑𝑞𝑖 𝑤𝑖 𝑁 𝑖=1 (2.10) where 𝑞𝑖 is the score assigned to the i-th qualitative parameters (e.g., elevation layout, type and quality of the masonry, state of conservation) which might remarkably affect the seismic behaviour of the building typology under investigation, 𝑤𝑖 is the weighted coefficients, and 𝑁 the number of qualitative parameters. The global vulnerability score is then usually normalised between 0 and 100, for comparative purposes. The data may be collected by means of specific survey forms tailored to each method. The number of the qualitative parameters depends on the methodology adopted. Most of the existing methodologies for seismic vulnerability assessment descends from the procedure proposed by (Benedetti & Petrini, 1984) for unreinforced masonry buildings. Several researchers and scholars have adapted this method to other contexts, such as urban centres (Vicente et al., 2011), masonry aggregates (Formisano et al., 2015) and specific macroelements such as the façade (Ferreira et al., 2017). The method was also officially adopted in Italy by the National Group for Earthquake Defence (GNDT-SSN, 1994) and it was the reference for the procedure proposed by (Sepe et al., 2008) specifically for historic masonry towers. The parameters associated to the original methodology and the approach tailored to towers in (Sepe et al., 2008) are listed in Table 2.7, including the scores for each class and corresponding weights. Eleven parameters are needed for compiling the vulnerability survey form adopted by the GNDT II Level (Met1), whereas the vulnerability form addressing historic masonry towers (Met2) is the same as the GNDT, with the exclusion of the plan layout and maximum span CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 40 where 𝑃(𝐷𝑘|𝑃𝐺𝐴𝑗) denotes the fragility function, namely the probability of reaching a damage state 𝐷𝑘 (from D1 to D5) for a given PGA value 𝑗 (or bin), 𝜙 represents the standard normal Cumulative Distribution Function, 𝜃 denotes the PGA median value, and 𝛽 constitutes standard deviation of the logarithms of the PGA. Figure 2.13: Fragility curves based on data fitting. The lognormal cumulative probability distribution function is widely considered a reliable and established method in the scientific community for several reasons. It is straightforward to apply, extensively used in earthquake engineering, and often provides a good fit to observed data. Specifically, CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 41 the lognormal distribution’s characteristic right-skewed shape can better represent observed probabilities, which tend to cluster at lower levels of ground shaking. The procedure, described and adopted in this study in Chapters 4 and 5, fits the lognormal cumulative probability distribution to the empirical data by means of the MLE method, as originally proposed by (Baker, 2015), and widely adopted in the literature in similar works. The MLE method allows for the estimation of the median and standard deviation of the lognormal cumulative distribution function, which are the two unknowns in Eq. (2.16). These parameters define the fragility function that most likely generated the observed probability data. It is important to note that, in common applications, the MLE optimisation process is iterated to derive multiple fragility functions, each corresponding to different sequential damage levels. Other optimisation procedures commonly found in the literature for data fitting are not recommended for this type of data. For example, the LSE method does not account for the variance of the observed probabilities. For further details on this topic, the reader is encouraged to refer to (Baker, 2015). Mathematically, the probability of attaining a certain damage level is given by the binomial distribution, expressed as follows: 𝑃(𝐷𝑘|𝑃𝐺𝐴𝑗)= (𝑛𝑗 𝑧𝑗) 𝑝𝑗𝑧𝑗(1−𝑝𝑗)𝑛𝑗−𝑧𝑗 (2.17) where 𝑛𝑗 denotes the number of total samples, 𝑧𝑗 the number of damaged samples, and 𝑝𝑗 the true probability of reaching a damage level for a given PGA value 𝑗 (or bin), calculated according to Eq. (2.16). To account for the different PGA values (or bins), the likelihood function is given by the product of the probabilities for each PGA value as: 𝐿𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑= ∏(𝑛𝑗 𝑧𝑗) 𝑚 𝑗=1 𝜙 (𝑙𝑛(𝑃𝐺𝐴𝑗 𝜃) 𝛽)𝑧𝑗 ( 1− 𝜙 (𝑙𝑛(𝑃𝐺𝐴𝐽 𝜃) 𝛽) ) 𝑛𝑗−𝑧𝑗 (2.18) To ensure the parameter estimation with the highest probability of reproducing the observed data, the maximum value of the maximum likelihood function is then computed as: {𝜃,𝛽}=𝑎𝑟𝑔𝑚𝑎𝑥 𝜃,𝛽 ∏(𝑛𝑗 𝑧𝑗) 𝑚 𝑗=1 𝜙 (𝑙𝑛(𝑃𝐺𝐴𝑗 𝜃) 𝛽)𝑧𝑗(1− 𝜙 (𝑙𝑛(𝑃𝐺𝐴𝑗 𝜃) 𝛽))𝑛𝑗−𝑧𝑗 (2.19) CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 42 The applicability of such procedure is very straightforward requiring a very few input parameters which are: (i) the number of total samples (𝑛𝑗); (ii) the number of damaged samples (𝑧𝑗); and (iii) the range (or bins) of PGA. It is important to note that fragility functions, when developed for distinct damage grades, may exhibit a cross-behaviour. This occurs when the estimated parameters for different curves cause their intersection. This behaviour may be due to the dataset being poorly distributed across the different PGA bins. However, when the damage grades are sequential, as in the case of the EMS-98 scale, ranging from D0 to D5, the intersection of the curves implies that the probability of exceeding a higher damage grade is greater than the probability of exceeding a lower damage grade, for certain values of PGA. Therefore, avoiding crossing of the fragility curves for sequential damage states is critical for ensuring the reliability of the resulting applications. To address the intersection of the fragility curves, two distinct approaches (Approach 1 and 2) have been suggested by (Porter, 2021). Both are tested and discussed in Chapters 4 and 5. The simplest approach (Approach 1) requires a revision of the parameters posterior to the separated fitting for each damage state. The procedure involves the computation of a new logarithmic standard deviation, equal for all damage states, and new adjusted median values, one for each damage state. In particular, the following equations are adopted: 𝛽𝑛𝑒𝑤= 1𝑘 ∑𝛽𝑘 𝑘 𝐷=1 𝑘=[1,..,5] (2.20) 𝜃𝑛𝑒𝑤= 𝜃𝑘·exp[ 0.842· (𝛽𝑛𝑒𝑤−𝛽𝑘)] (2.21) Approach 2 is considered more advanced and it involves developing the fragility curves simultaneously using the MLE, resulting in a constant logarithmic standard deviation across all damage states, while the median varies, with a distinct value for each damage state. To address the issue of fragility curve intersections, recent contributions worth mentioning include the work by (Sisti et al., 2023) for masonry churches and the work of (Tatangelo et al., 2024) for ordinary constructions. Nevertheless, the significance of this issue remains somewhat underappreciated within the scientific community. In the literature, applications based on simplified approaches to develop typological fragility curves are numerous (Chieffo et al., 2022; Lagomarsino et al., 2021). Although fragility functions have been mostly resorted to PGA as seismic intensity measure, alternative applications using the macroseismic intensity can also be found, as an example in (De Matteis, Brando, & Corlito, 2019; Lagomarsino & CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 43 Podestà, 2004c; Vicente et al., 2011). With the recent progresses in the field of monitoring and territorial scale assessment, especially in light of the recent seismic experiences, earthquake and damage data have become increasingly accessible. As a result, fitting procedures have become standard practice for deriving fragility functions for various types of buildings and structures. While a substantial body of work exists reporting empirical fragility curves for historic masonry churches (Ceroni et al., 2022; Cescatti et al., 2020; Hofer et al., 2018; Lagomarsino et al., 2021; Marotta et al., 2021; Sisti et al., 2023), limited attention has been given to historic masonry towers, with just a few exceptions (Marotta et al., 2021; Sisti et al., 2023). While alternative seismic intensity measures, such as the SA, are gaining increasing scientific interest for the development of fragility models for reinforced concrete (Rossetto & Elnashai, 2003) and unreinforced masonry (Gautam et al., 2018; Zucconi et al., 2020) buildings, their application to masonry churches and historic masonry towers remains relatively limited. 2.4 ANALYTICAL METHODOLOGIES Empirical methods rely on observations of actual damage from past earthquakes to develop vulnerability relationships or fragility curves. While the use of actual damage data makes the results realistic and directly linked to observed outcomes, this approach requires extensive, high-quality damage data, which may not be available in many regions. Moreover, this data, even when available, reflects local construction traditions, materials, and seismic characteristic, thus, could be hardly applicable to regions with different building practices or seismicity and/or could overlook evolving construction standards or retrofitting efforts. To overcome these limitations, adjustment to the methods or novel strategies based on expert judgment and qualitative assessments may be explored and applied across diverse regions with modifications to suit local conditions and tackle data-scarcity. However, these methods may be less precise and the variability in expert judgment can lead to inconsistent outcomes across studies. A valid alternative is offered by analytical methods. These involve detailed simulations or calculations of building responses to seismic actions assessing the vulnerability across a wide range of geographical areas, scenarios and structural types at various level of detail, accuracy and complexity. While these methods address all the limitations of the previous strategies, they are less simple to implement, requiring significant computational resources, advanced expertise in structural engineering and computational modelling. Moreover, their scalability is affected by the level of accuracy demanded to the models, reflected into the level of knowledge available or pursued of the building. The knowledge path affects both the modelling stage and the seismic demand evaluation. In general, for the development of a mechanical model, the representative features of the towers such as geometry (in CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 44 plan and elevation), material characteristics, boundary conditions, and state of conservation are necessary. Typically, those can be retrieved either from existing documentations and codes, or from observations and measurements acquired by in situ inspections. Regarding the seismic input characterisation, features such as location and soil type, topography and morphology need to be identified to extract the code spectra to be used for the estimation of the seismic demand. Therefore, highly reliable models are suitable for individual structures only but impractical for larger scale in a reasonable timeframe, due to resource constraints to collect the information required and carry out the analyses. To ensure their viability for territorial scale assessment, simplifications must be introduced at the risk of overlooking important aspects of the seismic response of the building. To this end, the Italian Guidelines (DPCM, 2011) for the assessment and mitigation of the seismic risk of the cultural heritage introduces a classification of the modelling strategies according to three increasing levels of complexity. Particular attention is given to the first level of evaluation (LV1), further described hereafter with specific focus on masonry towers, for its suitability to territorial scale analysis. However, a significant body of existing research in the literature employs more accurate evaluations (LV2 and LV3) for historic towers. At these levels, commonly each study investigates one or few case studies in detail. For the second level of evaluation (LV2), potential collapse mechanisms and local failures occurring in historic masonry towers have been analysed through limit analysis by (Bartoli, Betti, & Monchetti, 2017; Chisari et al., 2022; Curti et al., 2012; Faccio et al., 2011b; Sarhosis et al., 2018; Tanganelli et al., 2024; Torelli et al., 2020) and, more recently, using finite element models by (Degli Abbati et al., 2024; Mehrotra et al., 2023; Milani, 2019). Typical mechanisms in historic masonry towers included in these studies are presented and discussed in Chapter 3. For the third level of evaluation (LV3), illustrative applications resorting to finite element models are (Bartoli, Betti, & Monchetti, 2017; D’Ambrisi et al., 2012; Degli Abbati et al., 2024; Micelli & Cascardi, 2020; Milani & Clementi, 2021; Tanganelli et al., 2024; Torelli et al., 2020; Valente & Milani, 2016). Advanced 3D models are developed to accurately estimate the structural response, typically by means of nonlinear static analysis. Methodologies, advantages and limitations of these two levels of evaluations (LV2 and LV3) are not here addressed due to their relevance at the individual scale analysis. Although the first level of evaluation (LV1) reduces the number of input parameters when compared with the higher levels of evaluation, its practical use for seismic vulnerability assessment at the territorial scale remains still a field of research not fully explored, with just a few examples (Bartoli, Betti, & Monchetti, 2017; Chisari et al., 2022; Curti et al., 2008; Sarhosis et al., 2018). This is likely due to the level of knowledge required by this seismic safety evaluation, which is often constrained by the challenges of addressing CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 45 the unique characteristics of each tower, particularly those arising from geometry complexities and material variations within each asset (Degli Abbati et al., 2024; Faccio et al., 2011b; Lagomarsino et al., 2014; Tanganelli et al., 2024; Torelli et al., 2020). The first level of evaluation (LV1) proposed by the Italian Guidelines (DPCM, 2011), for masonry towers, analyses the seismic response based on a very simplified mechanical model, namely a cantilever masonry beam, subjected to both vertical loads simulating the dead loads and static horizontal forces simulating the earthquake effects. Each masonry tower is then subdivided in portions with uniform geometrical and constructive characteristics at different heights, taking into account peculiar structural features, including interaction with surrounding buildings, tapered sections and presence of significant openings. In this approach, the resultant of the lateral forces 𝐹ℎ, representative of the seismic demand, is expressed as follows: 𝐹ℎ=0.85𝑆𝑒(𝑇1)𝑊 𝑞𝑔 (2.22) where 𝑆𝑒(𝑇1) is the ordinate of the elastic response spectrum to be computed as a function of the fundamental vibration period (or natural frequency) of the tower, 𝑊 is the total weight of the tower, 𝑞 is the behaviour factor and 𝑔 denotes the gravitational acceleration. This force, corresponding to the base shear, is then applied statically in the form of horizontal concentrated forces 𝐹𝑖, whose distribution grows linearly along the height of the tower, and therefore: 𝐹𝑖= 𝑤𝑖𝑧𝑖 ∑𝑤𝑘𝑧𝑘 𝑛𝑘=1 𝐹ℎ (2.23) where 𝑤𝑖 and 𝑤𝑘 are the weights of the portion i and k respectively, 𝑧𝑖 and 𝑧𝑘 are the heights of the centre of mass of portions i and k with respect to the foundation. Subsequently, the resultant of the seismic forces 𝐹ℎ𝑖 acting on the i-th sector are determined as follows: 𝐹ℎ𝑖= ∑𝑤𝑘𝑧𝑘 𝑛𝑘=i ∑𝑤𝑘𝑧𝑘 𝑛𝑘=1 𝐹ℎ (2.24) The height 𝑧𝐹𝑖 to which 𝐹ℎ𝑖 is applied can be computed as: CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 46 𝑧𝐹𝑖= ∑𝑤𝑘𝑧𝑘2 𝑛𝑘=i ∑𝑤𝑘𝑧𝑘 𝑛𝑘=i − 𝑧𝐹𝑖∗ (2.25) where 𝑧ℎ𝑖∗ denotes the height of the verification portion with respect to the tower base. The behaviour of the masonry material is assumed to have no tensile resistance with a non-linear distribution of the compressive stresses. Under this assumption, structural failure is expected in a generic cross-section of the tower due to crushing of a compressed masonry region. The ultimate resisting bending moment 𝑀𝑅𝑑 at the base of each portion is: 𝑀𝑅𝑑,𝑖= 𝜎𝑖𝐴𝑖 2(𝑏𝑖−𝜎𝑖𝐴𝑖 0.85𝑎𝑖𝑓𝑑) (2.26) where 𝑓𝑑 is the compressive strength, 𝑎𝑖 and 𝑏𝑖 are the transversal and longitudinal dimensions of the i-th portion with respect to the direction of the horizontal forces, 𝐴𝑖 is the area of the cross-section of the i-th portion, 𝜎𝑖= 𝑊𝑖/𝐴𝑖 is the average compressive stresses of the i-th portion due to the gravity loads, and 𝑊𝑖 is the weight of the portion of the tower above the analysed cross-section. To account for all the uncertainties included in the modelling stage, the ultimate resisting bending moments are corrected through a confidence factor 𝐹𝐶 that varies according to the level of knowledge of the structure. A first preliminary evaluation to determine the safety level of the towers via simplified mechanical models can be performed by comparing the acting bending moment (demand) with the ultimate resisting bending moment (capacity), both computed along the tower height. The safety is ensured when the structural capacity is greater than the demand. The demand at each portion base is: 𝑀𝐸𝑑,𝑖=𝐹ℎ𝑖 𝑧𝐹𝑖 (2.27) However, for comparative assessment purposes, the safety level of the towers is more commonly expressed in terms of safety index, 𝐼𝑆,𝐿𝑆 and acceleration factor, 𝑓𝑎,𝐿𝑆. The safety index, 𝐼𝑆,𝐿𝑆, is the ratio between the return period 𝑇𝑈𝐿𝑆 of the seismic action that brings the structure to the considered Limit State, commonly the Life Safety Ultimate Limit State (ULS), and the expected return period of the earthquake of the site, corresponding to the same Life Safety ULS, 𝑇𝑅,𝑈𝐿𝑆, as follows: CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 47 𝐼𝑆,𝑈𝐿𝑆= 𝑇𝑈𝐿𝑆 𝑇𝑅,𝑈𝐿𝑆 𝑇𝑅,𝑈𝐿𝑆=475 𝑦𝑒𝑎𝑟 (2.28) The return period 𝑇𝑈𝐿𝑆 of the seismic action that brings the structure to the ULS can be computed as the return period that determines a spectral ordinate such that the weakest cross-section of the structure would meet the ULS threshold. By setting the equivalence between the acting bending moment and the ultimate resisting bending moment, the spectral ordinate for the i-th portion can be computed as follows: 𝑆𝑑,𝑈𝐿𝑆,𝑖(𝑇1)=𝑞𝑔𝑀𝑅𝑑𝑖∑𝑧𝑘𝑊𝑘 𝑛𝑖=1 0.85𝑊(∑ 𝑧𝑘2𝑊𝑘 𝑛𝑖=1 −𝑧𝑖∗∑𝑧𝑘𝑊𝑘 𝑛𝑘=i ) 𝐹𝐶 (2.29) Once the minimum SA is identified among the portions considered for the tower, the return period is computed according to an iterative procedure resorting to a linear interpolation. This interpolation relies on data reported in the Appendix of the Italian building code, where values needed for the calculation of the spectrum are reported for each point of the topographic network and for increasing return period, from 30 to 2475 years. On the other hand, the acceleration factor, 𝑓𝑎,𝐿𝑆, is defined as the ratio between the horizontal maximum ground acceleration that brings the structure to the ULS, 𝑎𝑈𝐿𝑆, and the reference ground acceleration associated with the ULS, 𝑎𝑔,𝑈𝐿𝑆, as follows:: 𝑓𝑎,𝑈𝐿𝑆=𝑎𝑈𝐿𝑆 𝑎𝑔,𝑈𝐿𝑆 (2.30) While the estimation of 𝑎𝑔,𝑈𝐿𝑆 is straightforward, 𝑎𝑈𝐿𝑆 being function of the response spectrum ordinate, can be obtained by inverting the elastic response spectrum formulas, as follows: 𝑎𝑈𝐿𝑆= { 𝑆𝑑,𝑈𝐿𝑆 𝑆𝐹0 𝑇𝐵≤𝑇1≤ 𝑇𝐶 𝑆𝑑,𝑈𝐿𝑆 𝑆𝐹0𝑇1 𝑇𝐶 𝑇𝐶≤𝑇1≤ 𝑇𝐷 𝑆𝑑,𝑈𝐿𝑆=𝑆𝑒,𝑈𝐿𝑆 𝑞 (2.31) A general framework for the development of simplified mechanical models is illustrated in Figure 2.14. Several applications of the LV1 approach to the seismic vulnerability assessment of masonry towers have been discussed in the literature, often to compare its effectiveness with methods presenting higher level of accuracy and to test and validate the strategies admitted by the Italian standard (Bartoli, Betti, & CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 48 Monchetti, 2017; Degli Abbati et al., 2024; Faccio et al., 2011a; Tanganelli et al., 2024; Torelli et al., 2020). A first illustrative application was given by (Faccio et al., 2011b) who assessed the seismic vulnerability of the bell tower of the St. Antonin church in Venice through the safety index and acceleration factor associated with ULS. Different assumptions in the boundary conditions to investigate the effect of adjacent buildings have been made when computing the safety value. Figure 2.14: LV1 approach suggested by the Italian Guidelines for slender masonry structures. Another noteworthy application was conducted within the scope of the research project RISEM, Seismic Risk of Monumental Buildings, (Bartoli, Betti, & Monchetti, 2017), that aimed at evaluating the seismic safety indexes and acceleration factors of four monumental masonry towers in San Gimignano. The evaluation included the influence of the adjacent buildings by considering different heights of the towers and by estimating the fundamental period of the towers using distinct simplified equation. In addressing the seismic vulnerability of the Cugnanesi tower, also located in San Gimignano, (Torelli et al., 2020) evaluated the safety index by examining the effect of the uncertainties in the visual identification of the masonry typology. Following the approach outlined in the Italian guidelines, this led to variations in the CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART 49 assumed masonry material properties, resulting in significant variability in the results. A more recent work, carried out by (Tanganelli et al., 2024) evaluated the safety indexes of the Giotto Bell tower in Florence using different simplified equations for the estimation of the first natural frequency and assessing their influence on the definition of the seismic demand. Finally, (Degli Abbati et al., 2024) adopted the LV1 approach for the global assessment of the bell tower of Saint Lawrence Cathedral relying on the experimentally identified natural frequencies through ambient vibration testing, and analysing the impact of different boundary conditions. Although the method has been extensively tested on Italian case studies, where it is recommended by the national standard, some applications to heritage assets in other countries can also be found in the literature, as in (Magalhães et al., 2012). Even though the LV1 approach in the literature mainly addresses the assessment of single towers, a few studies have discussed the applications of this simplified modelling strategy at the territorial scale (Chisari et al., 2022; Curti et al., 2008). 31 bell towers struck by the 1976 Friuli Earthquake were investigated through this method by (Curti et al., 2008). In particular, the acceleration factor was evaluated for each tower and the results were compared with the damage levels observed in the post-earthquake surveys. The comparison showed a clear discrepancy as towers with greater observed damage levels were characterised, by contrast, by higher values of safety indexes, highlighting the limitation of the simplified mechanical approach. Very recently, (Chisari et al., 2022) proposed a novel simplified vulnerability model suitable for territorial scale assessment of historic masonry towers. In particular, the development of this method required the construction of a parametric mechanical model, based on an enhanced version of the LV1 approach, to investigate the tower collapse for a wide number and range of parameters affecting it, in order to assess their influence and provide a reduced set of the most significant ones. The definition of the parameters and their range followed a detailed survey of 56 historic masonry towers in the area of the city of Naples, and the final method was tested against 31 towers found in the literature. The analysis of the existing studies shows that the seismic safety evaluation using LV1 benefits from employing multiple analytical models and sensitivity analysis, providing a more detailed understanding of the effects of uncertainties and varying assumptions on the estimated structural performance. This is often focused on the boundary conditions (Bartoli, Betti, & Monchetti, 2017; Faccio et al., 2011a), the first natural frequency (Faccio et al., 2011a; Tanganelli et al., 2024) or the masonry mechanical properties (Torelli et al., 2020), especially when their direct evaluation based on experimental tests is not available. CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 56 classification in five distinct groups for increasing levels of MCS intensity: Group I includes 40 towers with 5≤𝐼𝑀𝐶𝑆<6, Group II comprises 29 towers with 6≤ 𝐼𝑀𝐶𝑆<7, Group III is composed of 25 towers characterised by 7≤𝐼𝑀𝐶𝑆<8, Group IV includes 22 towers with 8 ≤𝐼𝑀𝐶𝑆<9, and finally Group V consists of 13 towers with 𝐼𝑀𝐶𝑆 equal to or greater than 9. It should be noted that whenever the MCS intensity associated with the tower location was not available in the sources listed in Table 3.2, the intensity associated with the Municipality where the tower is located was assigned. Figure 3.4: Map highlighting the tower distribution grouped by MCS intensity. To give more insight into the seismic behaviour of the historic masonry towers included in the database, reducing the scatter in their response and damage level, the investigated structures were classified according to the tower typology. In this regard, as mentioned in Chapter 2, the towers can be generally grouped into three categories according to their plan configuration (or plan layout): isolated towers, bounded towers, and gable bell towers. The bounded towers can be additionally classified into subgroups according to their location in the aggregate (i.e., confined and integrated) as in (Ceroni et al., 2022; Sepe et al., 2008). Still, due to the limited number of towers and information available for their characterisation, an excessive fragmentation of the dataset is avoided to ensure that all subsets are statistically representative. Therefore, the towers were grouped in two main classes only: isolated and bounded. This typological classification was conducted based on the gathered documentation and using the (Google Earth Pro, 2022) software in case of missing information in the investigated studies. It CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 57 should be noted that the bell gable typology was not considered, given the focus of the thesis on historic towers and slender structures. The damage mechanisms and extents for each historic masonry tower included in the present database were collected from the results of post-earthquake survey activities conducted by field engineers and researchers focusing on masonry churches using the macroelement approach, according to the Italian Guidelines for Cultural Heritage (DPCM, 2011). Macroelements can be activated during the ground shaking leading to different failure mechanisms. Therefore, different classifications of macroelements have been proposed and included into survey forms over time. Concerning the tower as a single macroelement, a division into mechanisms affecting its main components, namely shaft (or main body) and belfry, as shown in Figure 3.5, was introduced and systematically adopted, simplifying the collection of information from the sources of the database. (a) (b) Figure 3.5: Failure mechanisms associated with the tower macroelement: (a) tower mechanism (Mechanism A) and (b) belfry mechanism (Mechanism B). The two distinct failure mechanisms in Figure 3.5, highlighted as Mec27 and Mec28 in the last version of the A-DC survey form for the seismic damage to churches, provided by the Italian Civil Protection (DPCM, 2011), are in this work renamed Mechanism A and Mechanism B, respectively. A mechanism of non-structural emerging elements (listed as Mec26 of the same form) records damage to the spires, pinnacles, crenulations and other decorations of the tower or its belfry. This mechanism is not included in the data collection as the information available does not allow to clearly distinguish when the surveyors identified this damage in the tower and when it affected other emerging elements of the church. This is a well-known intrinsic limitation of the A-DC survey form that could not be addressed by the present work. Additionally, it is worth noting that, although historic towers are apparently regular structures typically with a heavy weight that may be beneficial for structural stability, experience shows that, when subjected to ground shaking, they exhibit damage patterns that can vary widely due to specific structural features (e.g., the slenderness, the presence of openings, the configuration of the belfries, the interaction with adjacent buildings and soil). CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 58 The aforementioned Mechanisms A and B, indeed, collect a series of typical failure mechanisms that occurred in historic masonry towers (Figure 3.6). as extensively discussed in (Doglioni et al., 1994). Figure 3.6: Typical damage mechanisms observed in historic masonry towers. In particular, several combinations of damage patterns can affect the main body of the tower. In-plane shear cracking may lead to a rocking mechanism, thus affecting the entire superstructure. The crack patterns and the subsequent formation of a hinge depend on the orientation of the shear planes, as CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 59 they can form a horizontal axis of rotation along one side, converge towards a corner or fragment the panels upon the formation of double-diagonal (X-type) cracks. Very rare is the case of failure due to sliding. Nonetheless, some examples of towers that suffered this type of damage exist, showing remarkable crack patterns but without causing total failure. Finally, the shaft may fail due to crushing. Cracking in the belfry is likely the most common damage observed in bell towers. In most cases, the damage is due to rocking of the upper part because of the formation of horizontal cracks in the masonry pillars of the belfry, often involving some detachment of masonry corners in the level below. In some cases, the tower can undergo a shear failure right below the belfry combining the two mechanisms. Other possible combinations of mechanisms depend on the distribution of the openings. Indeed, vertical cracks may occur in the presence of aligned openings and lead to splitting of the tower. The identification of the two mechanisms (i.e., Mechanism A and Mechanism B) for masonry bell towers dates back to the seminal studies on the macroelement behaviour of masonry churches, in the aftermath of the 1976 Friuli Earthquake (Doglioni et al., 1994). The masonry towers affected by such a severe event and included in the database have been investigated by (Curti et al., 2008), according to this distinction in the tower and the belfry mechanisms. For the 1997 Umbria-Marche Earthquake and the 2002 Molise Earthquake, information about the damage to the towers derived from technical reports, (Spence et al., 1998) and (Cifani et al., 2005), respectively. For the following events included in the database, the damage records are extracted from scientific publications. In particular, for the 2009 L’Aquila Earthquake, the main sources were (Augenti & Parisi, 2010; Brandonisio et al., 2013; Criber et al., 2015). For the towers affected by the 2012 Emilia Earthquake, relevant information is provided by (Ferrari, 2020; Paupério et al., 2012; Sorrentino et al., 2014; Valente et al., 2017). Finally, the damage observed in the towers after the 2016 Central Italy Earthquake was obtained from the works of (Acito et al., 2021; Clementi et al., 2020; De Matteis & Zizi, 2019; Giordano et al., 2019; Jain et al., 2020). Unlike the classification of the mechanisms recurring among different sources, regarding the extent of the damage, documents referring to different earthquakes typically adopt different discrete scales, until the diffusion and adoption of the EMS-98 six-level scale (Grünthal, 1998), with the damage ranging from 0 up to 5. Therefore, an essential task conducted during the formation of the present dataset consisted in the reclassification of the damage level, based on the grade assigned in the original sources and/or other available information, such as descriptions in the texts, photos or drawings. For around 65% of the identified case studies, the aforementioned sources provided the damage levels, e.g., (Criber et al., 2015; Curti et al., 2008; Ferrari, 2020; Sorrentino et al., 2014), while in the remaining 35% the damage level was assigned based on the visual inspection of the gathered CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 60 photographic documentation, e.g., (Augenti & Parisi, 2010; Brandonisio et al., 2013; Cifani et al., 2005), according to the EMS-98 (Grünthal, 1998). These cases are highlighted in the database with the letter c (Appendix A) . Finally, it should be noted that for a few towers included in the database no damage level was reported associated with one or both failure mechanism(s) (Appendix A). These cases for which no clear information was obtained from the sources are properly highlighted in the database table, since it was not possible to confirm the presence of the specific portion of macroelement or the activation of the mechanism and, in case, its severity. These towers belong to the 1976 Friuli and 2012 Emilia earthquakes affecting the entire range of MCS intensities, specifically between 5 and 9. The final number of towers is reported in Table 3.3, including the subgroups associated with each earthquake, each MCS intensity group and each failure mechanism (Mechanism A, Mechanism B). Table 3.3: Number of towers included in the database to which a damage score has been assigned, grouped by earthquake event, damage mechanism and macroseismic intensity. The distribution of the final number of towers obtained by considering their corresponding seismic events and MCS intensity is given in Figure 3.7, for the case of damage mechanism to the body of the tower (Mechanism A). The distribution of the towers affected by the damage mechanism to the belfry (Mechanism B) is very similar, thus, it is not reported for the sake of brevity. Many towers included in the database are associated with the Friuli, the Molise and the Emilia events, while a few of them are affected by the Umbria-Marche, the L’Aquila, and the Central Italy earthquakes. Additionally, many towers were affected by a MCS intensity range between 5 and 7.5. These towers refer mainly to the Umbria-Marche, the Molise, the Emilia, and the Central Italy events. In the remaining range (MCS Earthquake Damage Mechanism Macro-seismic Intensity MCS Total 5.0/5.5 6.0/6.5 7.0/7.5 8.0/8.5 9.0/9.5 10.0/10.5 Friuli (1976) Mechanism A 0 0 2 12 8 0 22 Mechanism B 0 0 2 13 8 0 23 Umbria-Marche (1997) Mechanism A 0 4 4 0 0 0 8 Mechanism B 0 4 4 0 0 0 8 Molise (2002) Mechanism A 7 5 5 1 0 0 18 Mechanism B 7 5 5 1 0 0 18 L’Aquila (2009) Mechanism A 0 0 1 5 0 0 6 Mechanism B 0 0 1 5 0 0 6 Emilia (2012) Mechanism A 23 13 9 0 0 0 45 Mechanism B 23 15 8 0 0 0 46 Central Italy (2016) Mechanism A 7 2 0 0 0 2 11 Mechanism B 7 2 0 0 0 2 11 Total Mechanism A 37 24 21 18 8 2 110 Mechanism B 37 26 20 19 8 2 112 CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 61 intensity 8-10.5), the number of towers is mainly given by the Friuli, the L’Aquila and the Central Italy earthquakes. Figure 3.7: Distributions of towers affected by Mechanism A according to earthquake event and MCS intensity. The distribution of the analysed towers distinguishing them in isolated and bounded is given in Figure 3.8, revealing that most, among the investigated building stock, are bounded towers and that the isolated towers data mainly belong to the Friuli and Emilia earthquakes; the distribution in Figure 3.8Figure 3.8:a refers to towers affected by the shaft mechanism (Mechanism A), whereas the distribution in Figure 3.8b refers to towers affected by the belfry mechanism (Mechanism B). Figure 3.8: Distributions of the towers according to earthquake event and structural typology: towers affected by Mechanism A (a), and towers affected by Mechanism B (b). The isolated and bounded towers were then sorted according to the MCS intensity. Table 3.4 outlines the total number of the isolated and bounded towers within each intensity group, together with the distinct damage mechanisms (Mechanisms A and B). As can be observed, similar distributions are Central Italy 2016 Emilia 2012 L'Aquila 2009 Molise 2002 Umbria-Marche 1997 Friuli 1976 0 5 10 15 20 MCS Intensity No. Towers Isolated Bounded 0 10 20 30 No Towers Isolated Bounded 0 10 20 30 No Towers (a) (b) CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 62 obtained for the towers affected by distinct damage mechanisms. Once again, it is worth highlighting that, due to missing information, the number of towers for which the actual damage level is known for Mechanisms A and B does not always correspond and both are less than the entire size of the dataset, as shown in the following table. Table 3.4: Number of towers included in the database. Plan Layout Macro-seismic MCS Intensity Total Known Total in Database 5.0/5.5 6.0/6.5 7.0/7.5 8.0/8.5 9.0/9.5 10.0/10.5 Isolated Mechanism A 8 4 2 4 7 1 26 32 Mechanism B 7 5 2 4 7 1 26 32 Bounded Mechanism A 29 20 19 14 1 1 84 97 Mechanism B 30 21 18 15 1 1 86 97 Total Mechanism A 37 24 21 18 8 2 110 129 Mechanism B 37 26 20 19 8 2 112 129 After the identification of the damage levels observed in the distinct mechanisms, the global assessment of the tower as a single macroelement was carried out using the method adopted in similar studies which follows the weighted sum criterion, as mentioned in Section 2.3 and in (Testa et al., 2024). The global damage level of the macroelement, hereafter called Macroelement damage, was computed only for the towers whose damage levels (ranging from 0 to 5) are available for both damage mechanisms (i.e., 106 towers out of the total 129). When the damage level related to one of the two distinct failure mechanisms was not known (i.e., empty cell in the database), the global damage level was not computed. It is stressed that an empty cell does not indicate absence of damage, instead indicates lack of information. 3.3 DAMAGE PROBABILITY MATRICES (DPMs) The DPMs were produced for each combination and grouping the towers discussed in the previous Section. The DPMs were also compared with the distribution of the probability of damage estimated using the Binomial Density Probability Function (BDPF) at a given level, 𝑝𝑘, based on the knowledge of the observed mean damage 𝜇, as follows: 𝑝𝑘=5! 𝑘! (5−𝑘)!∙( 𝜇5 )𝑘∙( 1−𝜇5 )5−𝑘 𝑘=[0,..,5] (3.1) 3.3.1 Entire dataset and individual events The observed DPMs and the related BDPFs are reported in Figure 3.9, considering the entire set (i.e., 5≤𝐼𝑀𝐶𝑆<11). Similar values emerge for the mean damage associated to the two damage CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 63 mechanisms and to the whole tower macroelement: 2.69 and 2.53 for Mechanism A and Mechanism B, respectively, and 2.77 for Macroelement damage. Mechanism A Mechanism B Macroelement damage ENTIRE SET (a) (b) (c) Figure 3.9: DPMs for distinct damage mechanisms (a and b), and for the entire macroelement (c). Although similar values of the observed mean damage are obtained, the distributions of the collected levels of damage are quite different. In particular, for Mechanism A, a unimodal (i.e., a single peak) almost symmetrical distribution of the damage with a peak at level 3 emerges, while Mechanism B exhibits a growing trend in the distribution with a large number of towers presenting a damage score equal to 5. This trend suggests a higher susceptibility to severe damage of the belfry when compared with the shaft. The last case, Macroelement damage, reveals an almost uniform distribution of damage with slightly larger occurrence of D3 and slightly more cases with low damage than severe damage. The displayed damage histograms for Mechanism A and Macroelement damage have an acceptable fit by the binomial distribution, whereas the distribution of Mechanism B does not seem to match the BDPFs. The value obtained for the global mean damage is larger than the individual mechanisms mean damage. To better investigate this unexpected trend, the analysis was repeated by taking into account the same number of towers (i.e., 106) instead of the 110 and 112 towers originally considered for Mechanism A and Mechanism B, respectively. Thus, all the towers for which no information was available for one of the two mechanisms were excluded from the calculation. Mechanism A Mechanism B Macroelement damage ENTIRE SET (a) (b) (c) Figure 3.10: DPMs for distinct damage mechanisms (a and b), and for the entire macroelement (c) considering 106 towers only. 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence µd = 2.53 µD = 2.77 µd = 2.69 µd = 2.63 µd = 2.47 µD = 2.77 CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 64 Mechanism A Mechanism B Macroelement damage FRIULI (1976) UMBRIA-MARCHE (1997) MOLISE (2002) Figure 3.11: DPMs of the investigated towers developed for Friuli, Umbria-Marche and Molise events, and by considering both the damage levels observed in each damage mechanism and macroelement. The new DPMs and the BDPFs are illustrated in Figure 3.10, which provide the same unexpected result. No significant variation in the mean damage values and corresponding damage distributions emerges and the mean global damage continues to be larger than the mean damage obtained for the single individual mechanisms. This is likely ascribed to the equation used to combine the damage levels into the global damage index (Eq. 2.3) and to the related correlation with the global damage level (Table 2.2). The overestimation of the global damage with respect to the average of the individual mechanisms’ values provided by this approach, when adopted for the case of towers, is likely influenced by the weight assigned to each mechanism and by the correlation used to transform the continuous global damage index into the discrete global damage level. Alternative approaches to the weighted sum criterion have been proposed in literature, especially for reinforced concrete structures. A study that focuses on the evaluation of how the global damage index using the existing approaches affect the damage level estimates is given in (Zucconi et al., 2022). However, in the present study, both weights and correlation are maintained as in the original approach by (Lagomarsino & Podestà, 2004b, 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence µd =3.18 No. 22 µd =2.91 No.23 µD =3.27 No.22 µd =1.63 No. 8 µd =1.63 No. 8 µD =1.75 No. 8 µd =1.33 No.18 µd =1.72 No.18 µD =1.83 No.18 CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 65 2004c) and in similar works as in (De Matteis & Zizi, 2019); variations to this approach are left as future scopes of the research. Considering possible differences in the towers behaviour due to the peculiar characteristics of each earthquake and local constructions, additional DPMs were developed by taking into account the earthquake events individually are shown in Figure 3.11 (Friuli, Umbria-Marche and Molise) and Figure 3.12 (L’Aquila, Emilia and Central Italy). Mechanism A Mechanism B Macroelement damage L’AQUILA (2009) EMILIA (2012) CENTRAL ITALY (2016) Figure 3.12: DPMs of the investigated towers developed for L’Aquila, Emilia and Central Italy events and by considering both the damage levels observed in each damage mechanism and the global damage levels. The DPMs developed by classifying the investigated towers based on the set of earthquakes aim to perform an in-depth examination of the towers' behaviour and condition against the corresponding seismic event they experienced. The observed damage, for the cases of the Umbria-Marche and the Molise earthquakes, is mostly characterised by occurrences of no damage or small damage extents. For the former event, the limited number of known towers in the database may affect the evaluation, although the histogram shows a clear trend. For the Central Italy event, the observed damage is likely normally distributed, especially for Mechanism A and Macroelement damage. For the remaining cases (i.e., the Emilia, the Friuli and L’Aquila earthquakes), the observed damage is left-skewed distributed, 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence µd =3.67 No.6 µd =2.33 No.6 µD =3.33 No.6 µd =3.11 No. 45 µd =2.67 No.46 µD =2.95 No.41 µd =2.55 No. 11 µd =2.91 No.11 µD =3.00 No. 11 CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 72 Mechanism A Mechanism B Macroelement damage 5≤𝐼𝑀𝐶𝑆<6 (a) (b) (c) 6≤𝐼𝑀𝐶𝑆<7 (d) (e) (f) 7 ≤𝐼𝑀𝐶𝑆<8 (g) (h) (i) 8 ≤𝐼𝑀𝐶𝑆<9 (l) (m) (n) 𝐼𝑀𝐶𝑆≥9 (o) (p) (q) Figure 3.14: DPMs for different MCS intensities. 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence 0,0 0,2 0,4 0,6 0,8 1,0 D0 D1 D2 D3 D4 D5 Damage occurrence µd = 2.68 µd = 2.35 µD = 2.75 µd = 1.92 µd = 2.00 µD = 2.00 µd = 2.86 µd = 2.75 µD = 2.83 µd = 3.44 µd = 3.11 µD = 3.56 µd = 2.90 µd = 3.00 µD = 3.20 CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 73 Table 3.9: DPMs for the tower mechanism (Mechanism A), grouped by macroseismic MCS intensity. Source Seismic Event Observed Damage Level (𝒅𝒌) D0 D1 D2 D3 D4 D5 𝐼𝑀𝐶𝑆=5 Authors’ database (see Section 3.2) 11% 14% 16% 22% 32% 5% (Canuti et al., 2021) Central Italy (2016) 60% 16% 24% 0% 0% 0% 𝐼𝑀𝐶𝑆=6 Authors’ database (see Section 3.2) 33% 8% 21% 21% 4% 13% (Canuti et al., 2021) Central Italy (2016) 58% 18% 12% 8% 4% 0% 𝐼𝑀𝐶𝑆=7 Authors’ database (see Section 3.2) 14% 5% 19% 24% 19% 19% (Canuti et al., 2021) Central Italy (2016) 50% 22% 10% 5% 2% 11% 𝐼𝑀𝐶𝑆=8 Authors’ database (see Section 3.2) 0% 6% 6% 44% 27% 17% (Canuti et al., 2021) Central Italy (2016) 9% 9% 0% 39% 22% 22% Table 3.10: DPMs for the belfry mechanism (Mechanism B), grouped by macroseismic MCS intensity. Source Seismic Event Observed Damage Level (𝒅𝒌) D0 D1 D2 D3 D4 D5 𝐼𝑀𝐶𝑆=5 Authors’ database (see Section 3.2) 27% 3% 22% 22% 11% 15% (Canuti et al., 2021) Central Italy (2016) 65% 21% 10% 4% 0% 0% 𝐼𝑀𝐶𝑆=6 Authors’ database (see Section 3.2) 35% 15% 12% 8% 15% 15% (Canuti et al., 2021) Central Italy (2016) 48% 21% 15% 12% 2% 2% 𝐼𝑀𝐶𝑆=7 Authors’ database (see Section 3.2) 25% 10% 5% 10% 25% 25% (Canuti et al., 2021) Central Italy (2016) 61% 10% 12% 8% 7% 2% 𝐼𝑀𝐶𝑆=8 Authors’ database (see Section 3.2) 16% 16% 5% 5% 21% 37% (Canuti et al., 2021) Central Italy (2016) 9% 19% 27% 9% 27% 9% 3.4 VULNERABILITY FUNCTIONS As an alternative to the DPMs, vulnerability curves are a powerful tool to rapidly estimate the expected mean damage, 𝜇, for a given level of macroseismic intensity, 𝐼𝑀𝐶𝑆. To build a vulnerability model, the function originally proposed by (Lagomarsino & Giovinazzi, 2006; Lagomarsino & Podestà, 2004c) was here adopted, as follows: 𝜇=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+6.25𝑉𝐼−13.1 𝑄)] (3.2) where 𝜇 is correlated with increasing levels of 𝐼𝑀𝐶𝑆 through two physical parameters intrinsic of a specific investigated asset typology, component or subclass, namely the ductility index, Q , and the vulnerability index, 𝑉𝐼. CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 74 In particular, values for vulnerability and ductility indexes for historic masonry bell towers were recommended by (Curti et al., 2008; Lagomarsino et al., 2004) and have been here used as a reference, see Table 3.11. Table 3.11: Vulnerability and ductility indexes for historic masonry towers. 𝑽𝑰 𝑸 Source Damage Mechanism Mechanism A 0.890 2.00 (Curti et al., 2008) (1) Mechanism B 0.940 1.49 (Curti et al., 2008) (2) Entire Macroelement Macroelement damage 0.776 2.30 (Lagomarsino et al., 2004) As already mentioned in the state of the art, (Lagomarsino et al., 2004) proposed a vulnerability function, also recently reported by (Despotaki et al., 2018) for masonry towers and slender buildings, not specifically for bell towers, here used as reference for the entire tower macroelement only, while (Curti et al., 2008) developed vulnerability functions specifically addressing all the damage mechanisms of towers. It should be noted that the modifier parameters of the vulnerability index, discussed in Chapter 2, have not been taken into account due to the lack of data. Figure 3.15 shows the vulnerability curves obtained by applying these existing equations against the present dataset. The curves are compared with the values of mean damage observed in the investigated towers for different groups of MCS intensity, further grouped by seismic events, through a colour key. It should be noted that the towers characterised by maximum MCS intensity have been distinguished into two subgroups: 9 ≤𝐼𝑀𝐶𝑆<10 and 10 ≤𝐼𝑀𝐶𝑆<11. The plots also report the values of the mean damage considering the full set of earthquakes for each group of MCS intensity as the Global Mean. This value was preferred for validation and calibration purposes to overcome a possible bias given by the different number of towers observed in distinct earthquakes. Nonetheless, this approach could not completely solve the unexpected high mean damage in the range of 5 ≤𝐼𝑀𝐶𝑆<6 and the extremely low mean damage in the range of 10 ≤𝐼𝑀𝐶𝑆<11. Samples in these ranges may be not representative of the expected real population of damaged towers; therefore, for the calibration of new vulnerability curves, they were discarded. The existing predictive functions for the individual damage mechanisms fit well the observed mean damage, especially in the range 6 ≤𝐼𝑀𝐶𝑆<10. A certain scatter between the predicted and the observed damage levels emerges but the same is reduced when the Global Mean values are CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 75 considered. The predictive function for the entire macroelement, instead, appears to be nonconservative, producing an underestimation of the mean damage. Mechanism A 𝜇𝑑=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+6.25·0.89−13.1 2.00 )] (3.3) (a) Mechanism B 𝜇𝑑=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+6.25·0.94−13.1 1.49 )] (3.4) (b) Macroelement damage 𝜇𝐷=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+6.25·0.776−13.1 2.30 )] 𝜇𝐷=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+7.70·0.776−13.1 2.30 )] (3.5) (3.6) (c) Figure 3.15: Comparison between the existing vulnerability functions and the observed mean damage for distinct mechanisms (a and b) and for the entire macroelement (c). In light of these considerations, further analyses were conducted to fit the observed mean damage using the well-known formulation of the vulnerability curve but recalibrating its parameters. Initially, the calibration focused on the parameters α , β , Q , as shown in Eq. (3.7), one by one, while keeping the vulnerability index from literature. These parametric analyses performed on the vulnerability equations CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 76 provided a graphic tool useful to identify the best curve according to specific data in hand. This procedure is similar to (De Matteis, Brando, & Corlito, 2019). 𝜇=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+𝛼𝑉𝐼−𝛽 𝑄)] (3.7) The results obtained for the vulnerability curves varying the parameters over a wide range are illustrated in Figure 3.16. The lower and upper bound for the parameters (α , β , Q ) of the three existing formulations and the intervals in their variation, considered for the definition of the curves are reported in Table 3.12 (here ∆ is the step adopted for the different curves shown). It can be observed that while the variations of the parameters α and β produce a shift of the vulnerability curves, where the effect on the curve of increasing α corresponds to the effect of reducing β and vice-versa, the variation of the ductility index Q generates a change in its slope. Among the calibration coefficients, α and β play a major role in fitting the curves to the present dataset. Mechanism A (a) (b) (c) Mechanism B (d) (e) (f) Macroelement damage (g) (h) (i) Figure 3.16: Parametric analyses of the existing vulnerability functions for distinct damage mechanisms, (a) to (f), and for the whole macroelement, (g) to (i). CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 77 Table 3.12: Constraints adopted in the analysis. Initial values Min Max Δ α 6.25 5.00 7.50 0.25 β 13.10 12.10 14.10 0.20 Q (Mechanism A) 2.00 1.00 3.00 0.20 Q (Mechanism B) 1.49 0.49 2.49 0.20 Q (Macroelement damage) 2.30 1.30 3.30 0.20 Upon this preliminary sensitivity analysis, a regression analysis was conducted to develop new functions fitting the observed Global Mean damage, by choosing α as a variable to calibrate. The results related to the individual damage mechanisms (Mechanism A and Mechanism B) show values similar to the ones proposed by (Curti et al., 2008), Eq. (3.3) and (3.4), confirming their capability for damage prediction. When fitting the dataset for the entire tower macroelement (Macroelement damage), instead, the obtained Eq. (3.6) produces a significant improvement with respect to the original formulation proposed by (Lagomarsino et al., 2004), given in Eq. (3.5), for the specific dataset in hand. Figure 3.17 shows the performance plots, namely predicted versus observed mean damage values, obtained by applying both the existing and proposed formulations, together with the Residual Sum of the Square (RSS) metric, expressed as follows: 𝑅𝑆𝑆= ∑(𝑦𝑖−𝑦𝑖 )2 𝑛 𝑖=1 (3.8) where n is the number of observations, 𝑦𝑖 denotes the actual observed damage extent, and 𝑦𝑖  denotes the predicted value estimated via the vulnerability function. This metric shows a significant improvement of the predictive capability upon recalibration of the vulnerability function coefficient α, considering both the Global Mean values (plotted as diamonds), very well predicted, and the mean values for the samples grouped by earthquake (plotted as circles) that, despite a larger dispersion, are better predicted as well. (a) (b) Figure 3.17: Performance plots before (a) and after (b) recalibration. RSS estimated for the Global Mean value. RSS = 4.2 RSS = 1.1 CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 78 Finally, to highlight the statistical relevance of the present database and to further validate the existing and proposed vulnerability functions, investigating the effect of the size and characteristics of the tower samples on their performance, multiple statistical analyses were performed by randomly generating several subsets of historic masonry towers. More than 200’000 subsets for each analysed instance (Mechanism A, Mechanism B and Macroelement damage) were generated following distinct sampling strategies to randomly exclude towers from the whole database, defining several groups of towers of different sizes. Exclusions based on the specific events or the specific sources of information, despite being potentially valid alternative approaches for the generation of the subsets, were discarded as they could introduce biases in the dataset at hand. Indeed, sources like Curti et al. (2008), Acito et al. (2021) or Jain et al. (2020) provide only information for high macroseismic intensity while others, like De Matteis and Zizi (2019) or Valente (2017), inform about low macroseismic intensity. Moreover, as demonstrated by the DPMs categorised by earthquake and by intensity, the information is likely not well distributed over the damage levels. Therefore, combining all sources together was deemed essential for a correct inference of the towers behaviour against real earthquake events. First, 10’000 distinct subsets of 100, 102, and 96 historic masonry towers were generated for Mechanism A, Mechanism B and Macroelement damage, respectively, by randomly removing, in each subset, 10 towers from the initial whole dataset. All the possible combinations of leave-10-out subsets were too numerous to be managed (i.e., in the order of tera). The Global Mean damage values were computed for all the subsets and then clustered accounting for each macroseismic intensity as shown in Figure 3.18(a, c and e). The resulting distribution of Global Mean damage values appear to be very narrow considering both the interquartile range and the spread of the whiskers in the boxplots, which account for a 99% coverage assuming a normal distribution of the data. Nonetheless, a few significant outliers emerge. It should be also noted that the distribution of the Global Mean damage for high macroseismic intensity features the largest spread. To statistically measure the performance of the existing and novel formulations, the RSS metric was evaluated over all the random subsets, and the obtained distributions, which were also fitted by a Gaussian probability density function, are illustrated in Figure 3.18(b, d and f). The RSS distributions present lower mean values for Mechanism A than for Mechanism B and Macroelement damage. In Figure 3.18e and f), beside the original formulation of the vulnerability function for Macroelement damage, the suggested recalibration and its performance in terms of RSS are shown. Here, the RSS comparative results confirm a significant improvement upon recalibration of CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 79 the curve for Macroelement damage, as their mean decreases and their distribution becomes narrower. A recalibration of the curve coefficients for Mechanism B data has been attempted as well but has led to a negligible improvement of the performance. Indeed Figure 3.18(c) suggests that a recalibration of the ductility factor Q would be more effective than the attempted calibration of the coefficient α. However, a tentative calibration of the factor Q led to a value too far from its acceptable physical range, likely due to the uncertainties in the dataset, and was discarded. Mechanism A (a) (b) Mechanism B (c) (d) Macroelement damage (e) (f) Figure 3.18: Validation of the existing (a-d) and the proposed (e-f) vulnerability functions against 10’000 subsets generated by randomly removing 10 towers each time. CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 80 In a second stage, randomly generated samples of different sizes were investigated. In particular, 3 and 20 towers at a time were randomly removed from the initial whole dataset. Mechanism A (a) (b) Mechanism B (c) (d) Macroelement damage (e) (f) Figure 3.19: Validation of the existing (a-d) and the proposed (e-f) vulnerability functions against more than 200’000 subsets generated by randomly excluding 3 towers each time. CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET 81 Mechanism A (a) (b) Mechanism B (c) (d) Macroelement damage (e) (f) Figure 3.20: Validation of the existing (a-d) and the proposed (e-f) vulnerability functions against 10’000 subsets generated by randomly removing 20 towers each time. Exclusion of 3 towers produced smaller subsets of 107, 109, and 103 towers for Mechanism A, Mechanism B and Macroelement damage, respectively. All the possible combinations of 3 samples out of the original database were considered, summing up to 215’820, 227’920, and 192’920 distinct random subsets for Mechanism A, Mechanism B and Macroelement damage, respectively (Figure 3.19). Exclusion of 20 towers produced smaller subsets of 90, 92 and 86 towers for Mechanism A, CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 88 4.1 INTRODUCTION The present Chapter addresses the most relevant shortcomings that emerged during the analyses discussed in Chapter 3 improving the reliability of territorial scale seismic vulnerability models for masonry towers and slender structures (see Figure 4.1). Figure 4.1: Flowchart methodology adopted in this Chapter. In the first stage, historic masonry bell towers affected by the 2016-2017 Central Italy seismic sequence have been collected, filtering the entire set of churches resulting from the Database of Observed Damage (Da.D.O.) from (Dolce et al., 2019). The final database with its statistical analysis is described in Section 4.2. Successively, vulnerability models (DPMs, vulnerability and fragility functions) have been produced in Section 4.3. The entire dataset was considered and divided in multiple subsets accounting for distinct seismic scenarios, namely single seismic shock and multiple seismic shocks, to provide an insight into the seismic risk of masonry bell tower under cumulative damage. Finally, additional models have been developed, in Section 4.4, taking into account key features such as location, geometry, interactions, and maintenance, seeking to identify towers with more likely homogenous behaviour. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 89 4.2 DESCRIPTION OF CENTRAL ITALY SEISMIC EVENTS 4.2.1 Seismic sequence The Central Italy seismic sequence struck a wide area at the boundary among Lazio, Abruzzo, Marche and Umbria regions with a series of moderate to strong normal-fault earthquakes. The first main shock occurred on August 24, 2016 with epicentre in the area of Monti della Laga near Accumoli, Lazio region, and a magnitude 𝑀𝑤 6.2, followed by several aftershocks recorded during remaining 2016 and 2017. Two severe aftershocks hit on October 26, 2016, in the area of Valnerina with epicentre near Castelsantangelo sul Nera, Marche region, with 𝑀𝑤 5.5 and 6.1. The main shock of the sequence occurred four days later on October 30, 2016, with 𝑀𝑤 6.6 and epicentre in Norcia, Umbria region. Finally, on January 17, 2017, three events occurred with the strongest having magnitude 𝑀𝑤 of 5.7, all with epicentre near Capitignano, Abruzzo region (Rovida et al., 2020; Sisti et al., 2023). The recovery plan and reconstruction works are still on-going almost 10 years after the events (PCM, 2025). Table 4.1 shows the general information of the main seismic events considered in this study, according to the Italian Earthquake Catalogue (Rovida et al., 2020). The epicentres of these events are illustrated over the Italian territory in Figure 4.2. Table 4.1: General information about the seismic sequence. No. Event Epicentral Area Date [dd-mm-yyyy] Time (UTC) Lat [°] Long [°] Depth [km] 𝑴𝒘 [-] 1st Monti della Laga 24/08/2016 01:36 42.698 13.233 08.1 6.2 2nd Valnerina 26/10/2016 17:10 42.874 13.124 08.1 5.5 3rd Valnerina 26/10/2016 19:18 42.904 13.090 09.6 6.1 4th Valnerina 30/10/2016 06:40 42.830 13.109 10.0 6.6 5th Aquilano 18/01/2017 10:14 42.531 13.283 09.6 5.7 Damage surveys were conducted in three main phases. Initially promoted after the first main shock on August 24, 2016, the activities were interrupted by the severe earthquakes in October to start again only in November. Finally, the third phase was triggered by the last strong shock in January 2017 (Carbonari et al., 2019). Second and third phases, in some cases, comprised also repetition of previous inspections. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 90 Figure 4.2: Map of the epicentres. 4.2.2 Dataset Data about the investigated bell towers were extrapolated from the Da.D.O., Database of Observed Damage (Dolce et al., 2019), within a collaboration with a research group from the University D’Annunzio of Chieti-Pescara. Da.D.O. is the Italian WEB-GIS platform that stores and catalogues the digitised inspection forms of unreinforced masonry and monumental buildings, namely masonry churches, among other assets struck by past seismic events. The extensive sample gathering is paramount and allows not only to assess the seismic risk at different scales based on past events, but also to develop models to forecast future possible damage scenarios, thus supporting the civil protection activities. The collected data were registered during field inspections by means of specific survey forms for each building typology (e.g., regular buildings, churches, palaces, etc.). The forms have been conceived by the Italian Civil Protection Department for post-earthquake damage assessment within the framework of the emergency technical activities and comprise information about: location (site characteristics, urban context, and address), architectural and structural features (construction date, plan layout vertical system, material, existing cracks, recent interventions, and state of maintenance), damage (observed damage levels, in the most recent versions according to the EMS-98 scale), seismic input CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 91 characterisation (macroseismic intensity and PGA) and relevant exposure-related features (use, time usage and peak time), among other details, such as the date and the level of accuracy of the survey. The platform allows filtering for distinct seismic events and building typologies. Information about the historic masonry towers are retrieved from Da.D.O. as part of the broader surveys of the churches affected by the seismic event. In the church survey form, specific features strictly referring to the bell towers or, in any case, relevant for their assessment, can be extrapolated, namely: location, geometry, tower typology, damage grade, and intensity measures characterising the seismic input. Beside the specific coordinates, the form directly reports the region and province/district where the church is located. The available geometrical data refer to the length of the two base sides (rectangular or squared cross-section) and height, measured or estimated. The tower typology indicates the type of interaction between the towers and the surrounding buildings, by identifying recurring plan configurations, namely isolated, confined, integrated, and bell gable, as introduced in Chapter 2. As for the previous applications of the present thesis, target mechanisms comprise damage to the shaft and belfry, neglecting the non-structural elements (e.g., pinnacles, spires and decorations), with their corresponding damage levels assigned in the form through expert judgment during site inspections. Finally, the intensity measures included in Da.D.O. for each asset are MCS intensity and PGA. These values, provided for the churches, are assumed to be the same for the towers macroelements. At the time of the consultation of the database, a total of 3356 churches were recorded for the Central Italy seismic sequence. However, different filters were applied to obtain a more reliable dataset focused on masonry bell towers only. In particular, to ensure that the same number of recorded damage levels was available for the shaft and belfry mechanisms, all the cases characterised by absence of one of the two elements or missing damage data were excluded. Moreover, the bell gable typology was not considered, given the focus of the thesis on historic towers and slender structures, as in Chapter 3. For the sake of clarity the remaining three classes of interaction are exemplified in Figure 4.3. Finally, to avoid issues related to completeness of the survey, towers located in areas with macroseismic intensity (𝐼𝑀𝐶𝑆) lower than 5 or no information available were discarded. Indeed, as the area of low and very low intensity expand drastically, far from the epicentre, the likelihood of missing survey data for many undamaged assets increases dramatically, with the consequence that most of the recorded surveys in this wide region may have addressed churches because of their peculiarities or of an anomalous damage level at such intensities. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 92 Figure 4.3: Tower typologies considered in the present Chapter. This issue, well-known in literature (Perelli et al., 2019; Rossetto & Ioannou, 2018; Tatangelo et al., 2024; Zucconi et al., 2022), may be tackled for residential buildings relying on census data. However, effective strategies for monumental buildings have not been developed or extensively applied yet and are outside the scope of the present thesis. The filtering process allowed to gather 794 historic masonry towers with potential activation of both damage mechanisms. After cleansing the data, the global damage index for the tower macroelement was computed as in Chapter 3 and in (Testa et al., 2024). These 794 towers constitute the entire dataset analysed hereafter, also indicated for simplicity GT0. Focusing on the date of the inspection, compared to the date of the main shocks of the sequence, it was possible to identify four groups of towers. Most of them were inspected between the first and second shocks, or after all main shocks of the seismic sequence. Very few towers were inspected either between the third and the fourth shocks, or between the fourth and the fifth shocks. Since two seismic shocks have occurred on October 26, 2016 and the available documentation does not allow to identify the hour of the inspection, it is assumed that the towers were inspected after both seismic events occurring in the same day. Figure 4.4(a) illustrates the spatial distribution of the epicentres of the main events during the seismic sequence and the towers, identified according to the last main event prior to the survey date. The recorded towers are mainly concentrated to the north and north-east of the epicentres. The long duration of the seismic sequence and the relationship between the date of inspection and the main shocks of the sequence require a careful definition of the intensity measure value adopted to characterise the input to the tower. In particular, the Da.D.O. database provides a single value of the macroseismic intensity for each tower location but five values of PGA, one per each main event. Following a well-established approach (Carbonari et al., 2019; Rossetto & Ioannou, 2018; Sisti et al., 2023), the PGA at the tower site that was considered for the analyses corresponds to the maximum CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 93 acceleration among the events previous to the inspection date. Figure 4.4(b) shows the spatial distribution of the towers, distinguished by the specific event, prior to inspection, in which the maximum PGA at the site was registered. In general, the maximum PGA at the site of the towers was induced by the seismic events with greater magnitudes, namely, the shocks occurred on August 24 and October 30, 2016. A non-negligible number of towers suffered the maximum PGA during the third shock, namely on October 26, also characterised by a rather strong intensity, due to their proximity to its epicentre. Based on the experienced PGA and the date of inspection, the investigated towers of the whole dataset (GT0) were grouped into four partially overlapping subset: (i) 219 towers inspected between first and second shocks (GT1); (ii) 487 towers inspected after last main shock (GT2); (iii) 349 towers inspected after fourth main shock for which the second highest PGA in the events prior to inspection is lower than 0.1g (GT3); (iv) 212 towers inspected after fourth main shock for which the second highest PGA in the events prior to inspection is higher than 0.1g (GT4). The characteristics of these datasets and their sampling are further described in the following analyses. (a) (b) Figure 4.4: Localisation of the epicentres with their magnitude and geographical distribution of the towers distinguished by: (a) last seismic event before the survey date; (b) seismic events causing the maximum PGA at site before survey date. Besides analysing the potential accumulation of seismic effects during the sequence, in the present Chapter an attempt is made to identify homogeneous classes of towers based on their attributes described in the inspection form. In Figure 4.5, a first classification based on the geographical location is carried out, illustrating the distribution of all inspected towers across the four regions and the ten provinces affected by the Central Italy seismic sequence. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 94 (a) (b) Figure 4.5: Towers geographical distribution: (la) regional-based; (b) province-based. Most of the towers are located in the Marche and Umbria regions, representing 61% and 26% of the dataset, respectively. A similar trend is observed at the provincial level, where the majority of towers are situated in the Macerata province of the Marche region and the Perugia province of the Umbria region, accounting for 37% and 23% of the dataset, respectively. The remaining regions and provinces contain fewer towers, with a maximum of 12% of the dataset found in any other province. Subsets based on geographical location were created, taking into account possible peculiarities in local construction traditions and historical events. For this purpose, the dataset was divided into four subsets, which include towers from the Marche and Umbria regions, and towers from the Macerata and Perugia provinces, respectively. Other regions and provinces were not considered due to the limited sample sizes, which is not statistically significant. A second classification focused on the geometry. Some records pertain to inspections conducted at different times for the same churches (e.g., Abbazia SS. Rufino & Vitale, Cathedral of San Catervo or Church of Madonna delle Grazie). A comparison of the geometric and typological data registered in the survey forms from different inspections revealed discrepancies, which were corrected by checking the original sources and remotely inspecting the towers. When the data could not be reliably corrected, the affected towers were excluded from the following subsets based on geometry. A similar process was applied to instances where available geometrical data appeared unrealistic. This reflects the well-known issue of potential errors in data collection, transmission and digitisation during field post-earthquake investigations, likely due to the limited number and experience of technical personnel, affecting the CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 95 accuracy and reliability of the information (Baggio et al., 2007; Rossetto & Ioannou, 2018). It is possible that other cases within the dataset may require attention due to errors. However, a detailed evaluation of these cases is beyond the scope of this study. Four continuous independent variables were identified: three directly extracted from the Da.D.O., namely, the minimum base side (𝐿𝑚𝑖𝑛), the maximum base side (𝐿𝑚𝑎𝑥), and the total height (𝐻𝑡𝑜𝑡), and one indirectly computed as the ratio of the total height to the minimum side length at the base, referred to as aspect ratio (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛). A summary of their statistics is reported in Table 4.2. Additionally, the number of samples with available data is reported for each variable. This number, 396 out of the 794 towers in the entire dataset, highlights a certain lack of completeness in the forms. This is a common issue when many assets must be quickly inspected by a few operators within a very short timeframe, a situation often seen in post-disaster surveys (Rossetto et al., 2015; Rossetto & Ioannou, 2018) The variable distributions are illustrated in Figure 4.6. The distributions are right-skewed and unimodal. 𝐿𝑚𝑖𝑛 and 𝐿𝑚𝑎𝑥 present very similar statistics, except for the maximum values that for 𝐿𝑚𝑎𝑥 reach 14 m, likely due to outliers in the distribution, since most of the towers have square cross-section and both side lengths are often shorter than 5 m, mostly in the range 2-4 m. The height ranges from a minimum of 8 m to a maximum of 53 m, with a concentration of towers below 20 m high. Finally, the aspect ratio varies between 2 and 11, with most of the samples presenting a ratio smaller than 9 and a high concentration in the range 3-6. Besides the maximum range of variation of the distribution, Table 4.2 reports the 5 and 95 percentiles of the geometrical data. This range provides a better insight into the parameter variations, reducing the effect of few extreme values especially in the lower bound, characterised by particularly small-sized and stocky configurations. At the same time, due to the rightskewness, the 95 percentile limits significantly the upper bound. The statistics here reported are in good agreement with those computed for other datasets of existing masonry towers analysed within the present thesis (see Chapter 5, Section 5.3.2). Table 4.2: Main statistics of the geometrical data. Variable No. Samples μ m Min Max σ 5% 95% 𝑳𝒎𝒊𝒏 396 3.67 3.50 1.50 9.80 1.32 2.00 6.00 𝑳𝒎𝒂𝒙 396 3.82 3.50 1.50 14.00 1.46 2.00 6.00 𝑯𝒕𝒐𝒕 396 18.11 16.00 8.00 53.00 7.37 10.00 34.00 𝑯𝒕𝒐𝒕𝑳𝒎𝒊𝒏 ⁄ 396 5.16 5.00 1.82 11.00 1.74 3.00 8.52 μ = mean; m = median; σ = standard deviation; 5% = fifth percentile; 95% = ninety-fifth percentile. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 96 Figure 4.6: Distributions of the geometrical characteristics of the towers. Among the geometric data, the aspect ratio is an indirect parameter that significantly affects the seismic behaviour of towers. A higher aspect ratio tends to result in a cantilever-like response, characterised by larger lateral deformations at the top. Conversely, lower aspect ratio values lead to an overall shear behaviour, primarily involving the resisting masonry walls. Although the aspect ratio is crucial for describing the response of this building typology, the literature lacks studies defining thresholds that explain the differing structural behaviours. Notably, (Sepe et al., 2008) classifies the towers by proposing an aspect ratio threshold of 4. This threshold was adopted in the present study to define two subsets, which include 120 towers (15% of the dataset) with aspect ratio equal or smaller than 4, and 276 towers (35% of the dataset) with ratio larger than 4, respectively. It is worth noting that for about 50% of the samples included in the dataset the available information does not allow an estimation of the aspect ratio. A third classification focused on the plan configuration due to its significant impact on the seismic response of historic masonry towers. While towers often exhibit a regular arrangement in plan due to their vertical development, lateral interactions can influence their seismic behaviour, particularly due to adjacent buildings, causing sudden variations of stiffness along the height. Although it is impossible to account for all the possible real-world scenarios in plan configuration, the inspection form groups the towers based on the type of interactions in three main categories: isolated (no interaction), confined (partial interaction), and integrated (full interaction). Therefore, two subsets were created, which include 312 confined towers (39% of the dataset), and 233 integrated towers (29% of the dataset), respectively. Isolated towers are very rare (3% of the dataset, 20 towers), thus they cannot be considered for a statistical analysis. Likewise for the geometry, a significant lack of completeness of the form was CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 97 observed, with information about the plan configuration unavailable for 29% of the dataset, namely 229 towers. Finally, the towers were classified according to their state of maintenance. The inspection form provides a qualitative assessment of the general condition of the churches in the Da.D.O., using the following categories: Good, Acceptable, Poor, Very Poor, and Work in Progress. Similar to the previous features, the state of maintenance is an indirect parameter expected to influence the seismic response of historic masonry towers. Although the assessment in the form does not directly refer to this macroelement but to the whole church, hereafter it is assumed that the same evaluation is applicable to the tower. Therefore, three subsets were created, which include 419 towers in a good state of maintenance (53% of the dataset), 234 towers in acceptable state of maintenance (29% of the dataset), and 85 towers in poor condition (11% of the dataset), respectively. Towers in very poor conditions were not enough to form an independent subset for statistical analysis, comprising only 22 samples (3% of the dataset). Similarly, a very small percentage of towers presented work in progress (1% of the dataset, 7 towers). Thus, none of them were grouped into any subset. Information about the state of maintenance of the remaining towers was not available (4% of the dataset, 29 towers). 4.2.3 Critical aspects Several assets in the area under study were previously damaged by the destructive 1997 UmbriaMarche earthquake, being then subjected to repairs and strengthening (Carbonari et al., 2019). In many churches that suffered extensive damage, effective mitigation measures were implemented before the new events, including light ring beams, tie roads and other metal restraints, limiting the formation of various mechanisms. During the Central Italy sequence, these low-impact interventions led to the migration of damage to the discontinuities caused by the repairs or to other areas of the structures. Thus, damage emerged according to mechanisms for which no intended strengthening was implemented and likely for a higher level of seismic action (Parisi et al., 2018; Sferrazza Papa & Silva, 2018). This peculiar characteristic of the churches in the area and their macroelements, including the bell towers, should be properly considered in the evaluation of the vulnerability towards a generalisation of the results. Additionally, it is worth mentioning that studies conducted in the area suggested a relevant contribution of site effects to the damage level, with both stratigraphic (e.g., localised alluvial and softer deposits in areas otherwise featuring outcropping rock and shallow bedrock) and topographical characteristics (e.g., ridges and sharp hills) likely causing amplifications in some of the municipalities closer to the epicentres (e.g., Accumuli, Amatrice, Visso or Camerino) (Sextos et al., 2018). CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 104 Figure 4.12: DPMs as functions of MCS intensity for GT0. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 105 Figure 4.13: DPMs as functions of PGA for GT0. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 106 In both cases, the DPMs are well distributed, showing an agreement with the expected behaviour, as the size and characteristics of the dataset allows to overcome several issues emerged with the open database analysed in Chapter 3. Indeed, while for lower seismic intensity the damage distribution is expected to be right-skewed generally unimodal and showing peaks for lower damage extents, for higher seismic intensity the damage distribution is expected to be left-skewed and showing peaks for greater damage extents. These trends are respected with a few exceptions. In particular, a high number of undamaged towers can be observed for lower seismic intensities, whereas the number of damaged towers grows for greater seismic intensities. This is also reflected in the values of mean damage which become significant for increasing intensity values. However, for the maximum values of the intensity measures, the observed damage is distributed quite uniformly or normally across the different damage grades. Only in the case of mechanism B, a clear left-skewed distribution appears. This reflects the tendency, more evident when considering the PGA rather than the MCS intensity, for the damage distributions and global mean damage values to increase more significantly for mechanism B than for mechanism A, starting from certain seismic intensity thresholds (e.g., 0.1/0.15g in terms of PGA). These results suggest a higher vulnerability of the belfry compared to the shaft of the towers. Additionally, this trend highlights the better correlation between PGA and observed damage. A condensed form of the DPMs as functions of the PGA, for the individual mechanisms and the overall macroelement damage, is reported in Figure 4.14(a, b, c) repeating the results for GT0 and comparing them with GT1 and GT2. GT2 samples confirm the findings discussed for GT0 and, in particular, the clear shift from a right skewed distribution of the damage levels for small intensity to a left skewed for high intensity in the case of mechanism B. Mechanism A, instead, is characterised by a lower percentage of collapses, even at the largest intensity, and an unimodal distribution with peaks in D3, for GT0, or in D4, for GT1. The DPMs for GT1, on the other hand, present a different behaviour, similar for most bins of PGA, except for the highest range. It is worth noting that the distribution of damage level for GT1 in the range 0.2-0.3g is influenced by a reduced number of samples. These obtained results suggest that the greater severity of damage in GT2 was primarily due to the cumulative effects of multiple seismic events. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 107 (a) (b) (c) Figure 4.14: DPMs as functions of the PGA for different subsets: (a) GT0; (b) GT1; (c) GT2. To explore this hypothesis, two new subsets, namely GT3 and GT4 were defined. Both were drawn from towers surveyed after either the 4th or 5th main shock. GT3 consists of towers likely affected by a single event. To this end, it contains 349 samples where the second-highest PGA recorded at each site is less than or equal to 0.1g. As discussed earlier, this threshold is used to expect a mean damage level above D1, suggesting that towers in GT3 likely experienced at most one shock capable of causing at least moderate damage. On the other hand, GT4 includes 212 towers where the second-highest PGA exceeds 0.1g, thus, impacted by at least two seismic events potentially damaging. It is important to note that while GT0 is large, the number of samples becomes too small for a meaningful statistical analysis when attempting to identify towers that experienced first a larger event and then a weaker one CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 108 (mainshock-aftershock sequence). Therefore, towers in GT4 are collected regardless of the chronological order of the seismic inputs. (a) (b) Figure 4.15: Geographical distribution of the towers categorised by the damage grades assigned to individual mechanisms and overall macroelement: (a) GT3, towers affected by a single seismic event (second highest PGA ≤ 0.1 g ) ; (b) GT4, towers affected by multiple seismic events (second highest PGA > 0.1 g ). Figure 4.15 shows the spatial distribution of these two subsets, while Figure 4.16 illustrates the correlation of both damage grade and maximum PGA with the epicentral distance, including the mean CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 109 values (red diamonds in the graphs) for increasing distance bins of 10 km. As for GT1 and GT2, the correlation is presented for the overall macroelement damage only. (a) (b) (c) (d) Figure 4.16: Decay of seismic effects with the epicentral distance in terms of damage mechanisms (a, b) and maximum PGA registered at the tower site (c, d) for GT3 (a, c) and GT4 (b, d). The spatial distribution of the two subsets relative to the epicentres differs significantly. In particular, the towers in GT3 are located at more than 10 km from the epicentre of the event that caused the highest PGA with only a few closer than 25 km. Instead, they are spread over a very wide area, ranging from about 10 km to 100 km. Moreover, with few exceptions, the towers present smaller damage grades. The average damage consistently remains below D1, showing an almost constant trend up to 60 km, followed by even lower values. In contrast, the towers in GT4 are concentrated in a narrow area, mostly within 35-40 km of the epicentre of the event that caused the highest PGA. Only a few cases are located between 40 and 45 km, with none beyond this distance. This distribution reflects the sampling strategy and the trend of PGA values with increasing epicentral distance. While the average PGA for GT3 is around 0.1g or less, GT4 exhibits a wider range of PGA values. The average PGA per 10 km bin constantly decreases with distance, from a peak of approximately 0.4g within 10 km to about 0.2g at 40 km. Beyond this point, no towers are affected by more than one major event. These moderate to CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 110 high PGA values induce significant damage, with an average damage grade above D3 within 10 km from the epicentre and above D2 up to 30 km. A noticeable reduction in average damage occurs only beyond 30 km. Finally, the two highest values of the PGA at each tower location are correlated in Figure 4.17, and the samples are grouped based on the damage grades, through a colour key, considering only the towers featuring a second-highest PGA above 0.1g (GT4). (a) (b) (c) Figure 4.17: First and second-highest PGA correlations for both individual damage mechanisms (a, b) and overall macroelement (c) for GT4, together with the damage grades. This correlation shows the sequential order between the events that caused the first and second highest PGA, since, in the plot, x-axis refers to the first chronological event and y-axis to the second. The results show that most of the towers included in this subset are characterised by highest PGA in the second event, as, for the Central Italy event, shocks with higher intensity occurred in the middle of the seismic sequence. The classification of the chronological order does not provide any evidence of trends or clustering of the tower damage grades. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 111 4.3.2 Vulnerability functions The values of the global mean damage obtained from the analysis of the DPMs for increasing seismic intensities (in terms of MCS and PGA) are plotted as indicators of the seismic vulnerability and compared against a set of existing vulnerability models, including the formulation derived in Chapter 3, developed specifically for historic masonry bell towers (Figure 4.18). (a) (b) Figure 4.18: Observed mean damage, plotted with circles, compared with existing and novel calibrated (proposed) vulnerability functions for the entire dataset (GT0) as functions of: (a) MCS macroseismic intensity; (b) PGA. The vulnerability functions are identified through the publication where they have been proposed (Ceroni et al., 2022; Curti et al., 2008; Lagomarsino et al., 2004), and the model developed in Chapter 3 is indicated as in (Testa et al., 2024). For the MCS intensity measure, the observed damage is comparable to other events at lower seismic intensities. However, at higher seismic intensities, the bell towers exhibited a better response than expected, as shown by the comparison with the vulnerability CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 112 curves proposed by (Curti et al., 2008; Testa et al., 2024), which were calibrated against various seismic events. The model proposed by (Lagomarsino et al., 2004) for masonry towers and slender buildings, not specifically bell towers, here applied to the entire microelement, presents a better predictive performance in the higher intensity range. Nevertheless, a slight underestimation can be observed in the lower range, where, in any case, the mean damage level remains very low. For the PGA intensity measure, the models identified in literature present a rather high predictive performance when compared with the observed mean damage values. This result is however expected since these models have been calibrated over a similar dataset of towers affected by the same sequence, except for a different filtering and sampling strategy leading to the final investigated set. In the absence of a vulnerability curve for the entire macroelement as a function of the PGA, a recalibration of the parameters a’ and b’ of the formulation suggested by (Ceroni et al., 2022), Eq. (4.1), is proposed, by fitting the global mean damage-PGA pairs, as given in Table 4.3. 𝜇𝑑=2.5 [1+𝑡𝑎𝑛ℎ(𝑎′log(𝑃𝐺𝐴)+𝑏′)] Eq. (4.1) Table 4.3: Calibration of vulnerability function coefficients. Mechanism a’ b’ Reference Mechanism A 0.89 -2.54 (Ceroni et al., 2022) Mechanism B 0.89 -2.22 (Ceroni et al., 2022) Macroelement damage 1.20 0.56 Proposed recalibration At this stage, the available information does not allow for a deeper investigation into the causes of the better performance of the towers compared to other events. It is plausible that this result is influenced by the characteristics of the assets in the area, many of which were retrofitted and strengthened following the Umbria and Marche earthquake. Additionally, the characteristics of the seismic input, at least for some of the shocks, with peaks in a period range that likely did not affect the towers’ first modes may have played a role. However, more detailed research is needed to draw definitive conclusions. In order to provide a better insight into the distinct behaviour of the towers affected by a single main shock and multiple shocks, Figure 4.19 shows the mean damage of GT1 and GT2 plotted against increasing seismic intensity values, in terms of MCS and PGA. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 113 (a) (b) Figure 4.19: Observed mean damage compared with newly calibrated vulnerability models as functions of the MCS macroseismic intensity (a) and PGA (b) for GT1 and GT2. A recalibration was conducted to propose new formulations for historic masonry towers to predict the expected damage caused by two distinct scenarios: single shock and multiple shocks. The fit of these new formulations with the observational data for both earthquake scenarios is shown in Figure 4.19, with the calibration coefficients provided in Table 4.4 using the same form of the previous equation, Eq. (4.1), in terms of PGA, and, Eq. (2.8), in terms of MCS. It should be noted that the formulation coefficients are calibrated by fitting the global mean damage (according to the EMS98 scale)-MCS pairs, differently from the Eq. (2.8) which instead links the MCS to the global damage index (in 0-1 scale). The intersection of the vulnerability functions in the lower range of the intensity measures likely results from the larger uncertainties in the available data for these input values. This outcome contradicts the physical interpretation of the two curves, as it suggests that the average damage from cumulative events could be lower than from a single event. However, as demonstrated by the decay curves in CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 120 Figure 4.26: Fragility functions for GT1 derived using Approach 1. 4.4 VULNERABILITY ASSESSMENT: KEY FEATURES The effectiveness of the risk analysis models (i.e., DPMs, vulnerability and fragility functions), namely their reliability and usefulness, is constrained by the ability to identify homogeneous asset classes. Within these classes, distinguishing the attributes and features that influence the overall susceptibility to damage and the peculiar behaviour against the hazard is critical (Lagomarsino et al., 2021; Rossetto & Ioannou, 2018). This is essential to reduce the large scatter in the response of assets within the same class under the same intensity measure. While ordinary buildings may exhibit greater homogeneity, monuments often cannot be adequately represented by broad asset classes, as these fail to capture the intrinsic variability within each typology. Key attributes of a typology may be reflected through vulnerability corrective factors, which adjust the vulnerability score to improve predictions of mean damage and better represent the susceptibility of each asset to seismic hazards. Typical vulnerability modifiers discussed in the literature include factors such as maintenance, material quality, structural regularity, size and aspect ratio, interaction with adjacent structures, and the presence of retrofitting interventions (Lagomarsino, 2006; Lagomarsino et al., 2004). In order to better understand the seismic response of the towers, this Section investigates specific attributes and features described in the inspection forms, seeking to identify subsets with more homogeneous behaviour. However, the dataset’s available information limits the number and type of features that can be analysed. Consequently, several subsets of the entire GT0 were derived, grouping the towers based on the following attributes, among the ones described in Section 4.2.2, due to their relevance to towers as potential sources of vulnerability and/or hazard amplification: (i) maintenance; (ii) geometry; (iii) tower typology and interaction with surrounding buildings; and (iv) geographical location. It is important to note that the state of maintenance, in the inspection form, refers to the entire masonry church. However, here, it is assumed to apply to the associated towers and is therefore included in the analysis. Given the utility of PGA in developing predictive models and its strong CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 121 correlation with the tower damage grades, the following Sections focus exclusively on this intensity measure. The effectiveness of vulnerability and fragility functions was explored for both damage mechanisms A and B and entire macroelement. In particular, the existing and proposed vulnerability functions were reported against the mean damage evolution observed for the distinct subsets, together with the mean damage evolution for the entire dataset (GT0). The fragility functions were compared, highlighting, for the sake of clarity, the damage levels D3 and D4 only. These two damage levels, in fact, ensure the availability of a sufficient number of samples across the PGA range, providing consistency and significance. At the same time, they represent a range of damage in which the collapse mechanism is clearly activated, preventing any possible misinterpretation. Hereafter, only the results of the calibration through the approach 2 by (Porter, 2021) are discussed. Table 4.7 summarises the median and standard deviation estimated by maximising the likelihood function for the development of the lognormal cumulative distribution functions representative of the best fit to the observed probabilities. However, for the sake of completeness, the DPMs and the fragility functions developed for all considered subsets, damage levels and calibration approaches, namely traditional approach and optimised one (approach 2), are individually presented in the Appendix B of the present thesis. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 122 Table 4.7: Median and standard deviation for damage states D3 and D4 of the distinct subsets accounting for the different key features considered in this study. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 123 4.4.1 Influence of maintenance The influence of the maintenance condition of the towers at the time they were affected by the earthquake on their susceptibility of damage was investigated referring to three states of maintenance (Figure 4.27): good , acceptable , and poor. (a) (b) (c) Figure 4.27: Observed mean damage compared with the vulnerability functions accounting for the maintenance states; (a) mechanism A; (b) mechanism B; (c) overall macroelement. Comparing the overall distribution of the mean damage observed for good, acceptable and poor states,, despite their similarities, an emerging difference in behaviour can be detected. In particular, the mean damage for good state is distributed among lower damage levels (with a maximum level not exceeding D3). For poor state, the mean damage consistently reaches higher levels (exceeding D3 for a PGA of approximately 0.25g, but presenting a lower value for higher PGA). Meanwhile, the mean damage for acceptable state falls between good and poor ranges. The different damage distribution can be better highlighted by examining the DPMs (see Appendix B). The obtained results revealed that the state of CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 124 maintenance of the analysed towers can be considered an explanatory variable which can explain the variations characterising the towers with different behaviour under similar intensity of the seismic action. In this light, the mean damage predicted for increasing values of PGA through the existing and proposed vulnerability functions has a limited potential, as it cannot fully capture the actual damage distribution characterising the towers with different maintenance states. Therefore, the formulation coefficients were recalibrated, allowing to propose novel formulations to better explain the different behaviour accordingly. The recalibrated coefficients are summarised in Table 4.8. Table 4.8: Recalibration coefficients for the three distinct subsets based on the state of maintenance. a’ b’ Good state Mechanism A 1.26 0.31 Mechanism B 1.32 0.49 Macroelement damage 1.22 0.44 Acceptable state Mechanism A 0.83 0.20 Mechanism B 1.23 0.57 Macroelement damage 0.94 0.41 Poor state Mechanism A 1.00 0.52 Mechanism B 1.89 1.20 Macroelement damage 1.36 0.88 To further check the obtained results, an additional analysis involved the comparison of the predictive performance between the vulnerability functions proposed specifically for the towers that feature an acceptable state of maintenance with the existing and novel vulnerability functions for the entire dataset, allowing to check if the formulation for acceptable maintenance state is sufficiently close to the average behaviour of the entire set of towers. The results, illustrated in Figure 4.28(a), show a negligible difference in the predictive performance. Based on the obtained results, the predictive performance of the formulations proposed specifically for subsets of towers characterised by poor and good states of maintenance can be considered as upper and lower bounds respectively, as shown in Figure 4.28(b). CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 125 (a) (b) Figure 4.28: Observed mean damage compared with the vulnerability functions for the acceptable state of maintenance and for the entire set (a); Predictive performance of the formulations, together with upper and lower bounds provided by formulations proposed for the poor and good states of maintenance (b). When comparing the fragility curves obtained for towers with different maintenance (Figure 4.29), the results of the vulnerability functions are confirmed, as significant differences can be observed in the probabilities of exceeding the considered damage states (D3 and D4) for increasing values of PGA. Analysing D3, the probabilities of exceeding the damage states for good (dashed curves) and acceptable states (solid lines) show similar increasing trends, peaking at around 50% for a maximum PGA of approximately 0.4g. In contrast, for poor state (dotted curves), the probabilities follow an increasing trend as well but reach a significantly higher peak of about 70% for the same maximum PGA. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 126 Figure 4.29: Fragility curves of D3 and D4 exceedance fitted to good (diamond points), acceptable (circle points), and poor (squared points) maintenance states. To support the conclusions regarding the influence of the state of maintenance on the vulnerability, a possible bias in the results was checked to ensure that differences in damage levels across the three subsets were not influenced by the varying presence of towers affected by a single main shock or cumulative effects. The results are summarised in Table 4.9. Towers in good, acceptable, and poor maintenance states represent different proportions of the entire dataset (53%, 29%, and 11%, respectively), reflecting a significant difference in the number of towers. However, when analysing the towers affected by single or multiple shocks, the subsets of towers with distinct maintenance states show comparable distributions. Specifically, the proportions of towers inspected after first main shock are 27%, 29%, and 31%, respectively, while, for multiple shocks, they are 55%, 61%, and 63%, respectively. This confirms that including the state of maintenance provides a reliable indicator for better understanding the vulnerability behaviour of this particular building typology. Table 4.9: Statistics of the subsets considering the maintenance states based on type of damage. Key Feature State of Maintenance Subsets Good Acceptable Bad 419 (53%) 234 (29%) 85 (11%) First Event 115 (27%) 68 (29%) 26 (31%) Last Event 230 (55%) 143 (61%) 54 (63%)) 4.4.2 Influence of geometry and interactions The towers were grouped according to two additional parameters accounting for their geometry and interaction with surrounding structures. Regarding the former, the ratio of the total height with the minimum side length at the base, referred to as aspect ratio (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛), was considered to distinguish two classes: the first class characterised by towers with ratios lower than or equal to 4 and the second class by towers with ratios greater than 4. The impact of the aspect ratio on the mean CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 127 damage evolution is presented in Figure 4.30. It can be generally observed that the mean damage values observed for the first class (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛≤4) are very similar to those observed for the second class (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛>4). (a) (b) (c) Figure 4.30: Observed mean damage compared with the vulnerability functions accounting for the aspect ratio threshold; (a) mechanism A; (b) mechanism B; (c) overall macroelement. This is confirmed by the comparison of the fragility curves obtained for the different aspect ratios (Figure 4.31), where no significant differences in the probabilities of exceeding the considered damage states (D3 and D4) for increasing values of PGA can be observed. Based on the results, the aspect ratio of the analysed towers can be considered an explanatory variable which has relatively low influence on the damage data at hand. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 128 Figure 4.31: Fragility curves of D3 and D4 exceedance fitted to towers with Htot/Lmin ≤ 4 (diamond points) and towers with Htot/Lmin > 4 (circle points). The interaction between the towers and the surrounding buildings, instead, was investigated by analysing two typologies of bounded towers, namely confined and integrated. The influence of the tower typology on the mean damage evolution is given in Figure 4.32. As for the previous case, the mean damage observed for confined towers is very comparable with the mean damage observed for integrated towers. (a) (b) (c) Figure 4.32: Observed mean damage compared with the vulnerability functions accounting for the tower typology (confined and integrated); (a) mechanism A; (b) mechanism B; (c) overall macroelement. CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES 129 However, when comparing the fragility curves (Figure 4.33), slight differences in the probabilities of exceeding the considered damage states (D3 and D4) for increasing values of PGA can be observed. For confined (dashed curves) and for integrated (solid curves), the probabilities are very close to each other until reaching values of PGA above 0.15g for which the probabilities are larger for integrated than for confined. The functions tend to diverge more for damage mechanism B. Despite the slight difference in the fragility data, the tower typology is at this stage considered an irrelevant explanatory variable in the behavioural response of the dataset, but more research is needed to provide a better insight into the influence of the configuration on the seismic behaviour. Figure 4.33: Fragility curves of D3 and D4 exceedance fitted to confined (diamond points) and integrated (circle points). 4.4.3 Influence of location Finally, two explanatory variables, namely the region and the province, were identified to account for the location feature. The effect of the location on the mean damage evolution is reported in Figure 4.34. The mean damage observed for Marche region is higher than the mean damage observed in Umbria region (Figure 4.34a). Similarly, the mean damage observed for Macerata province is greater than the mean damage observed in Perugia province (Figure 4.34b). The comparable trends can be justified by the similar statistics between the considered region datasets and the provinces that constitute their main subsets. The results suggest higher vulnerability of towers belonging to the Macerata province and Marche region in general, irrespective of the individual damage mechanism and macroelement, over the entire range of PGA. This can be better highlighted by examining the different DPMs developed for the different regions and provinces (see Appendix B). While the towers in Macerata province and Marche region present mean damage values, for both mechanism A and B, only slightly larger than the entire dataset (GT0), the towers in Perugia province and Umbria region exhibit a significantly better behaviour until 0.25-0.30g.