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

Uni e sidade do Minho
Escola de Engenha ia
F ancesco Tes a
De elopmen and Valida ion
o Empi ical Seismic Vulne abili y Models
o His o ic Mason y Towe s
Ma ch 2025
UMinho | 2025 F ancesco Tes a De elopmen and Valida ion o Empi ical Seismic
Vulne abili y Models o His o ic Mason y Towe s
F ancesco Tes a
De elopmen and Valida ion
o Empi ical Seismic Vulne abili y Models
o His o ic Mason y Towe s
Doc o al Thesis
Ci il Enginee ing
Wo k conduc ed unde he supe ision o
P o esso Paulo José B andão Ba bosa Lou enço
Doc o Albe o Ba on ini
Uni e sidade do Minho
Escola de Engenha ia
Ma ch 2025
Copy igh and Te ms o Use o Thi d Pa ies
This academic wo k is made a ailable unde he ollowing e ms o use, in acco dance wi h in e na ionally
ecognised s anda ds and bes p ac ices ega ding copy igh and ela ed igh s.
This wo k may be eely used, sha ed, and adap ed by hi d pa ies unde he condi ions speci ied in he
license below. I in ended use alls ou side he scope o his license, p io w i en pe mission mus be ob ained
om he au ho ia he Uni e si y o Minho’s Ins i u ional Reposi o y (Reposi ó iUM).
License
h ps://c ea i ecommons.o g/licenses/by/4.0/
Acknowledgemen s
The p esen hesis,
De elopmen and Valida ion o Empi ical Seismic Vulne abili y Models o His o ic Mason y
Towe s
, has been made as pa o my Ph.D. p og amme om Janua y 2021 o Ma ch 2025 a he Ins i u e
o Sus ainabili y and Inno a ion in S uc u al Enginee ing (ISISE), Depa men o Ci il Enginee ing, Uni e si y
o Minho, Po ugal.
The hesis has been suppo ed by he Po uguese go e nmen al agency, Founda ion o Science and
Technology (FCT), whose inancial con ibu ion is g a e ully acknowledged.
I would like o exp ess my since e g a i ude o my supe iso , P o . Paulo B. Lou enço, o gi ing me his
oppo uni y, guiding me, p o iding he necessa y ools, and sha ing knowledge, ad ice and sugges ions
h oughou my Ph.D. s udies. I am deeply g a e ul o my supe iso , D . Albe o Ba on ini, o p o iding
suppo , indispensable assis ance and ca e ul eading, as well as o engaging in insigh ul con e sa ions and
sha ing p ecious commen s on he esea ch.
Fu he mo e, I would like o hank D . Nicola Chie o, om he Depa men o Ci il Enginee ing a he
Uni e si y o Hudde s ield, o he collabo a ion in he esea ch p esen ed in Chap e 3. I am e y hank ul o
P o . Giuseppe B ando, D . Ma ia Gio anna Mascio a and D . Gio gia Cianchino, om he Depa men o Ci il
Enginee ing a he Uni e si y Chie i-Pesca a, o sha ing pos -ea hquake su ey da a, ins umen al in he
esea ch in ol ed in Chap e s 4 and 5, and pa icipa ing in ui ul discussions.
Las , bu no leas , I would like o hank mom and dad o always being he e when I needed hem mos , my
sis e , Pina, o being my sou ce o inspi a ion and Luisa o he endless lo e, pa ience and encou agemen .
Guima ães,
Ma ch 2025

S a emen o In eg i y
I he eby decla e ha ing conduc ed his academic wo k wi h in eg i y. I con i m ha I ha e no used plagia ism
o any o m o undue use o in o ma ion o alsi ica ion o esul s along he p ocess leading o i s elabo a ion. I
u he decla e ha I ha e ully acknowledged he Code o E hical Conduc o he Uni e si y o Minho.
Resumo
Es a ese em como obje i o a alia a ulne abilidade sísmica de o es his ó icas em al ena ia e
p e ende desen ol e mé odos empí icos com base em dados ecolhidos após sismos. Embo a exis am
modelos empí icos bem es abelecidos na li e a u a pa a a alia a ulne abilidade sísmica de es u u as,
poucos es udos abo dam sis ema icamen e as o es his ó icas em al ena ia, sendo ainda mais
limi ados aqueles que se ocam na sua alidação e na análise compa a i a. Assim, o p incipal obje i o
é analisa , alida e melho a os modelos de ulne abilidade exis en es, com especial oco na a aliação
do compo amen o dis in o e in e ligado dos á ios componen es, o us e e o campaná io, que
cons i uem as o es. São explo ados dois conjun os de dados dis in os. O p imei o esul a de uma
e isão bibliog á ica ex ensa, eunindo dados de danos causados po sismos an e io es em I ália,
compilados numa base de dados em acesso abe o. O segundo abo da algumas limi ações do conjun o
an e io , menos ab angen e, a a és da ecolha e il agem de in o mações ela i as à sequência
sísmica da I ália Cen al en e 2016-2017, disponí el na pla a o ma Da.D.O., Base de Dados de Danos
Obse ados, disponibilizada pelo Depa amen o de P o eção Ci il I aliano. Es e es udo analisa o
desempenho de á ias medidas de in ensidade sísmica, incluindo a in ensidade mac o-sísmica Me calli-
Cancani-Siebe g (MCS), a Acele ação Máxima do Solo (PGA) e a Acele ação Espec al (SA), de o ma a
melho a a p ecisão das p e isões de dano. A es ima i a da SA eco e a modelos de mo imen o do
solo exis en es, jun amen e com no as o mulações empí icas desen ol idas pa a es ima o pe íodo
undamen al das o es. Além disso, a in es igação conduzida analisa como a ulne abilidade e o ní el
de dano são in luenciados pelas ca ac e ís icas da ação sísmica, ais como e en os p incipais simples
ou múl iplos, e a o es elacionados com as o es, ais como geome ia, in e ação com es u u as
en ol en es, o ní el de manu enção e a localização geog á ica. A in es igação in eg a me odologias
como Ma izes de P obabilidade de Dano (DPMs), unções de ulne abilidade e de agilidade, e
in oduz no os modelos que conside am di e en es classes de o es e cená ios sísmicos, com especial
ên ase nos e ei os de danos cumula i os.
Pala as-cha e: A aliação empí ica de ulne abilidade; Dados pós-desas es; Danos cumula i os;
Pa imónio cons uído; To es em al ena ia.
Abs ac
This hesis aims o assess he seismic ulne abili y o his o ic mason y owe s and seeks o ad ance
cu en empi ical me hods de i ed om pos -ea hquake su ey da a. Al hough empi ical models o
e alua ing he seismic ulne abili y o s uc u es a e well-es ablished in he li e a u e, ew s udies
sys ema ically add ess his o ic mason y owe s, and e en ewe ocus on hei alida ion and
compa a i e analysis. The e o e, he main objec i e o he hesis is o analyse, alida e and enhance
exis ing ulne abili y models, wi h a pa icula ocus on assessing he dis inc and in e connec ed
beha iou o owe ’s sha and bel y componen s. Two dis inc da ase s a e exploi ed. The i s is
de i ed om an ex ensi e li e a u e e iew, ga he ing damage da a om pas ea hquakes in I aly wi hin
a da ase sha ed in open access. The second add esses some limi a ions o he p e ious, less
comp ehensi e, by collec ing and il e ing in o ma ion abou he 2016-2017 Cen al I aly seismic
sequence, hos ed in he Da.D.O., Da abase o Obse ed Damage, pla o m eleased by he I alian
Depa men o Ci il P o ec ion. This s udy examines he ole o a ious seismic in ensi y measu es,
including Me calli-Cancani-Siebe g (MCS) mac oseismic in ensi y, Peak G ound Accele a ion (PGA), and
Spec al Accele a ion (SA), in imp o ing he accu acy o damage p edic ions. SA es ima ion eso s o
exis ing g ound mo ion models alongside newly de eloped empi ical o mula ions o p edic he owe s’
undamen al pe iod. Addi ionally, his esea ch in es iga es how ulne abili y and damage le els a e
a ec ed by key ea u es o seismic ac ions, such as he occu ence o single o mul iple main shocks,
and ac o s ela ed o he owe s, such as geome y, in e ac ion wi h su ounding s uc u es,
main enance le el, and geog aphical loca ion. To his end, he esea ch in eg a es me hodologies such
as Damage P obabili y Ma ices (DPMs), ulne abili y and agili y unc ions and in oduce no el models
ha accoun o di e en owe classes and seismic scena ios, wi h pa icula emphasis on cumula i e
damage e ec s.
Keywo ds: Empi ical ulne abili y assessmen ; Pos -disas e da a; Cumula i e damage; Buil he i age;
Mason y owe s.
I
Con en s
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 P obabili y Ma ices (DPMs) 25
2.3.2 Vulne abili y unc ions 27
2.3.3 Vulne abili y index me hods 32
2.3.4 F agili y unc ions 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 En i e da ase and indi idual e en s 62
3.3.2 Typology o owe and indi idual in ensi y g oups 69
3.4 VULNERABILITY FUNCTIONS 73
3.5 FRAGILITY FUNCTIONS 82
3.6 CONCLUSIONS 84
Con en s
VIII
Figu e 4.32: Obse ed mean damage compa ed wi h he ulne abili y unc ions accoun ing o he
owe ypology (con ined and in eg a ed); (a) mechanism A; (b) mechanism B; (c) o e all mac oelemen .
...................................................................................................................................................... 128
Figu e 4.33: F agili y cu es o D3 and D4 exceedance i ed o con ined (diamond poin s) and
in eg a ed (ci cle poin s). ................................................................................................................ 129
Figu e 4.34: Obse ed mean damage compa ed wi h he ulne abili y unc ions: (a) egional-based
analysis and (b) p o incial-based analysis. ....................................................................................... 130
Figu e 4.35: F agili y cu es o D3 and D4 exceedance i ed o owe s in Ma che (diamond poin s) and
Umb ia (ci cle poin s). .................................................................................................................... 131
Figu e 4.36: F agili y cu es o D3 and D4 exceedance i ed o owe s in Mace a a (diamond poin s)
and Pe ugia p o ince (ci cle poin s). ............................................................................................... 131
Figu e 5.1: Flowcha o he me hodology adop ed in his Chap e . .................................................. 135
Figu e 5.2: Compa ison o GMMs p edic ions wi h Da.D.O. alues o he M6.2 Augus 24, 2016
seismic shock. ................................................................................................................................ 143
Figu e 5.3: Compa ison o GMMs p edic ions wi h Da.D.O. alues o he M5.5 Oc obe 26, 2016
seismic shock. ................................................................................................................................ 143
Figu e 5.4: Compa ison o GMMs p edic ions wi h Da.D.O. alues o he M6.1 Oc obe 26, 2016
seismic shock. ................................................................................................................................ 144
Figu e 5.5: Compa ison o GMMs p edic ions wi h Da.D.O. alues o he M6.6 Oc obe 30, 2016
seismic shock. ................................................................................................................................ 144
Figu e 5.6: Compa ison o GMMs p edic ions wi h Da.D.O. alues o he M5.7 Janua y 21, 2017
seismic shock. ................................................................................................................................ 144
Figu e 5.7: Co ela ions be ween he highes PGA p io o inspec ion and he co esponding epicen al
dis ance. Da.D.O. alues (da k blue ci cles) and GMM p edic ion (ligh blue ci cles) using: (a) SP96; (b)
ITA08. ............................................................................................................................................ 145
Figu e 5.8: Co ela ion be ween he highes PGA p io o inspec ion compu ed based on Da.D.O. alues
and he highes PGA p edic ed using: (a) SP96; (b) ITA08. .............................................................. 146
Figu e 5.9: Geome ic ea u es: (a) plan iew, hickness (
s
), maximum side leng h (
Lmax
) and minimum
side leng h (
Lmin
); (b) ele a ion, o al heigh (
H o
), e ec i e heigh (
He
). ...................................... 149
Figu e 5.10: Fundamen al equency, o al heigh and aspec a io co ela ions o he aining se . ... 151
Figu e 5.11: Co ela ions be ween i s na u al equency and geome ic cha ac e is ics o he aining.
...................................................................................................................................................... 151

Con en s
IX
Figu e 5.12: His og ams o he occu ence o he con inuous a iables o he en i e da abase (whi e
colou ) and o he aining se (blue colou ). .................................................................................... 152
Figu e 5.13: Dis ibu ion o he samples acco ding o he plan con igu a ion, mason y ma e ial and
c oss-sec ion shape: (a) aining se ; (b) alida ion se . .................................................................... 154
Figu e 5.14: P edic ion o he i s na u al equency o he en i e aining se : (a) models and aining
samples; (b) models, aining and alida ion samples. ..................................................................... 156
Figu e 5.15: His og ams o he occu ence o he undamen al pe iod o he analysed owe s. ........ 158
Figu e 5.16: Co ela ions o he owe heigh s wi h he undamen al pe iod (a), and wi h he
undamen al equency (b). ............................................................................................................. 158
Figu e 5.17: Spec al accele a ions o he analysed owe s as unc ion o he undamen al pe iod (a)
and epicen al dis ance (b). ............................................................................................................. 159
Figu e 5.18: Spec al accele a ions, epicen al dis ances, and undamen al pe iod co ela ions. ....... 159
Figu e 5.19: Compa ison o he spec al accele a ions wi h he esponse spec um (sou ce: INGV). . 160
Figu e 5.20: Decay o seismic e ec s wi h he epicen al dis ance (GT0): (a) damage g ades o
Mechanism A; (b) damage g ades o Mechanism B; (c) damage g ades o o e all mac oelemen ; (d)
maximum SA es ima ed a he owe si e. Sca e plo (blue poin s) and mean alues o 10 km bins
( ed diamonds). .............................................................................................................................. 162
Figu e 5.21: Decay o maximum SA wi h epicen al dis ance: (a) a e he i s e en (GT1), and (b) a e
he las e en (GT2). Sca e plo (blue poin s) and mean alues o 10 km bins ( ed diamonds). ...... 163
Figu e 5.22: DPMs as unc ions o SA. ............................................................................................ 164
Figu e 5.23: DPMs o he en i e se using wo dis inc in ensi y measu es: (a) SA and (b) PGA ........ 165
Figu e 5.24: PGA-SA co ela ions o bo h indi idual damage mechanisms (a, b) and o e all
mac oelemen (c). Damage g ades highligh ed h ough he colou scale. ......................................... 166
Figu e 5.25: Calib a ion using Eq. (5.15), (5.16), (5.17), o i he da a. .......................................... 167
Figu e 5.26: Obse ed mean damage, plo ed wi h ci cles, compa ed wi h he de eloped ulne abili y
unc ions o he en i e da ase (GT0) as unc ions o SA. ................................................................. 168
Figu e 5.27: Obse ed mean damage as unc ions o he SA: (a) GT1 (b) GT2. ................................ 171
Figu e 5.28: F agili y unc ions o GT0 using SA. ............................................................................ 172
Figu e 5.29: F agili y unc ions o GT0 using PGA. .......................................................................... 173
Figu e B.1: DPMs as unc ions o he PGA: (a) Good; (b) Accep able; (c) Poo . ................................. 191
Figu e B.2: F agili y unc ions o owe s wi h good, accep able and poo main enance s a es: (a, c, and
e) adi ional app oach; (b, d and ) op imised app oach. ................................................................. 192
Con en s
X
Figu e B.3: DPMs as unc ions o he PGA: (a)
H o /Lmin
≤ 4 ; (b)
H o /Lmin
> 4 . ......................... 194
Figu e B.4: F agili y unc ions o owe s cha ac e ised by aspec a io equal o lowe han 4: (a)
adi ional app oach; (b) op imised app oach. ................................................................................. 194
Figu e B.5: F agili y unc ions o owe s cha ac e ised by aspec a io g ea e han 4: (a) adi ional
app oach; (b) op imised app oach................................................................................................... 195
Figu e B.6: DPMs as unc ions o he PGA: (a) Con ined; (b) In eg a ed. ........................................... 195
Figu e B.7: F agili y unc ions o con ined owe s: (a) adi ional app oach; (b) op imised app oach. 196
Figu e B.8: F agili y unc ions o in eg a ed owe s: (a) adi ional app oach; (b) op imised app oach.
...................................................................................................................................................... 196
Figu e B.9: DPMs as unc ions o he PGA: (a) Ma che; (b) Umb ia; (c) Mace a a; and (d) Pe ugia. .. 198
Figu e B.10: F agili y unc ions o owe s loca ed in Ma che and in Umb ia: (a and c) adi ional
app oach; (b and d) op imised app oach. ........................................................................................ 199
Figu e B.11: F agili y unc ions o owe s loca ed in Mace a a and in Pe ugia: (a and c) adi ional
app oach; (b and d) op imised app oach. ........................................................................................ 200
Figu e C.1: Flowcha me hodology o he de elopmen and alida ion o empi ical o mula ion o
p edic ing he i s na u al equency o his o ic mason y owe s. ..................................................... 203
Figu e C.2: P edic ion o he i s na u al equency: (a) isola ed; (b) bounded owe s. ...................... 205
Figu e C.3: P edic ion o he i s na u al equency o b ick, s one and mixed his o ic owe s acco ding
o hei bounda y condi ion: (a) isola ed; (b) bounded. ..................................................................... 205
Figu e C.4: (a) Compa ison be ween he p edic ion based on he o al heigh and e ec i e heigh o
bounded owe s; (b) P edic ion o he i s na u al equency o b ick, s one and mixed bounded owe s.
...................................................................................................................................................... 205
Figu e C.5: P edic ion o he i s na u al equency based on he aspec a io (a) and on he o al heigh
and minimum side leng h (b) o he en i e aining se . ................................................................... 207
XI
Lis o Tables
Table 2.1: Typical ime able o echnical ac i i ies adop ed a e seismic e en s. ................................ 17
Table 2.2: Co ela ion be ween he global damage index and he global damage le el. ....................... 24
Table 2.3: A e age alues o
V0
and
Q
o be used in Eq. (2.6) o dis inc building ypologies. ............ 28
Table 2.4: Vulne abili y modi ie pa ame e s. .................................................................................... 30
Table 2.5: Co ec i e coe icien s p oposed o he sha and bel y mechanisms. .............................. 30
Table 2.6: Coe icien s o Eq. (2.8)-(2.9) p oposed o he sha and bel y mechanisms. .................... 32
Table 2.7: Exis ing lis o pa ame e s p oposed o mason y buildings (Me 1) and o his o ic mason y
owe s (Me 2), oge he wi h sco es and impo ance weigh s. ............................................................ 34
Table 3.1: Ea hquake e en s conside ed in he p esen s udy. .......................................................... 54
Table 3.2: Maximum MCS in ensi y associa ed o each ea hquake e en and ange o in ensi y a he
loca ion o he in es iga ed owe s. .................................................................................................... 55
Table 3.3: Numbe o owe s included in he da abase o which a damage sco e has been assigned,
g ouped by ea hquake e en , damage mechanism and mac oseismic in ensi y. ................................ 60
Table 3.4: Numbe o owe s included in he da abase. ..................................................................... 62
Table 3.5: Compa ison o he mean damage ound wi h li e a u e alues o mason y owe s. ........... 67
Table 3.6: DPMs o owe s as en i e mac oelemen s (Mac oelemen damage). ................................. 67
Table 3.7: DPMs o owe damage mechanism (Mechanism A). ....................................................... 68
Table 3.8: DPMs o bel y damage mechanism (Mechanism B). ....................................................... 69
Table 3.9 DPMs o he owe mechanism (Mechanism A), g ouped by mac oseismic MCS in ensi y .. 73
Table 3.10 DPMs o he bel y mechanism (Mechanism B), g ouped by mac oseismic MCS in ensi y 73
Table 3.11 Vulne abili y and duc ili y indexes o his o ic mason y owe s .......................................... 74
Table 3.12 Cons ain s adop ed in he analysis ................................................................................. 77
Table 4.1: Gene al in o ma ion abou he seismic sequence. ............................................................. 89
Table 4.2: Main s a is ics o he geome ical da a. ............................................................................. 95
Table 4.3: Calib a ion o ulne abili y unc ion coe icien s. .............................................................. 112
Con en s
XII
Table 4.4: Recalib a ion coe icien s o he wo dis inc subse s and o he seismic in ensi y measu es.
...................................................................................................................................................... 114
Table 4.5: Median and s anda d de ia ion o he dis inc damage g ades. ....................................... 116
Table 4.6: Cumula i e logno mal dis ibu ion pa ame e s o he agili y unc ions. .......................... 118
Table 4.7: Median and s anda d de ia ion o damage s a es D3 and D4 o he dis inc subse s
accoun ing o he di e en key ea u es conside ed in his s udy. .................................................... 122
Table 4.8: Recalib a ion coe icien s o he h ee dis inc subse s based on he s a e o main enance.
...................................................................................................................................................... 124
Table 4.9: S a is ics o he subse s conside ing he main enance s a es based on ype o damage. ... 126
Table 4.10: S a is ics o he subse s conside ing he loca ion based on he ype o damage. ............ 132
Table 5.1: SP96 a enua ion law coe icien s o he ho izon al maximum PGA (
g
). ........................... 137
Table 5.2: SP96 a enua ion law coe icien s o he ho izon al maximum 5% damped PSV (
cm/sec
) o
di e en undamen al pe iods (na u al equencies) o he in es iga ed s uc u es. ........................... 137
Table 5.3: ITA08 a enua ion law coe icien s o he ho izon al maximum PGA (
cm/sec2
). .............. 138
Table 5.4: ITA08 a enua ion law coe icien s o he ho izon al maximum 5% damped SA (
cm/sec2
) o
di e en undamen al pe iods (na u al equencies) o he in es iga ed s uc u es. ........................... 139
Table 5.5: ITA10 a enua ion law coe icien s o he ho izon al maximum PGA (
cm/sec2
). .............. 141
Table 5.6: ITA10 a enua ion law coe icien s o he ho izon al maximum 5% damped SA) (
cm/sec2
)
...................................................................................................................................................... 141
Table 5.7: Summa y o he cha ac e is ics o he GMMs conside ed in his s udy. ............................ 142
Table 5.8: Summa y o he R-squa ed alues. ................................................................................. 145
Table 5.9: S a is ics o he en i e mason y owe s da abase, he aining se and he alida ion se . .. 150
Table 5.10: Equa ions based on he o al heigh : exis ing and new o mula ions and hei pe o mance.
...................................................................................................................................................... 155
Table 5.11: Seismic s a ions and g ound mo ion ime his o ies cha ac e is ics (sou ce: INGV). ......... 160
Table 5.12: Co ela ion coe icien s be ween he in ensi y measu es and he damage le els. ............ 167
Table 5.13: Calib a ion o ulne abili y unc ion coe icien s and pe o mance me ics esul s. .......... 167
Table 5.14: Calib a ion o ulne abili y unc ion coe icien s. ............................................................ 168
Table 5.15: Pe o mance me ics o he ulne abili y models using SA and PGA. .............................. 170
Table 5.16: Recalib a ion coe icien s o he wo dis inc subse s and o he seismic in ensi y measu es
...................................................................................................................................................... 171
Table 5.17: Median and s anda d de ia ion o he dis inc damage g ades, SA-based unc ions. ...... 172
Con en s
XIII
Table 5.18: Median and s anda d de ia ion o he dis inc damage g ades, PGA-based unc ions. ... 173
Table 5.19: AIC alues o he agili y unc ions de i ed using he SA and he PGA. ......................... 174
Table A.1: A no el open da abase o s uc u al damage obse ed in his o ic mason y owe s .......... 184
Table B.1: Median and s anda d de ia ion o damage s a es ( om D1 o D5) o he dis inc subse s
accoun ing o he s a e o main enance. ......................................................................................... 193
Table B.2: Median and s anda d de ia ion o damage s a es ( om D1 o D5) o he dis inc subse s
accoun ing o he geome y and in e ac ions. ................................................................................. 197
Table B.3: Median and s anda d de ia ion o damage s a es ( om D1 o D5) o he dis inc subse s
accoun ing o he loca ion ( egion and p o ince). ............................................................................ 201
Table C.1: Summa y o he exis ing and no el empi ical o mula ions p oposed o p edic ing he
equency o his o ic mason y owe s. ............................................................................................. 203
Table C.2: Summa y o single independen pa ame e s o mulas, acco ding o Eq. (C.1), Eq. (C.2) and
Eq. (C.3). ....................................................................................................................................... 204
Table C.3: Summa y o wo independen pa ame e s o mulas, acco ding o Eq. (C.4) and Eq. (C.5).
...................................................................................................................................................... 206
Table C.4: Summa y o h ee independen pa ame e s o mulas, acco ding o Eq. (C.6) and Eq. (C.7).
...................................................................................................................................................... 207
Table C.5: Summa y o h ee independen pa ame e s o mulas, acco ding o Eq. (C.8) and Eq. (C.9).
...................................................................................................................................................... 208
Table C.6 Summa y o ou independen pa ame e o mulas, acco ding o Eq. (C.10) and Eq. (C.11).
...................................................................................................................................................... 209

XIV
Lis o Abb e ia ions/Ac onyms/Symbols
GEM Global Ea hquake Da abase
NSPP Na ional Seismic P e en ion Plan
INGV Na ional Ins i u e o Geophysics and Volcanology
DPMs Damage P obabili y Ma ices
SA Spec al Accele a ion
MCS Me calli-Cancani-Siebe g
EMS-98 Eu opean Mac oseismic Scale
Da.D.O. Da abase o Obse ed Damage
PGA Peak G ound Accele a ion
GMM G ound Mo ion Model
MM Modi ied Me calli
MSK Med eded-Sponheue -Ka nik
JMA Japan Me eo ological Agency
ITACA I alian Accele ome ic A chi e
GNDT Na ional G oup o Ea hquake De ence
PGV Peak G ound Veloci y
PGD Peak G ound Displacemen
SV Spec al Veloci y
PSV Pseudo Spec al Veloci y
SD Spec al Displacemen
BDPF Binomial Densi y P obabili y Func ion
LSE Leas Squa e Es ima ion
MLE Maximum Likelihood Es ima ion
ULS Ul ima e Limi S a e
DBMI15 I alian Mac oseismic Da abase
EC8 Eu ocode 8
Con en s
XV
RSS Residual Sum o he Squa e
𝑅2 Coe icien o De e mina ion
MSE Mean Squa ed E o
PCC Pea son Co ela ion Coe icien
AIC Aikake In o ma ion C i e ion
𝐻 Haza d
𝑉 Vulne abili y
𝐸 Exposu e
𝑑 Indi idual Damage Le el
𝜌 Impo ance Coe icien
𝑖𝑑 Global Damage Index
𝐷 Global Damage Le el
𝜇 Global Mean Damage Le el
𝑉𝐼 Vulne abili y Index
𝑄 Duc ili y Index
𝑉0 A e age ulne abili y Index
𝑉j Co ec i e ulne abili y Pa ame e s
𝐼𝑣 Global Vulne abili y Sco e
𝑞𝑖 Vulne abili y Sco e Pa ame e s
𝑤𝑖 Weigh Pa ame e s
𝑝 Damage P obabili y
𝜙 S anda d No mal Cumula i e Dis ibu ion Func ion
𝜃 Median Es ima e Pa ame e
𝛽 S anda d De ia ion Es ima e Pa ame e
𝑀 Magni ude
𝑅 Si e- o-sou ce Dis ance
𝑀𝑤 Momen Magni ude
𝐻𝑡𝑜𝑡 To al Heigh
𝐻𝑒𝑓𝑓 E ec i e Heigh
𝐿𝑚𝑖𝑛 Minimum Side Leng h
𝐿𝑚𝑎𝑥 Maximum Side Leng h
Con en s
XVI
𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛 Aspec Ra io
𝑠 Base Wall Thickness
𝑓1 Fi s Na u al F equency
𝑇1 Fundamen al Vib a ion Pe iod
1
CHAPTER 1
INTRODUCTION
1.1 MOTIVATIONS
Ea hquakes a e na u al haza ds ha can ha e ca as ophic e ec s on human li es, buil en i onmen ,
in as uc u e, economies and ecosys ems, hinde ing he de elopmen o he a ec ed egions. Se e e
e en s, such as he 2023 Mo occo ea hquake (Mw=6.9, esul ing in 2960 dea hs) and he 2023 Sy ia-
Tu key ea hquake (Mw=7.7, wi h o e 60000 dea hs), se e as ecen eminde s o he des uc i e
powe o seismic ac i i y wo ldwide. In he Eu opean con ex (Figu e 1.1), I aly s ands ou due o i s high
seismici y, which is he esul o i s geog aphical loca ion along he bounda y o he Eu asian and
A ican ec onic pla es. This geo ec onic se ing has led o se e al signi ican seismic e en s h oughou
i s his o y, including he 1976 F iuli ea hquake (Mw=6.5 and 978 a ali ies), he 1980 I pinia
ea hquake (Mw=6.9 and 4689 a ali ies), he 1997 Umb ia-Ma che ea hquake (Mw=6.0 and 14
a ali ies), he 2002 Molise ea hquake (Mw=5.7 and 32 a ali ies), he 2009 L’Aquila ea hquake
(Mw=6.3 and 295 a ali ies), he 2012 Emilia ea hquake (Mw=5.9 and 24 a ali ies) and he 2016
Cen al I aly ea hquakes (Mw=6.2 and 327 a ali ies).
Figu e 1.1: Eu opean a ali ies om ea hquakes o e he las wen y yea s (Sou ce: The In e na ional Disas e
Da abase, EM-DAT (Del o ge e al., 2023) - wi h majo p ocessing by Ou Wo ld in Da a).
CHAPTER 1. INTRODUCTION
8
 Chap e 1 p o ides an o e iew o he mo i a ions and objec i es o his esea ch, ou lining he
signi icance o assessing he seismic ulne abili y o his o ic mason y owe s and he goals o
he s udy.
 Chap e 2 p o ides an in oduc ion o seismic isk, including a e iew o s a e-o - he-a
me hodologies o assessing seismic ulne abili y, wi h a speci ic ocus on his o ic mason y
owe s. I o e s a comp ehensi e o e iew o he me hodologies commonly employed in la ge-
scale assessmen s, as applied in his esea ch. Addi ionally, his Chap e discusses he
p inciples ou lined in he I alian Guidelines o seismic sa e y assessmen and mi iga ion o
he i age s uc u es, pa icula ly wi h ega d o owe s.
 Chap e 3 compiles damage da a om an ex ensi e li e a u e e iew on I alian mason y bell
owe s a ec ed by pas seismic e en s, c ea ing an open-access da abase. This da abase is
used as a es bed o assess exis ing e i o ial scale ulne abili y models. The da a is ini ially
analysed h ough DPMs, wi h a compa ison o analogous DPMs a ailable in he li e a u e. The
obse ed mean damage is hen compu ed o a ious mac oseismic in ensi ies, allowing an
e alua ion o he p edic i e capabili ies o exis ing ulne abili y unc ions. De elopmen o new
ulne abili y unc ions is pe o med, and c oss- alida ion s a egies a e employed o assess he
e ec s o sample size on he eliabili y and he p edic i e pe o mance o hese models. Finally,
agili y unc ions a e de i ed using cumula i e p obabili y ep esen ed by he binomial
dis ibu ion, and hei alidi y is assessed by compa ing hem o obse ed p obabili ies.
 Chap e 4 analyses damage da a om his o ic mason y owe s ollowing he 2016-2017
Cen al I aly seismic sequence. The da a, sou ced om he Da.D.O. pla o m (Dolce e al.
2019), a e used o de elop ulne abili y models, including DPMs, ulne abili y unc ions, and
agili y unc ions, wi h PGA employed as he seismic in ensi y measu e. The models a e ini ially
de eloped and compa ed unde di e en scena ios ( owe s a ec ed by a single main shock o
mul iple shocks o he sequence) o in es iga e he impac o cumula i e damage. Key owe
ea u es, such as loca ion, geome y, con igu a ion, and s a e o main enance, a e conside ed
o assess hei in luence on ulne abili y. F agili y unc ions a e de i ed by i ing a logno mal
cumula i e dis ibu ion unc ion o he obse ed p obabili ies o damage exceedance using he
Maximum Likelihood Es ima ion (MLE) me hod.
 Chap e 5 p oposes no el ulne abili y models based on he da a analysed in Chap e 4, using
SA as an al e na i e seismic in ensi y measu e o PGA. Since SA is no included in he Da.D.O.
da abase, an exis ing GMM is applied o es ima e i . This equi es knowledge o he

CHAPTER 1. INTRODUCTION
9
undamen al pe iod o he s uc u es. While se e al empi ical ela ionships o p edic ing he
undamen al pe iod o his o ic mason y owe s a e a ailable in he li e a u e, his Chap e
in oduces no el empi ical laws de i ed om an open da ase compiled h ough an ex ensi e
li e a u e e iew. The newly de eloped ulne abili y and agili y unc ions a e compa ed o
hose based on PGA, using a ious pe o mance me ics o highligh he impac o in ensi y
measu e selec ion on he accu acy and e ec i eness o hese models.
 Chap e 6 summa ises he main inno a i e aspec s and ou comes o his wo k, o e ing
conclusions and ecommenda ions o u u e esea ch in he ield o seismic ulne abili y
assessmen o his o ic mason y owe s.
10
CHAPTER 2
SEISMIC RISK ASSESSMENT: STATE OF THE
ART
ABSTRACT
This Chap e p esen s an o e iew o seismic isk and sys ema ically e iews he me hodologies o
assessing he ulne abili y o mason y s uc u es, wi h a speci ic ocus on slende buildings and owe s.
I highligh s cu en , ad anced echniques o inc easing complexi y and de ail, p io i ising hose mos
sui able o la ge-scale e i o ial applica ions. While a ious well-es ablished classi ica ions a e
p esen ed, hese me hodologies a e p ima ily o ganised by dis inguishing be ween empi ical and
analy ical app oaches, e lec ing he emphasis on he o me in he subsequen Chap e s. Bo h
solu ions a e c i ically e alua ed, add essing hei ad an ages and limi a ions, and discussing ele an
illus a i e case s udies.
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
11
2.1 SEISMIC RISK
Seismic isk, 𝑅, may be de ined as he cha ac e isa ion o he po en ial consequences o an
ea hquake, commonly ep esen ed as an es ima e o he expec ed losses, such as damage, a ali ies
and economic losses, esul ing om he e en . Risk assessmen is ypically conce ned wi h a la ge
asse s ock a he han a single building, o en ca ego ising speci ic ypes (e.g., esiden ial dwellings,
public buildings, he i age, in as uc u es, c i ical acili ies) and i is a unc ion o how he seismic
haza d, 𝐻, combines wi h he exposu e, 𝐸, o he conside ed asse s and hei ulne abili y, 𝑉, hus
being a spa ial and empo al con olu ion o hese h ee speci ic models, as in Figu e 2.1.
𝑅 =𝑓(𝐻,𝐸,𝑉)
(2.1)
Figu e 2.1: Seismic isk componen s.
Mo e ecen ly, he capaci y as quali a i e pa ame e has been in oduced in he abo e equa ion by
(UNISDR, 2015) o a mo e comp ehensi e de ini ion o he isk. De eloping obus models which can
accu a ely desc ibe and p edic he cha ac e is ics o such en i ies and hei dynamic in e ac ions in
o de o assess he o e all seismic isk, is a challenging mul idisciplina y ask. I equi es he
collabo a ion among a ious expe s, including seismologis s, haza d modelle s, da a analys s and
s uc u al enginee s.
In gene al e ms, he seismic haza d 𝐻 e e s o he p obabili y o a seismic e en which has he
po en ial o cause a ali ies, inju ies, building damage, se ices down ime and socio-economic
in e up ions wi hin a speci ic e i o ial egion. The main pa ame e s cha ac e ising he seismic haza d
a e he equency o occu ence ( he e u n pe iod) and he o ce ( he se e i y) o he expec ed g ound
shaking in each a ea. Haza d assessmen in ol es e alua ing he seismici y, h ough app op ia e and
sui able seismic in ensi y measu es. Se e al in ensi y measu es a e oday a ailable in he li e a u e o
cha ac e ise s ong g ound mo ions a ec ing he buil en i onmen . Typical seismic in ensi y measu es
a e he mac oseismic in ensi y, magni ude, Peak G ound Accele a ion (PGA), Peak G ound Veloci y
(PGV), Peak G ound Displacemen (PGD), Spec al Accele a ion (SA), Spec al Veloci y (SV), Spec al
Displacemen (SD), A ias in ensi y, among o he s. Today, haza d maps used o design pu poses a e
mos ly gi en in e ms o PGA, p esen ing he spa ial dis ibu ion o he expec ed se e i y o a gi en
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
12
e u n pe iod. The e o e, he seismic haza d can be gene ally de ined as he p obabili y o occu ence o
an e en exceeding a ce ain alue o PGA wi hin a gi en ime span. Se e al s a egies can be
unde aken o assessing he seismic haza d, anging om de e minis ic (Sil a, 2016) o p obabilis ic
app oaches (Dolce e al., 2021). In his ield, signi ican ad ances ha e been made and he seismic
haza d can be nowadays compu ed by means o haza d cu es a any si e in Eu ope (Pagani e al.,
2014).
The exposu e, 𝐸,
e e s o he loca ion and quali a i e a ibu es (e.g., occupan s, building ype,
p ope ies, and a e ac s, collec ions, pain ings and sculp u es, i p esen ) o he asse s a isk,
desc ibing he possible consequences o a haza dous e en in economic, cul u al and social e ms.
Typically, da a ela ed o he loca ion, ypology classi ica ion, cons uc ion age, cos o he building s ock
and occupancy class, wi h po en ial numbe o occupan s, a e equi ed o he de elopmen o obus
exposu e models. This in o ma ion, s o ed in a building in en o y when a ailable, o g ea in e es o
planning decision and o sa egua ding human li e, allowing o ex apola e quali a i e es ima es o he
people a ec ed and economic losses caused by he e en (Dolce e al., 2021; Lagoma sino & Podes à,
2004b).
The ulne abili y, 𝑉, e e s o he suscep ibili y o buildings o damage when exposed o a seismic
e en . The damage depends on he esponse o buildings o he g ound shaking and hei capaci y. The
mo e a building is ulne able, he g ea e is he expec ed damage. The damage mechanism ype and
ex en can be ela ed p ima ily o key physical ea u es and cha ac e is ics o he in es iga ed building
s ock and hei su ounding ha go e n hei esponse o he seismic ac ion. These ea u es include bu
a e no limi ed o he ype o he s uc u al sys em, geome y egula i y, ma e ials and soil condi ions.
O he ele an cha ac e is ics, such as cons uc ion age and s a e o main enance, including exis ing
damage, may in luence he suscep ibili y. In isk assessmen , he damage can be modelled in di e en
ways. Sui able disc e e damage scales (e.g., he Eu opean Mac oseismic Scale, EMS-98) a e ypically
adop ed in empi ical s udies, especially in pos -ea hquake in es iga ions, o conduc he s a is ical
analysis o he e en and i s consequences and gene a e he dis ibu ions o obse ed damage.
Di e en ly, enginee ing demand pa ame e s, such as d i o c ack wid h, a e mainly used in nume ical
in es iga ions, o ealis ically ep oduce possible damage scena ios.
Among hese h ee componen s o he seismic isk, exis ing haza d and exposu e can be ha dly
modi ied and o en e ol e slowly o e ime. Dis inc le els o accu acy in hei assessmen can be
ensu ed wi h di e en model e inemen s, depending on he objec i es o he assessmen , expe ise
and economic budge . Cu en e o s a e p ima ily ocused on enhancing ulne abili y models o
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
13
seismic isk assessmen and mi iga ion, since educing he ulne abili y o buildings is a mo e iable
app oach o dec easing he seismic isk.
The e o e, unde s anding he seismic isk and de eloping e ec i e models o e alua ing i and i s
componen s ensu e consis en s uc u al condi ion sc eening, ha is pa amoun o suppo he
eme gency esponse and disas e planning, sa egua d c i ical buildings and, a he same ime,
minimise human and economic po en ial losses (Fe ei a e al., 2021). The s uc u al condi ion
sc eening, in seismic isk assessmen , is enabled by he iden i ica ion and p edic ion o adequa e loss
me ics (i.e., damage, casual ies, cos s) measu ed in co ela ion wi h he inc easing le els o one (o
mo e) seismic in ensi y measu es. Exis ing models designed o his pu pose a e comp ehensi ely
desc ibed in he ollowing Sec ions.
2.2 ASSESSMENT METHODOLOGIES: AN OVERVIEW
Al hough a ious c i e ia ha e been discussed in li e a u e o classi y isk analysis and especially
ulne abili y assessmen me hods, as o ins ance in (Cal i e al., 2006; Vicen e e al., 2011), he ea e
he ollowing axonomy is used o in oduce and discuss exis ing s a egies o mason y s uc u es a
e i o ial scale: (i) analy ical models; (ii) empi ical models; (iii) heu is ic models; (i ) hyb id models.
Analy ical models a e based on he idealisa ion o he in es iga ed s uc u al ypology h ough nume ical
simula ions, de ined o quan i a i ely es ima e he s uc u al esponse o expec ed local seismic
in ensi ies (Shabani e al., 2021; Zizi e al., 2021). As he le el o de ails o he in ol ed models may
a y signi ican ly depending on he scope, hese me hods can be u he subdi ided o accoun o he
accu acy, complexi y, compu a ional esou ces needed and inpu da a equi ed, add essing all di e en
scales om la ge e i o ial s udies o building scale assessmen s. Empi ical models a e di ec ly based
on expe imen al o obse a ional da a, ypically collec ed a e a seismic e en , eso ing o a da a-
d i en e alua ion o he s uc u al pe o mance h ough s a is ical me hods (Ro a e al., 2008). The
adop ion o eal obse a ional da a makes hem ex emely eliable, al hough, a he same ime, limi s
hei applicabili y o con ex s o which s a is ically ele an da a o pas e en s a e a ailable, and hei
scalabili y and gene alizabili y o o he con ex s may be ques ionable (Rosse o & Ioannou, 2018).
Heu is ic me hods ely on expe ise and echnical judgmen , quali a i e c i e ia, and ule-based decision-
making o es ima e he ulne abili y o a building ypology and/o o iden i y he main ac o s a ec ing
i s seismic pe o mance (Giuliani e al., 2021; Romão e al., 2016). These me hods a e pa icula ly
use ul when quan i a i e obse a ional da a o compu a ional simula ions a e no eadily a ailable no
easy o achie e. Focusing on expe insigh s and opinions a he han de ailed s uc u al analysis, hey
ha e been ex ensi ely used o apid e i o ial scale assessmen s wi hin simple amewo ks. Finally,

CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
14
hyb id models esul om possible combina ions among he models p e iously desc ibed (Ca dinali e
al., 2022; Kappos, 2016; Kappos e al., 2006; Sandoli e al., 2023). Wi hin his la e class, a ious
combina ions o app oaches a e included as o ins ance he use o obse ed da a o calib a e
nume ical models, enhancing hei eliabili y, and hen exploi hem o expand he in es iga ion o
unknown seismic scena ios.
Among hese me hodologies, he empi ical models and he analy ical models ha e been ex ensi ely
adop ed o assessing he seismic isk and ulne abili y o his o ic mason y owe s and slende
s uc u es, in some cases hyb idised wi h expe decision ega ding he weigh o in luen ial ac o s. Due
o he ele ance o he scopes o he p esen hesis, hese me hodologies a e sho ly p esen ed
he ea e , and mo e de ails a e p o ided in he ollowing Sec ion, alongside wi h hei p ac ical and
po en ial applica ions.
Examples o common ools adop ed by empi ical models a e he Damage P obabili y Ma ices (DPMs),
ulne abili y unc ions and agili y unc ions, as summa ised in Figu e 2.2. While hese models
s a is ically analyse pos -ea hquake damage da a o es ablish speci ic ela ionships be ween building
ypology, i s ea u es and obse ed damage ela i e o seismic in ensi y, ulne abili y index me hods,
epo ed in Figu e 2.2, o en ely on expe judgemen o enhance he models. This app oach allows o
he inclusion o in o ma ion ha may no be eadily a ailable om pas e en s.
Figu e 2.2: Empi ical models.
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
15
These models di e om each o he depending on how he da a a e used and which in o ma ion is
needed o ex ac . DPMs, o iginally p oposed by (B aga e al., 1982; Whi man e al., 1973), a e abula
ep esen a ions, ex ensi ely adop ed in empi ical s udies, ha show he equency o occu ence
(p obabili y) o di e en damage s a es acco ding o a gi en scale o a building o s uc u e ypology,
gi en a speci ic le el o seismic in ensi y. The ulne abili y unc ions a e ma hema ical o g aphical
ela ionships ha desc ibe he expec ed damage le el o loss (e.g., cos , down ime) as a unc ion o a
seismic in ensi y measu e (Lagoma sino & Gio inazzi, 2006). The ulne abili y index me hods a e
p ocedu es ha assign a single nume ical ulne abili y sco e o buildings based on he main
pa ame e s ha ema kably a ec hei seismic beha iou (Benede i & Pe ini, 1984). The de ini ion o
hese ele an pa ame e s and hei weigh ed con ibu ion o he o e all ulne abili y o en esul om
expe opinions and heu is ic app oaches. Finally, he agili y unc ions a e p obabilis ic models ha
ep esen he likelihood o a building o s uc u e eaching o exceeding a speci ic damage s a e as a
unc ion o seismic in ensi y measu es (Rosse o & Elnashai, 2003; Ro a e al., 2008). I espec i e o
he ool, nume ical da a can be used as su oga e o o in eg a e obse a ional ones, suppo ing model
de elopmen o speci ic building ypologies. Fu he de ails on empi ical me hods a e p o ided in
Sec ion 2.3.
Examples o analy ical app oaches o his o ic mason y buildings a a ious le el o de ail and
complexi y ha e been discussed in he I alian documen o he p ese a ion o he cul u al he i age,
namely he “Guidelines o he assessmen and mi iga ion o he seismic isk o he Cul u al He i age”
(DPCM, 2011). This documen es ablishes a amewo k o s uc u al analysis ailo ed o he peculia
cha ac e is ics o he buil he i age. In pa icula , h ee analy ical le els o e alua ion (LV1, LV2 and LV3)
wi h an inc easing complexi y a e sugges ed o speci ic s uc u al mason y building ypologies,
including mason y chu ches and mason y owe s, among o he s. A summa y o he h ee le els o
e alua ion p oposed speci ically o slende mason y s uc u es is ske ched in Figu e 2.3.
Va ying he le el o analysis, wi h i s co esponding complexi y, a ec s he easibili y o he me hods o
he wo dis inc scales o assessmen : he e i o ial and he indi idual building scale. Mo e speci ically,
he i s le el o e alua ion (LV1) is sugges ed o assessing he ulne abili y o a la ge numbe o
buildings o e a de ined e i o y, an u ban cen e o a egion, wi h he main goal o iden i ying he mos
c i ical buildings, es ablishing he p io i ies o u u e in e en ions and suppo ing he seismic isk
managemen (Ba oli, Be i, & Monche i, 2017; Casapulla e al., 2018; Fo misano & Ma zo, 2017).
The emaining wo le els o e alua ion (LV2 and LV3) a e ecommended o he ulne abili y
assessmen o indi idual buildings when mo e e ined analysis a e equi ed o be e unde s and he
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
16
building's esponse and design u u e p ese a ion s a egies in a cos -e ec i e manne (D’Ayala &
Spe anza, 2003; Lagoma sino & Ca a i, 2015; Tanganelli e al., 2024; To elli e al., 2020). In gene al,
he i s le el o e alua ion can be conside ed he p elimina y s ep owa d mo e accu a e and de ailed
me hods o analysis (DPCM, 2011). Fu he de ails on analy ical me hods a e p o ided in Sec ion 2.4.
Figu e 2.3: Summa y o LV1, LV2 and LV3 le els o e alua ion p oposed by he I alian Guidelines o slende
s uc u es.
2.3 EMPIRICAL METHODOLOGIES
The de elopmen o empi ical models equi es damage da a, in speci ic seismic e en s, o s uc u es
associa ed wi h speci ic building ypologies, spanning a a ious e i o ial scales (i.e., u ban, egional,
na ional). The de elopmen o hese models also equi es in o ma ion on he co esponding ea hquake
cha ac e is ics. Damage and ea hquake da a a e ypically collec ed and assessed ollowing he seismic
e en s, o en as pa o echnical ac i i ies aimed a enabling apid sa e y and usabili y e i ica ions and
suppo ing eme gency managemen e o s. A ypical ime able o hese ac i i ies in case o g ound
shaking is p o ided in Table 2.1. Gene ally, he echnical ac i i ies a e coo dina ed by he Depa men
o Ci il P o ec ion o a simila na ional agency, oge he wi h o he o ganisa ions, au ho ised p i a e
en i ies, esea ch cen es and uni e si ies.
Ea hquake da a gene ally includes he p ocessing o soil and building accele ome ic da a, as well as
he si e in es iga ions o de ine he mac oseismic Me calli-Cancani-Siebe g (MCS) in ensi y and o he
in ensi y measu es. To his end, mos seismically ac i e coun ies main ain a s a egically deployed
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
17
moni o ing ne wo k. Wi hou he knowledge o he ea hquake da a, i is no possible o assess he
seismic haza d and consequen ly he seismic isk. The MCS in ensi y scale, as o he simila in ensi y
scales like he Modi ied Me calli (MM), he Med eded-Sponheue -Ka nik (MSK), he Eu opean
Mac oseismic Scale (EMS-98), and he Japan Me eo ological Agency (JMA), is a p eda e mode n
ins umen al me hod o eco d and cha ac e ise seismic in ensi y (Dolce e al., 2019, 2021). The e o e,
i has been ex ensi ely used in he pas o cha ac e ise he loca ion and he in ensi y o an ea hquake.
Table 2.1: Typical ime able o echnical ac i i ies adop ed a e seismic e en s.
No.
Time
Main ac i i ies
Desc ip ion
1
om 2 min up o
5-30 min
Epicen e and Magni ude e alua ion
Collec ing and p ocessing seismome ic
ne wo k da a
2
om 10 min up
o 60 min
Simula ed damage scena ios and
da a p ocessing o moni o ing sys ems
So wa e simula ion o he ea hquake
impac on cons uc ions;
Collec ing and p ocessing soil and building
accele ome ic da a.
3
om 6 hou s up
o 7-14 days
Si e su eys o mac oseismic e ec s
Si e e alua ion o mac oseismic in ensi y;
Geological su eys o landslides, su ace
aul ing, and soil lique ac ion.
4
om 6 hou s up
o 6-12 mon hs
Tempo a y moni o ing o soil and
s uc u es
Ins alling empo a y soil and accele ome ic
s a ions and s uc u e moni o ing sys ems.
5
om 24 hou s up
o 6-12 mon hs
Pos -ea hquake damage and sa e y
assessmen
Building inspec ions o damage and
usabili y assessmen ;
Technical e alua ion o empo a y houses.
MCS in ensi y is based on obse ed e ec s o an ea hquake and hei quali a i e desc ip ion, namely
obse ed damage and human eac ion, a a pa icula loca ion (Musson, 2009). In ensi y da a can be
collec ed om ield su eys, eyewi ness epo s, and seconda y sou ces, making i use ul o egional o
na ional isk assessmen s, especially in a eas wi h limi ed ins umen a ion o o in es iga e pas e en s
p io o he in oduc ion o he moni o ing ne wo ks. Fo his eason, many adi ional ulne abili y
assessmen models ha e been o iginally de eloped by elying on hese in ensi y measu es.
None heless, se e al limi a ions o he assessmen acco ding o he MCS scale a e g adually leading o
i s eplacemen by o he in ensi y measu es. Among hese limi a ions, i is wo h men ioning he
subjec i i y o he e alua ion ha elies on human obse a ions, i s esolu ion, o en limi ed o coa se
spa ial scales and he impossibili y o calcula ing i in eal- ime, suppo ing eme gency esponse.
Addi ionally, as i e lec s he e ec s o he ea hquake, i is no solely ela ed o haza d, as an in ensi y
measu e ypically should be, bu i is also a ec ed by he ulne abili y o he local building s ock
(Musson, 2009). Wi h he ecen g ow h o ins umen al moni o ing, he use o PGA as seismic in ensi y
measu e has become e y popula (Dolce e al., 2019, 2021). The PGA is a measu e o he maximum
accele a ion expe ienced by he g ound du ing an ea hquake, as egis e ed by a seismic s a ion. The
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
24
A e he iden i ica ion o each occu ing mechanism and he e alua ion o i s damage le el acco ding o
he EMS-98 g ading sys em (𝑑𝑖), a global damage index o he whole owe is compu ed (𝑖𝑑) wi h a
maximum equal o he uni alue, based on he ollowing equa ion (weigh ed sum c i e ia):
𝑖𝑑= 15 ∑𝜌𝑖 𝑑𝑖
𝑛𝑖=1
∑𝜌𝑖
𝑛𝑖=1
(2.2)
whe e 𝜌𝑖 deno es he weigh o be assigned o each o he
n
possible damage mechanisms, based on
i s impo ance o he s uc u al beha iou o he whole building.
Focusing in his wo k on he owe mac oelemen only, an impo ance weigh equal o one is conside ed
o bo h damage mechanisms (see Chap e s 3, 4 and 5), as commonly done o he global assessmen
o chu ches, o ins ance in (De Ma eis & Zizi, 2019). The e o e, he exp ession becomes:
𝑖𝑑= 1
10 ∑ 𝑑𝑖
2
𝑖=1
(2.3)
An al e na i e would be o gi e mo e ele ance o he sha mechanism, Mechanism A, as he
consequences o ailu e a e likely o be g ea e . Howe e , his app oach was no pu sued in his wo k.
The global damage index (𝑖𝑑) is, hus, he a e age o he damage le els obse ed in he ac i a ed
damage mechanisms. This is a con inuous index wi hin he 0-1 ange. To ob ain a disc e e inal global
damage le el (𝐷𝑘) o each owe aligned wi h EMS-98 scale, he co ela ions p o ided in
(Lagoma sino & Podes à, 2004c) and epo ed in Table 2.2 a e used.
Table 2.2: Co ela ion be ween he global damage index and he global damage le el.
Global damage index (𝒊𝒅)
Global damage le el (𝑫𝒌)
Desc ip ion
𝑖𝑑≤0.05
0
No damage
0.05<𝑖𝑑≤0.25
1
Negligible o mode a e damage
0.25<𝑖𝑑≤0.40
2
Mode a e damage
0.40<𝑖𝑑≤0.60
3
Subs an ial o hea y damage
0.60<𝑖𝑑≤0.80
4
Ve y hea y damage
𝑖𝑑>0.80
5
Collapse
Wi h he ad en o digi al echnologies, signi ican e o s ha e been made o digi ise his essen ial
in o ma ion abou pas e en s. Some o he mos well-known da abases a ailable include he GEM

CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
25
Da abase, he Camb idge Ea hquake Damage and Casual y Da abase, he CATDAT Da abase, and he
EM-DAT In e na ional Disas e Da abase (Rosse o & Ioannou, 2018).
In I aly, he Da.D.O. (Da abase o Obse ed Damage) web pla o m has been concei ed by he I alian
Ci il P o ec ion Depa men and ealised by he Eu opean Cen e o T aining and Resea ch in
Ea hquake Enginee ing o allow a long las ing and easy s o age and e ie al o in o ma ion (Di Meo e
al., 2023; Dolce e al., 2019). The pla o m includes he digi isa ion o he su ey o ms om he
physical inspec ion ca ied ou du ing o ollowing eigh seismic e en s o na ional impo ance, namely,
he 1997 Umb ia-Ma che, he 2002 Molise, he 2003 Piedmon , he 2004 Salò, he 2009 L’Aquila, The
2012 Emilia, he 2016-2017 Cen al I aly, and he 2017 Ischia ea hquakes. In he cases o chu ches
a ec ed by hese e en s, he damage da a, ca alogued wi h e e ence o he p e-de ined damage
mechanisms, can be displayed oge he wi h he cha ac e is ics o he e en s in e ms o MCS in ensi y
and in e ms o PGA, elabo a ed by he Na ional Ins i u e o Geophysics and Volcanology (INGV).
The collec ed da a on haza ds and losses se e as ounda ional inpu s o s a is ical analyses, which a e
used o de i e damage dis ibu ions, de elop ulne abili y and isk models, and e alua e hei p edic i e
accu acy. Fo o dina y mason y buildings, ancien chu ches, and his o ic owe s, commonly adop ed
me hodologies include DPMs, ulne abili y unc ions, ulne abili y index me hods, and agili y unc ions,
each o which is discussed in de ail in he ollowing Sec ions.
2.3.1 Damage P obabili y Ma ices (DPMs)
Among he ea lies me hods de eloped o isk analysis a he e i o ial scale, DPMs play a pi o al ole
in in e p e ing he ulne abili y o asse ypologies based on obse ed damage da a. These ma ices
o m he ounda ion o p obabilis ic isk assessmen amewo ks, o e ing a s uc u ed app oach o
quan i y he occu ence o speci ic damage le els o a ious asse classes, gi en a ange o in ensi y
measu e. To cons uc DPMs, obse a ional da a is ypically agg ega ed in o anges o in ensi y
measu e, wi h he esul s isualised as ables o his og ams o easie in e p e a ion and applica ion.
Eme ging in he 1970s, a e he 1971 San Fe nando ea hquake (Whi man e al., 1973), DPMs gained
p ominence ollowing hei applica ion o analyse he damage om majo seismic e en s. In he case o
I alian ea hquakes, DPMs we e ex ensi ely adop ed since he sys ema ic s udy o da a om he 1980
I pinia ea hquake (B aga e al., 1982). In he case o mason y owe s, DPMs a e used o desc ibe he
disc e e p obabili y dis ibu ion o he exceedance o speci ic damage le els, conside ing he indi idual
collapse mechanisms (Mechanisms A and B) and he en i e owe mac oelemen . In pa icula , when
he DPMs a e de eloped o he indi idual damage mechanisms, he mean damage 𝜇𝑑 is gi en by:
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
26
𝜇𝑑= ∑𝑑𝑘,𝑖
𝑛𝑖=1𝑛 𝑘=[0,..,5]
(2.4)
whe e 𝑑𝑘,𝑖 is he damage le el a ibu ed o ei he he sha o he bel y o he i- h owe , depending on
he damage mechanism unde s udy, and 𝑛 deno es he numbe o owe s in es iga ed. When he
DPMs a e de eloped o he en i e mac oelemen , he mean damage 𝜇𝐷 is gi en by:
𝜇𝐷= ∑𝐷𝑘,𝑖
𝑛𝑖=1𝑛 𝑘=[0,..,5]
(2.5)
whe e 𝐷𝑘,𝑖 is he global damage le el ob ained o he i- h owe , and 𝑛 deno es he numbe o owe s
analysed.
As DPMs ep esen a s a is ical ela ionship be ween he in ensi y measu e o he haza d and he
obse ed damage le els o a speci ic asse ypology, hei eliabili y depends on he consis ency o
hese inpu da a. I he asse s wi hin a ypology sha e simila cha ac e is ics (e.g., cons uc ion
ma e ials, age, design s anda ds) and he haza d’s e ec s a e uni o m ac oss he a ea o obse a ion,
he damage dis ibu ions a e expec ed o exhibi ela i ely low sca e wi hin each in ensi y ange.
When applied o di e en building ypologies, DPMs se e as an e ec i e ulne abili y sc eening ool,
allowing o he p elimina y iden i ica ion o he mos c i ical asse s. Howe e , DPMs p o ide a single
ou pu (dis ibu ion) pe asse class, wi hou explici ly accoun ing o unique ea u es o he in es iga ed
building s ock in hei model de ini ion, excep as c i e ia o selec ing homogeneous classes.
Consequen ly, DPMs can be used o assess he seismic ulne abili y o buildings no included in he
o iginal da ase and o p edic hei expec ed damage le els o e en s wi h speci ic in ensi y measu es,
only assuming ha he buildings unde assessmen belong o he same ypology, sha e simila
cha ac e is ics, and will expe ience analogous haza d condi ions.
Nume ous s udies ha e epo ed DPMs o di e en ypes o mason y s uc u es, including chu ches
subjec ed o a ious seismic e en s, since he 1976 F iuli and he 1997 Umb ia-Ma che ea hquakes
(Doglioni e al., 1994; Lagoma sino & Podes à, 2004c). Recen s udies ha e also ocused on speci ic
ypologies o chu ches, as o ins ance, one-na e (Ce oni e al., 2022; De Ma eis & Zizi, 2019; Ruggie i
e al., 2022) o h ee-na e (De Ma eis e al., 2016; De Ma eis, B ando, & Co li o, 2019). Howe e ,
mos o he s udies ha e ea ed chu ches as a whole s uc u al sys em, a ely epo ing and discussing
he indi idual beha iou o each single mac oelemen . This p e en s c i ical insigh s in o he localised
ulne abili ies ha signi ican ly in luence he o e all seismic pe o mance o hese s uc u es.
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
27
DPMs o dis inc mac oelemen s, including mason y owe s, a e p o ided by (De Ma eis e al., 2016;
De Ma eis & Zizi, 2019), while (Canu i e al., 2021; Ho e e al., 2018; Ruggie i e al., 2022) also
in es iga ed dis inc damage mechanisms (i.e., sha and bel y). The e o e, he con ibu ion o he
dis inc componen s and hei damage mechanisms on he o e all owe beha iou is no ully assessed
in he li e a u e. Addi ionally, in he case o mason y owe s, he sou ces a ely p o ide a comple e se
o DPMs o a ious anges o in ensi y measu es, wi h jus a ew excep ions, e.g., (Canu i e al., 2021).
The sca ce sou ces a ailable a e ex ensi ely discussed in Chap e 3.
Finally, mos o he exis ing s udies de eloped DPMs using he MCS scale, which was adi ionally one
o he i s measu es adop ed o seismic in ensi y. None heless, ecen ad ancemen s in e i o ial
scale seismic isk assessmen ha e shi ed owa ds he use o al e na i e in ensi y measu es ha
p o ide mo e quan i iable and objec i e me ics. In pa icula , PGA has been inc easingly adop ed o
de elop DPMs o mason y chu ches in gene al and he owe mac oelemen s in pa icula (Ce oni e
al., 2022; Sis i e al., 2023).
2.3.2 Vulne abili y unc ions
While DPMs p o ide disc e e p obabili y dis ibu ions ha quan i y he equency o speci ic damage
le els o a gi en in ensi y measu e, ulne abili y unc ions agg ega e his in o ma ion in o a single
p edic i e cu e, o e ing a con inuous es ima ion o he expec ed damage, h ough a ep esen a i e
me ic exp essed as a unc ion o he in ensi y measu e. In his con ex , DPMs and ulne abili y
unc ions a e closely ela ed ools, wi h DPMs se ing as a ounda ion o cons uc ing he cu es.
Addi ionally, hey a e o en complemen a y in a s udy as he ulne abili y unc ions ocus on a single
me ic o he damage o he class, while he DPMs allow an insigh in o he a iabili y wi hin he asse
class. Commonly, he expec ed damage me ic adop ed is he mean damage, 𝜇𝑑 and he in ensi y
measu e ollows he mac oseismic scale, 𝐼𝑀𝐶𝑆.
Due o hei ease-o -use and in e p e a ion, ulne abili y unc ions a e likely he mos common me hods
used in he li e a u e o assessing he seismic ulne abili y o o dina y and monumen al mason y
buildings, enabling a apid sc eening o he s uc u al condi ion ac oss he conside ed s udy a ea. Fo
ins ance, hese unc ions ha e been ex ensi ely used wi hin he RISK UE P ojec (Mou oux & Le B un,
2008), “An ad anced app oach o ea hquake isk scena ios”, o p o ide he seismic isk scena ios in
some Eu opean ci ies, including Ba celona, Ca ania, Nice, and Thessaloniki. A simila app oach has
been adop ed in I aly o es ima e he expec ed damage o mason y chu ches and o he monumen s
(DPCM, 2011).
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
28
To build a ulne abili y model o a monumen al mason y building ypology, he unc ion o iginally
p oposed by (Lagoma sino & Gio inazzi, 2006; Lagoma sino & Podes à, 2004c) is usually assumed, as
ollows:
𝜇=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+𝛼𝑉𝐼−𝛽
𝑄)]
(2.6)
whe e 𝜇 is he expec ed mean damage 0≤𝜇≤5 co ela ed o he mac oseismic damage le els
de ined by he EMS-98 scale, 𝐼𝑀𝐶𝑆 is he MCS in ensi y a ying om 5 up o 12, 𝑄 is he duc ili y
index, 𝑉𝐼 is he ulne abili y index and 𝛼 and 𝛽 a e coe icien s ha assume cons an alues o 6.25
and 13.1, espec i ely, i espec i e o he building ypology.
The e o e, he expec ed mean damage 𝜇 is co ela ed wi h inc easing le els o mac oseismic in ensi y
𝐼𝑀𝐶𝑆 h ough he de ini ion o wo in insic pa ame e s o he monumen al building ypology, namely,
he duc ili y index,
Q
, and he ulne abili y index, 𝑉𝐼.
Q
accoun s o he s uc u al esponse in nonlinea
egime. I con ols he a e o inc ease o he damage wi h he in ensi y and is ypically de ined o he
building ypology. The ulne abili y index, 𝑉𝐼, may be assumed as equal o an a e age alue, 𝑉0, o he
en i e class o asse s, adjus ed, i needed, o ake in o accoun ele an ea u es o each speci ic
building ha in luence hei s uc u al esponse. A e age alues o he ulne abili y and he duc ili y
indexes ha e been p o ided in li e a u e o di e en building ypologies acco ding o expe judgemen
o by calib a ion o he eal obse ed da a collec ed om pas seismic e en s. Di e en pai s o 𝑄 and
𝑉0 alues o monumen al mason y s uc u es sugges ed by (Despo aki e al., 2018; Lagoma sino,
2006; Lagoma sino e al., 2004) a e shown in Table 2.3.
Table 2.3: A e age alues o
V0
and
Q
o be used in Eq. (2.6) o dis inc building ypologies.
Monumen al building ypology
Vulne abili y index 𝑉0
Duc ili y index Q
A ch b idges
0.296
2.30
Cas les
0.456
2.30
Chu ches
0.890
3.00
Columns
0.456
1.95
Monas e ies
0.736
2.30
Mosques
0.730
2.65
Obelisks
0.456
1.95
Palaces
0.616
2.30
Temples
0.500
1.95
Towe s
0.776
2.30
T ili hes
0.456
1.95
T iumphal a ches
0.456
2.30
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
29
The ulne abili y unc ions ob ained o he monumen al mason y building classes wi h he coe icien s
in Table 2.3 a e illus a ed in Figu e 2.9. Jus by looking a he ulne abili y unc ions, he expec ed
mean damage compu ed o mason y chu ches is he g ea es o e he en i e ange o he in ensi y.
His o ic owe s a e one o he mos ulne able building ypologies, oge he wi h mosques, palaces and
mason y a ch b idges. I is wo h no ing ha bell owe s a e no included in he mason y owe class
discussed by he au ho s.
Figu e 2.9: Simpli ied ulne abili y unc ions o dis inc monumen al mason y building ypologies.
In his amewo k, a lis o co ec i e ac o s 𝑉𝑗 we e o iginally sugges ed o mason y chu ches by
(Lagoma sino e al., 2004) and subsequen ly adop ed o o he his o ic s uc u al ypologies, including
owe s, by (Despo aki e al., 2018). These ac o s con ibu e o he es ima ion o he ulne abili y index
as:
𝑉𝐼=𝑉0+∑𝑉𝑗
(2.7)
Table 2.4 p esen s he pa ame e s and ca ego ies conside ed along wi h hei co esponding modi ie s.
Wi h espec o he o iginal lis o pa ame e s, he plan egula i y is neglec ed o his o ic owe s due o
i s minimal in luence, as sugges ed by (Sepe e al., 2008).
The co ec i e ac o s 𝑉𝑗 enable he adjus men o ulne abili y unc ions o an en i e asse class by
inco po a ing speci ic building cha ac e is ics. This app oach equi es minimal knowledge, making i
well-sui ed o la ge building s ock and e i o ial-scale assessmen s. Howe e , e en he limi ed
in o ma ion necessa y o accu a ely e alua e hese co ec i e ac o s is o en una ailable in la ge-scale
assessmen s based on exis ing da a sou ces. S anda dised ools o da a collec ion, such as inspec ion
o ms, in ac , equen ly lack ields o cap u e all he equi ed in o ma ion. In his case, he ulne abili y

CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
30
index can be assumed equal o he a e age alue o he class, as shown in Table 2.3, p oducing a less
e ined ulne abili y assessmen .
Table 2.4: Vulne abili y modi ie pa ame e s.
Pa ame e s
Classi ica ion
Vulne abili y sco e
S a e o p ese a ion
Poo
Medium
Good
+0.04
0
-0.04
Damage le el
Se e e
Ligh
None
+0.04
+0.02
0
A chi ec u al ans o ma ion
Yes
No
–
+0.02
0
–
Recen in e en ions
Yes
No
–
-0.02
+0.02
–
Si e mo phology
Ridge
Slope
Fla
g ound
+0.04
+0.02
0
Addi ional pa ame e s o chu ches
Posi ion
Included
Addi ions
Isola ed
-0.02
+0.02
0
La e al wall heigh
Low
(<6m)
Medium (>6 m
and < 12m)
High
(>12m)
-0.02
0
+0.04
Plan egula i y
(na e ypology)
Cen al
One
Th ee
-0.02
0
+0.02
Ele a ion egula i y (e.g.,
eme ging elemen s)
Yes
No
–
+0.04
0
–
Besides he de elopmen o ulne abili y unc ions o mason y owe s as independen s uc u al
ypology and ollowing he seminal con ibu ion p o ided by (Lagoma sino e al., 2004; Lagoma sino &
Podes à, 2004c), ew au ho s ocused on he analysis o he ulne abili y o he bell owe
mac oelemen s, le e aging a much la ge da ase s based on chu ches inspec ions ca ied ou in he
a e ma h o he main I alian ea hquakes and using he p e iously discussed inspec ion o m o he ci il
p o ec ion (Canu i e al., 2021; Ce oni e al., 2022; Cu i e al., 2008; Sepe e al., 2008).
A i s no ewo hy con ibu ion is he wo k o (Cu i e al., 2008), who exploi ed he collec ion o a wide
da ase o his o ic mason y owe s a ec ed by di e en e en s, namely he 1976 F iuli, he 1997
Umb ia-Ma che, he 2002 Molise and he 2004 Lomba dia ea hquakes. This was used o calib a e ad-
hoc ulne abili y and duc ili y indexes o he wo dis inc se o mechanisms (sha and bel y) by i ing
he ulne abili y cu e, Eq. (2.6), p oposed by (Lagoma sino & Gio inazzi, 2006; Lagoma sino &
Podes à, 2004c), o he obse ed da a. The wo pai s o ulne abili y and duc ili y indexes o he sha
and bel y mechanisms p oposed by (Cu i e al. 2008) a e summa ised in Table 2.5.
Table 2.5: Co ec i e coe icien s p oposed o he sha and bel y mechanisms.
Monumen al Building Typology
𝑉0
Q
Sou ce
Towe mac oelemen damage
0.776
2.30
(Lagoma sino e al., 2004)
Damage Mechanisms
Mechanism A (sha )
0.890
2.00
(Cu i e al., 2008)
Mechanism B (bel y)
0.940
1.49
(Cu i e al., 2008)
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
31
The o mula ions sugges ed by (Cu i e al., 2008) a e illus a ed in Figu e 2.10 in compa ison wi h he
cu e p oposed o he owe building ypology. F om he compa ison, he bell owe sha s (Mechanism
A) and bel ies (Mechanism B) a e expec ed o unde go g ea e expec ed mean damage han he owe
ypology. Among he wo componen s, bel y p esen s highe suscep ibili y han sha .
(a)
(b)
Figu e 2.10: Vulne abili y unc ions o he sha (a) and he bel y (b) mechanisms, compa ed wi h he
ulne abili y unc ion o he owe ypology o monumen s.
A mul i-le el me hodology o assessing he seismic ulne abili y o his o ic mason y owe s was
p oposed in (Sepe e al., 2008), analysing a se o 107 his o ic owe s. In he p elimina y s age, he
s uc u al ulne abili y was oughly es ima ed using limi ed in o ma ion o iden i y he mos c i ical
owe s wi hin he in es iga ed s ock. Ini ial insigh s we e gained by co ela ing he le el o conse a ion
o he owe s wi h he MCS in ensi y, and by compa ing he exis ing ulne abili y unc ion p oposed by
(Cu i e al., 2008) wi h he obse a ional da a. A e wa ds, he owe s ha de ia ed signi ican ly om
he expec ed end we e lagged o mo e de ailed s uc u al analysis.
(Canu i e al., 2021) exploi ed he su eys o 541 mason y chu ches in Ma che egion inspec ed in he
a e ma h o he 2016-2017 Cen al I aly seismic sequence o p o ide an o e iew on he occu ed
damage and s uc u al ulne abili y. Among he mos ecu en mechanisms, bell owe s ecei ed
pa icula a en ion. The Au ho s epo ed he DPMs o dis inc anges o mac oseismic in ensi y,
sepa a ing ailu e mechanisms o he sha and he bel y. Then, he dis ibu ions o he obse ed
damage we e compa ed wi h hose es ima ed by he exis ing ulne abili y models p oposed by (Cu i e
al., 2008), e ealing g ea consis ency be ween obse a ions and model p edic ions.
Al hough he use o mac oseismic in ensi y in p edic i e ulne abili y models is widesp ead, as al eady
discussed o DPMs, se e al me hodologies a e now explo ing he use o PGA as a g ound mo ion
in ensi y measu e. PGA-based app oaches a e gaining inc easing popula i y (Ce oni e al., 2022; De
Ma eis & Zizi, 2019). Fo mason y owe s, (Ce oni e al., 2022) p oposed mul iple ulne abili y cu es
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
32
as a unc ion o he MCS in ensi y and PGA by i ing he Eq. (2.8)-(2.9) o he obse ed damage o 633
single na e chu ches s uck by he 2016-2017 Cen al I aly sequence. The p oposed calib a ion
coe icien s a e shown in Table 2.6. Unlike he o iginal o mula ion o he ulne abili y unc ions, he
Au ho s exp essed he damage da a in e ms o mean damage index a he han mean damage le el,
hus ha ing a con inuous a iable in he 0-1 ange. Fu he mo e, he ulne abili y and duc ili y indexes
we e no included in he no el p edic i e o mula ions.
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: Coe icien s o Eq. (2.8)-(2.9) p oposed o he sha and bel y mechanisms.
MCS
PGA
Damage Mechanisms
𝒂
𝒃
𝒂′
𝒃′
Sou ce
Mechanism A (sha )
0.25
2.08
0.89
2.54
(Ce oni e al., 2022)
Mechanism B (bel y)
0.25
1.87
0.89
2.22
(Ce oni e al., 2022)
Besides hese ew men ioned cases, alida ions o exis ing ulne abili y unc ions o his o ic mason y
owe s on pos -ea hquake damage da a a e qui e a e in he li e a u e. The e o e, in he p esen hesis,
hese exis ing models a e ex ensi ely es ed on empi ical obse a ions, e alua ing hei p edic i e
pe o mance and sugges ing ecalib a ions o speci ic da ase s a hand.
2.3.3 Vulne abili y index me hods
While DPMs and ulne abili y unc ions ha e been ex ensi ely used o desc ibe he p obabili y
dis ibu ion o damage le els based on empi ical obse a ions o classes o asse s wi h expec ed
homogeneous beha iou , a widesp ead amily o me hods, he so-called ulne abili y index me hods,
ha e been de eloped by add essing he isk assessmen p oblem om a di e en pe spec i e. Indeed,
ulne abili y index me hods p o ide a way o es ima e he suscep ibili y o a building based on a se o
quali a i e o semi-quan i a i e c i e ia s essing he ea u es ha con ibu e he mos o i . Each
c i e ion is de ined by a speci ic pa ame e , which is assigned a ulne abili y sco e and an associa ed
weigh . These me hods calcula e he o e all nume ical ulne abili y index, 𝐼𝑣, o an asse agg ega ing
he sco es o each pa ame e , weigh ed acco ding o i s ela i e impo ance. The e o e, hese me hods
do no p o ide a di ec es ima ion o he expec ed damage, a he highligh asse s ha a e mo e
suscep ible o damage. Typical applica ions a e aimed a iden i ying he c i ical s uc u es o u he
de ailed e alua ion. These c i ical s uc u es a e gene ally de ec ed as he ones wi h highes
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
33
ulne abili y sco e ob ained om he compa ison among all he s uc u es included in he in es iga ed
building s ock (Sepe e al., 2008).
A he same ime, he assessmen o s uc u al pa ame e s ela ed o he seismic beha iou allows a
mo e e ined analysis, capable o cap u ing mo e di e ences wi hin a class o asse s. The numbe o
pa ame e s and he ype o assessmen enable apid applica ion o la ge building s ocks, wi h mos
pa ame e s equi ing only a isual inspec ion suppo ed by basic ools. None heless, when he building
s ock is ex emely la ge he ulne abili y index me hods can be compu a ionally and economically
expensi e. Fu he mo e, he judgmen o he expe eams in ol ed in he e alua ion plays a dominan
ole in he esul s.
Se e al ulne abili y index me hods exis in li e a u e. As p e iously men ioned, hei basic unc ioning
consis s in he de ini ion o a sco e 𝐼𝑣 as he weigh ed sum o a se o ele an ea u es:
𝐼𝑣= ∑𝑞𝑖 𝑤𝑖
𝑁
𝑖=1
(2.10)
whe e 𝑞𝑖 is he sco e assigned o he i- h quali a i e pa ame e s (e.g., ele a ion layou , ype and quali y
o he mason y, s a e o conse a ion) which migh ema kably a ec he seismic beha iou o he
building ypology unde in es iga ion, 𝑤𝑖 is he weigh ed coe icien s, and 𝑁 he numbe o quali a i e
pa ame e s. The global ulne abili y sco e is hen usually no malised be ween 0 and 100, o
compa a i e pu poses. The da a may be collec ed by means o speci ic su ey o ms ailo ed o each
me hod. The numbe o he quali a i e pa ame e s depends on he me hodology adop ed. Mos o he
exis ing me hodologies o seismic ulne abili y assessmen descends om he p ocedu e p oposed by
(Benede i & Pe ini, 1984) o un ein o ced mason y buildings. Se e al esea che s and schola s ha e
adap ed his me hod o o he con ex s, such as u ban cen es (Vicen e e al., 2011), mason y
agg ega es (Fo misano e al., 2015) and speci ic mac oelemen s such as he açade (Fe ei a e al.,
2017). The me hod was also o icially adop ed in I aly by he Na ional G oup o Ea hquake De ence
(GNDT-SSN, 1994) and i was he e e ence o he p ocedu e p oposed by (Sepe e al., 2008)
speci ically o his o ic mason y owe s. The pa ame e s associa ed o he o iginal me hodology and he
app oach ailo ed o owe s in (Sepe e al., 2008) a e lis ed in Table 2.7, including he sco es o each
class and co esponding weigh s. Ele en pa ame e s a e needed o compiling he ulne abili y su ey
o m adop ed by he GNDT II Le el (Me 1), whe eas he ulne abili y o m add essing his o ic mason y
owe s (Me 2) is he same as he GNDT, wi h he exclusion o he plan layou and maximum span
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
40
whe e 𝑃(𝐷𝑘|𝑃𝐺𝐴𝑗) deno es he agili y unc ion, namely he p obabili y o eaching a damage s a e
𝐷𝑘 ( om D1 o D5) o a gi en PGA alue 𝑗 (o bin), 𝜙 ep esen s he s anda d no mal Cumula i e
Dis ibu ion Func ion, 𝜃 deno es he PGA median alue, and 𝛽 cons i u es s anda d de ia ion o he
loga i hms o he PGA.
Figu e 2.13: F agili y cu es based on da a i ing.
The logno mal cumula i e p obabili y dis ibu ion unc ion is widely conside ed a eliable and
es ablished me hod in he scien i ic communi y o se e al easons. I is s aigh o wa d o apply,
ex ensi ely used in ea hquake enginee ing, and o en p o ides a good i o obse ed da a. Speci ically,

CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
41
he logno mal dis ibu ion’s cha ac e is ic igh -skewed shape can be e ep esen obse ed
p obabili ies, which end o clus e a lowe le els o g ound shaking.
The p ocedu e, desc ibed and adop ed in his s udy in Chap e s 4 and 5, i s he logno mal cumula i e
p obabili y dis ibu ion o he empi ical da a by means o he MLE me hod, as o iginally p oposed by
(Bake , 2015), and widely adop ed in he li e a u e in simila wo ks. The MLE me hod allows o he
es ima ion o he median and s anda d de ia ion o he logno mal cumula i e dis ibu ion unc ion,
which a e he wo unknowns in Eq. (2.16). These pa ame e s de ine he agili y unc ion ha mos
likely gene a ed he obse ed p obabili y da a. I is impo an o no e ha , in common applica ions, he
MLE op imisa ion p ocess is i e a ed o de i e mul iple agili y unc ions, each co esponding o
di e en sequen ial damage le els. O he op imisa ion p ocedu es commonly ound in he li e a u e o
da a i ing a e no ecommended o his ype o da a. Fo example, he LSE me hod does no accoun
o he a iance o he obse ed p obabili ies. Fo u he de ails on his opic, he eade is encou aged
o e e o (Bake , 2015).
Ma hema ically, he p obabili y o a aining a ce ain damage le el is gi en by he binomial dis ibu ion,
exp essed as ollows:
𝑃(𝐷𝑘|𝑃𝐺𝐴𝑗)= (𝑛𝑗
𝑧𝑗) 𝑝𝑗𝑧𝑗(1−𝑝𝑗)𝑛𝑗−𝑧𝑗
(2.17)
whe e 𝑛𝑗 deno es he numbe o o al samples, 𝑧𝑗 he numbe o damaged samples, and 𝑝𝑗 he ue
p obabili y o eaching a damage le el o a gi en PGA alue 𝑗 (o bin), calcula ed acco ding o Eq.
(2.16).
To accoun o he di e en PGA alues (o bins), he likelihood unc ion is gi en by he p oduc o he
p obabili ies o each PGA alue as:
𝐿𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑= ∏(𝑛𝑗
𝑧𝑗)
𝑚
𝑗=1 𝜙 (𝑙𝑛(𝑃𝐺𝐴𝑗
𝜃)
𝛽)𝑧𝑗
(
1− 𝜙 (𝑙𝑛(𝑃𝐺𝐴𝐽
𝜃)
𝛽)
)
𝑛𝑗−𝑧𝑗
(2.18)
To ensu e he pa ame e es ima ion wi h he highes p obabili y o ep oducing he obse ed da a, he
maximum alue o he maximum likelihood unc ion is hen compu ed 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 applicabili y o such p ocedu e is e y s aigh o wa d equi ing a e y ew inpu pa ame e s which
a e: (i) he numbe o o al samples (𝑛𝑗); (ii) he numbe o damaged samples (𝑧𝑗); and (iii) he ange
(o bins) o PGA.
I is impo an o no e ha agili y unc ions, when de eloped o dis inc damage g ades, may exhibi a
c oss-beha iou . This occu s when he es ima ed pa ame e s o di e en cu es cause hei
in e sec ion. This beha iou may be due o he da ase being poo ly dis ibu ed ac oss he di e en PGA
bins. Howe e , when he damage g ades a e sequen ial, as in he case o he EMS-98 scale, anging
om D0 o D5, he in e sec ion o he cu es implies ha he p obabili y o exceeding a highe damage
g ade is g ea e han he p obabili y o exceeding a lowe damage g ade, o ce ain alues o PGA.
The e o e, a oiding c ossing o he agili y cu es o sequen ial damage s a es is c i ical o ensu ing
he eliabili y o he esul ing applica ions.
To add ess he in e sec ion o he agili y cu es, wo dis inc app oaches (App oach 1 and 2) ha e
been sugges ed by (Po e , 2021). Bo h a e es ed and discussed in Chap e s 4 and 5. The simples
app oach (App oach 1) equi es a e ision o he pa ame e s pos e io o he sepa a ed i ing o each
damage s a e. The p ocedu e in ol es he compu a ion o a new loga i hmic s anda d de ia ion, equal
o all damage s a es, and new adjus ed median alues, one o each damage s a e. In pa icula , he
ollowing equa ions a e adop ed:
𝛽𝑛𝑒𝑤= 1𝑘 ∑𝛽𝑘
𝑘
𝐷=1
𝑘=[1,..,5]
(2.20)
𝜃𝑛𝑒𝑤= 𝜃𝑘·exp[ 0.842· (𝛽𝑛𝑒𝑤−𝛽𝑘)]
(2.21)
App oach 2 is conside ed mo e ad anced and i in ol es de eloping he agili y cu es simul aneously
using he MLE, esul ing in a cons an loga i hmic s anda d de ia ion ac oss all damage s a es, while
he median a ies, wi h a dis inc alue o each damage s a e. To add ess he issue o agili y cu e
in e sec ions, ecen con ibu ions wo h men ioning include he wo k by (Sis i e al., 2023) o mason y
chu ches and he wo k o (Ta angelo e al., 2024) o o dina y cons uc ions. Ne e heless, he
signi icance o his issue emains somewha unde app ecia ed wi hin he scien i ic communi y.
In he li e a u e, applica ions based on simpli ied app oaches o de elop ypological agili y cu es a e
nume ous (Chie o e al., 2022; Lagoma sino e al., 2021). Al hough agili y unc ions ha e been
mos ly eso ed o PGA as seismic in ensi y measu e, al e na i e applica ions using he mac oseismic
in ensi y can also be ound, as an example in (De Ma eis, B ando, & Co li o, 2019; Lagoma sino &
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
43
Podes à, 2004c; Vicen e e al., 2011). Wi h he ecen p og esses in he ield o moni o ing and
e i o ial scale assessmen , especially in ligh o he ecen seismic expe iences, ea hquake and
damage da a ha e become inc easingly accessible. As a esul , i ing p ocedu es ha e become
s anda d p ac ice o de i ing agili y unc ions o a ious ypes o buildings and s uc u es. While a
subs an ial body o wo k exis s epo ing empi ical agili y cu es o his o ic mason y chu ches (Ce oni
e al., 2022; Cesca i e al., 2020; Ho e e al., 2018; Lagoma sino e al., 2021; Ma o a e al., 2021;
Sis i e al., 2023), limi ed a en ion has been gi en o his o ic mason y owe s, wi h jus a ew
excep ions (Ma o a e al., 2021; Sis i e al., 2023). While al e na i e seismic in ensi y measu es, such
as he SA, a e gaining inc easing scien i ic in e es o he de elopmen o agili y models o ein o ced
conc e e (Rosse o & Elnashai, 2003) and un ein o ced mason y (Gau am e al., 2018; Zucconi e al.,
2020) buildings, hei applica ion o mason y chu ches and his o ic mason y owe s emains ela i ely
limi ed.
2.4 ANALYTICAL METHODOLOGIES
Empi ical me hods ely on obse a ions o ac ual damage om pas ea hquakes o de elop
ulne abili y ela ionships o agili y cu es. While he use o ac ual damage da a makes he esul s
ealis ic and di ec ly linked o obse ed ou comes, his app oach equi es ex ensi e, high-quali y
damage da a, which may no be a ailable in many egions. Mo eo e , his da a, e en when a ailable,
e lec s local cons uc ion adi ions, ma e ials, and seismic cha ac e is ic, hus, could be ha dly
applicable o egions wi h di e en building p ac ices o seismici y and/o could o e look e ol ing
cons uc ion s anda ds o e o i ing e o s. To o e come hese limi a ions, adjus men o he me hods
o no el s a egies based on expe judgmen and quali a i e assessmen s may be explo ed and applied
ac oss di e se egions wi h modi ica ions o sui local condi ions and ackle da a-sca ci y. Howe e ,
hese me hods may be less p ecise and he a iabili y in expe judgmen can lead o inconsis en
ou comes ac oss s udies. A alid al e na i e is o e ed by analy ical me hods. These in ol e de ailed
simula ions o calcula ions o building esponses o seismic ac ions assessing he ulne abili y ac oss a
wide ange o geog aphical a eas, scena ios and s uc u al ypes a a ious le el o de ail, accu acy and
complexi y. While hese me hods add ess all he limi a ions o he p e ious s a egies, hey a e less
simple o implemen , equi ing signi ican compu a ional esou ces, ad anced expe ise in s uc u al
enginee ing and compu a ional modelling. Mo eo e , hei scalabili y is a ec ed by he le el o accu acy
demanded o he models, e lec ed in o he le el o knowledge a ailable o pu sued o he building. The
knowledge pa h a ec s bo h he modelling s age and he seismic demand e alua ion. In gene al, o he
de elopmen o a mechanical model, he ep esen a i e ea u es o he owe s such as geome y (in
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
44
plan and ele a ion), ma e ial cha ac e is ics, bounda y condi ions, and s a e o conse a ion a e
necessa y. Typically, hose can be e ie ed ei he om exis ing documen a ions and codes, o om
obse a ions and measu emen s acqui ed by in si u inspec ions. Rega ding he seismic inpu
cha ac e isa ion, ea u es such as loca ion and soil ype, opog aphy and mo phology need o be
iden i ied o ex ac he code spec a o be used o he es ima ion o he seismic demand. The e o e,
highly eliable models a e sui able o indi idual s uc u es only bu imp ac ical o la ge scale in a
easonable ime ame, due o esou ce cons ain s o collec he in o ma ion equi ed and ca y ou he
analyses. To ensu e hei iabili y o e i o ial scale assessmen , simpli ica ions mus be in oduced a
he isk o o e looking impo an aspec s o he seismic esponse o he building.
To his end, he I alian Guidelines (DPCM, 2011) o he assessmen and mi iga ion o he seismic isk
o he cul u al he i age in oduces a classi ica ion o he modelling s a egies acco ding o h ee
inc easing le els o complexi y. Pa icula a en ion is gi en o he i s le el o e alua ion (LV1), u he
desc ibed he ea e wi h speci ic ocus on mason y owe s, o i s sui abili y o e i o ial scale analysis.
Howe e , a signi ican body o exis ing esea ch in he li e a u e employs mo e accu a e e alua ions
(LV2 and LV3) o his o ic owe s. A hese le els, commonly each s udy in es iga es one o ew case
s udies in de ail. Fo he second le el o e alua ion (LV2), po en ial collapse mechanisms and local
ailu es occu ing in his o ic mason y owe s ha e been analysed h ough limi analysis by (Ba oli,
Be i, & Monche i, 2017; Chisa i e al., 2022; Cu i e al., 2012; Faccio e al., 2011b; Sa hosis e al.,
2018; Tanganelli e al., 2024; To elli e al., 2020) and, mo e ecen ly, using ini e elemen models by
(Degli Abba i e al., 2024; Meh o a e al., 2023; Milani, 2019). Typical mechanisms in his o ic
mason y owe s included in hese s udies a e p esen ed and discussed in Chap e 3. Fo he hi d le el
o e alua ion (LV3), illus a i e applica ions eso ing o ini e elemen models a e (Ba oli, Be i, &
Monche i, 2017; D’Amb isi e al., 2012; Degli Abba i e al., 2024; Micelli & Casca di, 2020; Milani &
Clemen i, 2021; Tanganelli e al., 2024; To elli e al., 2020; Valen e & Milani, 2016). Ad anced 3D
models a e de eloped o accu a ely es ima e he s uc u al esponse, ypically by means o nonlinea
s a ic analysis. Me hodologies, ad an ages and limi a ions o hese wo le els o e alua ions (LV2 and
LV3) a e no he e add essed due o hei ele ance a he indi idual scale analysis. Al hough he i s
le el o e alua ion (LV1) educes he numbe o inpu pa ame e s when compa ed wi h he highe le els
o e alua ion, i s p ac ical use o seismic ulne abili y assessmen a he e i o ial scale emains s ill a
ield o esea ch no ully explo ed, wi h jus a ew examples (Ba oli, Be i, & Monche i, 2017; Chisa i
e al., 2022; Cu i e al., 2008; Sa hosis e al., 2018). This is likely due o he le el o knowledge
equi ed by his seismic sa e y e alua ion, which is o en cons ained by he challenges o add essing
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
45
he unique cha ac e is ics o each owe , pa icula ly hose a ising om geome y complexi ies and
ma e ial a ia ions wi hin each asse (Degli Abba i e al., 2024; Faccio e al., 2011b; Lagoma sino e
al., 2014; Tanganelli e al., 2024; To elli e al., 2020).
The i s le el o e alua ion (LV1) p oposed by he I alian Guidelines (DPCM, 2011), o mason y owe s,
analyses he seismic esponse based on a e y simpli ied mechanical model, namely a can ile e
mason y beam, subjec ed o bo h e ical loads simula ing he dead loads and s a ic ho izon al o ces
simula ing he ea hquake e ec s. Each mason y owe is hen subdi ided in po ions wi h uni o m
geome ical and cons uc i e cha ac e is ics a di e en heigh s, aking in o accoun peculia s uc u al
ea u es, including in e ac ion wi h su ounding buildings, ape ed sec ions and p esence o signi ican
openings.
In his app oach, he esul an o he la e al o ces 𝐹ℎ, ep esen a i e o he seismic demand, is
exp essed as ollows:
𝐹ℎ=0.85𝑆𝑒(𝑇1)𝑊
𝑞𝑔
(2.22)
whe e 𝑆𝑒(𝑇1) is he o dina e o he elas ic esponse spec um o be compu ed as a unc ion o he
undamen al ib a ion pe iod (o na u al equency) o he owe , 𝑊 is he o al weigh o he owe , 𝑞 is
he beha iou ac o and 𝑔 deno es he g a i a ional accele a ion.
This o ce, co esponding o he base shea , is hen applied s a ically in he o m o ho izon al
concen a ed o ces 𝐹𝑖, whose dis ibu ion g ows linea ly along he heigh o he owe , and he e o e:
𝐹𝑖= 𝑤𝑖𝑧𝑖
∑𝑤𝑘𝑧𝑘
𝑛𝑘=1 𝐹ℎ
(2.23)
whe e 𝑤𝑖 and 𝑤𝑘 a e he weigh s o he po ion
i
and
k
espec i ely, 𝑧𝑖 and 𝑧𝑘 a e he heigh s o he
cen e o mass o po ions
i
and
k
wi h espec o he ounda ion.
Subsequen ly, he esul an o he seismic o ces 𝐹ℎ𝑖 ac ing on he i- h sec o a e de e mined as ollows:
𝐹ℎ𝑖= ∑𝑤𝑘𝑧𝑘
𝑛𝑘=i
∑𝑤𝑘𝑧𝑘
𝑛𝑘=1 𝐹ℎ
(2.24)
The heigh 𝑧𝐹𝑖 o which 𝐹ℎ𝑖 is applied can be compu ed as:

CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
46
𝑧𝐹𝑖= ∑𝑤𝑘𝑧𝑘2
𝑛𝑘=i
∑𝑤𝑘𝑧𝑘
𝑛𝑘=i − 𝑧𝐹𝑖∗
(2.25)
whe e 𝑧ℎ𝑖∗ deno es he heigh o he e i ica ion po ion wi h espec o he owe base.
The beha iou o he mason y ma e ial is assumed o ha e no ensile esis ance wi h a non-linea
dis ibu ion o he comp essi e s esses. Unde his assump ion, s uc u al ailu e is expec ed in a
gene ic c oss-sec ion o he owe due o c ushing o a comp essed mason y egion. The ul ima e
esis ing bending momen 𝑀𝑅𝑑 a he base o each po ion is:
𝑀𝑅𝑑,𝑖= 𝜎𝑖𝐴𝑖
2(𝑏𝑖−𝜎𝑖𝐴𝑖
0.85𝑎𝑖𝑓𝑑)
(2.26)
whe e 𝑓𝑑 is he comp essi e s eng h, 𝑎𝑖 and 𝑏𝑖 a e he ans e sal and longi udinal dimensions o he
i- h po ion wi h espec o he di ec ion o he ho izon al o ces, 𝐴𝑖 is he a ea o he c oss-sec ion o
he i- h po ion, 𝜎𝑖= 𝑊𝑖/𝐴𝑖 is he a e age comp essi e s esses o he i- h po ion due o he g a i y
loads, and 𝑊𝑖 is he weigh o he po ion o he owe abo e he analysed c oss-sec ion. To accoun o
all he unce ain ies included in he modelling s age, he ul ima e esis ing bending momen s a e
co ec ed h ough a con idence ac o 𝐹𝐶 ha a ies acco ding o he le el o knowledge o he
s uc u e.
A i s p elimina y e alua ion o de e mine he sa e y le el o he owe s ia simpli ied mechanical
models can be pe o med by compa ing he ac ing bending momen (demand) wi h he ul ima e
esis ing bending momen (capaci y), bo h compu ed along he owe heigh . The sa e y is ensu ed
when he s uc u al capaci y is g ea e han he demand. The demand a each po ion base is:
𝑀𝐸𝑑,𝑖=𝐹ℎ𝑖 𝑧𝐹𝑖
(2.27)
Howe e , o compa a i e assessmen pu poses, he sa e y le el o he owe s is mo e commonly
exp essed in e ms o sa e y index, 𝐼𝑆,𝐿𝑆 and accele a ion ac o , 𝑓𝑎,𝐿𝑆.
The sa e y index, 𝐼𝑆,𝐿𝑆, is he a io be ween he e u n pe iod 𝑇𝑈𝐿𝑆 o he seismic ac ion ha b ings he
s uc u e o he conside ed Limi S a e, commonly he Li e Sa e y Ul ima e Limi S a e (ULS), and he
expec ed e u n pe iod o he ea hquake o he si e, co esponding o he same Li e Sa e y ULS,
𝑇𝑅,𝑈𝐿𝑆, as ollows:
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
47
𝐼𝑆,𝑈𝐿𝑆= 𝑇𝑈𝐿𝑆
𝑇𝑅,𝑈𝐿𝑆 𝑇𝑅,𝑈𝐿𝑆=475 𝑦𝑒𝑎𝑟
(2.28)
The e u n pe iod 𝑇𝑈𝐿𝑆 o he seismic ac ion ha b ings he s uc u e o he ULS can be compu ed as
he e u n pe iod ha de e mines a spec al o dina e such ha he weakes c oss-sec ion o he
s uc u e would mee he ULS h eshold. By se ing he equi alence be ween he ac ing bending
momen and he ul ima e esis ing bending momen , he spec al o dina e o he i- h po ion can be
compu ed as ollows:
𝑆𝑑,𝑈𝐿𝑆,𝑖(𝑇1)=𝑞𝑔𝑀𝑅𝑑𝑖∑𝑧𝑘𝑊𝑘
𝑛𝑖=1
0.85𝑊(∑ 𝑧𝑘2𝑊𝑘
𝑛𝑖=1 −𝑧𝑖∗∑𝑧𝑘𝑊𝑘
𝑛𝑘=i ) 𝐹𝐶
(2.29)
Once he minimum SA is iden i ied among he po ions conside ed o he owe , he e u n pe iod is
compu ed acco ding o an i e a i e p ocedu e eso ing o a linea in e pola ion. This in e pola ion elies
on da a epo ed in he Appendix o he I alian building code, whe e alues needed o he calcula ion o
he spec um a e epo ed o each poin o he opog aphic ne wo k and o inc easing e u n pe iod,
om 30 o 2475 yea s.
On he o he hand, he accele a ion ac o , 𝑓𝑎,𝐿𝑆, is de ined as he a io be ween he ho izon al
maximum g ound accele a ion ha b ings he s uc u e o he ULS, 𝑎𝑈𝐿𝑆, and he e e ence g ound
accele a ion associa ed wi h he ULS, 𝑎𝑔,𝑈𝐿𝑆, as ollows::
𝑓𝑎,𝑈𝐿𝑆=𝑎𝑈𝐿𝑆
𝑎𝑔,𝑈𝐿𝑆
(2.30)
While he es ima ion o 𝑎𝑔,𝑈𝐿𝑆 is s aigh o wa d, 𝑎𝑈𝐿𝑆 being unc ion o he esponse spec um
o dina e, can be ob ained by in e ing he elas ic esponse spec um o mulas, as ollows:
𝑎𝑈𝐿𝑆=
{
𝑆𝑑,𝑈𝐿𝑆
𝑆𝐹0 𝑇𝐵≤𝑇1≤ 𝑇𝐶
𝑆𝑑,𝑈𝐿𝑆
𝑆𝐹0𝑇1
𝑇𝐶 𝑇𝐶≤𝑇1≤ 𝑇𝐷 𝑆𝑑,𝑈𝐿𝑆=𝑆𝑒,𝑈𝐿𝑆
𝑞
(2.31)
A gene al amewo k o he de elopmen o simpli ied mechanical models is illus a ed in Figu e 2.14.
Se e al applica ions o he LV1 app oach o he seismic ulne abili y assessmen o mason y owe s
ha e been discussed in he li e a u e, o en o compa e i s e ec i eness wi h me hods p esen ing highe
le el o accu acy and o es and alida e he s a egies admi ed by he I alian s anda d (Ba oli, Be i, &
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
48
Monche i, 2017; Degli Abba i e al., 2024; Faccio e al., 2011a; Tanganelli e al., 2024; To elli e al.,
2020). A i s illus a i e applica ion was gi en by (Faccio e al., 2011b) who assessed he seismic
ulne abili y o he bell owe o he S . An onin chu ch in Venice h ough he sa e y index and
accele a ion ac o associa ed wi h ULS. Di e en assump ions in he bounda y condi ions o in es iga e
he e ec o adjacen buildings ha e been made when compu ing he sa e y alue.
Figu e 2.14: LV1 app oach sugges ed by he I alian Guidelines o slende mason y s uc u es.
Ano he no ewo hy applica ion was conduc ed wi hin he scope o he esea ch p ojec RISEM, Seismic
Risk o Monumen al Buildings, (Ba oli, Be i, & Monche i, 2017), ha aimed a e alua ing he seismic
sa e y indexes and accele a ion ac o s o ou monumen al mason y owe s in San Gimignano. The
e alua ion included he in luence o he adjacen buildings by conside ing di e en heigh s o he owe s
and by es ima ing he undamen al pe iod o he owe s using dis inc simpli ied equa ion. In add essing
he seismic ulne abili y o he Cugnanesi owe , also loca ed in San Gimignano, (To elli e al., 2020)
e alua ed he sa e y index by examining he e ec o he unce ain ies in he isual iden i ica ion o he
mason y ypology. Following he app oach ou lined in he I alian guidelines, his led o a ia ions in he
CHAPTER 2. SEISMIC RISK ASSESSMENT: STATE OF THE ART
49
assumed mason y ma e ial p ope ies, esul ing in signi ican a iabili y in he esul s. A mo e ecen
wo k, ca ied ou by (Tanganelli e al., 2024) e alua ed he sa e y indexes o he Gio o Bell owe in
Flo ence using di e en simpli ied equa ions o he es ima ion o he i s na u al equency and
assessing hei in luence on he de ini ion o he seismic demand. Finally, (Degli Abba i e al., 2024)
adop ed he LV1 app oach o he global assessmen o he bell owe o Sain Law ence Ca hed al
elying on he expe imen ally iden i ied na u al equencies h ough ambien ib a ion es ing, and
analysing he impac o di e en bounda y condi ions. Al hough he me hod has been ex ensi ely es ed
on I alian case s udies, whe e i is ecommended by he na ional s anda d, some applica ions o
he i age asse s in o he coun ies can also be ound in he li e a u e, as in (Magalhães e al., 2012).
E en hough he LV1 app oach in he li e a u e mainly add esses he assessmen o single owe s, a
ew s udies ha e discussed he applica ions o his simpli ied modelling s a egy a he e i o ial scale
(Chisa i e al., 2022; Cu i e al., 2008).
31 bell owe s s uck by he 1976 F iuli Ea hquake we e in es iga ed h ough his me hod by (Cu i e
al., 2008). In pa icula , he accele a ion ac o was e alua ed o each owe and he esul s we e
compa ed wi h he damage le els obse ed in he pos -ea hquake su eys. The compa ison showed a
clea disc epancy as owe s wi h g ea e obse ed damage le els we e cha ac e ised, by con as , by
highe alues o sa e y indexes, highligh ing he limi a ion o he simpli ied mechanical app oach. Ve y
ecen ly, (Chisa i e al., 2022) p oposed a no el simpli ied ulne abili y model sui able o e i o ial
scale assessmen o his o ic mason y owe s. In pa icula , he de elopmen o his me hod equi ed he
cons uc ion o a pa ame ic mechanical model, based on an enhanced e sion o he LV1 app oach, o
in es iga e he owe collapse o a wide numbe and ange o pa ame e s a ec ing i , in o de o assess
hei in luence and p o ide a educed se o he mos signi ican ones. The de ini ion o he pa ame e s
and hei ange ollowed a de ailed su ey o 56 his o ic mason y owe s in he a ea o he ci y o
Naples, and he inal me hod was es ed agains 31 owe s ound in he li e a u e.
The analysis o he exis ing s udies shows ha he seismic sa e y e alua ion using LV1 bene i s om
employing mul iple analy ical models and sensi i i y analysis, p o iding a mo e de ailed unde s anding
o he e ec s o unce ain ies and a ying assump ions on he es ima ed s uc u al pe o mance. This is
o en ocused on he bounda y condi ions (Ba oli, Be i, & Monche i, 2017; Faccio e al., 2011a), he
i s na u al equency (Faccio e al., 2011a; Tanganelli e al., 2024) o he mason y mechanical
p ope ies (To elli e al., 2020), especially when hei di ec e alua ion based on expe imen al es s is
no a ailable.
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
56
classi ica ion in i e dis inc g oups o inc easing le els o MCS in ensi y: G oup I includes 40 owe s
wi h 5≤𝐼𝑀𝐶𝑆<6, G oup II comp ises 29 owe s wi h 6≤ 𝐼𝑀𝐶𝑆<7, G oup III is composed o 25
owe s cha ac e ised by 7≤𝐼𝑀𝐶𝑆<8, G oup IV includes 22 owe s wi h 8 ≤𝐼𝑀𝐶𝑆<9, and inally
G oup V consis s o 13 owe s wi h 𝐼𝑀𝐶𝑆 equal o o g ea e han 9. I should be no ed ha whene e
he MCS in ensi y associa ed wi h he owe loca ion was no a ailable in he sou ces lis ed in Table 3.2,
he in ensi y associa ed wi h he Municipali y whe e he owe is loca ed was assigned.
Figu e 3.4: Map highligh ing he owe dis ibu ion g ouped by MCS in ensi y.
To gi e mo e insigh in o he seismic beha iou o he his o ic mason y owe s included in he da abase,
educing he sca e in hei esponse and damage le el, he in es iga ed s uc u es we e classi ied
acco ding o he owe ypology. In his ega d, as men ioned in Chap e 2, he owe s can be gene ally
g ouped in o h ee ca ego ies acco ding o hei plan con igu a ion (o plan layou ): isola ed owe s,
bounded owe s, and gable bell owe s. The bounded owe s can be addi ionally classi ied in o
subg oups acco ding o hei loca ion in he agg ega e (i.e., con ined and in eg a ed) as in (Ce oni e al.,
2022; Sepe e al., 2008). S ill, due o he limi ed numbe o owe s and in o ma ion a ailable o hei
cha ac e isa ion, an excessi e agmen a ion o he da ase is a oided o ensu e ha all subse s a e
s a is ically ep esen a i e. The e o e, he owe s we e g ouped in wo main classes only: isola ed and
bounded. This ypological classi ica ion was conduc ed based on he ga he ed documen a ion and using
he (Google Ea h P o, 2022) so wa e in case o missing in o ma ion in he in es iga ed s udies. I

CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
57
should be no ed ha he bell gable ypology was no conside ed, gi en he ocus o he hesis on his o ic
owe s and slende s uc u es.
The damage mechanisms and ex en s o each his o ic mason y owe included in he p esen da abase
we e collec ed om he esul s o pos -ea hquake su ey ac i i ies conduc ed by ield enginee s and
esea che s ocusing on mason y chu ches using he mac oelemen app oach, acco ding o he I alian
Guidelines o Cul u al He i age (DPCM, 2011). Mac oelemen s can be ac i a ed du ing he g ound
shaking leading o di e en ailu e mechanisms. The e o e, di e en classi ica ions o mac oelemen s
ha e been p oposed and included in o su ey o ms o e ime. Conce ning he owe as a single
mac oelemen , a di ision in o mechanisms a ec ing i s main componen s, namely sha (o main body)
and bel y, as shown in Figu e 3.5, was in oduced and sys ema ically adop ed, simpli ying he
collec ion o in o ma ion om he sou ces o he da abase.
(a)
(b)
Figu e 3.5: Failu e mechanisms associa ed wi h he owe mac oelemen : (a) owe mechanism (Mechanism A)
and (b) bel y mechanism (Mechanism B).
The wo dis inc ailu e mechanisms in Figu e 3.5, highligh ed as Mec27 and Mec28 in he las e sion
o he A-DC su ey o m o he seismic damage o chu ches, p o ided by he I alian Ci il P o ec ion
(DPCM, 2011), a e in his wo k enamed Mechanism A and Mechanism B, espec i ely. A mechanism
o non-s uc u al eme ging elemen s (lis ed as Mec26 o he same o m) eco ds damage o he spi es,
pinnacles, c enula ions and o he deco a ions o he owe o i s bel y. This mechanism is no included
in he da a collec ion as he in o ma ion a ailable does no allow o clea ly dis inguish when he
su eyo s iden i ied his damage in he owe and when i a ec ed o he eme ging elemen s o he
chu ch. This is a well-known in insic limi a ion o he A-DC su ey o m ha could no be add essed by
he p esen wo k. Addi ionally, i is wo h no ing ha , al hough his o ic owe s a e appa en ly egula
s uc u es ypically wi h a hea y weigh ha may be bene icial o s uc u al s abili y, expe ience shows
ha , when subjec ed o g ound shaking, hey exhibi damage pa e ns ha can a y widely due o
speci ic s uc u al ea u es (e.g., he slende ness, he p esence o openings, he con igu a ion o he
bel ies, he in e ac ion wi h adjacen buildings and soil).
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
58
The a o emen ioned Mechanisms A and B, indeed, collec a se ies o ypical ailu e mechanisms ha
occu ed in his o ic mason y owe s (Figu e 3.6). as ex ensi ely discussed in (Doglioni e al., 1994).
Figu e 3.6: Typical damage mechanisms obse ed in his o ic mason y owe s.
In pa icula , se e al combina ions o damage pa e ns can a ec he main body o he owe . In-plane
shea c acking may lead o a ocking mechanism, hus a ec ing he en i e supe s uc u e. The c ack
pa e ns and he subsequen o ma ion o a hinge depend on he o ien a ion o he shea planes, as
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
59
hey can o m a ho izon al axis o o a ion along one side, con e ge owa ds a co ne o agmen he
panels upon he o ma ion o double-diagonal (X- ype) c acks. Ve y a e is he case o ailu e due o
sliding. None heless, some examples o owe s ha su e ed his ype o damage exis , showing
ema kable c ack pa e ns bu wi hou causing o al ailu e. Finally, he sha may ail due o c ushing.
C acking in he bel y is likely he mos common damage obse ed in bell owe s. In mos cases, he
damage is due o ocking o he uppe pa because o he o ma ion o ho izon al c acks in he
mason y pilla s o he bel y, o en in ol ing some de achmen o mason y co ne s in he le el below. In
some cases, he owe can unde go a shea ailu e igh below he bel y combining he wo
mechanisms. O he possible combina ions o mechanisms depend on he dis ibu ion o he openings.
Indeed, e ical c acks may occu in he p esence o aligned openings and lead o spli ing o he owe .
The iden i ica ion o he wo mechanisms (i.e., Mechanism A and Mechanism B) o mason y bell
owe s da es back o he seminal s udies on he mac oelemen beha iou o mason y chu ches, in he
a e ma h o he 1976 F iuli Ea hquake (Doglioni e al., 1994). The mason y owe s a ec ed by such a
se e e e en and included in he da abase ha e been in es iga ed by (Cu i e al., 2008), acco ding o
his dis inc ion in he owe and he bel y mechanisms. Fo he 1997 Umb ia-Ma che Ea hquake and
he 2002 Molise Ea hquake, in o ma ion abou he damage o he owe s de i ed om echnical
epo s, (Spence e al., 1998) and (Ci ani e al., 2005), espec i ely. Fo he ollowing e en s included in
he da abase, he damage eco ds a e ex ac ed om scien i ic publica ions. In pa icula , o he 2009
L’Aquila Ea hquake, he main sou ces we e (Augen i & Pa isi, 2010; B andonisio e al., 2013; C ibe e
al., 2015). Fo he owe s a ec ed by he 2012 Emilia Ea hquake, ele an in o ma ion is p o ided by
(Fe a i, 2020; Paupé io e al., 2012; So en ino e al., 2014; Valen e e al., 2017). Finally, he damage
obse ed in he owe s a e he 2016 Cen al I aly Ea hquake was ob ained om he wo ks o (Aci o e
al., 2021; Clemen i e al., 2020; De Ma eis & Zizi, 2019; Gio dano e al., 2019; Jain e al., 2020).
Unlike he classi ica ion o he mechanisms ecu ing among di e en sou ces, ega ding he ex en o
he damage, documen s e e ing o di e en ea hquakes ypically adop di e en disc e e scales, un il
he di usion and adop ion o he EMS-98 six-le el scale (G ün hal, 1998), wi h he damage anging om
0 up o 5. The e o e, an essen ial ask conduc ed du ing he o ma ion o he p esen da ase consis ed
in he eclassi ica ion o he damage le el, based on he g ade assigned in he o iginal sou ces and/o
o he a ailable in o ma ion, such as desc ip ions in he ex s, pho os o d awings.
Fo a ound 65% o he iden i ied case s udies, he a o emen ioned sou ces p o ided he damage le els,
e.g., (C ibe e al., 2015; Cu i e al., 2008; Fe a i, 2020; So en ino e al., 2014), while in he
emaining 35% he damage le el was assigned based on he isual inspec ion o he ga he ed
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
60
pho og aphic documen a ion, e.g., (Augen i & Pa isi, 2010; B andonisio e al., 2013; Ci ani e al.,
2005), acco ding o he EMS-98 (G ün hal, 1998). These cases a e highligh ed in he da abase wi h he
le e
c
(Appendix A)
.
Finally, i should be no ed ha o a ew owe s included in he da abase no damage le el was epo ed
associa ed wi h one o bo h ailu e mechanism(s) (Appendix A). These cases o which no clea
in o ma ion was ob ained om he sou ces a e p ope ly highligh ed in he da abase able, since i was
no possible o con i m he p esence o he speci ic po ion o mac oelemen o he ac i a ion o he
mechanism and, in case, i s se e i y. These owe s belong o he 1976 F iuli and 2012 Emilia
ea hquakes a ec ing he en i e ange o MCS in ensi ies, speci ically be ween 5 and 9. The inal
numbe o owe s is epo ed in Table 3.3, including he subg oups associa ed wi h each ea hquake,
each MCS in ensi y g oup and each ailu e mechanism (Mechanism A, Mechanism B).
Table 3.3: Numbe o owe s included in he da abase o which a damage sco e has been assigned, g ouped by
ea hquake e en , damage mechanism and mac oseismic in ensi y.
The dis ibu ion o he inal numbe o owe s ob ained by conside ing hei co esponding seismic
e en s and MCS in ensi y is gi en in Figu e 3.7, o he case o damage mechanism o he body o he
owe (Mechanism A). The dis ibu ion o he owe s a ec ed by he damage mechanism o he bel y
(Mechanism B) is e y simila , hus, i is no epo ed o he sake o b e i y. Many owe s included in
he da abase a e associa ed wi h he F iuli, he Molise and he Emilia e en s, while a ew o hem a e
a ec ed by he Umb ia-Ma che, he L’Aquila, and he Cen al I aly ea hquakes. Addi ionally, many
owe s we e a ec ed by a MCS in ensi y ange be ween 5 and 7.5. These owe s e e mainly o he
Umb ia-Ma che, he Molise, he Emilia, and he Cen al I aly e en s. In he emaining ange (MCS
Ea hquake
Damage
Mechanism
Mac o-seismic In ensi y MCS
To al
5.0/5.5
6.0/6.5
7.0/7.5
8.0/8.5
9.0/9.5
10.0/10.5
F iuli (1976)
Mechanism A
0
0
2
12
8
0
22
Mechanism B
0
0
2
13
8
0
23
Umb ia-Ma che
(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
Cen al I aly (2016)
Mechanism A
7
2
0
0
0
2
11
Mechanism B
7
2
0
0
0
2
11
To al
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
in ensi y 8-10.5), he numbe o owe s is mainly gi en by he F iuli, he L’Aquila and he Cen al I aly
ea hquakes.
Figu e 3.7: Dis ibu ions o owe s a ec ed by Mechanism A acco ding o ea hquake e en and MCS in ensi y.
The dis ibu ion o he analysed owe s dis inguishing hem in isola ed and bounded is gi en in Figu e
3.8, e ealing ha mos , among he in es iga ed building s ock, a e bounded owe s and ha he
isola ed owe s da a mainly belong o he F iuli and Emilia ea hquakes; he dis ibu ion in Figu e
3.8Figu e 3.8:a e e s o owe s a ec ed by he sha mechanism (Mechanism A), whe eas he
dis ibu ion in Figu e 3.8b e e s o owe s a ec ed by he bel y mechanism (Mechanism B).
Figu e 3.8: Dis ibu ions o he owe s acco ding o ea hquake e en and s uc u al ypology: owe s a ec ed by
Mechanism A (a), and owe s a ec ed by Mechanism B (b).
The isola ed and bounded owe s we e hen so ed acco ding o he MCS in ensi y. Table 3.4 ou lines
he o al numbe o he isola ed and bounded owe s wi hin each in ensi y g oup, oge he wi h he
dis inc damage mechanisms (Mechanisms A and B). As can be obse ed, simila dis ibu ions a e
Cen al I aly 2016
Emilia 2012
L'Aquila 2009
Molise 2002
Umb ia-Ma che 1997
F iuli 1976
0
5
10
15
20
MCS In ensi y
No. Towe s
Isola ed
Bounded
0
10
20
30
No Towe s
Isola ed
Bounded
0
10
20
30
No Towe s
(a)
(b)

CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
62
ob ained o he owe s a ec ed by dis inc damage mechanisms. Once again, i is wo h highligh ing
ha , due o missing in o ma ion, he numbe o owe s o which he ac ual damage le el is known o
Mechanisms A and B does no always co espond and bo h a e less han he en i e size o he da ase ,
as shown in he ollowing able.
Table 3.4: Numbe o owe s included in he da abase.
Plan
Layou
Mac o-seismic MCS In ensi y
To al
Known
To al in
Da abase
5.0/5.5
6.0/6.5
7.0/7.5
8.0/8.5
9.0/9.5
10.0/10.5
Isola ed
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
To al
Mechanism A
37
24
21
18
8
2
110
129
Mechanism B
37
26
20
19
8
2
112
129
A e he iden i ica ion o he damage le els obse ed in he dis inc mechanisms, he global
assessmen o he owe as a single mac oelemen was ca ied ou using he me hod adop ed in simila
s udies which ollows he weigh ed sum c i e ion, as men ioned in Sec ion 2.3 and in (Tes a e al.,
2024). The global damage le el o he mac oelemen , he ea e called Mac oelemen damage, was
compu ed only o he owe s whose damage le els ( anging om 0 o 5) a e a ailable o bo h damage
mechanisms (i.e., 106 owe s ou o he o al 129). When he damage le el ela ed o one o he wo
dis inc ailu e mechanisms was no known (i.e., emp y cell in he da abase), he global damage le el
was no compu ed. I is s essed ha an emp y cell does no indica e absence o damage, ins ead
indica es lack o in o ma ion.
3.3 DAMAGE PROBABILITY MATRICES (DPMs)
The DPMs we e p oduced o each combina ion and g ouping he owe s discussed in he p e ious
Sec ion. The DPMs we e also compa ed wi h he dis ibu ion o he p obabili y o damage es ima ed
using he Binomial Densi y P obabili y Func ion (BDPF) a a gi en le el, 𝑝𝑘, based on he knowledge o
he obse ed mean damage 𝜇, as ollows:
𝑝𝑘=5!
𝑘! (5−𝑘)!∙( 𝜇5 )𝑘∙( 1−𝜇5 )5−𝑘 𝑘=[0,..,5]
(3.1)
3.3.1 En i e da ase and indi idual e en s
The obse ed DPMs and he ela ed BDPFs a e epo ed in Figu e 3.9, conside ing he en i e se (i.e.,
5≤𝐼𝑀𝐶𝑆<11). Simila alues eme ge o he mean damage associa ed o he wo damage
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
63
mechanisms and o he whole owe mac oelemen : 2.69 and 2.53 o Mechanism A and Mechanism
B, espec i ely, and 2.77 o Mac oelemen damage.
Mechanism A
Mechanism B
Mac oelemen damage
ENTIRE SET
(a)
(b)
(c)
Figu e 3.9: DPMs o dis inc damage mechanisms (a and b), and o he en i e mac oelemen (c).
Al hough simila alues o he obse ed mean damage a e ob ained, he dis ibu ions o he collec ed
le els o damage a e qui e di e en . In pa icula , o Mechanism A, a unimodal (i.e., a single peak)
almos symme ical dis ibu ion o he damage wi h a peak a le el 3 eme ges, while Mechanism B
exhibi s a g owing end in he dis ibu ion wi h a la ge numbe o owe s p esen ing a damage sco e
equal o 5. This end sugges s a highe suscep ibili y o se e e damage o he bel y when compa ed
wi h he sha . The las case, Mac oelemen damage, e eals an almos uni o m dis ibu ion o damage
wi h sligh ly la ge occu ence o D3 and sligh ly mo e cases wi h low damage han se e e damage. The
displayed damage his og ams o Mechanism A and Mac oelemen damage ha e an accep able i by
he binomial dis ibu ion, whe eas he dis ibu ion o Mechanism B does no seem o ma ch he BDPFs.
The alue ob ained o he global mean damage is la ge han he indi idual mechanisms mean
damage. To be e in es iga e his unexpec ed end, he analysis was epea ed by aking in o accoun
he same numbe o owe s (i.e., 106) ins ead o he 110 and 112 owe s o iginally conside ed o
Mechanism A and Mechanism B, espec i ely. Thus, all he owe s o which no in o ma ion was
a ailable o one o he wo mechanisms we e excluded om he calcula ion.
Mechanism A
Mechanism B
Mac oelemen damage
ENTIRE SET
(a)
(b)
(c)
Figu e 3.10: DPMs o dis inc damage mechanisms (a and b), and o he en i e mac oelemen (c) conside ing
106 owe s only.
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
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
Mac oelemen damage
FRIULI (1976)
UMBRIA-MARCHE
(1997)
MOLISE (2002)
Figu e 3.11: DPMs o he in es iga ed owe s de eloped o F iuli, Umb ia-Ma che and Molise e en s, and by
conside ing bo h he damage le els obse ed in each damage mechanism and mac oelemen .
The new DPMs and he BDPFs a e illus a ed in Figu e 3.10, which p o ide he same unexpec ed
esul . No signi ican a ia ion in he mean damage alues and co esponding damage dis ibu ions
eme ges and he mean global damage con inues o be la ge han he mean damage ob ained o he
single indi idual mechanisms. This is likely asc ibed o he equa ion used o combine he damage le els
in o he global damage index (Eq. 2.3) and o he ela ed co ela ion wi h he global damage le el (Table
2.2). The o e es ima ion o he global damage wi h espec o he a e age o he indi idual
mechanisms’ alues p o ided by his app oach, when adop ed o he case o owe s, is likely
in luenced by he weigh assigned o each mechanism and by he co ela ion used o ans o m he
con inuous global damage index in o he disc e e global damage le el. Al e na i e app oaches o he
weigh ed sum c i e ion ha e been p oposed in li e a u e, especially o ein o ced conc e e s uc u es. A
s udy ha ocuses on he e alua ion o how he global damage index using he exis ing app oaches
a ec he damage le el es ima es is gi en in (Zucconi e al., 2022). Howe e , in he p esen s udy, bo h
weigh s and co ela ion a e main ained as in he o iginal app oach by (Lagoma sino & Podes à, 2004b,
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
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 simila wo ks as in (De Ma eis & Zizi, 2019); a ia ions o his app oach a e le as
u u e scopes o he esea ch. Conside ing possible di e ences in he owe s beha iou due o he
peculia cha ac e is ics o each ea hquake and local cons uc ions, addi ional DPMs we e de eloped by
aking in o accoun he ea hquake e en s indi idually a e shown in Figu e 3.11 (F iuli, Umb ia-Ma che
and Molise) and Figu e 3.12 (L’Aquila, Emilia and Cen al I aly).
Mechanism A
Mechanism B
Mac oelemen damage
L’AQUILA (2009)
EMILIA (2012)
CENTRAL ITALY (2016)
Figu e 3.12: DPMs o he in es iga ed owe s de eloped o L’Aquila, Emilia and Cen al I aly e en s and by
conside ing bo h he damage le els obse ed in each damage mechanism and he global damage le els.
The DPMs de eloped by classi ying he in es iga ed owe s based on he se o ea hquakes aim o
pe o m an in-dep h examina ion o he owe s' beha iou and condi ion agains he co esponding
seismic e en hey expe ienced. The obse ed damage, o he cases o he Umb ia-Ma che and he
Molise ea hquakes, is mos ly cha ac e ised by occu ences o no damage o small damage ex en s. Fo
he o me e en , he limi ed numbe o known owe s in he da abase may a ec he e alua ion,
al hough he his og am shows a clea end. Fo he Cen al I aly e en , he obse ed damage is likely
no mally dis ibu ed, especially o Mechanism A and Mac oelemen damage. Fo he emaining cases
(i.e., he Emilia, he F iuli and L’Aquila ea hquakes), he obse ed damage is le -skewed dis ibu ed,
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
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
Mac oelemen damage
5≤𝐼𝑀𝐶𝑆<6
(a)
(b)
(c)
6≤𝐼𝑀𝐶𝑆<7
(d)
(e)
( )
7 ≤𝐼𝑀𝐶𝑆<8
(g)
(h)
(i)
8 ≤𝐼𝑀𝐶𝑆<9
(l)
(m)
(n)
𝐼𝑀𝐶𝑆≥9
(o)
(p)
(q)
Figu e 3.14: DPMs o di e en MCS in ensi ies.
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
0,0
0,2
0,4
0,6
0,8
1,0
D0 D1 D2 D3 D4 D5
Damage occu ence
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 o he owe mechanism (Mechanism A), g ouped by mac oseismic MCS in ensi y.
Sou ce
Seismic E en
Obse ed Damage Le el (𝒅𝒌)
D0
D1
D2
D3
D4
D5
𝐼𝑀𝐶𝑆=5
Au ho s’ da abase
(see Sec ion 3.2)
11%
14%
16%
22%
32%
5%
(Canu i e al., 2021)
Cen al I aly (2016)
60%
16%
24%
0%
0%
0%
𝐼𝑀𝐶𝑆=6
Au ho s’ da abase
(see Sec ion 3.2)
33%
8%
21%
21%
4%
13%
(Canu i e al., 2021)
Cen al I aly (2016)
58%
18%
12%
8%
4%
0%
𝐼𝑀𝐶𝑆=7
Au ho s’ da abase
(see Sec ion 3.2)
14%
5%
19%
24%
19%
19%
(Canu i e al., 2021)
Cen al I aly (2016)
50%
22%
10%
5%
2%
11%
𝐼𝑀𝐶𝑆=8
Au ho s’ da abase
(see Sec ion 3.2)
0%
6%
6%
44%
27%
17%
(Canu i e al., 2021)
Cen al I aly (2016)
9%
9%
0%
39%
22%
22%
Table 3.10: DPMs o he bel y mechanism (Mechanism B), g ouped by mac oseismic MCS in ensi y.
Sou ce
Seismic E en
Obse ed Damage Le el (𝒅𝒌)
D0
D1
D2
D3
D4
D5
𝐼𝑀𝐶𝑆=5
Au ho s’ da abase
(see Sec ion 3.2)
27%
3%
22%
22%
11%
15%
(Canu i e al., 2021)
Cen al I aly (2016)
65%
21%
10%
4%
0%
0%
𝐼𝑀𝐶𝑆=6
Au ho s’ da abase
(see Sec ion 3.2)
35%
15%
12%
8%
15%
15%
(Canu i e al., 2021)
Cen al I aly (2016)
48%
21%
15%
12%
2%
2%
𝐼𝑀𝐶𝑆=7
Au ho s’ da abase
(see Sec ion 3.2)
25%
10%
5%
10%
25%
25%
(Canu i e al., 2021)
Cen al I aly (2016)
61%
10%
12%
8%
7%
2%
𝐼𝑀𝐶𝑆=8
Au ho s’ da abase
(see Sec ion 3.2)
16%
16%
5%
5%
21%
37%
(Canu i e al., 2021)
Cen al I aly (2016)
9%
19%
27%
9%
27%
9%
3.4 VULNERABILITY FUNCTIONS
As an al e na i e o he DPMs, ulne abili y cu es a e a powe ul ool o apidly es ima e he expec ed
mean damage, 𝜇, o a gi en le el o mac oseismic in ensi y, 𝐼𝑀𝐶𝑆. To build a ulne abili y model, he
unc ion o iginally p oposed by (Lagoma sino & Gio inazzi, 2006; Lagoma sino & Podes à, 2004c) was
he e adop ed, as ollows:
𝜇=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+6.25𝑉𝐼−13.1
𝑄)]
(3.2)
whe e 𝜇 is co ela ed wi h inc easing le els o 𝐼𝑀𝐶𝑆 h ough wo physical pa ame e s in insic o a
speci ic in es iga ed asse ypology, componen o subclass, namely he duc ili y index,
Q
, and he
ulne abili y index, 𝑉𝐼.
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
74
In pa icula , alues o ulne abili y and duc ili y indexes o his o ic mason y bell owe s we e
ecommended by (Cu i e al., 2008; Lagoma sino e al., 2004) and ha e been he e used as a
e e ence, see Table 3.11.
Table 3.11: Vulne abili y and duc ili y indexes o his o ic mason y owe s.
𝑽𝑰
𝑸
Sou ce
Damage Mechanism
Mechanism A
0.890
2.00
(Cu i e al., 2008) (1)
Mechanism B
0.940
1.49
(Cu i e al., 2008) (2)
En i e Mac oelemen
Mac oelemen damage
0.776
2.30
(Lagoma sino e al., 2004)
As al eady men ioned in he s a e o he a , (Lagoma sino e al., 2004) p oposed a ulne abili y
unc ion, also ecen ly epo ed by (Despo aki e al., 2018) o mason y owe s and slende buildings,
no speci ically o bell owe s, he e used as e e ence o he en i e owe mac oelemen only, while
(Cu i e al., 2008) de eloped ulne abili y unc ions speci ically add essing all he damage mechanisms
o owe s. I should be no ed ha he modi ie pa ame e s o he ulne abili y index, discussed in
Chap e 2, ha e no been aken in o accoun due o he lack o da a.
Figu e 3.15 shows he ulne abili y cu es ob ained by applying hese exis ing equa ions agains he
p esen da ase . The cu es a e compa ed wi h he alues o mean damage obse ed in he
in es iga ed owe s o di e en g oups o MCS in ensi y, u he g ouped by seismic e en s, h ough a
colou key.
I should be no ed ha he owe s cha ac e ised by maximum MCS in ensi y ha e been dis inguished
in o wo subg oups: 9 ≤𝐼𝑀𝐶𝑆<10 and 10 ≤𝐼𝑀𝐶𝑆<11. The plo s also epo he alues o he
mean damage conside ing he ull se o ea hquakes o each g oup o MCS in ensi y as he Global
Mean. This alue was p e e ed o alida ion and calib a ion pu poses o o e come a possible bias
gi en by he di e en numbe o owe s obse ed in dis inc ea hquakes. None heless, his app oach
could no comple ely sol e he unexpec ed high mean damage in he ange o 5 ≤𝐼𝑀𝐶𝑆<6 and he
ex emely low mean damage in he ange o 10 ≤𝐼𝑀𝐶𝑆<11. Samples in hese anges may be no
ep esen a i e o he expec ed eal popula ion o damaged owe s; he e o e, o he calib a ion o new
ulne abili y cu es, hey we e disca ded.
The exis ing p edic i e unc ions o he indi idual damage mechanisms i well he obse ed mean
damage, especially in he ange 6 ≤𝐼𝑀𝐶𝑆<10. A ce ain sca e be ween he p edic ed and he
obse ed damage le els eme ges bu he same is educed when he Global Mean alues a e
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
75
conside ed. The p edic i e unc ion o he en i e mac oelemen , ins ead, appea s o be non-
conse a i e, p oducing an unde es ima ion o he 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)
Mac oelemen 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)
Figu e 3.15: Compa ison be ween he exis ing ulne abili y unc ions and he obse ed mean damage o dis inc
mechanisms (a and b) and o he en i e mac oelemen (c).
In ligh o hese conside a ions, u he analyses we e conduc ed o i he obse ed mean damage
using he well-known o mula ion o he ulne abili y cu e bu ecalib a ing i s pa ame e s. Ini ially, he
calib a ion ocused on he pa ame e s α
,
β
, Q
, as shown in Eq. (3.7), one by one, while keeping he
ulne abili y index om li e a u e. These pa ame ic analyses pe o med on he ulne abili y equa ions
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
76
p o ided a g aphic ool use ul o iden i y he bes cu e acco ding o speci ic da a in hand. This
p ocedu e is simila o (De Ma eis, B ando, & Co li o, 2019).
𝜇=2.5 [1+𝑡𝑎𝑛ℎ(𝐼𝑀𝐶𝑆+𝛼𝑉𝐼−𝛽
𝑄)]
(3.7)
The esul s ob ained o he ulne abili y cu es a ying he pa ame e s o e a wide ange a e illus a ed
in Figu e 3.16. The lowe and uppe bound o he pa ame e s (α
,
β
, Q
) o he h ee exis ing
o mula ions and he in e als in hei a ia ion, conside ed o he de ini ion o he cu es a e epo ed
in Table 3.12 (he e ∆ is he s ep adop ed o he di e en cu es shown). I can be obse ed ha while
he a ia ions o he pa ame e s α and β p oduce a shi o he ulne abili y cu es, whe e he e ec on
he cu e o inc easing α
co esponds o he e ec o educing β
and ice- e sa, he a ia ion o he
duc ili y index
Q
gene a es a change in i s slope. Among he calib a ion coe icien s, α and β play a
majo ole in i ing he cu es o he p esen da ase .
Mechanism A
(a)
(b)
(c)
Mechanism B
(d)
(e)
( )
Mac oelemen damage
(g)
(h)
(i)
Figu e 3.16: Pa ame ic analyses o he exis ing ulne abili y unc ions o dis inc damage mechanisms, (a) o ( ),
and o he whole mac oelemen , (g) o (i).
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
77
Table 3.12: Cons ain s adop ed in he analysis.
Ini ial alues
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 (Mac oelemen damage)
2.30
1.30
3.30
0.20
Upon his p elimina y sensi i i y analysis, a eg ession analysis was conduc ed o de elop new unc ions
i ing he obse ed Global Mean damage, by choosing α as a a iable o calib a e. The esul s ela ed o
he indi idual damage mechanisms (Mechanism A and Mechanism B) show alues simila o he ones
p oposed by (Cu i e al., 2008), Eq. (3.3) and (3.4), con i ming hei capabili y o damage p edic ion.
When i ing he da ase o he en i e owe mac oelemen (Mac oelemen damage), ins ead, he
ob ained Eq. (3.6) p oduces a signi ican imp o emen wi h espec o he o iginal o mula ion p oposed
by (Lagoma sino e al., 2004), gi en in Eq. (3.5), o he speci ic da ase in hand.
Figu e 3.17 shows he pe o mance plo s, namely p edic ed e sus obse ed mean damage alues,
ob ained by applying bo h he exis ing and p oposed o mula ions, oge he wi h he Residual Sum o
he Squa e (RSS) me ic, exp essed as ollows:
𝑅𝑆𝑆= ∑(𝑦𝑖−𝑦𝑖
)2
𝑛
𝑖=1
(3.8)
whe e
n
is he numbe o obse a ions, 𝑦𝑖 deno es he ac ual obse ed damage ex en , and 𝑦𝑖
 deno es
he p edic ed alue es ima ed ia he ulne abili y unc ion. This me ic shows a signi ican imp o emen
o he p edic i e capabili y upon ecalib a ion o he ulne abili y unc ion coe icien α, conside ing bo h
he Global Mean alues (plo ed as diamonds), e y well p edic ed, and he mean alues o he
samples g ouped by ea hquake (plo ed as ci cles) ha , despi e a la ge dispe sion, a e be e
p edic ed as well.
(a)
(b)
Figu e 3.17: Pe o mance plo s be o e (a) and a e (b) ecalib a ion. RSS es ima ed o he Global Mean alue.
RSS = 4.2
RSS = 1.1

CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
78
Finally, o highligh he s a is ical ele ance o he p esen da abase and o u he alida e he exis ing
and p oposed ulne abili y unc ions, in es iga ing he e ec o he size and cha ac e is ics o he owe
samples on hei pe o mance, mul iple s a is ical analyses we e pe o med by andomly gene a ing
se e al subse s o his o ic mason y owe s. Mo e han 200’000 subse s o each analysed ins ance
(Mechanism A, Mechanism B and Mac oelemen damage) we e gene a ed ollowing dis inc sampling
s a egies o andomly exclude owe s om he whole da abase, de ining se e al g oups o owe s o
di e en sizes.
Exclusions based on he speci ic e en s o he speci ic sou ces o in o ma ion, despi e being po en ially
alid al e na i e app oaches o he gene a ion o he subse s, we e disca ded as hey could in oduce
biases in he da ase a hand. Indeed, sou ces like Cu i e al. (2008), Aci o e al. (2021) o Jain e al.
(2020) p o ide only in o ma ion o high mac oseismic in ensi y while o he s, like De Ma eis and Zizi
(2019) o Valen e (2017), in o m abou low mac oseismic in ensi y. Mo eo e , as demons a ed by he
DPMs ca ego ised by ea hquake and by in ensi y, he in o ma ion is likely no well dis ibu ed o e he
damage le els. The e o e, combining all sou ces oge he was deemed essen ial o a co ec in e ence
o he owe s beha iou agains eal ea hquake e en s.
Fi s , 10’000 dis inc subse s o 100, 102, and 96 his o ic mason y owe s we e gene a ed o
Mechanism A, Mechanism B and Mac oelemen damage, espec i ely, by andomly emo ing, in each
subse , 10 owe s om he ini ial whole da ase . All he possible combina ions o lea e-10-ou subse s
we e oo nume ous o be managed (i.e., in he o de o e a). The Global Mean damage alues we e
compu ed o all he subse s and hen clus e ed accoun ing o each mac oseismic in ensi y as shown
in Figu e 3.18(a, c and e). The esul ing dis ibu ion o Global Mean damage alues appea o be e y
na ow conside ing bo h he in e qua ile ange and he sp ead o he whiske s in he boxplo s, which
accoun o a 99% co e age assuming a no mal dis ibu ion o he da a. None heless, a ew signi ican
ou lie s eme ge. I should be also no ed ha he dis ibu ion o he Global Mean damage o high
mac oseismic in ensi y ea u es he la ges sp ead. To s a is ically measu e he pe o mance o he
exis ing and no el o mula ions, he RSS me ic was e alua ed o e all he andom subse s, and he
ob ained dis ibu ions, which we e also i ed by a Gaussian p obabili y densi y unc ion, a e illus a ed
in Figu e 3.18(b, d and ).
The RSS dis ibu ions p esen lowe mean alues o Mechanism A han o Mechanism B and
Mac oelemen damage. In Figu e 3.18e and ), beside he o iginal o mula ion o he ulne abili y
unc ion o Mac oelemen damage, he sugges ed ecalib a ion and i s pe o mance in e ms o RSS
a e shown. He e, he RSS compa a i e esul s con i m a signi ican imp o emen upon ecalib a ion o
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
79
he cu e o Mac oelemen damage, as hei mean dec eases and hei dis ibu ion becomes na owe .
A ecalib a ion o he cu e coe icien s o Mechanism B da a has been a emp ed as well bu has led
o a negligible imp o emen o he pe o mance. Indeed Figu e 3.18(c) sugges s ha a ecalib a ion o
he duc ili y ac o
Q
would be mo e e ec i e han he a emp ed calib a ion o he coe icien α.
Howe e , a en a i e calib a ion o he ac o
Q
led o a alue oo a om i s accep able physical ange,
likely due o he unce ain ies in he da ase , and was disca ded.
Mechanism A
(a)
(b)
Mechanism B
(c)
(d)
Mac oelemen damage
(e)
( )
Figu e 3.18: Valida ion o he exis ing (a-d) and he p oposed (e- ) ulne abili y unc ions agains 10’000 subse s
gene a ed by andomly emo ing 10 owe s each ime.
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
80
In a second s age, andomly gene a ed samples o di e en sizes we e in es iga ed. In pa icula , 3 and
20 owe s a a ime we e andomly emo ed om he ini ial whole da ase .
Mechanism A
(a)
(b)
Mechanism B
(c)
(d)
Mac oelemen damage
(e)
( )
Figu e 3.19: Valida ion o he exis ing (a-d) and he p oposed (e- ) ulne abili y unc ions agains mo e han
200’000 subse s gene a ed by andomly excluding 3 owe s each ime.
CHAPTER 3. VALID. AND IMPROV. OF EXISTING VULN. MODELS THROUGH AN OPEN DATASET
81
Mechanism A
(a)
(b)
Mechanism B
(c)
(d)
Mac oelemen damage
(e)
( )
Figu e 3.20: Valida ion o he exis ing (a-d) and he p oposed (e- ) ulne abili y unc ions agains 10’000 subse s
gene a ed by andomly emo ing 20 owe s each ime.
Exclusion o 3 owe s p oduced smalle subse s o 107, 109, and 103 owe s o Mechanism A,
Mechanism B and Mac oelemen damage, espec i ely. All he possible combina ions o 3 samples ou
o he o iginal da abase we e conside ed, summing up o 215’820, 227’920, and 192’920 dis inc
andom subse s o Mechanism A, Mechanism B and Mac oelemen damage, espec i ely (Figu e
3.19). Exclusion o 20 owe s p oduced smalle subse s o 90, 92 and 86 owe s o Mechanism A,
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
88
4.1 INTRODUCTION
The p esen Chap e add esses he mos ele an sho comings ha eme ged du ing he analyses
discussed in Chap e 3 imp o ing he eliabili y o e i o ial scale seismic ulne abili y models o
mason y owe s and slende s uc u es (see Figu e 4.1).
Figu e 4.1: Flowcha me hodology adop ed in his Chap e .
In he i s s age, his o ic mason y bell owe s a ec ed by he 2016-2017 Cen al I aly seismic
sequence ha e been collec ed, il e ing he en i e se o chu ches esul ing om he Da abase o
Obse ed Damage (Da.D.O.) om (Dolce e al., 2019). The inal da abase wi h i s s a is ical analysis is
desc ibed in Sec ion 4.2. Successi ely, ulne abili y models (DPMs, ulne abili y and agili y unc ions)
ha e been p oduced in Sec ion 4.3. The en i e da ase was conside ed and di ided in mul iple subse s
accoun ing o dis inc seismic scena ios, namely single seismic shock and mul iple seismic shocks, o
p o ide an insigh in o he seismic isk o mason y bell owe unde cumula i e damage. Finally,
addi ional models ha e been de eloped, in Sec ion 4.4, aking in o accoun key ea u es such as
loca ion, geome y, in e ac ions, and main enance, seeking o iden i y owe s wi h mo e likely
homogenous beha iou .

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 Cen al I aly seismic sequence s uck a wide a ea a he bounda y among Lazio, Ab uzzo, Ma che
and Umb ia egions wi h a se ies o mode a e o s ong no mal- aul ea hquakes. The i s main shock
occu ed on Augus 24, 2016 wi h epicen e in he a ea o Mon i della Laga nea Accumoli, Lazio
egion, and a magni ude 𝑀𝑤 6.2, ollowed by se e al a e shocks eco ded du ing emaining 2016 and
2017. Two se e e a e shocks hi on Oc obe 26, 2016, in he a ea o Valne ina wi h epicen e nea
Cas elsan angelo sul Ne a, Ma che egion, wi h 𝑀𝑤 5.5 and 6.1. The main shock o he sequence
occu ed ou days la e on Oc obe 30, 2016, wi h 𝑀𝑤 6.6 and epicen e in No cia, Umb ia egion.
Finally, on Janua y 17, 2017, h ee e en s occu ed wi h he s onges ha ing magni ude 𝑀𝑤 o 5.7, all
wi h epicen e nea Capi ignano, Ab uzzo egion (Ro ida e al., 2020; Sis i e al., 2023). The eco e y
plan and econs uc ion wo ks a e s ill on-going almos 10 yea s a e he e en s (PCM, 2025). Table
4.1 shows he gene al in o ma ion o he main seismic e en s conside ed in his s udy, acco ding o he
I alian Ea hquake Ca alogue (Ro ida e al., 2020). The epicen es o hese e en s a e illus a ed o e
he I alian e i o y in Figu e 4.2.
Table 4.1: Gene al in o ma ion abou he seismic sequence.
No.
E en
Epicen al
A ea
Da e
[dd-mm-yyyy]
Time
(UTC)
La
[°]
Long
[°]
Dep h
[km]
𝑴𝒘
[-]
1s
Mon i della Laga
24/08/2016
01:36
42.698
13.233
08.1
6.2
2nd
Valne ina
26/10/2016
17:10
42.874
13.124
08.1
5.5
3 d
Valne ina
26/10/2016
19:18
42.904
13.090
09.6
6.1
4 h
Valne ina
30/10/2016
06:40
42.830
13.109
10.0
6.6
5 h
Aquilano
18/01/2017
10:14
42.531
13.283
09.6
5.7
Damage su eys we e conduc ed in h ee main phases. Ini ially p omo ed a e he i s main shock on
Augus 24, 2016, he ac i i ies we e in e up ed by he se e e ea hquakes in Oc obe o s a again
only in No embe . Finally, he hi d phase was igge ed by he las s ong shock in Janua y 2017
(Ca bona i e al., 2019). Second and hi d phases, in some cases, comp ised also epe i ion o p e ious
inspec ions.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
90
Figu e 4.2: Map o he epicen es.
4.2.2 Da ase
Da a abou he in es iga ed bell owe s we e ex apola ed om he Da.D.O., Da abase o Obse ed
Damage (Dolce e al., 2019), wi hin a collabo a ion wi h a esea ch g oup om he Uni e si y
D’Annunzio o Chie i-Pesca a.
Da.D.O. is he I alian WEB-GIS pla o m ha s o es and ca alogues he digi ised inspec ion o ms o
un ein o ced mason y and monumen al buildings, namely mason y chu ches, among o he asse s
s uck by pas seismic e en s. The ex ensi e sample ga he ing is pa amoun and allows no only o
assess he seismic isk a di e en scales based on pas e en s, bu also o de elop models o o ecas
u u e possible damage scena ios, hus suppo ing he ci il p o ec ion ac i i ies.
The collec ed da a we e egis e ed du ing ield inspec ions by means o speci ic su ey o ms o each
building ypology (e.g., egula buildings, chu ches, palaces, e c.). The o ms ha e been concei ed by
he I alian Ci il P o ec ion Depa men o pos -ea hquake damage assessmen wi hin he amewo k o
he eme gency echnical ac i i ies and comp ise in o ma ion abou : loca ion (si e cha ac e is ics, u ban
con ex , and add ess), a chi ec u al and s uc u al ea u es (cons uc ion da e, plan layou e ical
sys em, ma e ial, exis ing c acks, ecen in e en ions, and s a e o main enance), damage (obse ed
damage le els, in he mos ecen e sions acco ding o he EMS-98 scale), seismic inpu
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
91
cha ac e isa ion (mac oseismic in ensi y and PGA) and ele an exposu e- ela ed ea u es (use, ime
usage and peak ime), among o he de ails, such as he da e and he le el o accu acy o he su ey.
The pla o m allows il e ing o dis inc seismic e en s and building ypologies. In o ma ion abou he
his o ic mason y owe s a e e ie ed om Da.D.O. as pa o he b oade su eys o he chu ches
a ec ed by he seismic e en . In he chu ch su ey o m, speci ic ea u es s ic ly e e ing o he bell
owe s o , in any case, ele an o hei assessmen , can be ex apola ed, namely: loca ion, geome y,
owe ypology, damage g ade, and in ensi y measu es cha ac e ising he seismic inpu . Beside he
speci ic coo dina es, he o m di ec ly epo s he egion and p o ince/dis ic whe e he chu ch is
loca ed. The a ailable geome ical da a e e o he leng h o he wo base sides ( ec angula o squa ed
c oss-sec ion) and heigh , measu ed o es ima ed. The owe ypology indica es he ype o in e ac ion
be ween he owe s and he su ounding buildings, by iden i ying ecu ing plan con igu a ions, namely
isola ed, con ined, in eg a ed, and bell gable, as in oduced in Chap e 2. As o he p e ious
applica ions o he p esen hesis, a ge mechanisms comp ise damage o he sha and bel y,
neglec ing he non-s uc u al elemen s (e.g., pinnacles, spi es and deco a ions), wi h hei
co esponding damage le els assigned in he o m h ough expe judgmen du ing si e inspec ions.
Finally, he in ensi y measu es included in Da.D.O. o each asse a e MCS in ensi y and PGA. These
alues, p o ided o he chu ches, a e assumed o be he same o he owe s mac oelemen s.
A he ime o he consul a ion o he da abase, a o al o 3356 chu ches we e eco ded o he Cen al
I aly seismic sequence. Howe e , di e en il e s we e applied o ob ain a mo e eliable da ase ocused
on mason y bell owe s only. In pa icula , o ensu e ha he same numbe o eco ded damage le els
was a ailable o he sha and bel y mechanisms, all he cases cha ac e ised by absence o one o he
wo elemen s o missing damage da a we e excluded. Mo eo e , he bell gable ypology was no
conside ed, gi en he ocus o he hesis on his o ic owe s and slende s uc u es, as in Chap e 3. Fo
he sake o cla i y he emaining h ee classes o in e ac ion a e exempli ied in Figu e 4.3. Finally, o
a oid issues ela ed o comple eness o he su ey, owe s loca ed in a eas wi h mac oseismic in ensi y
(𝐼𝑀𝐶𝑆) lowe han 5 o no in o ma ion a ailable we e disca ded. Indeed, as he a ea o low and e y low
in ensi y expand d as ically, a om he epicen e, he likelihood o missing su ey da a o many
undamaged asse s inc eases d ama ically, wi h he consequence ha mos o he eco ded su eys in
his wide egion may ha e add essed chu ches because o hei peculia i ies o o an anomalous
damage le el a such in ensi ies.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
92
Figu e 4.3: Towe ypologies conside ed in he p esen Chap e .
This issue, well-known in li e a u e (Pe elli e al., 2019; Rosse o & Ioannou, 2018; Ta angelo e al.,
2024; Zucconi e al., 2022), may be ackled o esiden ial buildings elying on census da a. Howe e ,
e ec i e s a egies o monumen al buildings ha e no been de eloped o ex ensi ely applied ye and
a e ou side he scope o he p esen hesis. The il e ing p ocess allowed o ga he 794 his o ic mason y
owe s wi h po en ial ac i a ion o bo h damage mechanisms. A e cleansing he da a, he global
damage index o he owe mac oelemen was compu ed as in Chap e 3 and in (Tes a e al., 2024).
These 794 owe s cons i u e he en i e da ase analysed he ea e , also indica ed o simplici y GT0.
Focusing on he da e o he inspec ion, compa ed o he da e o he main shocks o he sequence, i
was possible o iden i y ou g oups o owe s. Mos o hem we e inspec ed be ween he i s and
second shocks, o a e all main shocks o he seismic sequence. Ve y ew owe s we e inspec ed ei he
be ween he hi d and he ou h shocks, o be ween he ou h and he i h shocks. Since wo seismic
shocks ha e occu ed on Oc obe 26, 2016 and he a ailable documen a ion does no allow o iden i y
he hou o he inspec ion, i is assumed ha he owe s we e inspec ed a e bo h seismic e en s
occu ing in he same day. Figu e 4.4(a) illus a es he spa ial dis ibu ion o he epicen es o he main
e en s du ing he seismic sequence and he owe s, iden i ied acco ding o he las main e en p io o
he su ey da e. The eco ded owe s a e mainly concen a ed o he no h and no h-eas o he
epicen es.
The long du a ion o he seismic sequence and he ela ionship be ween he da e o inspec ion and he
main shocks o he sequence equi e a ca e ul de ini ion o he in ensi y measu e alue adop ed o
cha ac e ise he inpu o he owe . In pa icula , he Da.D.O. da abase p o ides a single alue o he
mac oseismic in ensi y o each owe loca ion bu i e alues o PGA, one pe each main e en .
Following a well-es ablished app oach (Ca bona i e al., 2019; Rosse o & Ioannou, 2018; Sis i e al.,
2023), he PGA a he owe si e ha was conside ed o he analyses co esponds o he maximum
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
93
accele a ion among he e en s p e ious o he inspec ion da e. Figu e 4.4(b) shows he spa ial
dis ibu ion o he owe s, dis inguished by he speci ic e en , p io o inspec ion, in which he maximum
PGA a he si e was egis e ed. In gene al, he maximum PGA a he si e o he owe s was induced by
he seismic e en s wi h g ea e magni udes, namely, he shocks occu ed on Augus 24 and Oc obe
30, 2016. A non-negligible numbe o owe s su e ed he maximum PGA du ing he hi d shock,
namely on Oc obe 26, also cha ac e ised by a a he s ong in ensi y, due o hei p oximi y o i s
epicen e.
Based on he expe ienced PGA and he da e o inspec ion, he in es iga ed owe s o he whole da ase
(GT0) we e g ouped in o ou pa ially o e lapping subse : (i) 219 owe s inspec ed be ween i s and
second shocks (GT1); (ii) 487 owe s inspec ed a e las main shock (GT2); (iii) 349 owe s inspec ed
a e ou h main shock o which he second highes PGA in he e en s p io o inspec ion is lowe han
0.1g (GT3); (i ) 212 owe s inspec ed a e ou h main shock o which he second highes PGA in he
e en s p io o inspec ion is highe han 0.1g (GT4). The cha ac e is ics o hese da ase s and hei
sampling a e u he desc ibed in he ollowing analyses.
(a)
(b)
Figu e 4.4: Localisa ion o he epicen es wi h hei magni ude and geog aphical dis ibu ion o he owe s
dis inguished by: (a) las seismic e en be o e he su ey da e; (b) seismic e en s causing he maximum PGA a
si e be o e su ey da e.
Besides analysing he po en ial accumula ion o seismic e ec s du ing he sequence, in he p esen
Chap e an a emp is made o iden i y homogeneous classes o owe s based on hei a ibu es
desc ibed in he inspec ion o m. In Figu e 4.5, a i s classi ica ion based on he geog aphical loca ion
is ca ied ou , illus a ing he dis ibu ion o all inspec ed owe s ac oss he ou egions and he en
p o inces a ec ed by he Cen al I aly seismic sequence.

CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
94
(a)
(b)
Figu e 4.5: Towe s geog aphical dis ibu ion: (la) egional-based; (b) p o ince-based.
Mos o he owe s a e loca ed in he Ma che and Umb ia egions, ep esen ing 61% and 26% o he
da ase , espec i ely. A simila end is obse ed a he p o incial le el, whe e he majo i y o owe s a e
si ua ed in he Mace a a p o ince o he Ma che egion and he Pe ugia p o ince o he Umb ia egion,
accoun ing o 37% and 23% o he da ase , espec i ely. The emaining egions and p o inces con ain
ewe owe s, wi h a maximum o 12% o he da ase ound in any o he p o ince. Subse s based on
geog aphical loca ion we e c ea ed, aking in o accoun possible peculia i ies in local cons uc ion
adi ions and his o ical e en s. Fo his pu pose, he da ase was di ided in o ou subse s, which
include owe s om he Ma che and Umb ia egions, and owe s om he Mace a a and Pe ugia
p o inces, espec i ely. O he egions and p o inces we e no conside ed due o he limi ed sample
sizes, which is no s a is ically signi ican .
A second classi ica ion ocused on he geome y. Some eco ds pe ain o inspec ions conduc ed a
di e en imes o he same chu ches (e.g., Abbazia SS. Ru ino & Vi ale, Ca hed al o San Ca e o o
Chu ch o Madonna delle G azie). A compa ison o he geome ic and ypological da a egis e ed in he
su ey o ms om di e en inspec ions e ealed disc epancies, which we e co ec ed by checking he
o iginal sou ces and emo ely inspec ing he owe s. When he da a could no be eliably co ec ed, he
a ec ed owe s we e excluded om he ollowing subse s based on geome y. A simila p ocess was
applied o ins ances whe e a ailable geome ical da a appea ed un ealis ic. This e lec s he well-known
issue o po en ial e o s in da a collec ion, ansmission and digi isa ion du ing ield pos -ea hquake
in es iga ions, likely due o he limi ed numbe and expe ience o echnical pe sonnel, a ec ing he
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
95
accu acy and eliabili y o he in o ma ion (Baggio e al., 2007; Rosse o & Ioannou, 2018). I is possible
ha o he cases wi hin he da ase may equi e a en ion due o e o s. Howe e , a de ailed e alua ion
o hese cases is beyond he scope o his s udy.
Fou con inuous independen a iables we e iden i ied: h ee di ec ly ex ac ed om he Da.D.O.,
namely, he minimum base side (𝐿𝑚𝑖𝑛), he maximum base side (𝐿𝑚𝑎𝑥), and he o al heigh (𝐻𝑡𝑜𝑡),
and one indi ec ly compu ed as he a io o he o al heigh o he minimum side leng h a he base,
e e ed o as aspec a io (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛). A summa y o hei s a is ics is epo ed in Table 4.2.
Addi ionally, he numbe o samples wi h a ailable da a is epo ed o each a iable. This numbe , 396
ou o he 794 owe s in he en i e da ase , highligh s a ce ain lack o comple eness in he o ms. This
is a common issue when many asse s mus be quickly inspec ed by a ew ope a o s wi hin a e y sho
ime ame, a si ua ion o en seen in pos -disas e su eys (Rosse o e al., 2015; Rosse o & Ioannou,
2018) The a iable dis ibu ions a e illus a ed in Figu e 4.6. The dis ibu ions a e igh -skewed and
unimodal. 𝐿𝑚𝑖𝑛 and 𝐿𝑚𝑎𝑥 p esen e y simila s a is ics, excep o he maximum alues ha o 𝐿𝑚𝑎𝑥
each 14 m, likely due o ou lie s in he dis ibu ion, since mos o he owe s ha e squa e c oss-sec ion
and bo h side leng hs a e o en sho e han 5 m, mos ly in he ange 2-4 m. The heigh anges om a
minimum o 8 m o a maximum o 53 m, wi h a concen a ion o owe s below 20 m high. Finally, he
aspec a io a ies be ween 2 and 11, wi h mos o he samples p esen ing a a io smalle han 9 and a
high concen a ion in he ange 3-6. Besides he maximum ange o a ia ion o he dis ibu ion, Table
4.2 epo s he 5 and 95 pe cen iles o he geome ical da a. This ange p o ides a be e insigh in o
he pa ame e a ia ions, educing he e ec o ew ex eme alues especially in he lowe bound,
cha ac e ised by pa icula ly small-sized and s ocky con igu a ions. A he same ime, due o he igh -
skewness, he 95 pe cen ile limi s signi ican ly he uppe bound. The s a is ics he e epo ed a e in
good ag eemen wi h hose compu ed o o he da ase s o exis ing mason y owe s analysed wi hin he
p esen hesis (see Chap e 5, Sec ion 5.3.2).
Table 4.2: Main s a is ics o he geome ical da a.
Va iable
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 = s anda d de ia ion; 5% = i h pe cen ile; 95% = nine y- i h pe cen ile.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
96
Figu e 4.6: Dis ibu ions o he geome ical cha ac e is ics o he owe s.
Among he geome ic da a, he aspec a io is an indi ec pa ame e ha signi ican ly a ec s he seismic
beha iou o owe s. A highe aspec a io ends o esul in a can ile e -like esponse, cha ac e ised by
la ge la e al de o ma ions a he op. Con e sely, lowe aspec a io alues lead o an o e all shea
beha iou , p ima ily in ol ing he esis ing mason y walls. Al hough he aspec a io is c ucial o
desc ibing he esponse o his building ypology, he li e a u e lacks s udies de ining h esholds ha
explain he di e ing s uc u al beha iou s. No ably, (Sepe e al., 2008) classi ies he owe s by
p oposing an aspec a io h eshold o 4. This h eshold was adop ed in he p esen s udy o de ine wo
subse s, which include 120 owe s (15% o he da ase ) wi h aspec a io equal o smalle han 4, and
276 owe s (35% o he da ase ) wi h a io la ge han 4, espec i ely. I is wo h no ing ha o abou
50% o he samples included in he da ase he a ailable in o ma ion does no allow an es ima ion o he
aspec a io.
A hi d classi ica ion ocused on he plan con igu a ion due o i s signi ican impac on he seismic
esponse o his o ic mason y owe s. While owe s o en exhibi a egula a angemen in plan due o
hei e ical de elopmen , la e al in e ac ions can in luence hei seismic beha iou , pa icula ly due o
adjacen buildings, causing sudden a ia ions o s i ness along he heigh . Al hough i is impossible o
accoun o all he possible eal-wo ld scena ios in plan con igu a ion, he inspec ion o m g oups he
owe s based on he ype o in e ac ions in h ee main ca ego ies: isola ed (no in e ac ion), con ined
(pa ial in e ac ion), and in eg a ed ( ull in e ac ion). The e o e, wo subse s we e c ea ed, which include
312 con ined owe s (39% o he da ase ), and 233 in eg a ed owe s (29% o he da ase ), espec i ely.
Isola ed owe s a e e y a e (3% o he da ase , 20 owe s), hus hey canno be conside ed o a
s a is ical analysis. Likewise o he geome y, a signi ican lack o comple eness o he o m was
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
97
obse ed, wi h in o ma ion abou he plan con igu a ion una ailable o 29% o he da ase , namely 229
owe s.
Finally, he owe s we e classi ied acco ding o hei s a e o main enance. The inspec ion o m p o ides
a quali a i e assessmen o he gene al condi ion o he chu ches in he Da.D.O., using he ollowing
ca ego ies: Good, Accep able, Poo , Ve y Poo , and Wo k in P og ess. Simila o he p e ious ea u es,
he s a e o main enance is an indi ec pa ame e expec ed o in luence he seismic esponse o his o ic
mason y owe s. Al hough he assessmen in he o m does no di ec ly e e o his mac oelemen bu
o he whole chu ch, he ea e i is assumed ha he same e alua ion is applicable o he owe .
The e o e, h ee subse s we e c ea ed, which include 419 owe s in a good s a e o main enance (53%
o he da ase ), 234 owe s in accep able s a e o main enance (29% o he da ase ), and 85 owe s in
poo condi ion (11% o he da ase ), espec i ely. Towe s in e y poo condi ions we e no enough o
o m an independen subse o s a is ical analysis, comp ising only 22 samples (3% o he da ase ).
Simila ly, a e y small pe cen age o owe s p esen ed wo k in p og ess (1% o he da ase , 7 owe s).
Thus, none o hem we e g ouped in o any subse . In o ma ion abou he s a e o main enance o he
emaining owe s was no a ailable (4% o he da ase , 29 owe s).
4.2.3 C i ical aspec s
Se e al asse s in he a ea unde s udy we e p e iously damaged by he des uc i e 1997 Umb ia-
Ma che ea hquake, being hen subjec ed o epai s and s eng hening (Ca bona i e al., 2019). In
many chu ches ha su e ed ex ensi e damage, e ec i e mi iga ion measu es we e implemen ed be o e
he new e en s, including ligh ing beams, ie oads and o he me al es ain s, limi ing he o ma ion o
a ious mechanisms. Du ing he Cen al I aly sequence, hese low-impac in e en ions led o he
mig a ion o damage o he discon inui ies caused by he epai s o o o he a eas o he s uc u es.
Thus, damage eme ged acco ding o mechanisms o which no in ended s eng hening was
implemen ed and likely o a highe le el o seismic ac ion (Pa isi e al., 2018; S e azza Papa & Sil a,
2018). This peculia cha ac e is ic o he chu ches in he a ea and hei mac oelemen s, including he
bell owe s, should be p ope ly conside ed in he e alua ion o he ulne abili y owa ds a gene alisa ion
o he esul s.
Addi ionally, i is wo h men ioning ha s udies conduc ed in he a ea sugges ed a ele an con ibu ion
o si e e ec s o he damage le el, wi h bo h s a ig aphic (e.g., localised allu ial and so e deposi s in
a eas o he wise ea u ing ou c opping ock and shallow bed ock) and opog aphical cha ac e is ics
(e.g., idges and sha p hills) likely causing ampli ica ions in some o he municipali ies close o he
epicen es (e.g., Accumuli, Ama ice, Visso o Came ino) (Sex os e al., 2018).
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
104
Figu e 4.12: DPMs as unc ions o MCS in ensi y o GT0.

CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
105
Figu e 4.13: DPMs as unc ions o PGA o GT0.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
106
In bo h cases, he DPMs a e well dis ibu ed, showing an ag eemen wi h he expec ed beha iou , as
he size and cha ac e is ics o he da ase allows o o e come se e al issues eme ged wi h he open
da abase analysed in Chap e 3. Indeed, while o lowe seismic in ensi y he damage dis ibu ion is
expec ed o be igh -skewed gene ally unimodal and showing peaks o lowe damage ex en s, o highe
seismic in ensi y he damage dis ibu ion is expec ed o be le -skewed and showing peaks o g ea e
damage ex en s. These ends a e espec ed wi h a ew excep ions. In pa icula , a high numbe o
undamaged owe s can be obse ed o lowe seismic in ensi ies, whe eas he numbe o damaged
owe s g ows o g ea e seismic in ensi ies. This is also e lec ed in he alues o mean damage which
become signi ican o inc easing in ensi y alues. Howe e , o he maximum alues o he in ensi y
measu es, he obse ed damage is dis ibu ed qui e uni o mly o no mally ac oss he di e en damage
g ades. Only in he case o mechanism B, a clea le -skewed dis ibu ion appea s. This e lec s he
endency, mo e e iden when conside ing he PGA a he han he MCS in ensi y, o he damage
dis ibu ions and global mean damage alues o inc ease mo e signi ican ly o mechanism B han o
mechanism A, s a ing om ce ain seismic in ensi y h esholds (e.g., 0.1/0.15g in e ms o PGA).
These esul s sugges a highe ulne abili y o he bel y compa ed o he sha o he owe s.
Addi ionally, his end highligh s he be e co ela ion be ween PGA and obse ed damage.
A condensed o m o he DPMs as unc ions o he PGA, o he indi idual mechanisms and he o e all
mac oelemen damage, is epo ed in Figu e 4.14(a, b, c) epea ing he esul s o GT0 and compa ing
hem wi h GT1 and GT2.
GT2 samples con i m he indings discussed o GT0 and, in pa icula , he clea shi om a igh
skewed dis ibu ion o he damage le els o small in ensi y o a le skewed o high in ensi y in he
case o mechanism B. Mechanism A, ins ead, is cha ac e ised by a lowe pe cen age o collapses, e en
a he la ges in ensi y, and an unimodal dis ibu ion wi h peaks in D3, o GT0, o in D4, o GT1. The
DPMs o GT1, on he o he hand, p esen a di e en beha iou , simila o mos bins o PGA, excep o
he highes ange. I is wo h no ing ha he dis ibu ion o damage le el o GT1 in he ange 0.2-0.3g
is in luenced by a educed numbe o samples.
These ob ained esul s sugges ha he g ea e se e i y o damage in GT2 was p ima ily due o he
cumula i e e ec s o mul iple seismic e en s.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
107
(a)
(b)
(c)
Figu e 4.14: DPMs as unc ions o he PGA o di e en subse s: (a) GT0; (b) GT1; (c) GT2.
To explo e his hypo hesis, wo new subse s, namely GT3 and GT4 we e de ined. Bo h we e d awn om
owe s su eyed a e ei he he 4 h o 5 h main shock. GT3 consis s o owe s likely a ec ed by a single
e en . To his end, i con ains 349 samples whe e he second-highes PGA eco ded a each si e is less
han o equal o 0.1g. As discussed ea lie , his h eshold is used o expec a mean damage le el abo e
D1, sugges ing ha owe s in GT3 likely expe ienced a mos one shock capable o causing a leas
mode a e damage. On he o he hand, GT4 includes 212 owe s whe e he second-highes PGA
exceeds 0.1g, hus, impac ed by a leas wo seismic e en s po en ially damaging. I is impo an o
no e ha while GT0 is la ge, he numbe o samples becomes oo small o a meaning ul s a is ical
analysis when a emp ing o iden i y owe s ha expe ienced i s a la ge e en and hen a weake one
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
108
(mainshock-a e shock sequence). The e o e, owe s in GT4 a e collec ed ega dless o he
ch onological o de o he seismic inpu s.
(a)
(b)
Figu e 4.15: Geog aphical dis ibu ion o he owe s ca ego ised by he damage g ades assigned o indi idual
mechanisms and o e all mac oelemen : (a) GT3, owe s a ec ed by a single seismic e en (second highes PGA
≤ 0.1
g
) ; (b) GT4, owe s a ec ed by mul iple seismic e en s (second highes PGA > 0.1
g
).
Figu e 4.15 shows he spa ial dis ibu ion o hese wo subse s, while Figu e 4.16 illus a es he
co ela ion o bo h damage g ade and maximum PGA wi h he epicen al dis ance, including he mean
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
109
alues ( ed diamonds in he g aphs) o inc easing dis ance bins o 10 km. As o GT1 and GT2, he
co ela ion is p esen ed o he o e all mac oelemen damage only.
(a)
(b)
(c)
(d)
Figu e 4.16: Decay o seismic e ec s wi h he epicen al dis ance in e ms o damage mechanisms (a, b) and
maximum PGA egis e ed a he owe si e (c, d) o GT3 (a, c) and GT4 (b, d).
The spa ial dis ibu ion o he wo subse s ela i e o he epicen es di e s signi ican ly. In pa icula , he
owe s in GT3 a e loca ed a mo e han 10 km om he epicen e o he e en ha caused he highes
PGA wi h only a ew close han 25 km. Ins ead, hey a e sp ead o e a e y wide a ea, anging om
abou 10 km o 100 km. Mo eo e , wi h ew excep ions, he owe s p esen smalle damage g ades.
The a e age damage consis en ly emains below D1, showing an almos cons an end up o 60 km,
ollowed by e en lowe alues. In con as , he owe s in GT4 a e concen a ed in a na ow a ea, mos ly
wi hin 35-40 km o he epicen e o he e en ha caused he highes PGA. Only a ew cases a e loca ed
be ween 40 and 45 km, wi h none beyond his dis ance. This dis ibu ion e lec s he sampling s a egy
and he end o PGA alues wi h inc easing epicen al dis ance. While he a e age PGA o GT3 is
a ound 0.1g o less, GT4 exhibi s a wide ange o PGA alues. The a e age PGA pe 10 km bin
cons an ly dec eases wi h dis ance, om a peak o app oxima ely 0.4g wi hin 10 km o abou 0.2g a
40 km. Beyond his poin , no owe s a e a ec ed by mo e han one majo e en . These mode a e o

CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
110
high PGA alues induce signi ican damage, wi h an a e age damage g ade abo e D3 wi hin 10 km
om he epicen e and abo e D2 up o 30 km. A no iceable educ ion in a e age damage occu s only
beyond 30 km.
Finally, he wo highes alues o he PGA a each owe loca ion a e co ela ed in Figu e 4.17, and he
samples a e g ouped based on he damage g ades, h ough a colou key, conside ing only he owe s
ea u ing a second-highes PGA abo e 0.1g (GT4).
(a)
(b)
(c)
Figu e 4.17: Fi s and second-highes PGA co ela ions o bo h indi idual damage mechanisms (a, b) and o e all
mac oelemen (c) o GT4, oge he wi h he damage g ades.
This co ela ion shows he sequen ial o de be ween he e en s ha caused he i s and second highes
PGA, since, in he plo , x-axis e e s o he i s ch onological e en and y-axis o he second. The esul s
show ha mos o he owe s included in his subse a e cha ac e ised by highes PGA in he second
e en , as, o he Cen al I aly e en , shocks wi h highe in ensi y occu ed in he middle o he seismic
sequence. The classi ica ion o he ch onological o de does no p o ide any e idence o ends o
clus e ing o he owe damage g ades.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
111
4.3.2 Vulne abili y unc ions
The alues o he global mean damage ob ained om he analysis o he DPMs o inc easing seismic
in ensi ies (in e ms o MCS and PGA) a e plo ed as indica o s o he seismic ulne abili y and
compa ed agains a se o exis ing ulne abili y models, including he o mula ion de i ed in Chap e 3,
de eloped speci ically o his o ic mason y bell owe s (Figu e 4.18).
(a)
(b)
Figu e 4.18: Obse ed mean damage, plo ed wi h ci cles, compa ed wi h exis ing and no el calib a ed
(p oposed) ulne abili y unc ions o he en i e da ase (GT0) as unc ions o : (a) MCS mac oseismic in ensi y; (b)
PGA.
The ulne abili y unc ions a e iden i ied h ough he publica ion whe e hey ha e been p oposed (Ce oni
e al., 2022; Cu i e al., 2008; Lagoma sino e al., 2004), and he model de eloped in Chap e 3 is
indica ed as in (Tes a e al., 2024). Fo he MCS in ensi y measu e, he obse ed damage is
compa able o o he e en s a lowe seismic in ensi ies. Howe e , a highe seismic in ensi ies, he bell
owe s exhibi ed a be e esponse han expec ed, as shown by he compa ison wi h he ulne abili y
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
112
cu es p oposed by (Cu i e al., 2008; Tes a e al., 2024), which we e calib a ed agains a ious
seismic e en s. The model p oposed by (Lagoma sino e al., 2004) o mason y owe s and slende
buildings, no speci ically bell owe s, he e applied o he en i e mic oelemen , p esen s a be e
p edic i e pe o mance in he highe in ensi y ange. Ne e heless, a sligh unde es ima ion can be
obse ed in he lowe ange, whe e, in any case, he mean damage le el emains e y low. Fo he PGA
in ensi y measu e, he models iden i ied in li e a u e p esen a a he high p edic i e pe o mance when
compa ed wi h he obse ed mean damage alues. This esul is howe e expec ed since hese models
ha e been calib a ed o e a simila da ase o owe s a ec ed by he same sequence, excep o a
di e en il e ing and sampling s a egy leading o he inal in es iga ed se . In he absence o a
ulne abili y cu e o he en i e mac oelemen as a unc ion o he PGA, a ecalib a ion o he
pa ame e s a’ and b’ o he o mula ion sugges ed by (Ce oni e al., 2022), Eq. (4.1), is p oposed, by
i ing he global mean damage-PGA pai s, as gi en in Table 4.3.
𝜇𝑑=2.5 [1+𝑡𝑎𝑛ℎ(𝑎′log(𝑃𝐺𝐴)+𝑏′)]
Eq. (4.1)
Table 4.3: Calib a ion o ulne abili y unc ion coe icien s.
Mechanism
a’
b’
Re e ence
Mechanism A
0.89
-2.54
(Ce oni e al., 2022)
Mechanism B
0.89
-2.22
(Ce oni e al., 2022)
Mac oelemen damage
1.20
0.56
P oposed ecalib a ion
A his s age, he a ailable in o ma ion does no allow o a deepe in es iga ion in o he causes o he
be e pe o mance o he owe s compa ed o o he e en s. I is plausible ha his esul is in luenced
by he cha ac e is ics o he asse s in he a ea, many o which we e e o i ed and s eng hened
ollowing he Umb ia and Ma che ea hquake. Addi ionally, he cha ac e is ics o he seismic inpu , a
leas o some o he shocks, wi h peaks in a pe iod ange ha likely did no a ec he owe s’ i s
modes may ha e played a ole. Howe e , mo e de ailed esea ch is needed o d aw de ini i e
conclusions.
In o de o p o ide a be e insigh in o he dis inc beha iou o he owe s a ec ed by a single main
shock and mul iple shocks, Figu e 4.19 shows he mean damage o GT1 and GT2 plo ed agains
inc easing seismic in ensi y alues, in e ms o MCS and PGA.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
113
(a)
(b)
Figu e 4.19: Obse ed mean damage compa ed wi h newly calib a ed ulne abili y models as unc ions o he
MCS mac oseismic in ensi y (a) and PGA (b) o GT1 and GT2.
A ecalib a ion was conduc ed o p opose new o mula ions o his o ic mason y owe s o p edic he
expec ed damage caused by wo dis inc scena ios: single shock and mul iple shocks. The i o hese
new o mula ions wi h he obse a ional da a o bo h ea hquake scena ios is shown in Figu e 4.19,
wi h he calib a ion coe icien s p o ided in Table 4.4 using he same o m o he p e ious equa ion, Eq.
(4.1), in e ms o PGA, and, Eq. (2.8), in e ms o MCS. I should be no ed ha he o mula ion
coe icien s a e calib a ed by i ing he global mean damage (acco ding o he EMS98 scale)-MCS pai s,
di e en ly om he Eq. (2.8) which ins ead links he MCS o he global damage index (in 0-1 scale).
The in e sec ion o he ulne abili y unc ions in he lowe ange o he in ensi y measu es likely esul s
om he la ge unce ain ies in he a ailable da a o hese inpu alues. This ou come con adic s he
physical in e p e a ion o he wo cu es, as i sugges s ha he a e age damage om cumula i e
e en s could be lowe han om a single e en . Howe e , as demons a ed by he decay cu es in
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
120
Figu e 4.26: F agili y unc ions o GT1 de i ed using App oach 1.
4.4 VULNERABILITY ASSESSMENT: KEY FEATURES
The e ec i eness o he isk analysis models (i.e., DPMs, ulne abili y and agili y unc ions), namely
hei eliabili y and use ulness, is cons ained by he abili y o iden i y homogeneous asse classes.
Wi hin hese classes, dis inguishing he a ibu es and ea u es ha in luence he o e all suscep ibili y o
damage and he peculia beha iou agains he haza d is c i ical (Lagoma sino e al., 2021; Rosse o &
Ioannou, 2018). This is essen ial o educe he la ge sca e in he esponse o asse s wi hin he same
class unde he same in ensi y measu e. While o dina y buildings may exhibi g ea e homogenei y,
monumen s o en canno be adequa ely ep esen ed by b oad asse classes, as hese ail o cap u e he
in insic a iabili y wi hin each ypology. Key a ibu es o a ypology may be e lec ed h ough
ulne abili y co ec i e ac o s, which adjus he ulne abili y sco e o imp o e p edic ions o mean
damage and be e ep esen he suscep ibili y o each asse o seismic haza ds. Typical ulne abili y
modi ie s discussed in he li e a u e include ac o s such as main enance, ma e ial quali y, s uc u al
egula i y, size and aspec a io, in e ac ion wi h adjacen s uc u es, and he p esence o e o i ing
in e en ions (Lagoma sino, 2006; Lagoma sino e al., 2004).
In o de o be e unde s and he seismic esponse o he owe s, his Sec ion in es iga es speci ic
a ibu es and ea u es desc ibed in he inspec ion o ms, seeking o iden i y subse s wi h mo e
homogeneous beha iou . Howe e , he da ase ’s a ailable in o ma ion limi s he numbe and ype o
ea u es ha can be analysed. Consequen ly, se e al subse s o he en i e GT0 we e de i ed, g ouping
he owe s based on he ollowing a ibu es, among he ones desc ibed in Sec ion 4.2.2, due o hei
ele ance o owe s as po en ial sou ces o ulne abili y and/o haza d ampli ica ion: (i) main enance;
(ii) geome y; (iii) owe ypology and in e ac ion wi h su ounding buildings; and (i ) geog aphical
loca ion. I is impo an o no e ha he s a e o main enance, in he inspec ion o m, e e s o he en i e
mason y chu ch. Howe e , he e, i is assumed o apply o he associa ed owe s and is he e o e
included in he analysis. Gi en he u ili y o PGA in de eloping p edic i e models and i s s ong

CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
121
co ela ion wi h he owe damage g ades, he ollowing Sec ions ocus exclusi ely on his in ensi y
measu e.
The e ec i eness o ulne abili y and agili y unc ions was explo ed o bo h damage mechanisms A
and B and en i e mac oelemen . In pa icula , he exis ing and p oposed ulne abili y unc ions we e
epo ed agains he mean damage e olu ion obse ed o he dis inc subse s, oge he wi h he mean
damage e olu ion o he en i e da ase (GT0). The agili y unc ions we e compa ed, highligh ing, o
he sake o cla i y, he damage le els D3 and D4 only. These wo damage le els, in ac , ensu e he
a ailabili y o a su icien numbe o samples ac oss he PGA ange, p o iding consis ency and
signi icance. A he same ime, hey ep esen a ange o damage in which he collapse mechanism is
clea ly ac i a ed, p e en ing any possible misin e p e a ion. He ea e , only he esul s o he calib a ion
h ough he app oach 2 by (Po e , 2021) a e discussed. Table 4.7 summa ises he median and
s anda d de ia ion es ima ed by maximising he likelihood unc ion o he de elopmen o he
logno mal cumula i e dis ibu ion unc ions ep esen a i e o he bes i o he obse ed p obabili ies.
Howe e , o he sake o comple eness, he DPMs and he agili y unc ions de eloped o all
conside ed subse s, damage le els and calib a ion app oaches, namely adi ional app oach and
op imised one (app oach 2), a e indi idually p esen ed in he Appendix B o he p esen hesis.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
122
Table 4.7: Median and s anda d de ia ion o damage s a es D3 and D4 o he dis inc subse s accoun ing o he di e en key ea u es conside ed in his s udy.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
123
4.4.1 In luence o main enance
The in luence o he main enance condi ion o he owe s a he ime hey we e a ec ed by he
ea hquake on hei suscep ibili y o damage was in es iga ed e e ing o h ee s a es o main enance
(Figu e 4.27): good , accep able , and poo .
(a)
(b)
(c)
Figu e 4.27: Obse ed mean damage compa ed wi h he ulne abili y unc ions accoun ing o he main enance
s a es; (a) mechanism A; (b) mechanism B; (c) o e all mac oelemen .
Compa ing he o e all dis ibu ion o he mean damage obse ed o good, accep able and poo s a es,,
despi e hei simila i ies, an eme ging di e ence in beha iou can be de ec ed. In pa icula , he mean
damage o good s a e is dis ibu ed among lowe damage le els (wi h a maximum le el no exceeding
D3). Fo poo s a e, he mean damage consis en ly eaches highe le els (exceeding D3 o a PGA o
app oxima ely 0.25g, bu p esen ing a lowe alue o highe PGA). Meanwhile, he mean damage o
accep able s a e alls be ween good and poo anges. The di e en damage dis ibu ion can be be e
highligh ed by examining he DPMs (see Appendix B). The ob ained esul s e ealed ha he s a e o
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
124
main enance o he analysed owe s can be conside ed an explana o y a iable which can explain he
a ia ions cha ac e ising he owe s wi h di e en beha iou unde simila in ensi y o he seismic
ac ion. In his ligh , he mean damage p edic ed o inc easing alues o PGA h ough he exis ing and
p oposed ulne abili y unc ions has a limi ed po en ial, as i canno ully cap u e he ac ual damage
dis ibu ion cha ac e ising he owe s wi h di e en main enance s a es. The e o e, he o mula ion
coe icien s we e ecalib a ed, allowing o p opose no el o mula ions o be e explain he di e en
beha iou acco dingly. The ecalib a ed coe icien s a e summa ised in Table 4.8.
Table 4.8: Recalib a ion coe icien s o he h ee dis inc subse s based on he s a e o main enance.
a’
b’
Good s a e
Mechanism A
1.26
0.31
Mechanism B
1.32
0.49
Mac oelemen damage
1.22
0.44
Accep able s a e
Mechanism A
0.83
0.20
Mechanism B
1.23
0.57
Mac oelemen damage
0.94
0.41
Poo s a e
Mechanism A
1.00
0.52
Mechanism B
1.89
1.20
Mac oelemen damage
1.36
0.88
To u he check he ob ained esul s, an addi ional analysis in ol ed he compa ison o he p edic i e
pe o mance be ween he ulne abili y unc ions p oposed speci ically o he owe s ha ea u e an
accep able s a e o main enance wi h he exis ing and no el ulne abili y unc ions o he en i e
da ase , allowing o check i he o mula ion o accep able main enance s a e is su icien ly close o he
a e age beha iou o he en i e se o owe s. The esul s, illus a ed in Figu e 4.28(a), show a negligible
di e ence in he p edic i e pe o mance.
Based on he ob ained esul s, he p edic i e pe o mance o he o mula ions p oposed speci ically o
subse s o owe s cha ac e ised by poo and good s a es o main enance can be conside ed as uppe
and lowe bounds espec i ely, as shown in Figu e 4.28(b).
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
125
(a)
(b)
Figu e 4.28: Obse ed mean damage compa ed wi h he ulne abili y unc ions o he accep able s a e o
main enance and o he en i e se (a); P edic i e pe o mance o he o mula ions, oge he wi h uppe and lowe
bounds p o ided by o mula ions p oposed o he poo and good s a es o main enance (b).
When compa ing he agili y cu es ob ained o owe s wi h di e en main enance (Figu e 4.29), he
esul s o he ulne abili y unc ions a e con i med, as signi ican di e ences can be obse ed in he
p obabili ies o exceeding he conside ed damage s a es (D3 and D4) o inc easing alues o PGA.
Analysing D3, he p obabili ies o exceeding he damage s a es o good (dashed cu es) and accep able
s a es (solid lines) show simila inc easing ends, peaking a a ound 50% o a maximum PGA o
app oxima ely 0.4g. In con as , o poo s a e (do ed cu es), he p obabili ies ollow an inc easing
end as well bu each a signi ican ly highe peak o abou 70% o he same maximum PGA.

CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
126
Figu e 4.29: F agili y cu es o D3 and D4 exceedance i ed o good (diamond poin s), accep able (ci cle poin s),
and poo (squa ed poin s) main enance s a es.
To suppo he conclusions ega ding he in luence o he s a e o main enance on he ulne abili y, a
possible bias in he esul s was checked o ensu e ha di e ences in damage le els ac oss he h ee
subse s we e no in luenced by he a ying p esence o owe s a ec ed by a single main shock o
cumula i e e ec s. The esul s a e summa ised in Table 4.9.
Towe s in good, accep able, and poo main enance s a es ep esen di e en p opo ions o he en i e
da ase (53%, 29%, and 11%, espec i ely), e lec ing a signi ican di e ence in he numbe o owe s.
Howe e , when analysing he owe s a ec ed by single o mul iple shocks, he subse s o owe s wi h
dis inc main enance s a es show compa able dis ibu ions. Speci ically, he p opo ions o owe s
inspec ed a e i s main shock a e 27%, 29%, and 31%, espec i ely, while, o mul iple shocks, hey
a e 55%, 61%, and 63%, espec i ely. This con i ms ha including he s a e o main enance p o ides a
eliable indica o o be e unde s anding he ulne abili y beha iou o his pa icula building ypology.
Table 4.9: S a is ics o he subse s conside ing he main enance s a es based on ype o damage.
Key Fea u e
S a e o Main enance
Subse s
Good
Accep able
Bad
419 (53%)
234 (29%)
85 (11%)
Fi s E en
115 (27%)
68 (29%)
26 (31%)
Las E en
230 (55%)
143 (61%)
54 (63%))
4.4.2 In luence o geome y and in e ac ions
The owe s we e g ouped acco ding o wo addi ional pa ame e s accoun ing o hei geome y and
in e ac ion wi h su ounding s uc u es. Rega ding he o me , he a io o he o al heigh wi h he
minimum side leng h a he base, e e ed o as aspec a io (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛), was conside ed o
dis inguish wo classes: he i s class cha ac e ised by owe s wi h a ios lowe han o equal o 4 and
he second class by owe s wi h a ios g ea e han 4. The impac o he aspec a io on he mean
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
127
damage e olu ion is p esen ed in Figu e 4.30. I can be gene ally obse ed ha he mean damage
alues obse ed o he i s class (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛≤4) a e e y simila o hose obse ed o he second
class (𝐻𝑡𝑜𝑡/𝐿𝑚𝑖𝑛>4).
(a)
(b)
(c)
Figu e 4.30: Obse ed mean damage compa ed wi h he ulne abili y unc ions accoun ing o he aspec a io
h eshold; (a) mechanism A; (b) mechanism B; (c) o e all mac oelemen .
This is con i med by he compa ison o he agili y cu es ob ained o he di e en aspec a ios
(Figu e 4.31), whe e no signi ican di e ences in he p obabili ies o exceeding he conside ed damage
s a es (D3 and D4) o inc easing alues o PGA can be obse ed. Based on he esul s, he aspec a io
o he analysed owe s can be conside ed an explana o y a iable which has ela i ely low in luence on
he damage da a a hand.
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
128
Figu e 4.31: F agili y cu es o D3 and D4 exceedance i ed o owe s wi h
H o /Lmin
≤ 4 (diamond poin s) and
owe s wi h
H o /Lmin
> 4 (ci cle poin s).
The in e ac ion be ween he owe s and he su ounding buildings, ins ead, was in es iga ed by
analysing wo ypologies o bounded owe s, namely con ined and in eg a ed. The in luence o he owe
ypology on he mean damage e olu ion is gi en in Figu e 4.32. As o he p e ious case, he mean
damage obse ed o con ined owe s is e y compa able wi h he mean damage obse ed o
in eg a ed owe s.
(a)
(b)
(c)
Figu e 4.32: Obse ed mean damage compa ed wi h he ulne abili y unc ions accoun ing o he owe ypology
(con ined and in eg a ed); (a) mechanism A; (b) mechanism B; (c) o e all mac oelemen .
CHAPTER 4. VULN. MODELS FOR CENTRAL ITALY: CUMULATIVE DAMAGE AND KEY FEATURES
129
Howe e , when compa ing he agili y cu es (Figu e 4.33), sligh di e ences in he p obabili ies o
exceeding he conside ed damage s a es (D3 and D4) o inc easing alues o PGA can be obse ed.
Fo con ined (dashed cu es) and o in eg a ed (solid cu es), he p obabili ies a e e y close o each
o he un il eaching alues o PGA abo e 0.15g o which he p obabili ies a e la ge o in eg a ed han
o con ined. The unc ions end o di e ge mo e o damage mechanism B. Despi e he sligh di e ence
in he agili y da a, he owe ypology is a his s age conside ed an i ele an explana o y a iable in
he beha iou al esponse o he da ase , bu mo e esea ch is needed o p o ide a be e insigh in o
he in luence o he con igu a ion on he seismic beha iou .
Figu e 4.33: F agili y cu es o D3 and D4 exceedance i ed o con ined (diamond poin s) and in eg a ed (ci cle
poin s).
4.4.3 In luence o loca ion
Finally, wo explana o y a iables, namely he egion and he p o ince, we e iden i ied o accoun o he
loca ion ea u e. The e ec o he loca ion on he mean damage e olu ion is epo ed in Figu e 4.34.
The mean damage obse ed o Ma che egion is highe han he mean damage obse ed in Umb ia
egion (Figu e 4.34a). Simila ly, he mean damage obse ed o Mace a a p o ince is g ea e han he
mean damage obse ed in Pe ugia p o ince (Figu e 4.34b). The compa able ends can be jus i ied by
he simila s a is ics be ween he conside ed egion da ase s and he p o inces ha cons i u e hei
main subse s. The esul s sugges highe ulne abili y o owe s belonging o he Mace a a p o ince and
Ma che egion in gene al, i espec i e o he indi idual damage mechanism and mac oelemen , o e
he en i e ange o PGA. This can be be e highligh ed by examining he di e en DPMs de eloped o
he di e en egions and p o inces (see Appendix B). While he owe s in Mace a a p o ince and
Ma che egion p esen mean damage alues, o bo h mechanism A and B, only sligh ly la ge han he
en i e da ase (GT0), he owe s in Pe ugia p o ince and Umb ia egion exhibi a signi ican ly be e
beha iou un il 0.25-0.30g.