PSI - Issue 44

Romina Sisti et al. / Procedia Structural Integrity 44 (2023) 1380–1387 Romina Sisti et al. / Structural Integrity Procedia 00 (2022) 000–000

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Finally, the apse is present in 1321 churches (46%) and absent in 827, while in 736 cases the information is not available, and the side chapels are present in only 446 churches (15%), absent in 1679 and not available in the rest. 3. Statistical model and fitting procedure for deriving the fragility curves One of the purposes of the A-DC form is to represent the earthquake damage to each church. Thus, during the inspection, it is necessary to identify the collapse mechanisms that can be activated in the church (hereinafter called ‘possible mechanisms’) among all the 28 ones in the form. The damage associated to each mechanism is then assessed by attributing a value from 0 (no damage) to 5 (collapse). Considering all the possible mechanisms and the corresponding damage levels, the A-DC form provides the following global damage index for each church: ! = " # · ∑ ! "!#$ ! % (1) where n is the number of possible mechanisms and d k is the damage for the k -th mechanism (from 0 to 5). The global damage index i d is transformed into a discrete variable correlated with the global damage level D i ( i = 0 to 5) provided by the European macro-seismic (EMS) scale (Grunthal 1998), according to the correlation proposed by Lagomarsino and Podestà (2005). New indices of damage related to the four most recurring macro-elements in the database (i.e., façade, principal nave, apse and bell tower) are herein defined to analyze the damage of single macro elements, as summarized in Table 1 also indicating the related mechanisms for each macro-element. For each church, the index relating to a specific macro-element is calculated if at least one of the mechanisms that contribute to the definition of the index is ‘possible’. The macro-element damage indices are then transformed into discrete variables through the same correlation used for the global index.

Table 1. Damage mechanisms and macro-elements defined in the fragility functions.

Macro element

Mechanism considered in the definition of macro-element damage index

No. of possible mechanisms

Macro-element damage index ! = ∑ # " ! "$% 5 ∙ ! &# = ∑ # " "# "$% 5 ∙ &# ' = ∑ # " $ "$% 5 ∙ ' () = ∑ # " %& "$% 5 ∙ ()

1 ≤ ! ≤ 3

M1 M2 M3 M5 M6 M8

Overturning of the façade Overturning of the gable Shear mechanism façade

Façade

1 ≤ 1 ≤

&# ≤ 4 ' ≤ 4

Transversal vibration of the nave Shear mechanism in the nave lateral walls

Principal Naves

Vaults of central naves

M19 M16 M17 M18 M21 M26 M27 M28

Hammering and damage to the nave roof

Overturning of the apses

Shear failure in the apses and presbytery walls Vaults of the apses and of the presbytery Hammering and damage to the apses roof Overturning of the standing out elements

Apses

1 ≤ () ≤

Bell tower

Global collapse of the bell tower Mechanism in the bell cell

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First, statistical elaborations of damage can be done through the damage probability matrices (DPMs), introduced for the first time by Braga et al. (1982) after the 1980 Irpinia earthquake for developing vulnerability analysis and damage prediction. DPMs are herein used to provide a discrete relationship between the observed damage (in terms of global damage index, macro-element index or damage related to a single mechanism) and the worst seismic action in terms of PGA recorded before the data of the survey for each church. Fig. 3 shows the percentage distribution of damage levels for each PGA range and for the most recurring macro-elements herein analyzed. Once defined the DPMs, it is possible to derive the fragility curves according to a statistical method (Baker 2015) based on a lognormal probability cumulative distribution fitted on the observations distributed in the PGA ranges previously fixed. The probability that a variable PGA may cause a given level of damage D i can be defined through the probability function expressed as: