3.1.1. From Observed Damage to Predictive Models: Fragility and Vulnerability Functions
The lowest level of the seismic vulnerability assessment, EL0, also called the macroseismic approach, is based on empirical methods aimed at predicting the expected mean damage grade, due to a seismic event of a certain intensity, on a homogeneous population of buildings having similar geometrical and constructive features. The methods belonging to this level of accuracy are calibrated based on the real damage experienced by existing buildings after an earthquake and have been particularly suitable and adopted in the recent past for masonry churches. In particular, extensive surveys allow for assigning a given church to a specific damage level
Dk, which is estimated according to the EMS-98 scale ranging between 0–5 [
26] (see
Table 2).
The global damage level
Dk is assessed as a function of a global damage index
id (ranging between 0–1) as suggested in
Table 3.
In turn, the global damage index
id is calculated for each church based on the experienced damage according to Equation (1), which is proposed in the Italian Guidelines:
where:
dk,i (0 ÷ 5) is the damage level observed for the
i-th damage mechanism of
Table 1 and
ρk,i (0 ÷ 1) is the corresponding score factor indicating the importance that each damage mechanism has on the global safety of the church.
The score factors
ρk,i were firstly defined in [
8], and those values were assumed as referenced in the Italian Guidelines. Nonetheless, in some works such values are modified according to the expert judgment of the authors and thus are not fully consistent with the Guidelines’ provisions (e.g., [
17,
27]), while in other works the damage mechanisms are all assumed with the same importance, with
ρk,i values constant and equal to 1 (e.g., [
14,
15,
18]). In
Table 4, the values of
ρk,i adopted in some reference works are shown in comparison with the ranges provided by the Italian Guidelines.
The damage assessment allows for defining fragility curves, which relate the probability of damage being larger than a specified level to the intensity of the earthquake (measured with a macroseismic or peak ground acceleration scale) and are formulated based on an observational vulnerability model. One of the most adopted function for describing the probability of damage exceedance is the binomial probability function (BDPF) shown in Equation (2).
In the above equation
pk (with
k = 0 ÷ 5) is the probability of reaching a specific damage level
Dk, and
μD is the observed mean damage grade in the population calculated as in Equation (3), where
n represents the number of considered buildings:
The binomial distribution was firstly introduced for damage statistical analyses by Braga, Dolce and Liberatore [
28], who based their study on the damage distribution matrixes (DPMs) obtained for ordinary buildings after the Irpinia earthquake (1980). By means of this methodology, the fragility curve is assumed as the cumulative probability to reach a specific damage grade, as shown in Equation (4).
Although, to date, the binomial distribution of the damage of Equation (2) represents the most adopted fragility function [
7,
8,
12,
29,
30,
31,
32], it has the disadvantage of not allowing for an independent definition of the scatter of the expected damage, which is dependent on the only free parameter of the distribution, the mean damage
µD [
33]. Hence, continuous beta or lognormal distributions can be successfully adopted in order to statistically interpret the observed damage. For masonry churches, recent studies carried out after the Central-Italy 2016–2017 earthquake showed that the observed damage as a function of seismic intensity was well interpreted by a lognormal distribution.
In Equation (5) the function adopted in [
14,
15] is shown as an example.
In particular, in the above equations, P(D ≥ Dk|PGA = x) is the probability of exceeding a specific damage grade Dk as function of a certain seismic intensity PGA = x, ϕ is the normal cumulative distribution, μ and θ are the mean and median values, respectively, (which can be alternatively adopted) and β is the standard deviation. Hence, in these cases, the shape of lognormal fragility curves strongly depends on the parameter β, indicating the dispersion of the observed data. Thus, although unlike binomial functions they can provide several distributions of the damage, it should be noted that these formulations strongly depend on the parameter β.
Moving from an observational to a predictive approach, fragility curves can be adopted to estimate the probability of damage occurrence if the mean damage grade
μD is determined a-priori. Hence, this method is suitable in combination with vulnerability curves, which can return the mean damage grade according to the seismic intensity and the vulnerability of the buildings. Some of the first authors who proposed vulnerability curves by means of a hyperbolic function were Sandi and Floricel [
34]. In their study, they proposed a vulnerability function for ordinary buildings, as indicated in Equation (6):
where the expected damage
μD is evaluated as a function of the seismic intensity
I in macroseismic scale (I
MCS), the vulnerability index
Vi ranges between 0–1 according to the vulnerability classification in EMS-98 [
24] and the ductility factor,
Q, for ordinary buildings can be assumed as 2.3.
Then, based on the damage that occurred after the 1997 Umbria and Marche seismic event, Lagomarsino and Podestà [
29] calibrated vulnerability curves for existing masonry churches on the basis of a wide post-earthquake survey activity including about 2000 churches. The law proposed in this study is shown in Equation (7):
In particular, the vulnerability law proposed in this study involves the use of a new vulnerability measure
iv, again ranging from 0 to 1. Hence, in place of the vulnerability index
Vi, Lagomarsino and Podestà [
29] proposed adoption of a value higher than those estimated for ordinary buildings
Vi, their correlation being the one indicated in Equation (8):
By inserting Equation (8) in Equation (6), Equation (7) is s obtained, showing the equivalence between the two formulations. In practice, Equation (7) can be considered as the result of Equation (8) introduced in Equation (6).
Moreover, in this case,
Q was assumed equal to 3. It is worth mentioning that, according to Equation (7), a higher ductility factor than the one adopted for ordinary buildings (i.e.,
Q = 2.3) means a higher mean damage grade for low-intensity (i.e.,
IMCS < VII) earthquakes, with less damage for high-intensity seismic event [
32].
After this study, in the recent years, and with reference to different seismic events occurring in Italy and all over the world, many research activities have been carried out with the aim of verifying the reliability of these methodologies for predictive purposes by comparing the observed damage with the predicted damage [
35,
36]. While agreement on the type of vulnerability function is generally present, specific regional and typological features may be better represented by slight modifications in the coefficients. For instance, De Matteis, Brando and Corlito [
32] proposed a modification to the vulnerability function provided in Equation (7) for three-nave masonry churches damaged by the L’Aquila 2009 earthquake (Equation (9)):
Again, a ductility factor Q = 3 was assumed, while the argument numerator of the hyperbolic tangent function was modified to obtain a better correspondence with the observed data.
The use of intensity measures in the vulnerability function implies a conceptual short-circuit since vulnerability depends on intensity, defined based on the effects of the ground motion on the built environment. However, those effects depend in turn on the vulnerability of the stock. In this context, De Matteis and Zizi [
17] recently proposed the adoption of vulnerability functions based on a PGA-approach. In their work, the authors studied 68 one-nave churches damaged after the 2016–2017 Central Italy earthquake and highlighted good correspondence between observed and predicted damage obtained if Equation (7) is modified by means of the empirical correlation between
IMCS and PGA proposed by Faenza and Michelini [
37].
In recent years, research has moved toward the adoption of PGA-based approaches, solving the issue highlighted above and increasing the feasibility of applying this low-detail level of accuracy (i.e., EL0) for predictive purposes. In this sense, one of the most notable examples of application is the national MaRS project promoted by the Department of Civil Protection and the consortium ReLUIS [
38]. The main aim of this activity is the realization of national seismic risk maps related to several building typologies, among them churches. Although for these kinds of structures the results are still in an embryonic phase, an interesting method for deriving fragility curves from observational data has been proposed. In particular, the MaRS fragility curves are assumed with a lognormal distribution and are derived from two parameters only: (i) the median value of the PGA
D2 related to a damage level D2 assigned to the specific vulnerability class, and (ii) the free parameter
α (in the range 0.36–0.66 indicating the brittle or ductile behaviour, respectively) for determining the PGA values related to the damage level
Dk according to Equation (10).
Also in this case, the fragility curves assigned to a specific building typology strongly depend on the dispersion β assumed for the lognormal distribution.
Nonetheless, to date, the vulnerability function of Equation (7) still appears the most adopted and robust law for fitting the observed damage, and thus could be adopted for predictive purposes, too. A graphical representation of the vulnerability curves obtained by applying Equation (7), and the consequent generic fragility curves (with a binomial distribution, see Equation (4)), are provided in
Figure 3 and
Figure 4, respectively.
3.1.2. Vulnerability Assessment: EL0 Methods and Applications
In the past decades, the application of vulnerability and fragility curves has seen wide application not limited to masonry churches. Nowadays, the research world is still moving to corroborate such empirical formulations to adopt them for predictive (and thus preventive) aims within a territorial approach.
It is clear, now, that one of the most complex issues in this field is the definition of a vulnerability parameter, which should be based on few typological characteristics to allow a fast large-scale assessment. In the following, some relevant literature works, not limited to the Italian context and addressing this issue, are reported. It must be pointed out that these methodologies have been adopted in combination with various vulnerability and/or fragility functions. Thus, it must be admitted that the different vulnerability models suggested in each work only make sense within the scope of the specific framework to which they refer.
Generally, within the low-detail level EL0, the vulnerability parameter is estimated according to an initial value (
V0), which is modified accounting for typological and geometrical characteristics as indicated in Equation (11), where the choice of the modifying parameters (
Vk) and relative scores are empirically determined on the basis of statistical analyses and expert judgements [
1,
30].
In the literature, many examples of similar EL0 vulnerability models are present, which are based on Equation (11) or its modifications. A notable example of this method has been implemented within the European Risk-UE project [
39], which involved seven European cities (Barcelona, Bitola, Bucharest, Catania, Nice, Sofia and Thessaloniki). Therein, a vulnerability model suitable for several ancient masonry construction typologies is provided and, in particular,
V0 = 0.89 is defined for churches. The value is then modified according to Equation (11) and by accounting for seven parameters: (i) state of maintenance, (ii) quality of materials, (iii) regularity in plan, (iv) regularity in elevation, (v) position in the urban context, (vi) retrofitting interventions, and (vii) site morphology. As shown in
Table 5, these modifiers may have an increasing or decreasing effect on the vulnerability, based on the quality of the feature. Examples of application of this methodology can be found, among others, in [
1,
30,
40].
It is worth specifying that the main aim of the European Risk-UE project was to provide unified vulnerability functions regardless of the investigated structural typologies (e.g., towers, bridge, churches) and with this method a vulnerability score in the range 0.63–1.22 can be obtained, which is consistent with Equation (8).
A similar approach is proposed in [
27]. In this work the authors proposed a simplified seismic risk model assessed by means of hazard and vulnerability scores. Vulnerability is estimated by examining thirteen parameters, ten of which are derived from the Italian GNDT II vulnerability datasheet [
41]. In this case, the vulnerability model assumes the form of Equation (12):
where
vk,i is the score value
i of the class selected for the generic
k-th parameter, while
ρk is the weight, representing the importance that each parameter has on the global vulnerability of the church.
Geometrical-based simplified methods, which consider geometrical features to obtain approximate vulnerability indexes, are worth mentioning, too. For example, Lourenço and Roque [
42] proposed three simplified safety indexes based on a study concerning 58 Portuguese churches: (i) in-plan area ratio, (ii) area-to-weight ratio, and (iii) base shear ratio. In this work, a vulnerability score is suggested as an indicator for fast screening aimed at prioritizing deeper assessment studies. Similarly, in the work of Salzano et al. [
18], vulnerability classes for 27 churches damaged after the Ischia earthquake (2017) are defined according to a fictitious slenderness parameter, namely nave height to square root of the plan area ratio. In this case, the authors examined a wide set of vulnerability functions in order to obtain the best correlation with the observed data.
Another significant study carried out by Palazzi et al. [
43] deals with 106 masonry churches that experienced the 2010 Maule earthquake (Central Chile). In this work, the authors tried to find a correlation between the experienced seismic intensity and damage level, and four typological parameters, i.e., (i) masonry type, (ii) architectural layout, (iii) architectural style, and (iv) foot-print area.