4.1 Basics of the adopted approaches
The statistical model selected to represent the relationship between the ground motion intensity measure and the probability of exceeding the damage grades is the cumulative lognormal distribution. According to this model, only two parameters are necessary to describe the fragility curve associated to the given DSk (k = 1,..5): the median value of the intensity measure (in this work the Peak Ground Acceleration - PGA) that induces a damage equal or greater than DSk (PGADSk) and the corresponding dispersion βDSk, which depends on the record-to-record variability and the inhomogeneity of buildings in the same class. This model is worldwide used for seismic risk analyses (HAZUS 1999, Rossetto et al. 2014, Silva et al. 2019, Silva et al. 2020, Baraschino et al. 2020, Spence et al. 2021) and is also coherent with the framework issued by the Italian Civil Protection Department in 2018 (ICPD 2018) for the most recent National Risk Assessment in Italy (Dolce and Prota, 2021; Dolce et al., 2021).
In this study three different approaches are used to derive fragility curves taking advantage of the post-earthquake damage data illustrated in section 2:
-
a direct empirical approach where an optimization procedure is directly applied to observational data allowing obtaining the unknown parameters;
-
a hybrid approach (named empirical-binomial approach) that exploits the simulated damage probability matrix by means of probability density functions, ensuring a regular distribution (i.e. binomial) of damage states, derived from the mean damage µD evaluated from the counts of buildings suffering the observed damage states;
-
a heuristic approach based on the expertise that is implicit in the European Macroseismic Scale (EMS98), which assumes a regular increase of mean damage with the earthquake intensity (vulnerability curves derived with fuzzy assumptions on the binomial damage distribution - Lagomarsino and Giovinazzi 2006), directly fitted on the available post-earthquake damage data.
Details on each of these three methods are provided in the following.
Fragility curves obtained by the empirical approach are obtained by fitting the assumed statistical model to observational data. For comparability purposes, this approach is quite similar to that adopted within the ICPD 2018 framework for residential buildings (Del Gaudio et al., 2020; Rosti et al., 2021a; Rosti et al., 2021b). The parameters of the fragility curves are herein estimated by maximizing the likelihood function (Maximum Likelihood Estimation -MLE, e.g. Baker, 2015) through an optimization algorithm, where the counts of buildings suffering a given damage grade belonging to the ith-PGA bin is assumed to follow a multinomial distribution (Charvet et al., 2014):
\(\text{L}\left({\text{n}}_{\text{i},\text{D}\text{S}},{\text{P}}_{\text{i},\text{D}\text{S}}\right)={\prod }_{\text{D}\text{S}=0}^{5}{\prod }_{\text{i}=1}^{\text{m}}\frac{{\text{N}}_{\text{i}}!}{{\text{n}}_{\text{i},\text{D}\text{S}}!}{{\text{P}}_{\text{i},\text{D}\text{S}}}^{{\text{n}}_{\text{i},\text{D}\text{S}}}\) (1)
where the term Pi,DS represents the conditional probability of suffering a given DS, for the ith-bin. This probability is herein evaluated as a function of lognormal cumulative function:
\({\text{P}}_{\text{i},\text{D}\text{S}}=\left\{\begin{array}{c}1-\text{l}\text{o}\text{g}\text{n}\text{c}\text{d}\text{f}(\text{P}\text{G}{\text{A}}_{\text{i}},{{\lambda }}_{\text{D}\text{S}+1},\beta )\\ \text{l}\text{o}\text{g}\text{n}\text{c}\text{d}\text{f}(\text{P}\text{G}{\text{A}}_{\text{i}},{{\lambda }}_{\text{D}\text{S}},\beta )-\text{l}\text{o}\text{g}\text{n}\text{c}\text{d}\text{f}(\text{P}\text{G}{\text{A}}_{\text{i}},{{\lambda }}_{\text{D}\text{S}+1},\beta )\\ \text{l}\text{o}\text{g}\text{n}\text{c}\text{d}\text{f}(\text{P}\text{G}{\text{A}}_{\text{i}},{{\lambda }}_{\text{D}\text{S}},\beta )\end{array}\right.\)
\(\begin{gathered} DS=0 \hfill \\ 1 \leqslant DS<5 \hfill \\ DS=5 \hfill \\ \end{gathered}\) (2)
The above optimization procedure is set to simultaneously fit all the 5 (DSs) fragility curves for a given building class to observational data assuming a common value for logarithmic standard deviation \({\beta }\) for all DSk and different values of logarithmic mean λDS (where PGADSk= exp(λDS)).
The use of the empirical-binomial approach is addressed to partially solve the irregularity and sparseness of damage data shown in DPM of Fig. 9 by assuming a continuous trend arising from observational data. Indeed, several studies showed that observational damage frequencies are well reproduced by a binomial probability density function (Braga et al., 1984; Sabetta et al., 1998; Lagomarsino and Giovanazzi, 2006; Rosti et al., 2018; Lagomarsino et al., 2021) or with beta probability density function (Giovinazzi and Lagomarsino 2005; Lallemant and Kiremidjian, 2015; Rosti et al., 2020). The term \({\text{n}}_{\text{i},\text{D}\text{S}}\) of Eq. 2 could be substituted by the following:
\(\tilde{{\text{n}}_{\text{i},\text{D}\text{S}}}=\frac{5!}{\text{D}\text{S}!\left(5-\text{D}\text{S}\right)!}{{\mu }}_{\text{D}}^{\text{D}\text{S}} {\left(1-{{\mu }}_{\text{D},\text{i}}\right)}^{5-\text{D}\text{S}}\) (3)
where:
(4)
represents the mean damage evaluated in the ith-PGA bin. The updated DPMs adopted for the fitting are those illustrated in Fig. 11.
Finally, the third method (classified as heuristic) aims guaranteeing a fairly well fitting with actual damage but, at the same time, ensuring physically consistent results for both low and high values of the seismic intensity (for which observed data are incomplete or lacking). This approach starts from the original proposal of Lagomarsino and Giovinazzi (2006) but it has been recently further developed by Lagomarsino et al. (2021) thanks to the valuable calibration supported by the use of data on URM residential buildings collected in Da.D.O.. In particular, according to this approach, the fitting of observed damage data is carried out in the domain given by the mean damage grade (µD) and the macroseismic intensity (I) with the aim of deriving the free parameters (V and Q) of the macroseismic vulnerability curve expressed by the following:
\({{\mu }}_{\text{D}}=2.5\left[1+\text{t}\text{a}\text{n}\text{h}\left(\frac{\text{I}+6.25\text{V}-\text{Q}-10.8}{\text{Q}}\right)\right]\) (5)
where V is the vulnerability index and Q the ductility index. To this aim, it is necessary converting, for each bin of the intensity measure, the PGA into macroseismic intensity I, by adopting a I-PGA correlation law. In particular, the following relationship has been adopted: \(\text{P}\text{G}\text{A}= {\text{c}}_{1} {\text{c}}_{2}^{\text{I}-5}\), where c1 represents the PGA for intensity I = 5, while c2 is the factor of increase of PGA due to an increase of 1 of the macroseismic intensity. In this work, c1 and c2 have been assumed equal to 0.047 and 1.7, respectively. However, the adoption of such law is only functional to operate the fitting in the domain coherent with Eq. (5), but it doesn’t alter the numerical consistence of each bin; indeed, the conversion has been operated by referring to the central value of each bin. The same correlation has been then applied to come back in the PGA domain; therefore, the final result is not sensitive to the choice of the I-PGA correlation law.
According to the original macroseismic method proposed in Lagomarsino and Giovinazzi (2006), Eq. (5) assumes that the completion of DPMs is made according to the binomial probability distribution.
The fitted points and the resulting V and Q values obtained by considering the whole sample of URM and RC schools are illustrated in Fig. 12. Once the V and Q values are fitted, according to the procedure described in Lagomarsino et al. (2021), it is possible to convert the vulnerability curve into the corresponding fragility curve. For values of V > 0.32 the following expression can be adopted:
\({\text{P}\text{G}\text{A}}_{\text{D}\text{S}\text{k}}\left(\text{V},\text{k}\right)={\text{c}}_{1}{\text{c}}_{2}^{\left({\text{I}}_{\text{D}\text{k}}-5\right)}={\text{c}}_{1}{\text{c}}_{2}^{\left[6.7-3.45\text{V}+\text{Q}\text{a}\text{t}\text{a}\text{n}\text{h}\left(0.36\text{k}-1.08\right)\right]}\) (6)
It could be demonstrated that also the dispersion βDk may be analytically derived as a function of Q, c1 and c2.
Figure 13 represents the DPMs obtained after the fitting of the vulnerability curve.
4.2 Fragility curves for RC school buildings
The section presents the fragility curves of RC school buildings obtained using the three approaches described in the previous section. Figures 14 and 15 show the resulting curves for the whole stock and the groupings, respectively; they also report the total number of schools for each group. Note that the percentiles reported in Figs. 14 and 15 are those obtained using observational data (see Fig. 9).
The fragility curves obtained by using the three approaches are in good accordance with empirical percentiles both for the whole stock of schools and the proposed groupings. For the heuristic approach, Fig. 15 also reports the values of V and Q resulting from the fitting of the vulnerability curve.
Table 3 summarizes the values of two parameters that identify each fragility curve. For all the three methods, within the same grouping, the fitting has been carried out by assuming a constant value of the dispersion in all cases. In the case of the pure empirical approach, for the derivation of curves associated to DS5 of the POST80 grouping the empirical data are statistically not sufficient to fit the parameters of the fragility curves (this problem is overcome by the other two approaches, thanks to the use of the binomial distribution to complete the missing data).
Table 3 shows that in general the median values of PGADSk obtained from the empirical and empirical-binomial approaches are higher than those from the heuristic one. Moreover, the dispersion values β associated to the empirical and empirical-binominal approaches (0.98 and 0.94, respectively) and higher than those associated to the heuristic one. Therefore, the heuristic approach appears to be more fragile, in terms of median values, but less disperse than the empirical and empirical-binomial ones.
Regarding the dispersion, the three approaches account for the same sources of the uncertainty: the seismic input; the inter and intra buildings variability; and, finally, the epistemic uncertainty possibly resulting from a subjectivity degree of surveyors in attributing the damage rating in the AeDES form (although strongly reduced in last Italian earthquakes thanks to the valuable effort of Italian Department of Civil Protection in increasing the training of surveyors). Therefore, the reasons of such differences in the β values are due to the differences in the hypotheses on which the methods are based. In particular, it is useful recalling that the heuristic model assumes a regular trend of µD with IM (i.e by fitting the observed damage data as a function of macroseismic intensity), with a rate of increase (ruled by the parameter Q) that cannot be too flat (Fig. 12) and a direct derivation of fragility curves through the binomial damage distribution. Conversely, the pure empirical approach relies completely on observed data, that are more irregular and disperse. Finally, the use of the binomial distribution in the empirical-binomial approach (applied separately to each empirical µDcomputed in each PGA-bin) slightly reduces the dispersion with respect to the pure empirical one.
Table 3
–RC school buildings: median PGA (PGADSk, with k = 1…5), and dispersion β)
Approch
|
Group
|
PGADS1
|
PGADS2
|
PGADS3
|
PGADS4
|
PGADS5
|
β
|
Empirical
|
ALL
|
0.11
|
0.32
|
0.57
|
1.13
|
2.73
|
1.06
|
PRE80 - L
|
0.10
|
0.30
|
0.59
|
1.39
|
2.66
|
0.98
|
POST80 - L
|
0.16
|
0.45
|
0.77
|
1.60
|
-
|
0.98
|
PRE80 - M
|
0.08
|
0.18
|
0.30
|
0.61
|
1.52
|
0.98
|
POST80 - M
|
0.13
|
0.31
|
0.50
|
1.91
|
-
|
0.98
|
Empirical-binomial
|
ALL
|
0.09
|
0.28
|
0.66
|
1.45
|
3.42
|
0.96
|
PRE80 - L
|
0.09
|
0.28
|
0.66
|
1.51
|
3.67
|
0.94
|
POST80 - L
|
0.15
|
0.40
|
0.92
|
2.06
|
4.83
|
0.94
|
PRE80 - M
|
0.07
|
0.19
|
0.36
|
0.71
|
1.66
|
0.94
|
POST80 - M
|
0.12
|
0.31
|
0.60
|
1.13
|
2.43
|
0.94
|
Heuristic
|
ALL
|
0.12
|
0.26
|
0.46
|
0.80
|
1.41
|
0.66
|
PRE80 - L
|
0.12
|
0.25
|
0.43
|
0.75
|
1.29
|
0.64
|
POST80 - L
|
0.19
|
0.35
|
0.54
|
0.85
|
1.33
|
0.56
|
PRE80 - M
|
0.08
|
0.17
|
0.29
|
0.49
|
0.84
|
0.63
|
POST80 - M
|
0.14
|
0.28
|
0.47
|
0.78
|
1.30
|
0.61
|
The fragility curves evaluated through the three approaches show a clear and appropriate hierarchy with the construction age and the number of storeys as depicted in Fig. 16, where the curves are grouped (PRE80-L, POST80-L, PRE80-M, POST80-M) for each damage state. It is worth noting that only the first three DSs are represented because more significant in the range of interest of the PGA; anyhow, these curves are represented in Fig. 15 and the relative parameters are in Table 3. In particular, for each DS: i) the fragility decreases with the construction age (passing from PRE-80 to POST-80), given the building height, consistently with the design evolution compliant to seismic codes; ii) the fragility increases as the height of the building increases (passing from L to M), given the construction age.
4.3 Fragility curves for URM school buildings
In this section, the fragility curves of URM school buildings are presented by using the same format adopted for RC school buildings.
Figure 17 shows as, in the case of the whole sample, the fragility curves are altogether able to well capture the trend of real data.
Passing from Fig. 17 to Figs. 18 and 19 it is evident how splitting the observed data in classes by construction age and building height leads to quite sparse percentiles, sometimes characterized by very few samples; in particular, this is evident for medium-rise buildings.
This results into a greater difficulty in fitting the data, particularly when the pure empirical approach is adopted. Indeed, from Table 4 it emerges how, for this method, almost all parameters are missing in the medium-rise case. Moreover, for this case, differences higher than those founded for RC school buildings arise also among the other two approaches.
In fact, the empirical-binomial approach solves the intrinsic inability of the empirical one associated to the almost complete absence of data, but it is anyway anchored to points which are few robust for the fitting. As matter of fact, the empirical-binomial approach works well when at least the estimate of the mean damage on each single PGA-bin may be considered reliable; for the URM medium-rise case, the mean is sometimes based on very few data (see the last PGA-bins in Fig. 10b).
Conversely, the heuristic one compensates the lack of sample by directly fitting the parameters (V and Q) that define the vulnerability curve, thus considering all together the data and assuming an heuristic increasing trend of the mean damage with the intensity; this allows to manage scattered irregularities in the trend of µD (e.g. as depicted in Fig. 12b in the case of the point referred to the last bin).
Regarding the values of the dispersion β, the comments reported in section 4.2are still valid. Moreover, it is worth noting that β values associated to the empirical and empirical-binomial approaches are slightly higher than those of RC school buildings confirming the highest scatter of URM sample, also due to a bigger variability of structural features within the same class.
Table 4
– URM school buildings: median PGA (PGADSk, with k = 1…5), and dispersionβ
Approach
|
Group
|
PGADS1
|
PGADS2
|
PGADS3
|
PGADS4
|
PGADS5
|
β
|
Empirical
|
ALL
|
0.16
|
0.40
|
0.63
|
1.21
|
2.17
|
1.15
|
PRE45 - L
|
0.13
|
0.23
|
0.46
|
0.61
|
1.16
|
1.07
|
45–61 - L
|
0.12
|
0.28
|
0.36
|
1.50
|
1.50
|
1.07
|
POST61 - L
|
0.18
|
0.95
|
1.32
|
-
|
-
|
1.07
|
PRE45 - M
|
0.10
|
0.20
|
0.45
|
0.45
|
-
|
1.07
|
45–61 - M
|
0.17
|
0.25
|
-
|
-
|
-
|
1.07
|
POST61 - M
|
0.13
|
-
|
-
|
-
|
-
|
1.07
|
Empirical-binomial
|
ALL
|
0.11
|
0.35
|
0.80
|
1.79
|
4.32
|
0.98
|
PRE45 - L
|
0.10
|
0.25
|
0.47
|
1.00
|
2.45
|
1.02
|
45–61 - L
|
0.10
|
0.30
|
0.66
|
1.43
|
3.47
|
1.02
|
POST61 - L
|
0.18
|
0.56
|
1.30
|
2.83
|
6.45
|
1.02
|
PRE45 - M
|
0.09
|
0.25
|
0.40
|
0.53
|
0.97
|
1.02
|
45–61 - M
|
0.12
|
0.50
|
1.59
|
4.64
|
15.48
|
1.02
|
POST61 - M
|
0.15
|
0.48
|
1.24
|
3.13
|
8.31
|
1.02
|
Heuristic
|
ALL
|
0.14
|
0.29
|
0.51
|
0.86
|
1.48
|
0.64
|
PRE45 - L
|
0.11
|
0.21
|
0.33
|
0.53
|
0.87
|
0.59
|
45–61 - L
|
0.11
|
0.23
|
0.39
|
0.67
|
1.15
|
0.61
|
POST61 - L
|
0.19
|
0.36
|
0.56
|
0.87
|
1.37
|
0.60
|
PRE45 - M
|
0.07
|
0.12
|
0.19
|
0.28
|
0.42
|
0.52
|
45–61 - M
|
0.11
|
0.21
|
0.34
|
0.54
|
0.86
|
0.63
|
POST61 - M
|
0.15
|
0.27
|
0.41
|
0.64
|
0.99
|
0.55
|
Finally, Fig. 20 illustrates the variation of seismic fragility across the examined groupings (PRE45-L, 46-61-L, POST61-L, PRE45-M, 46-61-M, POST61-M), with reference to the first three damage states only. Focusing the attention to the empirical-binomial and heuristic approaches, due to the limitation of the pure empirical approach when applied to the URM sample, it emerges that: i) for the low-rise buildings, both the approaches highlight a decrease in the fragility from the oldest to the modern age (confirmed by all DSs). The decrease is more evident passing from 45–61 to POST61 rather than passing from PRE45 to 45–61. Such a result may be ascribed to the expected increase in the use of modern units (i.e. with the evolution of the concept of regular masonry) passing from 45–61 to POST 61 (see also Fig. 6a), together with a bit increase also in the percentage of HQD (even not negligible even in the 45–61 age). Also, in the case of medium-rise buildings, for the heuristic approach, this trend is confirmed even if the distance among the ages is in some case a bit different; ii) varying the height-class (i.e. by comparing the dotted and the continuous lines), the heuristic approach estimates an increase of the vulnerability passing from the low-rise to medium-rise buildings.