The macroseismic vulnerability method, briefly described in the previous section, was converted in terms of PGA by assuming different I-PGA correlations taken from literature (Faccioli and Cauzzi 2006; Murphy and O’Brien 1977) and comparing the results with those from mechanical-based models (Lagomarsino and Cattari 2014). However, a systematic validation of the method with observed data is still lacking. This validation and further developments of the method were made within the ambit of a research project promoted by the Italian Civil Protection Department (DPC) and that involved the ReLUIS consortium and EUCENTRE (Dolce and Prota 2020), aimed to develop the Italian risk assessment for the residential building stock (Italian Civil Protection Department 2018). Many research groups were involved and different approaches for the development of fragility curves were adopted, all of them calibrated and validated through a comprehensive database of observed damage (Da.D.O.), which collects the survey forms collected after the main earthquakes occurred in Italy in the last 40 years (Dolce et al. 2017, 2019).

The inventory of the building stock used for the risk assessment comes from the census of the population made in 2001 (ISTAT 2001), within which for any building a form was filled, with information on: structural material; age of construction; number of stories. Data are available in aggregated form for each municipality, in terms of number of buildings, number of flats, built surface, together with the number of inhabitants.

In the following, the new heuristic vulnerability model is illustrated. As aforementioned, it is derived from the macroseismic one through a calibration/validation taking profit of the observed damage in Da.D.O. and considering the characteristics of the IRMA platform (Borzi et al. 2018), which was used for the risk calculation.

Da.D.O. (http://egeos.eucentre.it/danno_osservato/web/danno_osservato) is a web-gis platform that collects the observed damage to residential buildings after nine different earthquakes in Italy: Friuli (1976), Irpinia (1980), Abruzzo (1984), Umbria-Marche (1997), Pollino (1998), Molise e Puglia (2002), Emilia (2003), L'Aquila (2009), Emilia (2012). Table 4 shows the number of buildings (distinguishing masonry and RC) that are present in the database, together with the range of the macroseismic intensity for which damage data are available.

The survey form and procedure were not the same all along more than 40 years, so a first critical issue is how to convert the available damage information into the EMS98 damage grade (from D₀ to D₅). This conversion was made within the Da.D.O. platform according to specific rules described in Dolce et al. (2019), but it is also possible to process the original data according to alternative proposals.

To each damage record, referred to a specific building, the seismic input should be associated. For vulnerability models that considers the macroseismic intensity, like the one described in Sect. 2, the intensity assigned to that location by the macroseismic survey was assumed. Other models require a physically-based intensity measure, like the PGA. Shake maps derived from recorded accelerations in specific points in the area should be used; however, it is worth noting that reliable shake maps are available only for the last earthquakes, thanks to the increasing presence of many accelerometric stations in the area (http://itaca.mi.ingv.it, Accelerometric National Network—RAN).

The use of observed damage requires a preliminary assessment of the quality of data, both in terms of completeness and potential errors. First of all, it is evident that for a robust calibration/validation of fragility curves it is necessary to have a large number of data, distributed in locations that suffered different seismic intensities, up to those that induce a significant damage level. For this reason, some of the earthquakes in Table 4, although interesting, are not useful for a systematic calibration of the heuristic vulnerability model. For example, considering the main earthquakes, the Friuli (1976), Umbria-Marche (1997) and Emilia (2012) earthquakes present lack of data in the area far from the epicenter. The check of completeness was made by ISTAT census, by comparing the available damage records with the number of buildings in each location. It emerged that only Irpinia (1980) and L’Aquila (2009) have a robust rate of completeness, which slightly decreases moving far from the epicenter; thus, the estimated number of lacking damage records was used to complete the damage histogram in each location, by assuming that those buildings were not damaged. Indeed, this is consistent with the fact that far from the epicenter the survey was made only to the damaged buildings under specific request of the owner.

Table 5 shows the number of buildings with available damage data for different seismic intensities. For the case of L’Aquila earthquake, the completeness of the damage survey was checked by comparing the number of buildings expected from ISTAT census (2001); it emerges that for the highest values of the macroseismic intensity (greater than 6) the damage survey was complete, while far from the epicenter only buildings that suffered some damage have been assessed. It is worth noting that for intensity greater than 8 the number of surveyed buildings is even more than what is expected from ISTAT inventory; the reasons may be related to the identification of buildings during the AeDES damage survey, which, in particular in the aggregates of historical centres, leads to a more detailed subdivision with respect to what was done during the ISTAT census. Therefore, in the case of L’Aquila earthquake, the missed buildings in the area far from the epicenter are considered without damage. Regarding the Irpinia earthquake, not having an ISTAT database prior to 1980, reference was made to the 1991 database. The latter has no information about the number of buildings, but it contains the number of dwellings for each district and consequently municipality. This parameter has been adopted for the analysis of the completeness of the database. From the comparison between the number of dwellings from Da.D.O. and ISTAT database emerged a good consistency, since in each municipality in Da.D.O. the number of dwelling surveyed is greater than the 80% of the ISTAT ones, even in the case of low intensity.

The damage assessment was made with different forms in the two considered seismic events. The Irpinia form classified the damage into 8 levels, while the AeDES form (Baggio et al. 2002) was used after the L’Aquila earthquake. Regarding the conversion from damage information to the EMS98 damage degree, slightly different assumptions were made in this paper with respect to the Da.D.O. proposals.

In the case of Irpinia earthquake, a direct correspondence was established in Da.D.O. among the damage levels (see Table 6), while for the calibration of the macroseismic method a continuous damage level was evaluated, by simply assuming that the eight levels of the Irpinia form represented a gradually homogeneous increase of damage.

This choice was made to not introduce a bias in the correlation, because the calibration of the method considers a continuous damage measure, the mean damage grade µ_D, representative of the damage distribution of buildings in areas that suffered the same macroseismic intensity.

Differently, the AeDES form adopted for L’Aquila earthquake contains, in Sect. 4, information on the damage to primary and secondary structural elements, classified as: (1) vertical, (2) horizontal, (3) stairs, (4) roof, and (5) infills. Then for each one, the local damage is measured by three levels (Light—D1; Moderate to heavy—D2/D3; Very heavy—D4/D5) and the extension within the building is indicated (A—spread on more than 2/3; B—between 1/3 and 2/3; C—< 1/3). Therefore, an overall damage grade is not defined. The one in Da.D.O. is based only on the damage occurred in vertical structural elements, by assuming proper conversion rules as a function of its extension. However, this approach tends to overestimate the damage grade that would be assigned by applying the EMS98 scale and does not consider other important elements. Thus, in the present paper, a weighted average of damage in all elements is evaluated, by considering the diffusion and assuming proper weights for the different elements. In particular, more importance to the damage to vertical structural elements, roof and also to horizontal floors (when the survey was made also inside the building, information that is present in Da.D.O.) has been assumed. The damage grade is then calculated by the following relation:

where w_i is the weight given to the 5 different elements (Table 7), v_i,j is percentage of the elements of type i in which it was observed the damage j. According to the extension previously indicated, v_i,j was tentatively assumed equal to: 1 (A); 2/3 (B), 1/3 (C), 0 (when no option is indicated). It is evident that must be less than or equal to one (if it is less than one it means that some elements were not damaged); when it results greater than 1 but values in the form are compatible with the intervals, the values of v_i,j were normalized. Moreover, specific checks were made to single out clear errors and the most plausible actual damage was defined. By using Eq. (7) the damage level results a continuous value between 0 to 5, which may be discretized by assuming, for example, the damage equal to 1 if 0.5 < D_AeDES < 1.5. However, analogously to the Irpinia earthquake, the definition of a discrete damage level to any single building is not necessary for the calibration of the method.

Figure 5 shows the increase of the mean damage grade with the macroseismic intensity for the damage data from all the available earthquakes, according to both: the Da.D.O. (a) and the proposed damage measure (b). The latter presents, as expected, lower values and a slightly more regular and similar trend for all earthquakes. It is evident how the mean damage grade remains almost constant for intensity lower than 6: this is because the lack of buildings in that locations, which are probably undamaged. Their presence would correctly reduce the mean damage grade, but in the case of L’Aquila earthquake it has been avoided an arbitrary completion of the database (therefore, they were neglected in the calibration). These observational macroseismic vulnerability curves follow the typical trend of the ones derived from EMS98 for the vulnerability classes (Fig. 3); however, it is worth noting that masonry buildings are here considered altogether while a better trend is observed by grouping buildings according to homogenous types (see Sect. 3.2).

In particular, focusing the attention to Irpinia and L’Aquila earthquakes data, masonry buildings seem to be globally a bit less vulnerable in the case of L’Aquila. It is useful specifying that curves are referred to all masonry buildings, without a classification in sub-types, but the same trend would be observed in the different ages of construction. It is worth noting that the increase of damage with intensity is lower than that foreseen by the macroseismic method (Eq. 3), because in low intensity areas the damage is overestimated for different reasons: (i) the database is not complete (because in the municipalities far from the epicenter undamaged buildings were not surveyed); (ii) when damage is low there is an attitude of surveyors to record any light damage, often neglected if the damage is higher, and sometimes the surveyors prefer to take precautions. A strange trend may be observed in the case of L’Aquila for the higher intensity values (8.5–9); these data are referred to the epicentral area where most of the buildings are from L’Aquila town and are constituted by the good quality palaces of the historical centre (expected to be better than the poor masonry buildings in the surrounding areas) and by a bigger percentage of modern buildings.

In order to check the validity of binomial damage distribution, the damage histograms have been evaluated for different intensity values and aggregation of building types. Focusing to L’Aquila earthquake, Fig. 6 shows the observed damage histogram of all masonry buildings in areas with macroseismic intensity 6 and 8, compared with the binomial distribution evaluated for the same mean damage grade. It is worth noting that the observed damage presents a bigger dispersion with respect to that of the binomial distribution: this occurs because very different buildings are treated altogether. Figure 7 shows that by dividing masonry buildings into sub-types, by age and number of stories, the dispersion is significantly reduced and the binomial distribution fits quite well the observed damage. This confirms the relevance of these parameters for the classification of masonry buildings. It is worth noting that while the age influences significantly the vulnerability (Fig. 7b), the height of the building is not particularly relevant (the mean damage grade is almost constant).

The macroseismic vulnerability method is represented by Eq. (3) as a function of one free parameter, the vulnerability index V; then, the fragility curves in macroseismic intensity may be derived from Eqns. (5) and (6). The calibration with observed data for Italian masonry buildings was made by using Da.D.O. and considering only the Irpinia (1980) and L’Aquila (2009) earthquakes, as motivated by the good level of completeness and good distribution in terms of intensity degree as documented in Sect. 3.1. For these two earthquakes, an accurate check of reliability of the information in Da.D.O. was made; in particular, in the case of L’Aquila earthquake, due to missing data in the low intensity areas, only locations with intensity greater or equal to 6 have been considered.

The damage database was treated by grouping all buildings subjected to the same macroseismic intensity and splitting them into sub-types of masonry buildings, according to the information in ISTAT census (ISTAT 2001):

Tables 8 and 9 show the number of available data with the different intensity degrees, for the Irpinia and L’Aquila earthquake respectively. It is worth noting that in the case of Irpinia earthquake the intervals related to the age of construction are a bit different and, obviously, the A5 is not present. It emerges that the number of buildings reduces for the highest intensity degree, because the epicentral area is smaller. Moreover, very few data are available for high-rise buildings; as they are not statistically significant, the model was not calibrated for high-rise buildings and their vulnerability was assumed by expert judgement through the comparison with that of the low and mid-rise buildings.

For each sub-type, the empirical points of the macroseismic vulnerability curve were obtained by collecting all values of the damage level for the same macroseismic intensity and evaluating the mean damage grade (Fig. 8). The trend looks similar to that of Fig. 3 but the increase of the mean damage grade with the macroseismic intensity (slope of the curve) is not the same in all the sub-types. It is worth noting that EMS98 gives a general framework of the vulnerability of all building types, with the aim of giving the maximum objectivity to the macroseismic intensity survey; in other words, the EMS98 scale is the result of an expert elicitation that cannot consider the different behavior of any building type. With the aim of a calibration with observed damage, a second free parameter, the ductility index Q, is introduced in the macroseismic vulnerability curve:

The ductility index Q determines the slope of these curves, and the value Q = 2.3 is strictly related to the assumption of the EMS98, for which in each vulnerability class the increment of the intensity by one degree determines an increase of one in the damage distribution. Higher values of Q correspond to a slope decrease, which means that it is necessary to increase the intensity of more than one degree in order to have the increase of one in the damage distribution (ductile behavior). Figure 8 shows both the best fit of V by using the original macroseismic model (with Q = 2.3) and the vulnerability V and ductility Q indexes of the new proposed formulation.

The fitted values were represented as a function of the building age, both for low and mid-rise buildings, distinguishing the Irpinia and L’Aquila earthquake (Fig. 9). Regarding the vulnerability index V, as expected, it increases with the age of the buildings, while a less clear influence of the building height is observed. In particular, in the case of L’Aquila earthquake, mid-rise buildings turn out to be less vulnerable than the low-rise ones: this may be due to the better quality of mid-rise buildings with respect to the poor and less important low-rise ones, mainly located out of the urban areas. The same trend is observed in Irpinia, but only for the sub-types before 1945. The vulnerability results a bit higher in the case of Irpinia earthquake (Fig. 9a); this result may be coherent with the distinctive features of masonry buildings in the two areas, but also by the uncertainties related to the attribution of damage levels (from the different survey forms adopted in the two earthquakes) and of the macroseismic intensity.

Another interesting outcome is that the ductility index Q is not constant for the different ages and a correlation with the vulnerability index V is evident: Q decreases when V decreases (that is for modern engineered buildings). In order to limit the number of free parameters of the model, the following correlation has been obtained:

where a lower bound for Q is assumed, compatible with the values fitted by the observed data. Therefore, the macroseismic vulnerability curve is thus modified:

The original (see Sect. 2) and the new calibrated macroseismic vulnerability curves are compared in Fig. 10; it is worth noting that in order to represent as much as possible the EMS98 vulnerability classes, the values of V (Table 10) are slightly modified with respect to the original ones (Table 3), in order to be compatible with the EMS98 for mean damage grades up to 2 (those encountered most frequently for medium intensity earthquakes and also in the case of strong events in the wide area surrounding the epicentral one).

It is worth noting that the value Q = 2.3, implicitly derivable from EMS98, seems to be correct for modern masonry buildings (Classes C and D), while for the traditional ones a bigger value is observed. The bigger ductility observed for ancient masonry buildings may be due to the fact that, even if modern buildings have a bigger shear strength their ductility is lower, due to the brittle behavior of hollow blocks and to the weak-piers & strong-spandrels collapse mechanism, which may be induced by the presence of RC tie beams. Moreover, it should be considered that these curves represent the performance of a set of buildings and the bigger ductility should be interpreted as a wider variability of the architectural configurations of the buildings in the set, typical of ancient buildings more than engineered modern ones.

Finally, the values of the vulnerability index V derived from the calibration with the observed data range between 1 and 0.4, values which are characteristic of vulnerability classes from A to D, being fully coherent with the EMS98 (Fig. 2); indeed, very modern masonry buildings (built after 1981 in L’Aquila area) may be even better (Class E).

The proposed correlation between V and Q may be considered valid for masonry buildings, at least in Italy. In the case of other building types, e.g. in RC, the correlation should be verified/updated analogously from observed damage, starting from Eq. (8).

In the previous sections the macroseismic vulnerability method was validated by the observed damage data collected in Da.D.O.. In particular, the general framework of EMS98 turned out to be very effective to describe the vulnerability of the different building types both in terms of progression of damage with the macroseismic intensity and of damage levels distribution. A simple refinement to the formulation has been proposed, in order to better calibrate the method with observed data.

Indeed, the macroseismic vulnerability method has the drawback of being formulated in terms of macroseismic intensity I, while seismic risk and scenario analyses usually adopt as input the PGA (or other physically based intensity measures). The conversion of the method was already proposed (Lagomarsino and Cattari 2014) by using I-PGA correlation taken from literature. However, the dispersion of the available correlations (Murphy and O’Brien 1977; Guagenti and Petrini 1989; Margottini et al. 1992; Faccioli and Cauzzi 2006; Faenza and Michelini 2010) is huge, because derived in different countries and, sometimes, considering data from different earthquakes.

In this paper the conversion is made by fitting a new correlation with the shake maps of the events for which the observed damage was available. In particular the shake map of L’Aquila earthquake (2009) was used, because it is based on a significant number of records and on updated models; on the contrary, the shake map of the Irpinia earthquake (1980) was neglected due to some inconsistencies that have been detected in many towns with the reliable values of the assigned macroseismic intensity. The use of a correlation directly derived on the earthquake for which the damage data are available reduces the possibility of accumulation of errors. The Authors have named the method as “heuristic vulnerability model” because it combines, in a no rigorous and intuitive way, the general framework of the macroseismic method and the available damage and seismic input data.

Most of the I-PGA correlations are formulated in this form:

but may be transformed as follows:

where:

are coefficient to be fitted from the available data; in particular, 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. Figure 11a shows the statistic values of the PGA, in locations with the same macroseismic intensity I, derived from the L’Aquila earthquake shake map (median values, 16% and 84% quantiles) and the linear least square regression; that allowed to provide the coefficients of Eq. (11). Figure 11b shows all the available data (grey bubbles), with the obtained correlation, compared with the above-mentioned correlations from the literature. As expected, the dispersion is huge but a general trend is evident, except for I = 6: it is worth noting that the PGA values from the shake map have the ambition to take also into account of site effects, while the macroseismic intensity is associated to the whole urban centre (or, even to the municipality). Anyhow, the median correlation (c₁ = 0.05 g, c₂ = 1.66) is quite close to all the available correlations (except the one from Faenza and Michelini 2010), despite the huge dispersion.

The heuristic vulnerability model identifies any sub-type of masonry buildings (defined by a combination of attributes from the available inventory) by the vulnerability index V. By assuming the correlation of Eq. (12), it is possible to derive the median value PGA_Dk for the fragility curve of each damage level (k = 1,...5), passing through the corresponding intensity I_Dk, given by Eq. (6) in the original macroseismic method; however, the latter have been here evaluated using the calibrated method, represented by Eq. (10). The following equation is derived for , as this heuristic vulnerability model is proposed for masonry buildings:

The complete fragility curve is directly obtained numerically, by using the eqns. (1), (10) and (12), but the trend is very well fitted by the lognormal cumulative distribution, by calibrating the values of the dispersion β_Dk. The dispersion implicitly results from: (i) the new calibrated macroseismic vulnerability curve; (ii) the binomial distribution of damage levels; (iii) the assumed I-PGA correlation (in particular through the parameter c₂).

Figure 12 shows the fragility curves representative of the vulnerability classes A (V = 1) and C (V = 0.6). For each class, the best fitting is obtained by slightly different values of the dispersion for each damage level; however, the use of different values with the lognormal distribution leads to the intersection of the corresponding fragility curves for low PGA values (when β_Dk < β_Dk+1) or high PGA values (when β_Dk > β_Dk+1). Even if these intersections may occur out of the significant range of PGA (the one used for the risk calculation by the convolution integral), it is better to assume an average constant value β_D of the dispersion for the all set of fragility curves (k = 1,…5). On the contrary, the influence of V on the dispersion should be considered: this means that each vulnerability class has a proper dispersion.

A good fitting is provided by the following equation:

which may be simplified in the following, for the model derived from the L’Aquila shake map:

Therefore, the heuristic vulnerability model for masonry buildings provides analytically the set of fragility curves of a set of buildings, classified by a proper taxonomy through the attribution of the specific vulnerability index V (or range of values), and assuming the proper I-PGA correlation for the study area (in terms of the parameters c₁—PGA for intensity I = 5—and c₂—factor of increase of PGA due to an increase of 1 of the macroseismic intensity). The median values PGA_Dk for the different damage grades and the dispersion β_D (assumed constant for all damage levels) are given by eqns. (14) and (15).

For the six vulnerability classes of EMS98, the white values V_i (i = A,B,..F) of the vulnerability index have been assumed as reference (see Table 10). The classification of buildings only in terms of the masonry type (Fig. 2) implies that within the set there are buildings of different vulnerability class. Each subset is represented by the above-introduced fragility curves, while the behavior of the whole masonry type may be obtained through a combination of the response of the vulnerability classes that are present, weighted by a coefficient w_i (i = A,B,..F). The latter may be estimated from the table in Fig. 2 and by a specific knowledge of the buildings in the study area. As an alternative, one single set of fragility curves may be evaluated as representative for the whole masonry type. The median values may be evaluated with Eq. (14) by considering the representative vulnerability index V^*, which may be evaluated as:

The dispersion should be increased with respect to the one related to a single value of the vulnerability index, given by Eq. (16), by considering two additional contributions due to:

The first contribution may be evaluated by concentrating the variation of V in the class at the extremes of the plausible range. After simple mathematical steps the following formula is obtained:

which is simplified, assuming the I-PGA correlation from L’Aquila earthquake, as follows:

The second contribution may be evaluated as follows:

By assuming all these contributions as independent, the dispersion β^* of the whole masonry type is obtained as follows:

where the last two contributions depend both by the damage level k. As they decrease for the higher damage grades, the total dispersion decreases and the possibility of an intersection of the fragility curves for high values of PGA cannot be excluded and should be checked in the range adopted for the risk integration.

By way of example, the fragility curves for two masonry types are evaluated, by assuming a combination of the vulnerability classes coherently with EMS98 (Fig. 2):

Table 11 shows the assumed weights w_i and the consequent parameters of the fragility curves. Regarding the dispersion, it emerges that the contribution of β₁ is almost negligible, while that of β₂ is more significant, in particular for M6 type that is formed by buildings of three different vulnerability classes. The resulting dispersion β^* is not constant but decreases for the highest damage levels: this implies intersections between fragility curves, but actually they do not occur for the PGA of interest. Figure 13 shows the fragility curve of damage level 3 for M6 type, obtained from the parameters in Table 11, compared with those of the vulnerability classes B, C and D, as well as the one obtained by a weighted combination of the latter (dotted line). Conceptually, the fragility curve obtained by the combination is the correct one but the definition of an accurate single lognormal curve is very helpful.