Processing Italian damage data to derive typological fragility curves

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Soil Dynamics and Earthquake Engineering 28 (2008) 933–947 www.elsevier.com/locate/soildyn

Processing Italian damage data to derive typological fragility curves M. Rotaa, A. Pennab,, C.L. Strobbiab a

European School for Advanced Studies in Reduction of Seismic Risk, via Ferrata 1, Pavia, Italy European Centre for Training and Research in Earthquake Engineering, Via Ferrata 1, Pavia, Italy

b

Received 31 May 2006; received in revised form 25 July 2007; accepted 11 October 2007

Abstract Typological fragility curves have been derived from post-earthquake survey data on building damage, collected in the areas affected by the most relevant Italian earthquakes of the last three decades. A complex and time consuming codification and reinterpretation work has been done on a set of about 150,000 survey building records, in order to define empirical damage probability matrices for several building typologies, characteristic of the Italian building stock. The obtained data have then been processed by advanced nonlinear regression methods in order to derive typological fragility curves. These curves, organised in five damage levels, provide useful information both for relative comparisons among typologies and for seismic risk analyses at different scales. By combining hazard definitions, fragility curves and inventory data, complete earthquake risk scenario studies can be performed, but even the single convolution of hazard and fragility allows to obtain typological risk maps, both for single damage state definitions and for concise average loss parameters. The very high potential of these results is shown by some applications reported in the paper. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fragility curves; Seismic vulnerability; Post-earthquake surveys; Nonlinear inversion; Bootstrap; Existing buildings

1. Introduction The assessment of seismic vulnerability of buildings is one of the most relevant activities in earthquake engineering; this can be done at different levels, ranging from the evaluation of vulnerability of a single building to the approximated characterisation of the behaviour of classes of buildings for urban or regional risk analyses. Researchers, engineers, decision makers and other figures involved in risk prediction and management know very well that building vulnerability is the part of seismic risk depending on human actions and hence it is the main aspect when men can intervene, in order to reduce risk. The only other alternative, when it is not possible to change building exposure (as in the case of existing buildings) would be a relocation of the public functions eventually hosted in the buildings, in order to mitigate the seismic risk for people. Corresponding author. Tel.: +39 0382 516924; fax: +39 0382 529131.

E-mail addresses: [email protected] (M. Rota), [email protected] (A. Penna), [email protected] (C.L. Strobbia). 0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.10.010

Both a first evaluation of a single building and the characterisation of a whole building population can start from a typological identification. Building typologies are defined based on an expected common seismic behaviour: the probability of reaching a certain damage state for a given seismic input can vary significantly depending on building construction material and technology, on structural configuration and on several constructive details. A convenient and widely adopted way for defining typological seismic vulnerability is the use of fragility curves; they can be derived using different methods and based on various types of data, and they directly provide the probability of exceeding a damage state threshold, as a function of a selected seismic input parameter. Different classifications of the methods used to derive fragility curves have been proposed. One of the most commonly used is based on the main source of information and allows identifying four different groups of procedures: judgement-based, analytical, empirical and hybrid methods [1]. Judgement-based vulnerability curves are derived from statistical treatment of the information obtained from a

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team of experts, which are asked to provide an estimate of the average damage level (previously clearly defined), for various types of structures, subjected to different levels of seismic motion [2]. The answers are then circulated to all the experts, which are asked to eventually review their judgement in light of the opinion of the others, until a consensus is reached. The damage estimates are then fitted with some known probability distribution and plotted versus the ground motion severity, to obtain vulnerability curves, which can hence be used for damage scenarios and predictions. No problems of lack of data or limited range of ground motion, typical of other methods, are associated to this approach; experts can in fact be asked to provide damage estimates for any structural type, ground motion severity and so on. Moreover, the curves derived from expert opinion can easily include all the factors affecting the seismic response of different structures. This technique may be very useful for generating vulnerability curves or damage probability matrices (DPMs) for classes of structures which are reasonably well defined in structural terms, but for which other methods cannot be applied, due for example to limited availability of empirical damage data, or lack of information necessary to apply analytical methods. However, the reliability of judgement-based curves is questionable due to their dependence on the individual experience of the experts, which may determine a non-uniform assessment. Moreover, to be statistically sound, all estimates should be independent, which is often not the case, since experts tend to influence each other. Analytical methods derive vulnerability curves from statistical elaboration of the results of numerical analyses carried out on structural models. For large scale vulnerability analyses, capacity and demand are often determined using simplified analytical methods [3–8], since refined models would be too expensive from the computational point of view and would not be appropriate due to lack of knowledge of all necessary input parameters. Structural capacity, in particular, is often determined using equivalent single degree of freedom (SDOF) systems, while seismic demand may be represented by means of a spectral analysis. For the vulnerability analysis of a single building or of few particular cases, more refined models can be used. An important issue of analytical methods concerns the similarity of the model to the real structure, which strongly influences the reliability of the results and which is affected by modelling capabilities. Despite the very rapid development of numerical analysis techniques and solution procedures, which allow more and more realistic representation of the structures at a reduced computational time, some limitations still exist. This issue is particularly important when using simplified models of the structure: in this case, only structural elements are modelled, neglecting the contribution of all non-structural elements to the seismic response. In addition, many existing analysis environments show convergence problems when structures are subjected to large demands, with numerical collapse frequently preceding structural failure.

Analytical methods are intended to be used primarily when more detailed information than what may be offered by the empirical methods, is needed. Moreover, they can be required in case of analysis of a specific structure or type of structure, for which no empirical data is available, either because they are new or because they are too complex to be assigned to typological classes. Compared for example to judgement-based methods, analytical methods can result in reduced bias and increased reliability of the vulnerability estimate for different structures. The main disadvantage of this type of methods is that they require a substantial computational effort and hence are more demanding and usually more expensive than other types of approach. Empirical methods, which will be discussed in detail in the next section, use data collected during post-earthquake surveys as the main source of information and elaborate these data according to statistical procedures. Finally, hybrid approaches try to overcome some of the main limitations of the various methods previously described, making use of different sources of information combined together. Examples can be the macroseismic methods, which are based on both empirical data and expert judgement [9,10], or the method proposed by Kappos et al. [11], which combines empirical data and analytical results. 2. Empirical curves: limitations and advantages As already anticipated, empirical curves are derived from statistical elaboration of data collected during post-earthquake surveys. The main advantage of this method is quite evident and consists in the fact that real observed data are obviously the most realistic source of information, since they intrinsically take into account all the characteristics of the building stock and of the ground motion. Among others factors, they accurately represent structure, soil effects, site profile characteristics, source and path of the earthquake. One of the limitations of this method, on the other hand, is related to the heterogeneity of the empirical data from which curves are derived and of the post-earthquake surveys from which data are obtained. Intuitively, data should be as much as possible homogeneous both in time and space, since significant variations of construction technique may occur during time and from place to place, and this may compromise the homogeneity of building classes. On the other hand, surveys are performed using different forms and procedures from country to country and also from time to time and are carried out by people of very different experience and background knowledge. The need for using heterogeneous data is related to another issue of empirical methods: the amount of data available. A large amount of data is in fact needed to obtain realistic vulnerability curves and to reduce the scatter in the results. The use of homogeneous, but small, data sets, indeed, is likely to produce biased results, which are not significant from a statistical point of view and hence cannot be

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extended to cover different ranges of ground motion or constructional typologies. Hence, the choice becomes whether to use data coming from a single event, or combining data from different events. The first option (e.g. [12,13]) will provide a data set usually small, limited in ground motion range and specific to that given area, but homogeneous. The second option [1,14,15] allows solving the problems due to the limited amount of data, at the expenses of their homogeneity. The related risk is to loose the accuracy of the estimate, since very different information are combined together. In any case, this approach will often not solve the problem related to the limited range of ground motion, due to the (lucky!) infrequency of large magnitude events affecting densely populated areas. The issues described above draw the attention to one of the main advantages of the current study, which is based on a very large data set of empirical data, all collected during Italian post-earthquake surveys. This allows to significantly increase the data homogeneity, even if there still are some regional differences in the surveyed building stocks. The availability of such a rich database has also allowed to carry out severe choices on the quality of the information derived from the surveys, even if this has required disregarding some of the data, as will be discussed in a following section of this paper. Data have been hence homogenised and allocated to classes of ground motion severity and to building typologies, accurately selected based on a priori information and on the analysis of data. In the derivation of fragility curves from these empirical data, different issues have been considered, with a particular consideration for the evaluation of sample accuracy. 3. Elements of fragility curves Fragility curves represent the conditional probability of a class of buildings reaching or exceeding a specific level of damage, for a given level of ground motion. Thus, the development of fragility curves requires to identify a parameter to accurately represent the ground motion, to select the levels of damage which are of interest, and to choose the building typologies for which fragility curves are going to be developed. 3.1. Seismic input characterisation On the horizontal axis of a fragility curve the level of ground motion associated to a certain value of probability of damage is represented. Among the different parameters used in the literature to characterise the severity of ground motion, the most commonly encountered are macroseismic intensity and peak ground acceleration (PGA). It has been demonstrated that both of them have advantages and drawbacks and there is a rather heated argument on the literature on which one is more suitable for use in the derivation of vulnerability curves

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(e.g. [16,17]). On the other hand, none of the parameters which have been used for this scope has been universally accepted and hence, in any case, a compromise is needed. Until very recently, macroseismic intensity was used nearly exclusively to characterise ground motion. This parameter has the advantage that historical data on earthquakes are available, being the only possible type of data at the time when seismic instruments were not yet available. Actually, in many seismic regions of the world, there are still no instruments (or an insufficient number of them) to measure ground motion and hence only few data are available. However, this descriptive parameter is based on observations of the effects of the earthquake on the environment and hence it incorporates elements of structural vulnerability; therefore, its use for evaluating damage may appear as a nonsense, since it means actually using damage to evaluate damage. Indeed, being a subjective parameter, strongly dependent on the characteristics of the building stock under observation, it is very likely that different people would come up with different values of intensity at the same site and, also, it is possible for the same intensity to be recorded in two different regions for very different levels of ground motion [18]. Another drawback comes from the fact that intensity is a step function defined only at discrete levels, while the development of vulnerability curves requires the use of attenuation equations that predict arithmetical values of intensity as if it were a continuous variable. Moreover, it is rather difficult to relate macroseismic intensity to a physical parameter descriptive of the performance of the building, in terms of force capacity or displacement, as derived from structural analysis. To this end most often PGA is used, as it has a clear mechanical meaning. In addition, PGA is an objective measure of the severity of ground motion, with no problems related to its interpretation and use for different areas than where recorded. A common shortcoming of both intensity and PGA is that they ignore the relationship between the frequency content of ground motion and the dominant period of buildings. Other parameters have been proposed, such as peak ground velocity, spectral acceleration, spectral displacement, Housner intensity and many others. For a detailed discussion on most of them, the reader is referred to Rota [17]. However it should be noted that, for many applications, most of these parameters can be correlated with PGA, for example assuming predefined spectral shapes. PGA has been chosen in this work for representing ground motion in the derivation of fragility curves. The main reason for this choice, in addition to the issues already discussed, is that the Italian hazard map [19] is described in terms of PGA and hence vulnerability curves in PGA are needed in order to develop risk scenarios. The authors are aware that many relationships exist in the literature, which allow to transform intensity in PGA and

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vice versa (e.g. [20–22]); however, the scatter associated to these relationships is really large. A single mean value of PGA has been defined for each municipality affected by one of the considered earthquakes, evaluated using the attenuation relationship proposed by Sabetta and Pugliese [23], with the magnitude and the epicentre of the corresponding earthquake. The mean value of the attenuation relationship has been used, with the PGA evaluated on rock. The influence on results, in terms of lognormal fragility curves, of the uncertainty associated to the PGA evaluated with the attenuation law has been investigated. In order to do that, the PGA values have been perturbed by applying a randomly generated error, obtained from a normal distribution of PGA values, defined within the range of the value of PGA plus or minus the same value multiplied by a selected error. The comparison of the original (unperturbed) values of PGA and the values obtained by applying a random error of 50% (perturbed) is shown in Fig. 1. For one of the considered building typologies (IMA1), the corresponding fragility curves for the different levels of damage have been calculated, using the parameters of the lognormal distribution reported in Table 3 and the values of PGA obtained from the attenuation law. Subsequently the curves have been recalculated using the perturbed PGA values. Several values of the error have been applied. It has been demonstrated that even the application of an error of 7100% on the estimated value of PGA does not produce a significant variation in the lognormal curves. Fig. 2 shows the results, in terms of lognormal fragility curves for the five considered damage levels and for the typology IMA1, for an error of 750%. The results demonstrate that, even if the scatter in PGA is very significant, the difference in the results, in terms of fragility curves, is limited.

Fig. 2. Comparison of the fragility curves for the typology IMA1, obtained using the mean PGA from Sabetta and Pugliese, and using this value perturbed by an error of 50%. The curves refer to the following damage level, from top to bottom: red ¼ DS1, green ¼ DS2, blue ¼ DS3, black ¼ DS4 and magenta ¼ DS5.

It is well known that the value of PGA on rock may be amplified due to stratigraphical and/or topographical effects. These site effects can be incorporated into the evaluation of risk in different ways, either in the hazard part or in the vulnerability, since risk is obtained from the convolution of hazard, vulnerability and exposure. When using post-earthquake survey data, it is not possible to know the different nature of the site where each building is located, since the only information available concerns the municipality. It is not meaningful to evaluate site effects at the municipality level, because they can significantly vary from building to building. Hence, the only possibility is to evaluate the hazard on rock and assume that the vulnerability description incorporates also the actual variability of local site effects. Since we are analysing thousands of buildings, we consider our results as representative of the variability of real combinations of vulnerability and local amplification effects. 3.2. Damage scale

Fig. 1. Comparison of mean PGA values from the attenuation law (unperturbed) and values obtained by applying an error of 750%, as defined above (perturbed).

Fragility curves are constructed for selected damage states, which need to be defined in a clear and unambiguous way. Defining different levels of structural damage also means identifying different performance levels for the structure, allowing to predict other consequences of the earthquake, such as potential human casualties, interruption of building occupancy and facility function, cost of repair or eventually of replacement [24,25]. This representation of vulnerability allows to evaluate in a more accurate way the consequences of possible events, both in economical terms and in terms of civil protection.

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In this work, a damage scale corresponding to the one defined in the European Macroseismic Scale [26] has been adopted. It is articulated in five levels of damage (plus the case of no damage, DS0): negligible to slight damage DS1, moderate damage DS2, substantial to heavy damage DS3, very heavy damage DS4 and destruction DS5. Since post-earthquake surveys were carried out using forms varying in time and hence they refer to different damage scales from earthquake to earthquake, in order to derive fragility curves based on data from all the available events, it has been necessary to convert the different scales into a unique one. The scheme reported in Table 1 has been followed, which is only slightly different from the one adopted by Dolce et al. [27]. When interpreting the level of damage associated in the forms to each surveyed building, only structural damage has been considered, ignoring at this stage nonstructural damage. In particular, the maximum level of damage observed among vertical bearing structure, horizontal structure and roof has been used for assigning the building to either one class or the other of the adopted damage scale. The choice of using the maximum damage, especially in cases where multiple choice is allowed, is dictated by the necessity of using an unambiguous criterion. Moreover, the maximum observed damage is usually the main element influencing the usability evaluation of the building and its cost of repair. The information regarding non-structural damage has been used only for those cases in which there was no indication about the structural damage, to be able to separate cases of no damage, from cases of incompletely filled forms. In particular, if some type of non-structural damage was marked on the form, it has been assumed that the lack of indication of structural damage actually corresponds to a case of null structural damage. If instead, neither structural nor non-structural damage was indicated on the form, this is a clear case of incompletely filled form and hence damage has been considered as non-identified and the corresponding form has been disregarded, as discussed in a following section.

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3.3. Building typologies Fragility curves are defined for selected building typologies, grouping together structures which are expected to have a similar seismic behaviour. The introduction of a clear and well defined building classification is also necessary to ensure a uniform interpretation of the results. When developing a building classification, a compromise is always needed between usability and accuracy. A too detailed subdivision among different classes may lead to very specific results, but may be unpractical both for what concerns the derivation of the curves (many curves and very reduced sample for each of them) and for the use of the curves (need many information to be able to assign a building to a class). On the other hand, broad categories may group together buildings with a completely different seismic behaviour, leading to an average vulnerability which is not actually representative of any typology. It is obvious to anyone that the type of vertical and horizontal bearing structure is the most important factor affecting the seismic response of a building, as most studies of earthquake damage have demonstrated. In some instances, the vertical structure is a sufficient definition, but in other cases the horizontal structure used for floors and roof may be equally important [25]. However, a potentially strong building type can be very weak if the configuration or design details are badly considered, and conversely, basically weak systems can be greatly strengthened by careful design and good construction. This is the reason why the date or period of construction of a building is a crucial factor influencing the vulnerability evaluation, since it determines constructive details and design criteria enforced in a given area. The number of storeys also plays an essential role in defining the seismic behaviour of a structure, being directly related to the height of the building and hence to the period of vibration. Thus, the number of storeys has a strong effect on both structural capacity and demand. Many other aspects have been shown to have an influence over buildings performance in earthquakes, such

Table 1 Scheme of association of the different damage levels used in the various earthquakes to the five damage states Damage state

Damage description Irpinia (1980)

Abruzzo (1984)

Marche (1997)

Pollino (1998) and Molise (2002)

DS1

Irrelevant—repair is not urgent Slight—to be repaired

Slight

Null or slight 1323 Null or slight423

DS2 DS3

Significant—to be partially evacuated-repairable Severe—to be evacuated—repairable

Significant Severe

DS4

Very severe—to be evacuated and demolished

Very severe

DS5

Partially collapsed—to be demolished Destroyed

Destruction

Medium-severeo13 Medium-severe 13  23 Medium-severe423 Very severe-collapseo13 Very severe-collapse 13  23 Very severe-collapse 423

Slighto13 Slight 13  23 Slight423 Severeo13 Severe 13  23 Severe423 Very severeo13 Very severe 13  23 Very severe423

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as for example the structural form (regularity in plan and elevation, foundations), the site planning (pounding with adjacent buildings, slope effects, local ground failures), the construction quality (materials and work) and the history (pre-existing damage, repair, strengthening or maintenance interventions, modifications to the structure). However, it would not be easy to include all these aspects in a typological classification, especially because finer and finer subdivisions of the building classification would require correspondingly more vulnerability relationships to be defined, and quantitative measures of the separate influence of these factors are difficult to obtain. Nevertheless, it is very important to point out that the choice of the building typologies to be adopted will depend not only on the characteristics which are expected to influence the earthquake performance of the structure, but also on the extent and type of available data. In this study, building typologies have been identified with reference to the RISK-UE [7] building typology matrix, modified on the basis of the information available from all the databases (each one corresponding to a different post-earthquake survey). The selected typological classification is reported in Table 2. It can be noticed that buildings have been firstly subdivided according to type of vertical bearing structure: reinforced concrete, masonry, steel and mixed (masonry and reinforced concrete). Data concerning steel structures were actually quite limited in number, and even not all the forms used in postearthquake surveys included this structural typology. For this reason, no further subdivision has been introduced for this typology of structures. For what concerns mixed structures, two categories have been identified, based on the number of storeys. Reinforced concrete structures have been distinguished considering both the number of storeys and the year or period of construction. The latter information, together with the year of seismic classification of the municipality in which the building is located, allows to determine whether the building has been designed according to seismic regulations or not. Buildings have been considered to be seismically designed when constructed after 1975, in a municipality that was already classified as ‘‘seismic’’ at the time of construction. The year 1975 has been selected as a reference, according also to Di Pasquale et al. [37], since it corresponds to the first applicative decree of the Italian law No. 64 of 1974, which introduced a new seismic zonation and specific regulations for constructions in seismic areas. Regarding masonry, in addition to number of storeys and period of construction, the type of horizontal structure has been considered, distinguishing between rigid floors (reinforced concrete slabs and vaults) and flexible floors (wood, steel, etc.). Moreover, the layout and quality of masonry has been considered, with the class called ‘‘regular masonry’’ including buildings with a regular layout of good quality masonry and the class ‘‘irregular masonry’’ including buildings with irregular layout or bad quality masonry. The reference year of construction used to

Table 2 Adopted building typological classification Label

Description

No. of storeys

MX1 MX2 RC1 RC2 RC3 RC4 IMA1

Mixed Mixed Reinforced concrete—seismic design Reinforced concrete—no seismic design Reinforced concrete—seismic design Reinforced concrete—no seismic design Masonry—irregular layout—flexible floors—with tie rods or tie beams Masonry—irregular layout—flexible floors—w/o tie rods and tie beams Masonry—irregular layout—rigid floors—with tie rods or tie beams Masonry—irregular layout—rigid floors-w/o tie rods and tie beams Masonry—irregular layout—flexible floors—with tie rods or tie beams Masonry—irregular layout—flexible floors –w/o tie rods and tie beams Masonry—irregular layout—rigid floors—with tie rods or tie beams Masonry—irregular layout—rigid floors—w/o tie rods and tie beams Masonry—regular layout—flexible floors—with tie rods or tie beams Masonry—regular layout—flexible floors—w/o tie rods and tie beams Masonry—regular layout—rigid floors—with tie rods or tie beams Masonry—regular layout—rigid floors—w/o tie rods and tie beams Masonry—regular layout—flexible floors—with tie rods or tie beams Masonry—regular layout—flexible floors—w/o tie rods and tie beams Masonry—regular layout—rigid floors—with tie rods or tie beams Masonry—regular layout—rigid floors—w/o tie rods and tie beams Steel

1–2 X3 1–3 1–3 X4 X4 1–2

IMA2 IMA3 IMA4 IMA5 IMA6 IMA7 IMA8 RMA1 RMA2 RMA3 RMA4 RMA5 RMA6 RMA7 RMA8 ST

1–2 1–2 1–2 X3 X3 X3 X3 1–2 1–2 1–2 1–2 X3 X3 X3 X3 All

identify masonry buildings with tie rods and tie beams, in those databases where this information was not directly available, is 1909, i.e. the year after the terrible Reggio and Messina earthquake, corresponding to the first seismic classification of the Italian territory. It should be pointed out that for what concerns the number of storeys, basement floors have not been considered, assuming they are stiff enough so that their influence on the seismic response of the building is negligible. 4. Available empirical data The data set used to derive fragility curves consisted initially of approximately 163,000 buildings (163,479), which have been surveyed after the main recent Italian earthquakes. In particular, data collected after the

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following events have been used: Irpinia (1980), Abruzzo (1984), Umbria-Marche (1997), Pollino (1998) and Molise (2002). It should be pointed out that, for what concerns the earthquake of Umbria-Marche (1997), only data observed in the Marche region are included in this work, since data of the Umbria region are still being processed for the difficulties of homogenisation of this data set with the others. Since each survey has been carried out using a different form and, after the survey, data have been computerised in databases following different criteria, a very complex and time-consuming work was needed. In fact, it has been necessary to first understand the meaning of each field of each database, and then to homogenise the information among the different databases, in order to derive uniform damage scales, building typologies, etc. This process has obviously involved several interpretative assumptions, which are described in detail by Rota [17]. Due to the very rapid nature of the survey, not all the forms were adequately compiled: in some cases, some necessary data were missing and hence the corresponding forms have not been used for deriving fragility curves. This happened in cases when one of the following information was missing: vertical structure typology, horizontal structure typology (only for masonry buildings), municipality in which the building is located (necessary to associate a PGA value). After this first processing of the databases, less than 6% of the data has been disregarded, corresponding to 9713 buildings. 83 additional buildings have been disregarded, because they had a not well defined vertical structure, including for example steel, masonry and reinforced concrete at the same time. Moreover, some additional buildings have been discarded, since no information concerning the level of damage was associated to them. For this latter reason, 3476 buildings have been ignored (approximately 2.3% of the original amount of data). After all the processing just described, 150,207 buildings remained, subdivided among the different building typologies (described in Table 2) as indicated in Fig. 3.

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In order to directly derive fragility curves, it is necessary to know the distribution of damage into the different damage states, for a population of buildings exposed to a certain PGA level. For each PGA class, the number of buildings reaching a given level of damage has then to be normalised to the total number of buildings in the population. However, due to the nature of post-earthquake surveys, which are often carried out only on request and hence on the damaged buildings, the total number of buildings exposed to a selected PGA interval is often unknown. Only few municipalities, located in the epicentral areas are known to be completely surveyed. The use of data coming from incompletely surveyed samples is likely to lead to an overestimation of the probability of damage, since buildings are normalised with respect to an underestimated dimension of the sample. Moreover, it can be expected that the distribution of damage of non-surveyed buildings is different from that of surveyed buildings since, most likely, the lost observations refer to buildings without any damage. It is clear then that using all the data without any consideration on the sample completeness would provide biased results. One possible solution could be to use only data coming from municipalities which are declared as completely surveyed. However, this is not a suitable choice, since it would dramatically reduce the sample. On the other hand, it is very important to increase the sample as much as possible, both to improve the reliability of the results and to avoid small and regional samples, especially for some building typologies. Hence the following procedure has been used: for each considered municipality, the number of surveyed buildings has been compared to the total number of buildings according to census data [28]. This method may be questionable, since the definition of building can be slightly different in the two approaches, but it has been considered appropriate, in order to define a threshold for considering a municipality as completely surveyed and hence correct the normalisation. Two different samples have been defined, including data of different levels of completeness and both of them have been processed. The

Fig. 3. Subdivision of the surveyed buildings among the considered building typologies.

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comparison of the results obtained from the two samples, corresponding to only complete survey data and to an extended data set including municipalities surveyed for at least 60% (a reference percentage used also by Bramerini et al. [29]) of the buildings, as compared to census data, shows that the difference between the two can be significant for building typologies with only few data, while it is quite small for typologies characterised by a large sample. Fig. 4 shows the histogram of the percent difference between the experimental frequencies of the two samples. The value of the frequency of occurrence of each PGA level and each damage level of one sample has been compared with the corresponding value of the other sample. The histogram represents hence, in the horizontal axis, the percent value of the absolute difference and, in the vertical axis, the frequency of occurrence of that particular difference value. This figure shows that, even if some significant values of

Fig. 4. Percentage difference (absolute value) among the results of two samples: only completely surveyed data and extended set of data.

difference can be observed (up to 40%), 76% of the cases have a value of absolute difference lower than 10%. After these considerations, the data set used to derived fragility curves consists of about 90,000 surveyed buildings (91,394). The subdivision of these data among the different earthquakes, the various PGA intervals and the types of vertical structure is illustrated in Fig. 5.

5. Derivation of fragility curves The first step for defining fragility curves consists in the derivation of the DPMs, representing, for each building typology and for each PGA interval, the experimental probability of occurrence of the different damage states [30]. Then, the probability of reaching or exceeding a given damage level can be easily obtained by simply cumulating the experimental frequencies from the highest to the lowest level of damage. An example of the results obtained for five PGA classes and for the three typologies of RC2, MX1 and IMA2 is shown in Fig. 6. It can be observed that the frequency corresponding to null damage is not plotted, since its cumulative value is always equal to 1. Once DPMs have been extracted, the experimental data have been fitted using an analytical function, to be able to describe, for each value of PGA, the probability of exceeding a given level of damage, with a simple analytical expression. The lognormal cumulative distribution has been selected for this aim, since it has been very frequently used for representing fragility relationships (e.g. [5,31,32]). For a discussion on alternative analytical models for fragility curves, the reader is referred for instance to [17]. The cumulative lognormal distribution, F(X) is computed as the integral of the lognormal probability density function f(x) as follows: Z X 2 1 2 F ðX Þ ¼ f ðxÞ dx; where f ðxÞ ¼ pffiffiffiffiffiffi eðln xmÞ =2s . xs 2p 0 (1)

Fig. 5. Subdivision of data among PGA intervals (left) and building typologies (right), with identification of data coming from the different earthquakes.

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Fig. 6. Cumulated DPMs for five PGA intervals and for three selected building typologies: RC2, MX1 and IMA2.

The probability that a certain damage state is exceeded, given a value of PGA, depends hence on the two parameters m and s of the lognormal distribution, which have to be inferred by fitting the experimental data with a lognormal curve: for each typology, the model corresponding to the best fitting with empirical data is assumed as fragility curve. To find the two unknown model parameters (m and s) an inverse problem has to be solved: a standard local search approach has been chosen for the non-linear least-squares optimisation. The merit function is defined as the mean square error between observed and theoretical damage, and with a Levenberg–Marquardt method [33,34] the two parameters are identified, iteratively decreasing the mean square error by following gradients, downhill in the model parameter space. The partial derivatives are computed numerically at each iteration to build the Jacobian matrix, and convergence is reached in few steps, with a damped least-squares approach: the damping matrix, together with the weights, guarantees the stability and robustness of the inversion. Alternative methods for the solution of an inverse problem exist and could be applied, including for example local and global search approaches. However,

considering that only two parameters need to be estimated, the influence on the results of the inversion algorithm is likely to be negligible. Nevertheless, since the data quality is not homogenous, this simple approach can pose several problems. The subdivision of the available data into 23 building typologies and 10 PGA classes has the effect of reducing the size of some samples, hence reducing also the reliability of the estimated damage distribution. For each typology and each PGA, a point estimate of the probability has been obtained from the DPM, but the reliability of this elementary information is not constant and can be strongly dependent on the sample size. To reduce the effect of ‘‘bad’’ points on the estimated parameters, appropriate weights can be introduced. The first step in defining weights for the weighted inversion is the assessment of uncertainties of the damage distribution (i.e. probability of each damage state) for each typology and each PGA. Since we use as a point estimate of the vulnerability the experimental distribution of damage within the observed sample, the bootstrap technique [35] has been implemented to evaluate this uncertainty: each building typology data set is resampled with substitution to generate several samples of the same

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size, which are then analysed in order to estimate the standard deviation of each damage state probability. The estimated uncertainties will of course be larger for smaller data sets and low probability damage states. The inverses of the estimated standard deviations are then used as weights in the inversion: the merit function is hence defined as mean square error, standardised for the standard deviations. The effect of the weights is less important for the most common typologies, characterised by a large amount of data, while it can strongly improve the quality of results for typologies with smaller observed samples. Through this procedure, a fragility curve is directly derived for each building typology and for each damage state, unambiguously defined through the corresponding values of m and s. The effect of the variations of the values of m and s on the shape of the lognormal curve is shown in Fig. 7.

Fig. 7. Effect of the variations of m and s on the shape of the lognormal curve.

Examples of the fragility curves obtained using this approach are shown in Fig. 8, which illustrates the fitting and the effect of weights for the two typologies IMA2 and RMA2. It is important to note that the only difference between these two typologies is the type of masonry layout, which is irregular for IMA2 and regular for RMA2, while all the other characteristics are the same. It can be noted that, as expected, the difference between non-weighted and weighted curves is more pronounced for the smaller and lower quality samples. In the right part of Fig. 8, in particular, it can be observed that the curves of the highest damage states are strongly affected by the weights, which determine a change of curvature with respect to the nonweighted case. The observation of Fig. 8 also allows to draw some conclusions about the relative vulnerability of the two considered typologies: obviously, a regular layout of masonry (RMA2) significantly reduces the vulnerability, with respect to the case of irregular layout (IMA2), in particular for the higher damage levels. The parameters m and s of the lognormal distributions of each considered building typology and of each damage state, are summarised in Table 3. It can be observed that the results of some of the building typologies described in Table 2 are missing in Table 3, as well as the parameters of the DS5 level of damage for few typologies. This is due to the reduced amount of data, relative in particular to the typologies RC3 and ST, for which it has not been possible to obtain statistically reliable parameters of the lognormal distribution. The same occurred for DS5, corresponding to collapse of the building, which for some typologies had only very few observations. This problem can be possibly solved by just adding more data to the analysed data set, relative to strong events, in areas where high values of PGA have occurred and hence it is more likely to observe cases of building collapse. Since the bootstrap technique is based on the random generation of samples, its application produces different

Fig. 8. Comparison of non-weighted and weighted curves for the two building typologies of IMA2 (left) and RMA2 (right).

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Table 3 Parameters of the lognormal distribution for the adopted building typologies Label

MX1 MX2 RC1 RC2 RC4 IMA1 IMA2 IMA3 IMA4 IMA5 IMA6 IMA7 IMA8 RMA1 RMA2 RMA3 RMA4 RMA5 RMA6 RMA7 RMA8

DS1

DS2

DS3

DS4

DS5

m

s

m

s

m

s

m

s

m

s

4.83 5.05 0.17 1.34 1.95 10.48 9.96 4.20 7.32 10.61 9.35 3.64 6.89 5.13 10.76 1.12 6.66 4.77 11.12 2.02 6.03

4.73 5.11 2.80 2.06 0.86 8.01 5.95 4.74 5.74 7.45 5.35 2.89 4.26 6.82 9.84 5.67 12.63 6.34 10.53 2.51 9.12

0.81 0.43 0.64 0.18 1.16 0.04 3.02 1.12 1.48 2.50 2.91 1.48 2.39 6.28 1.30 5.55 4.57 0.98 0.67 0.18 0.82

4.28 2.32 1.42 0.98 0.44 10.81 5.30 4.72 3.60 8.20 4.36 2.59 2.08 8.44 9.89 5.54 7.17 6.34 8.99 1.78 3.43

3.31 0.15 0.38 0.26 1.04 2.13 1.12 2.54 0.22 0.51 1.36 0.75 1.67 5.18 5.48 4.97 2.92 1.79 0.80 0.01 0.35

5.24 1.74 1.08 0.64 0.40 4.99 4.77 4.09 3.41 2.98 3.05 1.86 1.67 5.11 9.33 4.16 3.96 4.17 5.13 1.48 2.33

2.76 0.97 0.23 0.31 0.82 3.06 0.21 3.73 1.12 0.74 0.41 0.67 0.69 3.70 5.42 3.90 3.44 0.46 1.66 0.13 0.36

3.33 2.05 0.85 0.91 0.43 3.81 2.95 3.52 3.21 2.46 2.10 2.16 1.64 2.90 6.18 2.94 3.22 1.45 3.48 1.11 1.29

2.07 1.56 – 0.67 0.47 2.68 0.04 1.93 2.03 – 0.15 1.99 0.53 – 3.53 – 2.33 – 4.13 0.11 3.49

2.24 1.83 – 0.93 0.52 2.51 1.84 1.89 2.65 – 1.34 2.00 1.79 – 3.41 – 1.94 – 3.54 0.55 2.83

values of the estimated standard deviation from one run to the other. Obviously the weights used in the regression affect the final estimate of the lognormal parameters and hence it is important to stabilise the results in terms of m and s. For this scope, the number of bootstrap samples has been increased, in order to reduce the variability of the resulting lognormal parameters, until the differences in the calculated standard deviations were lower than 0.01 (which is so small that it does not influence at all the final results and it is visually imperceptible on the curve). 6. Discussion of results and applications in typological risk analysis As already discussed in a previous work [36], a scalar function mD(PGA) of mean damage probability has been calculated, combining the probabilities of exceeding the different levels of damage as P5 PDSi ðPGAÞ mD ðPGAÞ ¼ i¼1 , (2) 5 where PDSi ðPGAÞ is the probability of exceeding the damage state DSi for a given PGA. This function allows relative comparisons among different typologies, as shown in Fig. 9, where the average value is plotted for some of the considered typologies, in the PGA range of main interest for Italy. All the typologies plotted in Fig. 9 are low rise buildings (p2 storeys for masonry and mixed structures, p3 storeys for concrete). From the relative comparison among curves, it can be noted that the expected mean damage of IMA2 (masonry with irregular layout and

Fig. 9. Expected mean damage probability, as a function of PGA, for some of the considered building typologies.

without tie rods and tie beams), at a PGA of 0.2 g, is almost six times higher than that of RC1 (reinforced concrete designed according to seismic regulations). Mixed structures (MX1) appear to be less vulnerable than buildings of masonry with irregular layout, but more vulnerable than those in reinforced concrete or masonry with regular layout. This parameter could be used for example to predict loss scenarios and to define prioritisation strategies for deciding whether retrofitting and/or replacing existing buildings. For this aim it may be necessary to recalculate this mean damage parameter using different weights to take into account the realistic contribution of each damage level to the economic loss of the building [37]. An interesting application of the proposed fragility curves is the development of GIS-based maps in terms of regional typological risk. As well known, seismic risk is

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obtained from the convolution of hazard, vulnerability and exposure. Since reliable data on exposure for the entire national territory are lacking, we decided to consider typological risk, which can be defined as the unconditional probability of exceeding a given limit state, for a selected typology of buildings. Typological risk can hence be obtained from the convolution of fragility with hazard curves (leaving aside exposure), according to the following formula (e.g. [31]):  Z 1  dH    Pf ¼ (3) dPGAPf ðPGAÞ dPGA, 0 where Pf is the typological risk, H is the hazard, expressed as the annual probability of exceeding a given level of PGA and Pf(PGA) is the fragility, representing the probability of damage given a certain PGA. The Abruzzo Italian region has been selected for an example of application, because of the significant variation of hazard from the internal Apennines to the Adriatic seaside and also because of the large amount of building damage data collected after the 1984 earthquake and included in the derivation of the DPMs. The hazard curves used to construct typological risk scenarios for Abruzzo have been specifically derived for the region of interest, following two of the sixteen branches of the logic tree of the official Italian hazard map [38]. An example of the derived hazard curves is reported in Fig. 10 for the city of L’Aquila: the curve represents the annual probability of exceedance (H) of any value of PGA, reported on the x-axis. The value corresponding to the probability of 1/475 has been compared with the median value of the official map and a good agreement has been observed between the obtained values. For any further detail on the derivation of hazard curves, the reader is referred to the work of Dall’Ara [39]. As well known, hazard curves are defined based on the seismicity rates of the different seismogenetic zones,

Fig. 10. Seismic hazard curve computed for the city of L’Aquila, in Abruzzo.

Fig. 11. Seismic hazard map for the region of Abruzzo, Italy [19].

making use of attenuation laws, and hence hazard varies from point to point. It is hence possible to create a plot of the relative severity of hazard for a selected area representing, for a given return period, an equal probability PGA map. As an example, the Italian hazard map [38] for Abruzzo is reported in Fig. 11. Once hazard and fragility are defined, the application of Eq. (3) allows to obtain the value of risk for each point in which hazard is defined and for each typology and damage level to which the fragility curve refers to. These typological risk values have been plotted for the selected region of Abruzzo and for different fragility curves, obtaining some typological risk maps, which are reported in Fig. 12. In particular, maps have been obtained for two selected building typologies: IMA2 and RMA2. As already pointed out, the only difference between these two typologies consists in the layout of masonry, which is irregular for IMA2 and regular for RMA2. All other parameters are the same. Fig. 12 shows the maps of typological risk for these two typologies (in the two columns) and for the different levels of damage considered (in the rows). It can be easily noted that the annual typological risk associated to IMA2 is significantly higher than the one associated to RMA2, for any damage level. Even if this was expected, since it has been shown that the vulnerability is rather higher for IMA2, while the hazard is only dependent on the geographical point, a quantitative assessment of typological risk can be very useful in order to develop mitigation strategies or insurance evaluations. An immediate and more concise risk comparison between the two typologies can also be obtained simply observing the maps of the mean damage probability, previously defined, as shown in Fig. 13. The significant annual probability of exceedance of low PGA values, combined with the vulnerability of some

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Fig. 12. Typological risk maps obtained for the Abruzzo region for the typologies IMA2 and RMA2.

typologies, produces very high typological risk values in some areas. This is also related to the fact that for some poor typologies, a certain level of damage can be observed at very low levels of PGA or even without any seismic event, due to lack of maintenance and pre-existing damages [40]. These already existing structural or non-structural defects can then be easily aggravated by the earthquake.

7. Conclusions Fragility curves have been directly derived, for several building typologies characteristic of the Italian building stock, starting from damage data collected during postearthquake surveys, following Italian earthquakes of the last 30 years. These curves have been obtained from a very

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Fig. 13. Typological risk comparison between the two typologies IMA2 and RMA2 in terms of mean damage probability maps.

complete database, consisting of many data, which are also reasonably homogeneous, since they have all been collected in Italy. This is a significant advantage over most other data sets used in the literature, which are either small [13] or obtained assembling data from earthquakes in different areas of the world [1]. Other authors in Italy have worked on very complete data sets of empirical data [41,42] but they did not use the observed post-earthquake information for directly deriving vulnerability functions. The curves have been obtained by fitting the experimental points with a lognormal distribution, using appropriate weights in order to account for the reliability of each experimental point, which depends on the size and quality of the sample. In particular, for each typology and each PGA level, a point estimate of the probability has been obtained from the DPM and the standard deviation of each point has been evaluated using the bootstrap technique. The inverse of this standard deviation is then used as a weight. The curves obtained through this robust procedure are very significant, at least for what concerns Italy, and can be used for representing vulnerability in risk scenario studies at different scales. As an example of application, they have been convoluted with specifically derived hazard curves, in order to obtain typological risk maps for the region of Abruzzo. This area has been chosen both because a significant amount of damage data has been collected in this zone after the 1984 earthquake and because of the interesting variations in its hazard. These typological risk scenarios can be useful in order to develop mitigation strategies or to make insurance evaluations. The obtained fragility curves cannot accurately represent the expected damage of a single building belonging to one typology but, since they have been derived collecting the observed behaviour of several thousands of buildings in different regions and with slightly different constructive details, they provide a reliable estimate of the mean vulnerability of classes of structures. This may also be the first step in the assessment process of single buildings or an useful tool to support decision making for selecting, in a large set of structures (e.g. school buildings, hospitals, strategic structures or buildings located in areas of new seismic classification), the ones to be analysed in detail and possibly retrofitted.

The use of empirical data takes into account, in the direct derivation of fragility curves, all the elements affecting damage, such as for example site effects and contribution of non-structural components to the seismic response, which are implicitly included in the vulnerability description. Since the curves are derived from correlations established between observed damage and calculated PGA, they can be consistently used, with proper definitions of hazard and inventory data, for assessing seismic risk.

Acknowledgements The authors would like to express sincere thanks to the Italian Civil Protection Department for providing the data collected during post-earthquake surveys and in particular to Dr. Giacomo Di Pasquale for his very precious suggestions in interpreting those data. The derivation of risk maps would not have been possible without the fundamental contribution of Dr. Andrea Dall’Ara, who developed and provided the hazard curves. The valuable help of Mr. Mauro Onida in the development of the visual basic codes necessary to manage the databases is greatly acknowledged. It is also important to recall that the first idea of this work has been originally hinted by Prof. Gian Michele Calvi and Dr. Barbara Borzi.

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