The first step to derive empirical fragility curves is to quantify the severity of seismic shaking for each building of the dataset (e.g., De Luca et al. 2018). In this way, georeferenced damage information can be correlated with spatial distribution of ground motion IMs such as PGA. This requires the adoption of “shake maps” which are the combined result of instrumental measurements, information about local geology, earthquake location and magnitude (Wald et al. 2006). Once damage and shake intensity information are paired, fragility curves are estimated through statistical regression procedures (Rossetto et al. 2014). Obviously, the validity of the results relies on the quality of input data, which is particularly challenging when working on data-scarce regions like Nepal (Robinson et al. 2018).
Shake maps of the 2015 Nepal earthquake sequence
At the time of the 2015 mainshock only a limited number of seismic recording stations were operating in Nepal, mainly concentrated in the city of Kathmandu. Therefore, it is problematic: (1) to evaluate the extent of the area affected by ground shaking and (2) to estimate the variation of the seismic excitation within the area (McGowan et al. 2017). In addition, the limited knowledge on the country’s geology (Gilder et al. 2020) and the lack of representative Ground Motion Prediction Equations (GMPEs) for the region (Bajaj and Anbazhagan 2019) result in large uncertainties on shake maps. In the last five years, the United States Geological Survey (USGS) has released a set of shake maps for the Gorkha sequence, progressively updated to include more advanced studies (McGowan et al. 2017). In this work the following USGS maps are considered:
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M 7.8—36 km E of Khudi, Nepal earthquake (mainshock) occurred on April 25th 2015 (USGS 2017a);
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M 7.3—19 km SE of Kodari, Nepal earthquake (aftershock) occurred on May 12th 2015 (USGS 2017b).
These two shake maps are illustrated in Fig. 9a, b, where it is evident that the mainshock affected a larger area with respect to the aftershock. In terms of intensity, the April 25 shake map had a maximum PGA of 1.0 g, while the value for the May 12 event reached 0.85 g. Given the absence of disaggregated information on the damage generated by mainshock and aftershock (SIDA surveying activities began after May 12), PGA values at individual building locations are extracted from the maximum envelope of the two shake maps (Fig. 9c). This simplification, though inevitable due to lack of data, cannot take properly into account the effect of aftershocks (Dong and Frangopol 2015) and most importantly, the sequence of nonlinear incremental damage (particularly relevant for masonry buildings (Sextos et al. 2018)).
In this study, PGA is selected as the IM for the derivation of empirical curves. This choice is motivated by four reasons. First, PGA is usually a good predictor for stiff structures (Silva et al. 2019) such as one-story buildings [85% of SIDA (Fig. 4c)]. Second, PGA is preferred to spectral quantities (such as the spectral acceleration at the fundamental period of the structure Sa(T1)) given the consistent lack of attenuation equations of spectral ordinates for the Himalayan region (e.g., Bajaj and Anbazhagan 2019). Third, the use of spectral values would require the estimation of an average fundamental period for each SIDA building class through empirical equations, which would subsequently add a degree of epistemic uncertainty to the problem (Rossetto et al. 2014). Forth, PGA is a commonly used IM in vulnerability models (Calvi et al. 2006) and was selected in previous fragility studies for Nepal (Didier et al. 2017; Gautam et al. 2018; Giordano et al. 2019).
Empirical fragility estimates for masonry school buildings
Once damage and seismic intensity at building locations are defined, fragility curves that express the probability of exceedance of different DSs can be derived adopting statistical regression methods (Rossetto et al. 2014). As a first step, building data must be aggregated in representative building classes which are characterized, for instance, by same structural typology and number of stories. The definition of classes should strike a balance between granularity (i.e., more classes result in a better description of the building stock vulnerability) and size of the corresponding sub-datasets (statistical regressions on small datasets can lead to meaningless results). To guarantee sufficiently large sub-datasets of building classes, four classes are defined to match the four main SIDA structural typologies. Another aspect to be considered when deriving fragilities from observational data is the spatial distribution of the dataset. Several studies have shown that spatially inhomogeneous datasets (i.e., buildings concentrated in areas where the range of variation of PGA is limited) can lead to inconsistent results (De Luca et al. 2018). In these cases, Bayesian updating procedure of existing fragility curves should be adopted (Singhal and Kiremidjian 1998; Miano et al. 2016). Given the observations presented in 3.2., only masonry school data is sufficient to derive fragilities with standard statistical methods, while fragility estimates for other structural typologies are made by means of the Bayesian method as discussed in the following section.
Observational fragilities for masonry school buildings are calculated with the Maximum Likelihood Estimation (MLE) method. MLE is one of the most widely adopted techniques for deriving empirical fragility curves and it has been used to assess the performance of numerous structural types in different regional contexts (Shinozuka et al. 2000; Colombi et al. 2008; De Luca et al. 2015; Del Gaudio et al. 2019). The first step of MLE consists of subdividing the dataset in ranges of PGA (bins) for which the Damage State exceedance probabilities are computed (Fig. 10a). Each of these bins should contain the same amount of buildings (Porter et al. 2007). Subsequently, given a probability distribution function (PDF) model, MLE permits the evaluation of the PDF parameters that maximize the probability of occurrence of the observed data (Lallemant et al. 2015). If the fragility curve is represented by a lognormal model, the estimates \(\hat{\mu }_{log}\) and \(\hat{\beta }\) of the logarithmic mean \(\mu_{log}\) and standard deviation \(\beta\) are given by the following expression:
$$\hat{\mu }_{\log } ,\hat{\beta } = \arg \mathop {\hbox{max} }\limits_{{\mu_{\log } ,\beta }} \mathop \sum \limits_{i = 1}^{m} \left[ {n_{i} \ln \left( {\varPhi \left( {\frac{{\ln \left( {{\text{PGA}}_{i} } \right) - \mu_{\log } }}{\beta }} \right)} \right) + \left( {N_{i} - n_{i} } \right)\ln \left( {1 - \varPhi \left( {\frac{{\ln ({\text{PGA}}_{i} ) - \mu_{\log } }}{\beta }} \right)} \right)} \right]$$
(1)
where m is the number of bins, PGAi is the average peak ground acceleration of the ith bin, ni is the number of buildings reaching or exceeding the considered DS in the ith bin, Ni is the total number of buildings in the ith bin, Φ(·) is the standard normal cumulative distribution function. The result of the MLE method can be graphically represented in the form of a linear regression as shown in Fig. 10b. Particularly, x is the logarithm of PGAi and y is the inverse standard normal distribution of (ni + 1)/(Ni + 1). The resulting fragility curves are shown in Fig. 11 (DS#-MLE), while corresponding statistical parameters are summarized in Table 2.
Table 2 Empirical fragility curves derived from SIDA damage data [η (g), β (–)] From Fig. 10b it is observed that the MLE method can lead to inconsistent results when fragility curves of consecutive DS cross with each other. These situations can be avoided by adopting a constant lognormal standard deviation (Porter et al. 2007):
$$\beta ' = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \hat{\beta }$$
(2)
and updated values for the median PGAs:
$$\eta ' = \exp \left( {1.28\left( {\beta^{\prime} - \hat{\beta }} \right) + \hat{\mu }_{\log } } \right)$$
(3)
Resulting fragility curves are shown in Fig. 11 (DS#-MLE n.c.), while relative statistical parameters are included in Table 2.
Bayesian updating of existing fragility models for different structural typologies
Bayesian Updating (BU) techniques are effective alternatives to traditional statistical methods when dealing with small or spatially inhomogeneous observational damage datasets. Several studies have suggested the adoption of Bayesian techniques to update preexisting fragility curves (e.g., Singhal and Kiremidjian 1998; Miano et al. 2016; De Risi et al. 2017; De Luca et al. 2018). As mentioned in the introduction, a Bayesian approach has been also used by Didier et al. (2017) in the context of Nepal, but the analysis was solely focused on residential buildings and did not include schools. Two main set of information are required to perform a Bayesian updating: the prior probabilistic model and the likelihood function of the empirical data (Singhal and Kiremidjian 1998). Subsequently, by applying the Bayes theorem, the probabilistic parameters of the posterior model are estimated. In details, by referring to the Bayesian regression analysis procedure reported by Faber (2012), the updated regression coefficients \(\varvec{B^{\prime\prime}} = \left( {\begin{array}{*{20}c} {b_{0}^{''} } & {b_{1}^{''} } \\ \end{array} } \right)^{T}\) of a generic linear model \(y = b_{0}^{''} + b_{1}^{''} x\) (such as the ones shown in Fig. 10b) are calculated as follow:
$$\varvec{B''} = \varvec{V}_{\varvec{B}}^{{\varvec{''}}} \left( {\left( {\varvec{V}_{\varvec{B}}^{\varvec{'}} } \right)^{ - 1} \varvec{B^{\prime}} + \hat{\varvec{X}}^{T} \hat{\varvec{y}}} \right)$$
(4)
$$\left( {\varvec{V}_{\varvec{B}}^{{\varvec{''}}} } \right)^{ - 1} = \left( {\varvec{V}_{\varvec{B}}^{\varvec{'}} } \right)^{ - 1} + \hat{\varvec{X}}^{T} \varvec{\hat{X} }\quad {\text{and}}\quad \varvec{V}_{\varvec{B}}^{\varvec{'}} = \left( {\hat{\varvec{X}}^{T} \hat{\varvec{X}}} \right)^{ - 1}$$
(5)
where \(\varvec{B^{\prime}} = \left( {\begin{array}{*{20}c} {b_{0}^{'} } & {b_{1}^{'} } \\ \end{array} } \right)^{T}\) are the regression coefficients of the prior model and \(\hat{\varvec{X}}\), \(\hat{\varvec{y}}\) are matrixes with the coordinates of the new empirical data as defined by Faber (2012).
The selection of the prior fragility models represents a crucial point of the BU procedure. When several structural typologies are considered, it is fundamental to adopt a consistent set of prior curves. Unfortunately, this information is lacking in data-scarce regions like Nepal. For instance, the fragility models presented by Didier et al. (2017) and Gautam et al. (2018) cover exclusively the unreinforced masonry and RC typologies. In addition, these fragilities do not represent a general baseline since were extracted from damage data of residential buildings. Previous vulnerability and risk assessment studies in low-to-middle income contexts have adopted the HAZUS models (Federal Emergency Management Agency 2015) as general reference for building fragility curves. Gentile et al. (2019) have used HAZUS fragilities to define the baseline score of a seismic risk index for school buildings in Indonesia while Sevieri et al. (2020) have extended the approach to the Philippines. HAZUS models have also been used in loss assessment studies in Nepal (Robinson et al. 2018). By analogy with the Uniform Building Code 1994 (ICBO 1994), HAZUS models are subdivided into four seismic code levels: high code, moderate code, low code and pre-code (Gentile et al. 2019). In countries where the building standards have followed the evolution of the UBC, these four levels can be used in full (Sevieri et al. 2020). This is not the case of Nepal where: (1) most of the constructions have been realized according to mandatory rules of thumb rather than engineering design, (2) the first building standard, the Nepal National Building Code (Department of Urban Development and Building Construction 1994) has been effectively enacted in 2003 (Giri et al. 2019). For these reasons, this study refers to low-code and pre-code fragilities exclusively. In details:
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Masonry unreinforced masonry bearing wall, low-rise (URML), pre-code;
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RC Frame concrete frame building with unreinforced masonry infill walls, low rise (C3L), low-code;
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Steel Frame steel light frame (S3), low-code;
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Timber Frame wood light frame (W1), pre-code.
To account for the poor construction quality of traditional buildings in Nepal, pre-code models (i.e., construction prior to seismic code enforcement) have been considered for masonry and timber frame typologies. RC frame school buildings are generally newer constructions with minimum seismic detailing. Therefore, low (i.e., older) code HAZUS fragilities are considered as prior. Lastly, low-code fragility curves are selected for the steel frame typology. In fact, most of these constructions are fairly designed since were built from 1992 to 1995 under the World Bank’s Earthquake-Affected Areas Reconstruction and Rehabilitation Project (NSET 2000) or, more recently, by the Japan International Cooperation Agency (JICA 2009). It should be noted that the steel light frame typology considered in HAZUS does not exactly correspond to the one of SIDA. In Nepal, perimeter walls are not realized with lightweight panels but with stone/brick units in mud/cement mortar depending on the local availability of materials. Figure 12 reports prior and posterior fragility curves for the four structural typologies. The corresponding probabilistic parameters are given in Table 2. It can be observed that HAZUS fragility curves are defined for four damage states, namely: Slight, Moderate, Extensive and Collapse. Therefore, to execute the BU procedure, the following equivalences with DS#SIDA are considered: DS2SIDA ≈ Slight, DS3SIDA ≈ Moderate, DS4SIDA ≈ Extensive, DS5SIDA ≈ Collapse.
Discussion of the fragility results
To facilitate the discussion of these results with respect to the existing observational fragilities by Didier et al. (2017) and Gautam et al. (2018), the DS equivalences given in Table 3 are considered.
Table 3 DS equivalences adopted for fragility comparisons General comments:
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1.
Relative fragility of masonry school buildings due to different statistical methods. As expected, for DS1SIDA, the MLE method provides consistently larger median PGA and standard deviation (0.021 g, 2.76) with respect to the MLE n.c. technique (0.006 g, 1.71). Conversely, for DS5SIDA, the MLE method gives conservative and less scattered results (0.97 g, 0.94) with respect to the MLE n.c. (2.59 g, 1.71). The BU method provides the lowest estimates of η and β at any DS. As expected, the intrinsic uncertainties of observational data and the inaccuracy of the shake map (Sect. 4.1.) affect the results of standard statistical methods. Median PGA for high DS appears unrealistically higher than previous analytical results available in the literature (Giordano et al. 2019). In this sense, the BU methodology seems to be an effective way to utilize the valuable set of observational data collected by the World Bank, while maintaining reasonable values for median PGA. Based on this observation the following comparisons focus on the BU method.
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2.
Masonry buildings fragility (BU method) and comparison with Didier et al. (2017): DS2-3SIDA fragilities are characterized by comparable values of η (0.14 g, 0.18 g) with respect to DS2D brick-mud URM (0.14 g). On the contrary, the median value of DS2D brick-cement URM is consistently higher (0.77 g). Dispersion of DS2D is about two times larger than DS2-3SIDA. Median PGAs of DS4-5SIDA (0.39 g, 0.55 g) are considerably lower than DS2D brick-mud URM (1.26 g) and brick-cement URM (1.90 g). Corresponding β (0.82, 0.76) are comparable with the brick-cement value (0.93) while considerably lower than brick-mud dispersion (1.96). These large discrepancies likely derive from a combination of factors such as data inhomogeneity and differences in prior models. What appears unrealistic is that median PGAs for residential buildings (Didier et al. 2017) are systematically higher than the corresponding estimates for school buildings. In Nepal, residential URMs are usually constructed by homeowners, non-engineered, non-compliant to building regulations and without basic seismic detailing (Gautam et al. 2016). On the contrary, schools are subjected to a stricter code enforcement and the required level of safety is higher with respect to residential structures. The non-uniformity (i.e., non-uniform distribution of buildings in the full range of PGAs) of the database used by Didier et al. (2017) could be a reason for the high dispersions and median values with respect to the SIDA fragilities.
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3.
Masonry buildings fragility (BU method), comparison with Gautam et al. (2018). DS2-3SIDA present comparable median values with respect to DS1-2G Brick URM (0.13 g, 0.16 g). On the contrary, there is little agreement with respect to the results of DS1-2G Stone URM (0.29 g, 0.32 g). Median values of DS4-5SIDA are fairly similar to DS3G Stone URM (0.39 g) but consistently different from DS3G Brick URM (0.22 g). Theoretically, the empirical curves by Gautam et al. (2018) should be considered the best fragility estimate since they account for damage variability from five past earthquake events. However, these historical datasets inevitably come with large uncertainties on damage data quality, shake maps, and geographical accuracy. It is also surprising that, despite these large uncertainties, dispersion values of DS#G are unexpectedly lower than DS#SIDA.
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4.
RC Frame buildings fragility (BU method), comparison with Didier et al. (2017). DS2-3SIDA median PGAs (0.19 g, 0.27 g) are systematically lower than the corresponding value for DS2D (1.67 g). The corresponding β (1.08, 1.04) are consistently lower than the value for DS2D (1.73). DS4-5SIDA (0.77 g, 1.13 g) provides a more conservative estimate of η with respect to DS3D (1.95 g). The related dispersions are instead comparable (0.88, 0.84 versus 0.71). Comments of point (2) can be extended to the case of RC.
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5.
RC Frame buildings fragility (BU method) comparison with Gautam et al. (2018). DS2-3SIDA median PGAs are consistently lower than DS1-2G (0.29 g, 0.63 g). Analogously, median values of DS4-5SIDA are conservative with respect to DS3G (1.29 g). Comments of point (3) can be extended to this comparison.
A further way to compare the existing empirical fragilities with the ones derived in this study is to assess the similarity of two probability distributions with information theory measures. Looking at the literature, some studies in the field of earthquake engineering have adopted the Kullback‐Leibler divergence as a measure of the statistical distance between a real distribution and its approximation (e.g., De Luca et al. 2015; Tsioulou and Galasso 2018). For instance, Tsioulou and Galasso (2018) have compared distributions of IMs from recorded and simulated ground motions. Since the fragility models presented in this study are all “approximations of the reality”, this works adopts the Bhattacharyya distance, DB, (Bhattacharyya 1946) instead of the Kullback‐Leibler divergence. This quantity, which relates to the amount of overlap between two statistical models, is expressed by the following equation:
$$D_{B} = - \ln \int \sqrt {p_{1} \left( x \right)p_{2} \left( x \right)} {\text{d}}x$$
(6)
where p1(x) and p2(x) are two probability distribution functions. From Eq. 6 it can be observed that DB is a nonnegative parameter. Additionally, DB, unlike the Kullback–Leibler divergence, is a symmetric quantity, i.e., DB (p1, p2)= DB (p2, p1). Figure 13 presents a comparison of the three empirical studies in terms of DB at each damage level. Particularly Fig. 13a, b refers to masonry and RC respectively. In the context of this work, the absolute value of DB is informative only when relatively compared with the full set of distances. This last aspect differs from the study by Tsioulou and Galasso (2018) where a procedure to assess the similarity of two distributions from the absolute value of the Kullback‐Leibler divergence is reported.
The differences in DB values between the three models can be attributed to (i) the non-uniform quality and extension of the damage data, (ii) the different definition of damage states, and (iii) the difference in building characteristics (e.g., school buildings are mainly single-storied, while residential are usually multi-storied), (iv) the different shake map accuracy. The results show that the discrepancy between the empirical models is smaller for RC buildings than for URMs, especially when assessing higher damage states. This is probably an inevitable result of the intrinsic larger uncertainties around the response of non-engineered masonry structures.