A methodology to assess and select seismic fragility curves: Application to the case of Costa Rica

Abstract

Different seismic risk studies have been carried out in the Central American region to quantify the human and economic losses caused by seismic events. The appropriate use of seismic fragility curves (FCs) allows a better approximation of the level of performance of a structural system in the face of seismic hazards. In this respect, an inadequate selection of FCs can lead to a notably unreliable estimation of damages and losses. This research contributes a methodology for assessing and selecting FCs for seismic risk studies from a catalog of existing proposals available in the literature. The methodology has been applied to the case study of Costa Rica, encompassing a bibliographic search, meetings with Central American experts, and the compilation of a catalog with the FCs used in different vulnerability projects. The conclusion drawn is that FCs with very different probabilities of damage are being used in the diverse vulnerability projects for buildings with the same attributes. Moreover, the proposed methodology allows knowing the reliability level of the FCs obtained depending on which class the curve was classified into based on its score. Therefore, it allows the adequacy and uncertainty related to the selected FCs to be established, a subject that researchers should consider in seismic vulnerability and risk studies.

Keywords

Seismic vulnerability fragility curves structural performance typologies seismic risk

Introduction

Central America is one of the greatest seismic hazard zones in the world. In addition, it is a region characterized by a high physical and social vulnerability (Quesada-Román and Campos-Durán, 2022). The most common types of buildings in the area are adobe, unreinforced masonry, and wooden ones (Calderón et al., 2021). According to the literature assessment and the comments from local engineers, there is ineffective enforcement of contemporary seismic laws, which unavoidably results in a stock of buildings that is quite vulnerable (Calderón et al., 2021). It has been precisely the combination of these two negative factors that has caused an extremely high number of fatalities and economic losses in the past when a seismic event occurred. The most destructive earthquake in the history of Central America occurred in Guatemala in 1976, when the Motagua–Polochic Fault caused an earthquake with a magnitude of surface levels of M_s = 7.5 (Benito et al., 2010). This earthquake left a balance of more than 22,700 deaths and close to 76,000 injuries and generating losses of around 1.1 trillion dollars, representing 18% of Gross National Product at that time (Davis, 1978; Espinosa, 1976). In April 1991, an M_w 7.7 earthquake devastated the Caribbean region of Costa Rica and Panama. In addition, in September 1992, an earthquake off the coast of Nicaragua caused the worst tsunami in the region. More recently, two earthquakes affected El Salvador on 13 January and 13 February 2001, in which more than 1000 people lost their lives (Benito et al., 2010; Rose et al., 2004). Furthermore, recently, the Cinchona earthquake occurred in Costa Rica (M_W = 6.2) on 8 January 2009, seriously affecting rural populations and hydroelectric works (Barquero, 2009; Benito et al., 2010). For that reason, in recent decades, several seismic risk studies have been carried out to identify the vulnerability of the existing buildings and the potential economic costs that a strong-motion earthquake could cause in the region (Cardona, 2009; Lang et al., 2009; Miyamoto International, 2016). Moreover, the research studies being performed aim to introduce mitigation measures such as the retrofitting of buildings or the improvement of the regulations in force.

Within this context, seismic risk studies must deal with the treatment of different types of uncertainty, whose assessment methods are still under development, such as those associated with seismic demand, or with the variability in the properties of the structural models developed and calibrated for the building portfolio of interest. Another important source of uncertainty in such studies is usually the selection of the FCs for each building type, from an existing catalog (Rossetto et al., 2014). The fragility curve (FC) is defined as the probability of reaching or exceeding a specific damage state (DS) under earthquake excitation. In this respect, an inadequate selection of the FCs can mean a notably unreliable estimation of damages and losses. Therefore, the estimation of seismic risk levels for Central America may turn out to be more inaccurate when the FCs used have been developed for regions of the world with construction techniques and quality of construction materials highly dissimilar to those under study, such as North America, Italy, or other countries with a greater Gross Domestic Product. However, previous studies have taken this approach due to the limited availability of FCs for Central America. The FC establishes the crucial connection between seismic risk at the location and the ensuing effects on any exposed components, such as buildings (Maio et al., 2017). Therefore, FC can be used in community resilience frameworks (Abdelhady et al., 2022).

Within this framework, based on the proposals and attributes identified in previous studies to select FCs, this article outlines a new method for selecting potential FCs using an index or ranking system based on relevant dimensions and attributes identified in prior research. While previous studies have identified variables that impact the suitability of FCs for various case studies, the rankings or ratings provided have been more qualitative than quantitative. Therefore, the proposed methodology is innovative because it organizes all the relevant variables according to the literature, assigns scores to them and aggregates them in a quantitative and multidimensional index, enabling researchers to rank FCs based on their appropriateness for the specific case study.

Moreover, the presented methodology has been applied to the capital of Costa Rica, San José. The case of Costa Rica is a very interesting example within Central American countries. A significant number of specific vulnerability studies have been carried out in the country, and its seismic codes are notably advanced and detailed, which has led to a significant decrease in poor-quality construction techniques in formal buildings that comply with current planning and building regulations. The FCs for several types of buildings in San José of Costa Rica have been assessed applying the new methodology proposed in this article, based on a multidimensional index developed to evaluate FCs from a broad set of variables, which involves comparisons between the available proposals.

State of the art

Seismic FCs have become an essential tool for seismic risk studies and vulnerability analysis. These curves allow a relationship to be established between seismic hazard at a particular site and the effects of a ground motion on built infrastructure (Minas et al., 2014). FCs for large-scale seismic risk assessments have been proposed for a few world regions. One of the best-known platforms available for determining building damage and loss in urban contexts is HAZUS-MH (Federal Emergency Management Agency (FEMA), 2001) for the United States. In Europe, several projects have been developed, and within this framework, a great number of fragility and vulnerability functions have been generated for different construction types. These may include these projects: RISK-UE (2001–2004), LESS-LOSS (2004–2007), SYNER-G (2009–2012), and SERA (2017–2020) (Martins et al., 2021). For Central America, the developed projects have analyzed exposure, vulnerability, and seismic risk at a regional level or solely for some specific countries. Among the most notable projects are RESIS II (2007–2010) (Benito et al., 2012; Lang et al., 2009), CAPRA 2009–2015 (Cardona et al., 2012), and more recently, the PREPARE II (2020–2022) project (Miyamoto International, 2016, 2020; Miyamoto International and USAID, 2021). As part of the Global Earthquake Model (GEM) initiative, hundreds of empirical and analytical fragility and vulnerability functions were collected and made publicly available through the OpenQuake-platform (Martins et al., 2021). While FCs specify the probabilities of exceeding a number of DSs conditional on a ground shaking intensity measure (IM), vulnerability functions establish the likelihood of loss ratio conditional on a ground shaking IM (Martins et al., 2021; Yepes-Estrada et al., 2017). In South America, the main projects carried out to evaluate seismic risk in the region are SARA (2013–2015) (Villar-Vega et al., 2017; Yepes-Estrada et al., 2017) and CCARA (2018) (Garcia et al., 2018).

In Costa Rica, the National Insurance Institute (INS) requested that the first relevant study on vulnerability is carried out on a national scale (Sauter and Shah, 1978). In the study, the dwellings were classified into seven types according to wall materials, seismic design, and the state of the building. The materials considered were adobe, concrete, reinforced masonry, and steel and wood frame. The study concluded that under seismic actions, on the Modified Mercalli Intensity (MMI) scale,¹ the typologies that suffer a higher percentage of damage are adobe, construction with low-quality materials, and reinforced concrete structures without seismic design (Calderón, 2017).

In 2003, seismic risk was evaluated in the San José Metropolitan Area (Salas Alvarado, 2003). In this area, six types of basic structural systems were identified depending on the material of the load-bearing walls: masonry, a combination of masonry and wood, wood, pre-cast concrete, waste material, and others (bahareque and adobe). Moreover, based on the criteria of specialists, 152 typologies were generated for the vulnerability analysis using the HAZUS methodology. The quantification of human losses due to damage to buildings was based on spatial distribution and structural typology using the Global Earthquake Safety Initiative methodology. This study was completed by Solórzano Arias (2005), which analyzed the seismic vulnerability of reinforced and confined masonry housing structures of one and two floors in a pilot area of the Central Valley. The construction sequence and system present key differences between confined masonry and reinforced concrete frames with masonry infill walls. In confined masonry construction, the masonry walls carry the seismic loads, while the reinforced concrete confining elements are used to confine individual walls. The confining elements are effective in enhancing the stability and integrity of the masonry walls for in-plane and out-of-plane earthquake loads by confining damaged masonry walls, as well as in enhancing the strength and ductility of masonry walls under lateral earthquake loads, which improves their earthquake performance. Reinforced and confined masonry, also known as integrally reinforced masonry, is one of the most used composite materials used in Costa Rica to construct 1- or 2-story single-family homes, thanks to its good acoustic, thermal, and humidity insulation. The use of ductile steel as reinforcement placed vertically and horizontally provides better quality, durability, and resistance to lateral loads of the constructions (Hidalgo-Leiva, 2017).

Likewise, in addition to the seismic vulnerability analyses carried out for Costa Rica in the RESIS II, CAPRA, and PREPARE projects, several research studies have developed capacity and FCs for different building typologies. Table 1 summarizes the main references and characteristics of these studies.

Table 1.

Principal studies and projects on seismic vulnerability in Costa Rica

Author(s)	Study year	Project type	Study type	Geographic area studied	Institutions involved	Analyzed variable(s)	Analyzed typologies	Use of fragility curves to assess vulnerability? Source of FC
Sauter and Shah (1978)	1978	National	Project: Earthquake Insurance Study	Costa Rica	National Insurance Institute, National Emergency Commission	Material, lateral load–resisting system (LLRS), year of construction	Adobe (mudbrick), unreinforced masonry (low quality), reinforced concrete frames (without seismic design), steel frames (without proper design), reinforced masonry (medium quality without seismic design), reinforced concrete frames (without seismic design), shear walls (with seismic design), wooden frame dwellings, steel frames (with seismic design), reinforced masonry (high-quality seismic design)	Yes. FCs given in MMI values
Montoya (2002)	2002	National	Doctoral thesis	Cartago	Geo-Information Science and Earth Observation (ITC) (doctoral thesis)	Material, LLRS, height, number of floors, type of foundation	Adobe walls, unreinforced masonry, reinforced concrete frame lacking seismic design, steel frames lacking seismic design, reinforced masonry lacking seismic design, reinforced concrete with seismic design, shear walls with seismic design, wooden frames, steel frames with seismic design, reinforced masonry with seismic design	Yes. FCs are obtained from Sauter and Shah (1978)
Climent et al. (2003)	2003	Regional	Capacity Building for Natural Disaster Reduction for Central America	Cañas (Province of Guanacaste)	Central America Disaster Prevention Center (CEPREDENAC), UNESCO, ITC	Qualitative and preliminary analysis of seismic vulnerability	Concrete buildings, block houses reinforced or confined concrete, prefabricated houses, mixed-blocks and wood, wooden houses	No. A qualitative and preliminary analysis of the vulnerability of buildings is carried out
Lang et al. (2009)	2008	Regional	Project: RESIS II	Costa Rica	Norwegian Seismic Array, Universidad Politécnica de Madrid, Universidad de Costa Rica (UCR), CEPREDENAC	Material, LLRS, height, level of code compliance	Minifalda (1 story), Adobe and Tapial (1 story), Taquezal and Bahareque (1–2 stories), Calicanto (1 story), unreinforced fired-brick masonry (1–2 stories), fired-brick masonry reinforced with internal steel bars (1–2 stories), fired-brick masonry confined with reinforced concrete frames (1–3 stories), non-reinforced concrete block masonry including quarried stone (1 story), concrete block masonry reinforced with internal steel bars (1–3), confined concrete block masonry with reinforced concrete frames (1–3 stories), panel block (1 story), light steel frames including die-cast laminate (1 story), steel frames with unreinforced masonry infill walls (1–3 stories; 4–7 stories, 8+ stories), reinforced concrete frame or structural frame (1–3 stories, 4–7 stories, and 8+ stories), concrete with shear wall (1–3 stories, 4–7 stories and 8+ stories), reinforced concrete frames with unreinforced masonry infill walls (1–3 stories, 4–7 stories, 8+ stories), steel frames with unreinforced masonry infill walls	Yes. FCs are obtained from different sources
Evaluación de Riesgos Naturales America Latina (ERN) (2009)	2009	Regional	Project: CAPRA	San José	Evaluación de Riesgos Naturales de América Latina (ERN), Ingeniería Técnica y Científica Ltda. Centro Internacional de Métodos Numéricos en Ingeniería (CIMNE), World Bank, Interamerican Development Bank	LLRS and number of stories	Structural masonry, simple masonry, wood, concrete frames	No. Vulnerability is analyzed using the physical and human loss vulnerability functions
Miyamoto International (2020)	2016	Regional	Project: PREPARE	San José	United States Agency for International Development (USAID), Miyamoto International, Inc.	Material, LLRS and height	Light structure without design (different height), unreinforced masonry (different heights, low code), confined-reinforced masonry (different heights, moderate code), reinforced concrete moment frame (different heights, moderate code), concrete shear wall reinforced (different heights, moderate code), steel moment frame (different heights, moderate code), braced steel frame (different heights, moderate code), unreinforced masonry in informal area (low height, pre-code)	Yes. Adaptation of the FCs proposed by Federal Emergency Management Agency (FEMA) (2001)
Calderón (2018)	2018	National	Graduate thesis		UCR	Material, LLRS, height, number of stories, vertical structural regularity and plan regularity	Cold Rolled Steel Frame (2–3 stories), steel with structure Intermediate Moment Frames (IMFs), Ordinary Moment Frames (OMFs), Special Moment Frames (SMFs) (4 stories), steel with structure IMF, OMF, SMF (10 stories), steel with Special Concentrically Braced Frames (SCBF) (4–10 stories), reinforced concrete dual structure (4–7 stories, 8–18 stories), reinforced concrete with frame (4–7 stories), reinforced concrete wall structure (6–8, 9–18 stories), masonry with wall structure (2 stories), reinforced concrete with dual structure (4–10 stories), reinforced concrete with frame structure (6–10 stories)	Yes. Own FCs generated from the capacity curves of UCR final degree projects
Calderón and Silva (2019)	2019	National	Project: Globalquakemodel	Costa Rica	Instituto Geográfico Nacional, Banco Central de Costa Rica, Cámara de la Construcción, Colegio Federado de Ingenieros y Arquitectos de Costa Rica, Instituto Nacional de Estadística y Censo, UCR	Material, LLRS, ductility, height, year of construction	Conf. masonry, wall system, (low to high ductility and 1–6 stories, post-1980); reinforced masonry, wall system (low to high ductility and 2–12 stories, post-1980); wall system (none to high ductility, 1-story, pre-1980); informal unknown non-ductile 1-story unknown; pre-cast concrete, wall system, (low to high ductility, 1-story, post-1980); reinforced concrete, frames (low to high ductility, 1–6 stories, post-1980); wood, wall system, (none to low ductility, 1–2 stories, pre-1980); reinforced concrete, dual system (low to high ductility, 3–6 stories, post-1980); reinforced concrete, wall system, (low to high ductility, 6–12 stories, post-1980); pre-cast concrete, wall system (low to high ductility, 2–3 stories, post-1980); unreinforced masonry, wall system (non-ductile, 1-story); unreinforced masonry, adobe blocks, wall system (non-ductile, 1-story); steel, cold form steel members, frames (low to high ductility, 1-story); steel, hot-rolled steel members, frames (low to high ductility, 1–2 stories); wood unknown, post and beam, low, 1 story; heavy wood, wall system (high ductility, 1–2 stories)	Yes. Own FCs for ductile confined masonry and pre-cast reinforced concrete structures and FCs for other typologies are from Villar-Vega et al. (2017)
Hidalgo-Leiva (2017)	2017	National	Doctoral thesis		Universidad Politécnica de Cataluña	Material, LLRS	Reinforced and confined masonry known as “Integrally reinforced Masonry” and reinforced concrete with dual system	Yes. Own FCs for both typologies analyzed
Esquivel-Salas (2020)	2019	National	Master thesis	Área metropolitana de San José	Laboratorio de Ingeniería Sísmica (LIS) de la UCR y German Research Centre for Geosciences	Material, LLRS, number of stories, structural regularity	Regular rigid reinforced and confined masonry, flexible confined and reinforced masonry, confined masonry, reinforced concrete with dual system (1–5 stories, 6–11 stories and more than 11 stories), reinforced concrete with infilled frame (1–5 stories), reinforced concrete with moment frame (1–5 stories, 6–11 stories, and more than 11 stories), wood with moment frame (1–2 stories), wood with wall system (1–2 stories), structural steel with moment frame (1 story), cold-rolled steel members with moment frame (1–5 stories), hot-rolled steel members with moment frame (1–5 stories), informal (1 story)	Yes. FCs from Calderón, (2017, 2018); Villar-Vega et al. (2017)
Calderón et al. (2021)	2021	Regional	Research paper	Costa Rica	Global Earthquake Model Foundation, University Fernando Pessoa, Universidad de Nicaragua, Universidad de Panamá, UCR, Universidad Mariano Gálvez de Guatemala, Universidad de El Salvador	Material, LLRS, ductility, stories, capacity	Wood and light wood members with shear wall LLRS (low and moderate ductility, 1–2 stories); wood and wattle and daub with shear wall LLRS (low ductility, 1 story); unreinforced masonry with shear wall LLRS (low ductility, 1–2 stories), unreinforced masonry and adobe blocks with shear wall LLRS (low ductility, 1–2 stories), reinforced masonry with shear wall LLRS (low, moderate, and high ductility, 1–3 stories); confined masonry with shear wall LLRS (non-ductile, moderate and high ductility, 1–3 stories); reinforced concrete with moment frame LLRS (non-ductile and with high ductility), reinforced concrete with moment frame LLRS with soft story (non-ductile, 2 stories); reinforced concrete with infilled frame LLRS (low, moderate, and high ductility, 1–3 stories and 4–6 stories); reinforced concrete with dual frame LLRS (high ductility, 6–12 stories); steel and hot-rolled steel members with braced frame LLRS (moderate ductility with 1 story and high ductility with 6–12 stories), steel and cold-formed steel members with moment frame LLRS (low, moderate, and high ductility, 1 story), waste/other materials with unknown LLRS (low ductility, 1 story)	Yes. FCs from Martins and Silva (2020)

FC: fragility curve; MMI: Modified Mercalli Intensity.

Furthermore, some direct antecedents of the present research are the studies that developed a procedure to select suitable seismic FCs by firstly assessing how well they represent the needs of the future application (Binda et al., 2006; Kaynia, 2013; Maio et al., 2017; Maio and Tsionis, 2015; Meslem et al., 2014; Stone et al., 2017). Maio et al. (2017) proposed qualitative evaluation criteria based on four main categories to cover fundamental characteristics related to capacity, demand, methodology for fragility analysis, and uncertainty. The aim of this report was to provide a set of qualitative criteria to support the selection process of the most appropriate seismic FCs for the European building stock.

The study by Stone et al. (2017) gathers expert opinion to determine which factors are most important for selecting FCs. The study suggests a selection framework for existing functions based on nine attributes. Some of them are related to what the authors call the quality of the FCs based on the quality of inputs, the rationality of derivation procedures, and the document quality. Other attributes are concerned with the relevance of the FCs, which include the appropriateness of the DSs, building classes, and ground motion IMs used in the original functions for the new application.

The framework adopted by Rossetto et al. (2014) for evaluating seismic FCs is based on four fundamental attributes proposed by Porter (2011). These are the curve representativeness of the characteristics of the buildings and seismicity in the location being assessed, the quality of the inputs used to generate the FCs, the rationality of the procedures followed to construct the curves, and the documentation quality.² The four attributes are also sub-divided into evaluation criteria sets that are different for empirical and analytical FCs. Each criterion is assigned a high, medium, or low rating to facilitate the evaluation and comparison of FCs.

As part of the SYNER-G project, Kaynia (2013) collected, reviewed, and validated FCs against observed damage and harmonized proposals for typologies in Europe. The authors also developed software to store, harmonize, and estimate the uncertainty of the FCs.

This article presents a method for assessing candidate FCs from an index or ranking obtained from the most pertinent dimensions and attributes, based on the suggestions and aspects established in prior studies. The previous contributions propose the series of variables that influence the adequacy of the FC to the different case studies but present rankings or rating systems more qualitative than quantitative. Therefore, the novelty of the proposal lies in organizing practically all the variables involved according to the literature and giving them scores so that the application of the methodology provides the researcher with a quantitative and multidimensional index to rank FCs according to their adequacy to the case study—an outcome for which there was no methodology available up until now. Likewise, an application to the case of Costa Rica is included.

Methodology to assess and select seismic FCs for seismic risk studies

The methodology proposed herein consists of the following steps. First, the typologies of buildings under study must be identified and characterized. Second, a deep search for an important number of FCs in the scientific literature available must be carried out (research projects, scientific articles, graduate and post-graduate theses, and conference papers, among others), focusing especially on those which characterize buildings of the types existing in the area analyzed.

Third, a rating index is obtained from a set of variables proposed to assess relevant aspects of the FCs identified as potential candidates for the typology under study. From this index, a classification is derived to define the adequacy of the FCs for each type of buildings identified (Figure 1). The Final Index, with a maximum value of 100 points, is obtained from a multidimensional index (Global Index) adjusted by a reduction coefficient (Building Class Similarity). The Global Index comprises two dimensions: the Technical Suitability of the FC (which includes three sub-dimensions: Capacity, Fragility, and Quality) and the Suitability for the Local System. In turn, each of these dimensions and sub-dimensions comprises a set of variables that allow evaluating different aspects of FCs. The maximum score an FC can obtain in each index, dimension, sub-dimension, and variable is indicated in brackets. The score of each of the two mentioned dimensions is obtained by adding the scores assigned to all the different variables involved. Moreover, the bottom of Figure 1 shows a graphical summary (in the form of a horizontal bar) of the influence of each element involved, along with an example. In the example, the Global Index (Technical Suitability + Suitability for the Local System) sums up 78 points. Next, the reduction coefficient (green arrow) reduces the Global Index to the Final Index. This final value determines the class of the FC assessed (from A—best to F—worst), that is, its level of adequacy within the following ranges of points defined: Class A: (85, 100], Class B: (65, 85], Class C: (55, 65], Class D: (47, 55], Class E: (40, 47], and Class F: (0, 40] points. Therefore, the FC used as an example is Class D.

As a fourth step, the set of appropriate FCs in the highest classes for each typology under study is selected and their coherence, must be checked typology by typology and globally (between typologies), preferably with local experts, according to the level of seismic vulnerability that is expected a priori of each one. This can be done by graphing the curves together or comparing their parameters or damage exceedance probabilities for similar values of seismic action intensity. Therefore, all the curves must be expressed in the same IM.

Finally, the FCs in the highest classes for each typology that show coherence (reasonable damage exceedance probability levels in comparative terms with other typologies) are selected for each typology. If two or more curves end up classified in the same highest class for a typology, it is recommended to consolidate them into a new curve using the conflation method. This method is a probability-averaging alternative that has several benefits and none of the drawbacks of existing averaging techniques. The conflation of several input distributions is the probability distribution with density equal to the normalized product of the input densities, which optimally and unbiasedly summarizes the data (Hill and Miller, 2011). Considering that FCs are log-normal conditional cumulative probability distributions, the basic principles of conflation, derived by Hill and Miller (2011), to the special case of normally distributed (Gaussian) data have been adapted and applied in this case. Specifically, let $m_{i}$ and $σ_{i}$ denote the mean and standard deviation of FC_i with i = 1,2. The mean $m_{p}$ and the standard deviation $σ_{p}$ of the new FC proposed for consolidation FC1 and FC2 can be calculated from the following equations:

m_{p} = \frac{σ_{2}^{2} m_{1} + σ_{1}^{2} m_{2}}{σ_{2}^{2} + σ_{1}^{2}}

(1)

σ_{p}^{2} = \frac{σ_{2}^{2} σ_{1}^{2}}{σ_{2}^{2} + σ_{1}^{2}}

(2)

While the methodology proposed herein has been developed to evaluate analytical FCs, it could easily be adapted to assess empirical FCs.

Variable selection and evaluated dimension

The variables used to evaluate the FCs available for a given typology were selected from among those proposed by Maio and Tsionis (2015); Rossetto et al. (2014); Stone et al. (2017) (Table 2). Moreover, other variables considered relevant in the FC development were included. The factors identified as central to the proposed FC selection procedure were grouped into the following three relevant dimensions: (i) technical suitability, (ii) suitability for the local system studied, and (iii) building class similarity. Two indexes are computed from the first two dimensions to provide a quantitative assessment for each and then summed up to obtain a global index. From the last dimension, an adjustment coefficient of the global index is proposed.

Table 2.

Dimensions, variables, and categories of the index to evaluate FCs and their corresponding scores

FC: fragility curve; SB: specific building; EDP: engineering demand parameter; BPR: building prototype; PGA: peak ground acceleration; PGD: peak ground displacement; MMI: Modified Mercalli Intensity; EMS: European Macroseismic Scale; SDOF: single degree of freedom; IDR: inter-story drift ratio; GEM: Global Earthquake Model.

Moreover, an example of the evaluation of three FCs in the case of 2-story reinforced confined masonry buildings for Costa Rica.

The three sources of uncertainty of FC parameters are: (i) uncertainty in the seismic demand, (ii) uncertainty in capacity of the building, and (iii) uncertainty in the definition of damage thresholds (Kaynia, 2013; Rossetto et al., 2014).

Furthermore, the first dimension contributes a measure of the technical characteristics of the FCs evaluated. The second contributes a starting point for selecting FCs to include in the study from all the curves available in the literature, and the third indicates how well the curve suits a particular building type within the region.

Technical suitability dimension

This first dimension comprises three blocks of variables or sub-dimensions related to the capacity curve from which the FC is derived, characteristics of the evaluated FC, and variables related to the quality and the general credibility of the study that proposes the analyzed curve.

Capacity curves sub-dimension’ variables

The first block is related to the capacity curve, including method and model, analysis type and the engineering demand parameter (EDP) considered. Concerning the method and model found in the reviewed capacity curve, the following were identified from highest to lowest score: experimental 3D, experimental 2D, analytical 3D, analytical 2D, combined single degree of freedom (SDOF) and SDOF. The SDOF option refers to a mechanical approach based on a simplified capacity curve derived from a SDOF model. The “combined SDOF” would correspond to the SDOF that has also been calibrated or validated with other method; combined SDOF was used, for example, by Lagomarsino and Giovinazzi (2006). The curves are also scored on the level of complexity of the analysis carried out for structural assessment. The types of analysis identified in the literature were grouped, in decreasing order of complexity and therefore score, as follows: (i) nonlinear dynamic (NLD), (ii) nonlinear static (NLS or pushover), and (iii) simplified methods/direct capacity curve definition (Rossetto et al., 2014). The last variable considered regarding the capacity curve is the EDP, which measures the structural response obtained from the structural analysis results. The EDPs were ranked from the highest to lowest as follows: inter-story drift ratio (IDR) or global inter-story drift (IDRG); maximum displacement; and roof or top displacement commonly utilized from NLS or pushover (Maio and Tsionis, 2015).

FCs’ sub-dimension variables

The second block of variables refers to characteristics directly related to the FCs evaluated, such as the IM used, the year of publication of the curve, the treatment given to the different sources of uncertainties associated with FCs, the level of popularity, and the damage thresholds considered. The IM is related to the quality of the information on the seismic demand used in the FC derivation process. The choice of the IM is relevant because it is the variable that connects the aspects related to the seismic hazard with the structural evaluation. A desirable IM should be practical (IM for which robust and modern ground motion prediction equations are available), efficient (the structural response should exhibit a low record-to-record variability at any given level of the IM), and sufficient (relevant record-specific seismological parameters, such as magnitude, distance, epsilon, should be represented without introducing any bias in the results). The IMs were ordered from highest to lowest score as follows: (i) advanced IMs or combination-type ground motion IMs based on $S_{a} (T_{1})$ , for example, critical spectral acceleration, $S_{a}^{c} = (S_{a} (T_{1})^{1 - α}) (S_{a} (T_{f})^{α}), T_{f} : softened period$ , critical IMs that consider the effect of the softened period $I M_{CR} = S_{a} (T_{1})^{1 - α} S_{a} (\sqrt[3]{R_{IM}} T_{1})^{α}$ , and $I M_{SR} = S_{a} (T_{1})^{1 - α} S_{a} (\sqrt{R_{IM}} T_{1})^{α}$ , where $R_{IM}$ is a self-adaptive parameter or multiplier defined as the lateral strength required to maintain the system elastic (Mehanniy, 2009; Zhang et al., 2022), (ii) spectral IMs, for example, spectral acceleration at the first vibration period, $S_{a} (T_{1})$ , spectral displacement at the first vibration period $S_{d} (T_{1})$ , (iii) scalar IMs, for example, peak ground acceleration (PGA = max|a(t)|), peak ground displacement (PGD = max|d(t)|), and (iv) discrete IMs, for example, MMI, European Macroseismic Scale (EMS-98) (Stone et al., 2017). The year of publication of the curve captures some other relevant characteristics, such as whether the functions were derived from aged empirical data or modeled on older building types and are therefore probably less suitable for more recent seismic hazard and vulnerability analyses. The development of FCs brings with it a set of sources of variability and uncertainties that can be grouped into the following three: (i) uncertainty in the seismic demand, (ii) variability in the properties of the structural model (building-to-building uncertainty or uncertainty in capacity), and (iii) uncertainty in the definition of damage thresholds (Kaynia, 2013; Rossetto et al., 2014). Considering this, the FCs that were derived considering the three sources of uncertainties explicitly scored higher, while the FCs derived considering only some or none of these sources of uncertainty scored lower (see Table 2). Pre-set or custom criteria can define the threshold of DSs. Defining capacities for each building provides the flexibility to tailor damage threshold values to a given building or sub-class of building typologies. Alternatively, the proposed simplified formula from the literature commonly expressed as a function of yield and ultimate roof displacement could be employed to directly estimate global DS thresholds (Maio and Tsionis, 2015). Custom damage status thresholds are considered preferable to pre-established ones, as they allow the definition of thresholds to be adapted to the typology whose seismic vulnerability is being evaluated. Furthermore, it introduces capacity-demand correlations that can greatly influence the vulnerability analysis results (Maio et al., 2017; Maio and Tsionis, 2015). Therefore, FCs derived from adjusted DS thresholds are given a higher score than those obtained from pre-set damage thresholds.

Source quality sub-dimension variables

The last block or sub-dimension included in the technical suitability dimension comprises a new combined variable that evaluates whether the capacity and FCs were derived from existing buildings or from building prototypes and the size of the sample of buildings used. It is expected that the larger the sample size, the greater the precision and the possibility of generalizing the results obtained to the typology evaluated. The second variable included in this fourth dimension refers to the authenticity and credibility of the study from which the FC was derived. To assess this characteristic, the type of publication in which the analyzed FC is found is considered. Those FCs from published scientific articles or doctoral theses, for example, are therefore understood to have higher credibility (and therefore a higher score) than those derived from a conference paper, master’s or degree final projects, or unpublished reports. The last variable in this third block measures the popularity of the FCs, that is, the number of times the function was used for seismic risk assessments. For this, the number of citations in Google Scholar presents the study in which the evaluated FCs was used as a proxy variable that captures the use of the FCs although imperfect; it is easier to measure (see Table 2). These last variables could be more subjective than those included in the other dimensions and sub-dimensions. However, several authors propose considering them to evaluate FCs (Kaynia, 2013; Maio and Tsionis, 2015; Meslem et al., 2014; Stone et al., 2015, 2017). As these variables present a degree of subjectivity, relatively low scores were assigned to them, so they do not determine the final classification of the FCs. In this way, the robustness of the method is maintained; that is, the class of the curve will not be affected by small variations from the assumptions of these subjective variables.

Suitability for the local system dimension

In the second dimension, the suitability for the local system seeks to assess the level of adequacy of the FCs evaluated for the typologies analyzed and the study to be carried out based on the following variables: similarity in construction techniques and IM similarity. Similarity in the construction techniques is assessed based on the geographical applicability in the case of some typologies, such as masonry, and on the criteria of the researcher for the high-rise reinforced concrete and steel typologies as these typologies are more standardized globally and are likely to be built by foreign engineers. The similarity in the expertise of the engineers that usually work in the country and in the level of compliance of the buildings with good-practice codes regarding seismic-resistant design could be reasonable parameters so that the researcher can assess this variable in the case of high-rise buildings. The geographic applicability assesses whether the curve evaluated was derived for the locality, state, or country of the typology under investigation or was obtained for another country in the same sub-region, in the same region or from another region. It is assumed that the functions derived from the typologies of the country itself will share similar characteristics, such as building design, construction materials, and building practices (Stone et al., 2017). In this case, the definition of country or state, region, and sub-region is associated with the main factors determining the construction techniques and existing typologies in a zone. Specifically, the construction typologies are mainly influenced by climate, sociocultural factors, income per capita, level of technological advancement and seismic-resistant regulation development, as well as its degree of application (Brzev et al., 2013). Therefore, a country or state is considered to be one delimited by political or natural borders that share the four factors. A sub-region comprises one or several neighboring countries or states with similarities in at least three of the above-mentioned factors. A region comprises one or several neighboring countries or states that share similarities in at least two factors. In the case study, Costa Rica is considered a country, Central America is the sub-region, and Latin America is the region. Thus, an FC not for a country or area in Latin America is classified as outside the region.

The IM similarity captures the suitability between the IM of the FC and that corresponding to the hazard information (Stone et al., 2017). Therefore, the FC will be ranked according to whether its IM matches the IM included in seismic hazard information or according to how easily the IM can be translated into the IM needed based on the hazard information. Based on Stone et al. (2017), the degree of similarity between the IM of the FC and that of the seismic hazard information and, therefore, the difficulty in converting one into the other is expressed in the high, medium, and low scores of Table 2 in the following way: (i) the high score is assigned when both IMs coincide, (ii) the medium score is assigned when they are similar but not the same, but the conversion does not introduce a significant source of uncertainty; this is the case when one of the IMs is pseudo-acceleration (Sa) and the other is pseudo-displacement (Sd), and (iii) the low score is assigned when the dissimilarity between the two IMs is high, and the conversion introduces a high uncertainty, as occurs between the discrete IMs (MMI, EMS-98, for example) or scalar IM (PGA or PGD) and the rest.

Building class similarity dimension

The third dimension refers to the similarity between the building class of the candidate functions and the building class of the function that must be applied. To evaluate this similarity, the number and type of common attributes between both building classes are considered. In particular, the basic attributes and those that can be considered more important in view of the seismic vulnerability of the examined typology are lateral load–resisting system, number of stories, ductility, year of construction, and compliance with seismic regulations. It must be noted that the FC must correspond to a typology of the same material as the one analyzed. Given the importance of the number of common attributes between the building class studied and the one for which the evaluated FC has been derived, this dimension is the most weighted one for determining the suitability of the FC through an adjustment coefficient of the global index. Table 3 indicates the attributes relevant to this dimension and its score according to the material of the FCs under study.

Table 3.

Attributes considered and their corresponding scores for the adjustment coefficient related to building class similarities

Number of attributes in common	Score	Type of attributes in common by typology
Number of attributes in common	Score	Wood	Reinforced concrete	Masonry	Steel	Informal/precarious
None beyond basic attributes to qualify as candidate FC	0.55	Similar primary construction material (wood) and height range	Similar primary construction material and height range	Similar primary construction (masonry) and height range	Similar primary construction material (steel) and height range	Informal/precarious and similar height range
One shared attribute^a	0.70	(+) (i) Structural system (LLRS), (ii) identical number of stories, (iii) ductility, (iv) other	(+) (i) Structural system (LLRS: walls, frames, dual), (ii) identical number of stories, (iii) ductility, (iv) other	(+) (i) Structural system (LLRS: unreinforced, reinforced, confined), (ii) identical number of stories, (iii) ductility, (iv) other	(+) (i) Structural system (LLRS: frames, frames with steel brace, steel light frames), (ii) identical number of floors, (iii) ductility, (iv) other	(+) (i) Similar material, (ii) identical number of floors, (iii) other
Two shared attributes^a	0.85
Three or more shared attributes^a	1

FC: fragility curve; LLRS: lateral load–resisting system.

Height ranges: low, 1–3 stories; mid, 4–7 stories; and high, 8 or more stories based on FEMA (2001).

When the typology studied and that of the FC present one, two, or three, or more of any of the attributes listed, it receives a score of 0.70, 0.85, or 1, respectively, in this dimension.

For all typologies analyzed, the FCs must be selected to present the same material of lateral load–resisting system and height range (low, medium, high)³ among those corresponding to the same building class. The height range is considered a basic attribute in selecting the candidate, as the natural vibration period of a building is highly conditioned by its height and is a crucial property for assessing the seismic base shear, which is controlled by its mass and stiffness. As the height of the building increases, its mass increases but its overall stiffness decreases. Therefore, the natural period of the building grows (Bhuskade and Sagane, 2015). Moreover, the empirical formulas used, for example, in construction codes, to estimate the fundamental time period of buildings are generally expressed as a function of the height of the building (Kwon and Kim, 2010). In the case of the masonry, reinforced concrete, wood, and steel typologies, a score from minimum to maximum is assigned in this coefficient when the FC of the candidate building class shares from one to all the following attributes for which information is generally available: the structural system, the number of floors, and the ductility. This ranking system avoids giving more value to the FCs of the candidate typologies that do not share the first basic attributes and others considered important for seismic vulnerability in the analyzed building class. In that case, neither candidate FCs will receive more scores when they share other attributes or constructive features that differ from those previously mentioned for each material.

A score assigned to each of the dimensions and variables analyzed (Table 2) considers the comparative importance of each to determine the suitability of the curve analyzed based on the framework for FC evaluations proposed by Maio and Tsionis (2015), Rossetto et al. (2014), and Stone et al. (2017).

A clear advantage in terms of practicality of the sub-division into dimensions is that some dimensions need not be particularized for each specific area or structural system (see Figure 2). In other words:

The technical dimension is independent of the area under study, or the type studied, that is, it depends only on the method used to elaborate the FCs. Therefore, once done for a specific FC, the value is valid for any seismic risk study. Moreover, this is the most time-consuming dimension.

The local system dimension depends only on the features of the town/city/area under study. The value is therefore valid for any structural type within the area.

The building class similarity-reduction coefficient is the only dimension that must be particularized for any structural type according to its attributes (material, structural system, number of stories and ductility).

Classification of seismic FCs

Once the FCs have been assessed and points assigned to each curve, the curves are classified (Figure 3) to give a rapid and clear idea of the adequacy of each curve to the study under development. Said sorting in classes is based on the methodology of classification that stems from the field of energy consumption, which is already present in several fields such as structural engineering (Mazzotti et al., 2013). In the classification, each color and letter represent a level of adequacy to describe classes or categories that differ in their assessment qualitatively rather than quantitatively. In this framework, FCs of Class A are those whose total score is higher than 85, Class B corresponds to FCs with a total score higher than 65, Class C to those higher than 55, Class D to those higher than 47, Class E to those higher than 40, and Class F to all other FCs with a total score under 40. The values assigned to the consecutive steps have been defined considering 85% the value of the step immediately above (Class A: (85, 100]; Class B: (65, 85]; Class C: (55, 65]; Class D: (47, 55]; Class E: (40, 47], Class F: (0, 40] points).

After that, the curves are drawn and those pertaining to the highest-scored classes are highlighted by type and state of damage. In addition, if considered pertinent, the FCs for different types can be drawn together.

At this point, if most of the FCs with higher scores for a particular type resemble each other, this consensus is an additional indicator of suitability. The resemblance between the FCs can be easily analyzed visually from the graph that represents them jointly. It is also possible to determine the similarity between the FCs by comparing their parameters (median or mean and standard deviation) or the corresponding damage exceedance probabilities for similar values of intensity of the seismic movement. Nonetheless, the evaluation tool must be assisted by the criteria of local experts, as they may establish whether the results are coherent between different typologies.

Therefore, the methodology can be considered a tool that permits a well-founded evaluation of the FCs available in the literature, whose suitability of results or output would depend on the quality of the inputs and the knowledge of the local experts involved.

Application to Costa Rica: results and discussion

In the following, the application of the methodology elaborated to assess FCs is tested in San Jose (Costa Rica). The GEM building taxonomy (Silva et al., 2022) is used to characterize the typologies of FCs evaluated according to their attributes in a uniform classification scheme.

The work developed by Esquivel-Salas (2020) presents an analysis of the type of buildings included in the catalog of exposure of Costa Rica, considering the total building stock, that is, constructions of all uses (residential, commercial, etc.): 20 types of buildings within which MRC (reinforced confined masonry) and MCF (confined masonry), RC (reinforced concrete), S (steel), W (wood), and informal constructions are included. Among the most relevant ones according to their frequency, the following typologies have been considered for the application of the methodology proposed to San José of Costa Rica: (i) 2-story, reinforced confined masonry with concrete blocks hollow, lateral wall system (LWAL), regular structure (IRRE), and no elevated or suspended floor material (FN) (MRC + CBH/LWAL/HEX:2/IRRE/FN) (38.7% of total built area), (ii) 1-story, reinforced confined masonry with wall system, regular structure and concrete floor (FC) (MRC + CBH/LWAL/HEX:1/IRRE/FC) (24.8% of total built area), (iii) more than 11-story reinforced concrete with dual system, regular structure, and concrete floor (RC/LDUAL/HBET:11+/IRRE/FC)(6.9% of total built area), (iv) 1- to 5-story reinforced concrete with wall system, regular structure and concrete floor (RC/LWAL/HBET:5,1/IRRE/FC) (3.3% of total built area), and (v) 1- to 5-story reinforced concrete with dual system, regular structure and concrete floor (RC/LDUAL/HBET:5,1/IRRE/FC) (2.5% of total built area).

In particular, the RC types have been selected in a way that permits a comparative assessment of the methodology, depending on the number of stories of buildings with the same structural system and on the structural system itself, either dual or just with shear walls. For each of these types, several FCs have been selected and scored. As an example, we will show the most extended type: 2-story reinforced confined masonry with wall system (Table 2).

The FCs selected for comparison have been taken from very different sources: big sources of data, such as HAZUS or GEM, regional or national seismic risk projects, scientific articles, PhD studies, and final master projects. Among these databases, the references used for Costa Rica consider the FCs included in the following works and developed or adapted from other sources by the researchers specified (see Table 1 for more detail): Calderón (2017, 2018), Calderón et al. (2021), Cattari et al. (2004), Esquivel-Salas (2020), Lang et al. (2009), Martins and Silva (2020), Miyamoto International (2016), and Villar-Vega et al. (2017).

Table 4 shows the scores for FCs assessed in the case of Costa Rica following the methodology proposed. Particularly, the results of the following cases are shown graphically in detail in Figure 4: (i) 2-story, reinforced confined masonry with wall system (MRC + CBH/LWAL/HEX:2/IRRE/FN) and 1-story, reinforced confined masonry with wall system (MRC + CBH/LWAL/HEX:1/IRRE/FC) and (ii) 1- to 5-story reinforced concrete with dual system (RC/LDUAL/HBET:5,1/IRRE/FC) and 1- to 5-story reinforced concrete with wall system (RC/LWAL/HBET:5,1/IRRE/FC). The candidate curves for the first two typologies (Figure 4a) are as follows: (i) FC1: MRC + CBH/LWAL + DUC/HEX:1/IRRE/FN by Calderón (2018) based on Hidalgo-Leiva (2017), (ii) FC2: MRC + CBH/LWAL + DUC/HEX:2/IRRE/FN by Calderón (2018) based on Hidalgo-Leiva (2017); (iii) FC3: MR/LWAL + DUC/HEX:1 by Calderón and Silva (2019); (iv) FC4: MR/LWAL + DUC/HEX:2 by Calderón and Silva (2019); (v) FC5: MR/LWAL + DNO/HEX:1 by Calderón and Silva (2019); and (vi) MCF/LWAL + DUC/HEX:1 by Villar-Vega et al. (2017). In the case of the 1- to 5-story reinforced concrete with wall system and dual system (Figure 4b), only the following FCs developed in Central America or used as inputs in previous seismic risk studies were analyzed: (i) FC1: RC + CIP/LDUAL/HBET:4,7 by Calderón (2018) based on Labarca (2006) and Ávila (2011); (ii) FC2: RC + CIP/LDUAL/HBET:11,6 by Esquivel-Salas (2020) based on Calderón (2018); (iii) FC3: RC + CIP/LWAL/HBET:1,3 by Lang et al. (2009); (iv) FC4: RC + CIP/LWAL/HBET:4+ by Miyamoto International (2016); (v) FC5: RC\LDUAL + DUC\HEX:5 by Martins and Silva (2020); and (vi) FC6: RC\LDUAL + DNO\HEX:5 by Martins and Silva (2020). The first bar for each FC shows the value of the global index and its component dimensions: those included in the technical suitability sub-dimension (the three dimensions related to the capacity curve, the FC, and the general quality of the study) and the suitability for the local system sub-dimension. The second and third bars show the final index value for each typology corrected according to the building class similarity coefficient. As previously mentioned, this coefficient adjusts the global index, evaluating the similarity between the typology under study and the one for which the FC evaluated was built or proposed. The horizontal lines in each figure show the thresholds set for each of the seven classes proposed to classify the FC according to its score in the final index score.

As can be seen in the mentioned figure, the application of the methodology proposed renders it very easy to identify the most appropriate FC for a particular structural type (Figure 4a). Moreover, it reveals the cases where the FCs assessed are not suitable and where more FCs should be included in the catalog or where efforts in the development of an enhanced FC for a particular type should be focused (Figure 4b—1- to 5-story reinforced concrete with dual system). Furthermore, the aspects in which the curves present greater efficiencies or deficiencies are quickly identifiable.

According to the results of the methodology, therefore, the quest for FCs should continue to include FCs developed for the same building type all over the world. It is worth mentioning that construction techniques present fewer differences between countries in the case of RC buildings, mostly medium- or high-rise than in the case of masonry dwellings, mostly low-rise.

In Figure 5, scores for each sub-dimension and dimensions of all the FC assessed are graphed from the highest to the lowest final score. This graph shows that the different sub-dimensions and dimensions present distinct results, and there is no clear or strong correlation between all of them. This fact suggests that the variables included in the index capture different aspects related to the FC evaluated; therefore, the dimensionality (quantity of variables) of the index should not be reduced.

Given that most of the variables in the proposed index refer to the three sub-dimensions of the technical suitability dimension, evaluating the degree of existing correlation between them is of interest. Furthermore, as previously mentioned, the technical dimension depends on the process used to elaborate the FC and is independent of the typology studied or the area under investigation. As a result, the value is applicable to all seismic risk studies once it has been completed for a particular FC. Figure 6 graphically presents the correlation matrix between these sub-dimensions calculated from the scores obtained for the FCs evaluated in the case of Costa Rica. Similar to what is observed in Figure 5, the estimated coefficients indicate low levels of correlation between the three sub-dimensions. This result again suggests that the aspects valued concerning the capacity curve, the FC, and the quality of the study that proposes them provide different information that, for the reasons previously stated in section “Variable selection and evaluated dimension,” must be considered when assessing an FC for a given typology.

This classification gives rise to Figures 7 and 8, in which the different FCs have been represented. All the FCs have been graphed into the same IM to facilitate the comparison. The IM selected to represent the FCs is PGA, since it is the most widely used, and it allows the alteration of the original curve developed by the different researchers to be avoided. Moreover, the seismic risk studies usually give as outputs seismic hazard maps in PGA or Sa, while other IM such as Sd cannot be applied without an intermediate step, the calculation of the performance point, which hinders their comparison. The regulation in force in Costa Rica (Colegio Federado de Ingenieros y de Arquitectos (CFIA), 2011, 2018) and the performance point identification method (Shibin et al., 2010) have been employed for the conversion of the IM.

The FCs have been represented in a manner that permits the differentiation into DSs and classes. Moreover, in the cases where there are two different FCs in the highest class, the conflation method has been applied for consolidating data from these different FCs.

A specific FC has been established as the most suitable one in the following cases:

(i) For 1-story, reinforced confined masonry with wall system typology (Figure 7a): (A) MRC + CBH/LWAL + DUC/HEX:1/IRRE/FN, Calderón (2018) based on Hidalgo-Leiva (2017).

(ii) For 2-story, reinforced confined masonry with wall system typology (Figure 7b): (A) MRC + CBH/LWAL + DUC/HEX:2/IRRE/FN, Calderón (2018) based on Hidalgo-Leiva (2017).

(iii) For more than 11-story reinforced concrete with dual system typology (Figure 8a): a ponderation of these two FCs applying the conflation method for consolidating data from different probability distributions: (B) RC + CIP/LDUAL/HBET:11+, Esquivel-Salas (2020) based on Calderón (2018) and (B) RC\LDUAL + DUC\HEX:11, Martins and Silva (2020).

(iv) For 1- to 5-story reinforced concrete with wall system (Figure 8b) and with dual system (Figure 8c) typologies: the methodology has identified the need for further research.

To this must be added the cases of RC buildings subject to different irregularities that also require a reliable FC for evaluating their compounded seismic vulnerability (Rajeev and Tesfamariam, 2012). There are other issues to consider when dealing with RC structures as, for example, the type of nonlinearity by which the response is characterized (Schotanus et al., 2004).

Conclusions and future lines of research

In this article, a methodology for assessing seismic FCs is presented. This method permits the well-founded selection of the most appropriate FCs for a particular type of building in a specific seismic-prone area for seismic risk studies from a set of curves available in the literature. A series of specific criteria are proposed to facilitate the creation of a rational ranking of FCs in terms of adequacy. The assessment process is based on several variables divided into three key dimensions: technical suitability, suitability for the local system, and building class similarity. The novelty of the proposal, therefore, lies in organizing all the variables involved according to the literature in a practical way and giving them scores so that the application of the methodology provides the researcher with a classification of the assessed FCs according to their adequacy to the area under study. The results have proven to be robust in the application case of reinforced confined masonry in Costa Rica because small changes in the score of the variables involved in the multidimensional index do not significantly affect the classes in which the FCs evaluated were classified.

Moreover, the proposed methodology permits the reliability level of the FC obtained to be identified depending on the class the curve was classified into because of its score. Therefore, it allows the adequacy related to the selected FC to be established and to reduce the uncertainty or error involved in that selection, which is a subject that researchers should consider in seismic vulnerability and risk studies.

In addition, the classification and a systematized representation of the FCs allow the researcher to easily compare the results and draw conclusions. For example, in the cases of 1- and 2-story reinforced confined masonry with ductile wall systems of Costa Rica, the candidate FCs analyzed show different variations in terms of PGA depending on the dissimilar attribute they present with respect to the Costa Rican typology. Moreover, in the case of 2-story reinforced confined masonry with ductile wall system, however, the curves pertaining to higher classes seem to yield more conservative values than the other curves.

Furthermore, the proposed index can be broken down into its three dimensions and component variables, allowing the researcher to recognize those aspects in which each of the evaluated FCs has obtained better and worse scores. In addition, in this way, a researcher can weigh the dimensions or variables differently in the case of his or her study that requires them. Another possible use of a quantitative assessment of the different dimensions evaluated in the FCs examined is to use these scores to obtain a weighted average of the different FCs classified in the higher classes from a logical tree in which those scores act as weighting elements.

In addition, when the FC introduced as inputs are insufficient or their scores too low, the method allows the identification of possible future lines of research in seismic vulnerability, as has been the case of 1- to 5-story reinforced concrete with dual system and with an RC wall system.

Future lines of research to improve the methodology should perform a formal survey of a greater number of experts in the field aiming to widen the consensus in relation to the scores assigned to the different components of the proposed index. Finally, it is worth noting that the methodology presents the advantage of being flexible enough to admit the implementation of new dimensions, sub-dimensions or variables, such as one in which it can be evaluated whether or not the FC has been calibrated with post-earthquake observations.

Supplemental Material

sj-pdf-1-eqs-10.1177_87552930231171177 – Supplemental material for A methodology to assess and select seismic fragility curves: Application to the case of Costa Rica

Supplemental material, sj-pdf-1-eqs-10.1177_87552930231171177 for A methodology to assess and select seismic fragility curves: Application to the case of Costa Rica by Laura Navas-Sánchez, Maribel Jiménez-Martínez, Beatriz González-Rodrigo, Orlando Hernández-Rubio, Luis Diego Dávila-Migoya, Belén Orta-Rial and Diego Hidalgo-Leiva in Earthquake Spectra

Footnotes

Acknowledgements

The authors thank Dr Jaime Cervera Bravo (UPM, ORCID 0000-0002-1060-7397) for his expertise and assistance throughout this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the support from KUK AHPAN grant RTI 2018-094827-B-C22 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe.”

ORCID iDs

Maribel Jiménez-Martínez

Beatriz González-Rodrigo

Diego Hidalgo-Leiva

Supplemental material

Supplemental material for this article is available online.

Notes

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