Methodologies for the evaluation of seismic vulnerability in masonry structures

For the evaluation seismic risk, there are evermore complex and elaborate procedures, the functions of fragility and vulnerability a fundamental tool. In the last 30 years, various methods have been developed for the analysis of seismic vulnerability in masonry structures. In this article, some of the most common empirical and analytical methodologies have been revised and are explained in a concise manner, which will help other investigators in the field to decide which method is the most suitable, depending on the available information in the study that is being carried out

In the great variety of methodologies regarding risk evaluation, to the damage or loss for a determined building classification (Silva et al., 2014) (Yamin et al., 2017) is its drawback.
Generally two different focuses are employed: fragility and vulnerability functions.The fragility functions express the probability of exceeding a damage state, for a predetermined value of the intensity measures (IM), generally represented by the PGA or the spectral displacements (Chieffo et al., 2019).
The damage states are normally defined in qualitative and descriptive terms (Park, Ang and Ween, 1987) (for example S: slight, M: moderate, E: extensive, C: collapse).The formulation of the fragility functions requires the consideration of N different damage states, for a structural system.Thus, the probability of reaching or exceeding the ith damage state given a danger intensity is normally estimated through a normal logarithmic probability distribution function with different parameters for each damage state.
The vulnerability functions provide all the necessary information to calculate the probability of reaching or surpassing a loss value, given the selected intensity of the ground motion (Borzi, Crowley and Pinho, 2008).The formulation of vulnerability functions requires the definition of the loss as a random variable.In this case, the vulnerability function describes the variation of the statistical moments of loss (average and variance) for different values of threat intensity.Loss is defined using numeric scales in place of qualitative scales as for the damage states (for example, the relationship Ciencia Latina Revista Científica Multidisciplinar, Ciudad de México, México.ISN 2707-2207 / ISSN 2707-2215 (en línea),noviembre-diciembre,2022,Volumen 6,Número 6 p 14011 between the cost of repairs and the value of replacing the component, also known as the Mean Damage Ratio (MDR), which allows its direct use in probabilistic risk and loss calculations.Generally, it is supposed that a Beta probability distribution function for the loss calculation is used (Yamin et al., 2017).It is possible to obtain the vulnerability function through the formulation of fragility of any given component once an expected loss value is assigned to each damage state (Silva et al., 2014).

Vulnerability methods: empirical and analytical
For the development of fragility and vulnerability functions, we will focus on the following methods: empirical and analytical (Figure 1).Empirical methods are based on the results of laboratory tests or the compiled information of earthquake damage evaluation reports (Yamin et al., 2017).They can have some disadvantages, such as the subjectivity of the assignment of a damage state for each building, or a lack of accuracy in the determination of ground motion which affects a region (Kassem, Mohamed Nazri and Noroozinejad Farsangi, 2020).In order to overcome these limits, analytical methods can be employed for a single structure which is believed to represent a class of buildings, or for a combination of buildings generated randomly, modeled through structural analysis techniques and subject to lateral load patterns or specific accelerograms (Silva et al., 2014).

Empirical Methods
Pioneering empirical methods of the 70's and 80's, as well as some of those from the past 15 years, used for the evaluation of seismic vulnerability of masonry structures are presented by way of summary in Table 1, although only some of those are described hereunder.

Vulnerability Index Methods
In the Italian method (Benedetti, Benzoni and Parisi, 1988), the vulnerability of a building is defined through a vulnerability index (Iv), which is obtained through the evaluation and deliberation of different parameters related to structural and non-structural components, which have an important role in the seismic behaviour of the building.
Basically, the method consists of the evaluation of 11 parameters through fieldwork.For unreinforced masonry structures, there are categories A, B, C, and D, which go respectively from favourable to unfavourable conditions and their evaluation allows the assignment of numeric value (Ki), which varies from 0 to 45.The weight (Wi) of the parameters varies between 0.25 and 1.0.The vulnerability index is obtained through the equation 1, which is the deliberated sum of the numeric values which express the quality of each one of the eleven parameters.The bigger the vulnerability index, the worse the resistant capacity of the building (Lantada, 2007).
Eq. 1 The macroseismic method (Lagomarsino and Giovinazzi, 2006) allows the evaluation of vulnerability for a group of buildings, up to the evaluation of vulnerability of a single building.The vulnerability is measured in terms of an index of vulnerability (V) and an index of ductility (Q), with both evaluations taking into account the typology of the building and their constructive characteristics.This index varies between 0 and 1; the closer to 1, the more vulnerable the building.The danger is described in terms of macroseismic intensity, according to the European Macroseismic Scale EMS-98 (EMS, 1998), which is considered, in the framework of macroseismic focus, as a continuously evaluated parameter regarding the condition of rigid ground; the possible effects of amplification owing to the different conditions of the ground are within the parameter of V.For physical damage to the building, the EMS-98 damage grades are used, describing the observed damage for structural and non-structural components.Five grades of damage (Dk) (k = 0/5) are identified: D1 light, D2 moderate, D3 heavy, D4 very heavy, D5 destruction, plus the absence of damage, D0 no damage (Bernardini and Lagomarsino, 2008) .
Some current studies carried out in residential buildings allowed the estimation of damages by seismic phenomenon, such as that by (Serrano-Lanzarote and Temes-Córdovez, 2015), who presented a study on the seismic vulnerability in the Valencian Community, Spain, for the assignment of vulnerability to each type of building.They carried it out according to the opinion of experts and contrasted said information through the application of the vulnerability index method.(Pavel et al., 2018) utilised the microseismic method for housing in Bucharest, the capital of Romania.They compared the evaluation of seismic damages and losses using the KOERILoss software, with the results of a previous study based in the Hazard of United State (HAZUS) manual.
Ciencia Latina Revista Científica Multidisciplinar, Ciudad de México, México.ISN 2707-2207 / ISSN 2707-2215 (en línea),noviembre-diciembre,2022,Volumen 6,Número 6 p 14013 The results of the analysis demonstrated that the economic loss and the average damage grade are smaller than those obtained using HAZUS methodology.However, the number of people affected in this study is higher.
The Vulnerability Index Method (VIM) has been simplified in order to evaluate the seismic vulnerability in masonry building facades, like the work carried out by (Ferreira et al., 2017) on the Azores archipelago, Portugal; additionally VIM has been adapted with a total of fourteen parameters by (Vicente et al., 2011) and utilised by (Catulo et al., 2018) for the seismic evaluation of buildings in Pombalino in the city of Lisbon.

Rapid Visual Screening, Rvs
Rapid visual screening is a qualitative evaluation procedure used in the evaluation of the seismic vulnerability of buildings.The procedure can be implemented with relative speed in a large inventory of buildings in order to identify potentially dangerous structures without the high costs of detailed seismic analysis of individual buildings.The RVS method involves a curbside survey which is used in order to compile information on the main parameters which influence the seismic vulnerability of the buildings (FEMA P-155, 2015).
The Federal Emergency Management Agency (FEMA) developed a number of directives for the evaluation and rehabilitation of the seismic risk of buildings.FEMA 310 provides a process of tree levels for the seismic evaluation of existing buildings in any seismic zone.
In accordance with FEMA 310, before using the three provided methodologies in the guidelines, a Rapid visual screening of the building should be carried out in order to decide if a seismic evaluation is necessary (FEMA 310, 1998).
The RVS procedure proposed in FEMA 154, utilises a scoring system which requires that the user identify the primary structure resistant to the lateral load and the building's attributes which modify the expected seismic behaviour.The results are recorded in a data collection form according to the seismicity of the region in question.The method assigns a basic structural score based on the structural typology and uses score modifiers in order to consider the effect of the number of floors, the type of ground, the vertical and floor irregularities, and details of the previous or subsequent code to the reference code (FEMA P-155, 2015).ISN 2707-2207 / ISSN 2707-2215 (en línea),noviembre-diciembre,2022,Volumen 6,Número 6 p 14014 In Egypt, (El-Betar, 2018), with the RVS procedure FEMA P-154, evaluated the seismic vulnerability of school buildings from the 1960's and schools built after the 1990's, designed in accordance with the country's code, and they determined the capacity functions with a pushover analysis, using the IDARC version 6 software.They concluded that buildings from the 60's tend to be more vulnerable under high seismic loads, while those designed in accordance with the code have a large capacity to resist earthquakes.Some authors present other alternatives to the RVS method, such as (Achs and Adam, 2012) in Vienna, Austria, who carried out the seismic evaluation of historic brick masonry buildings, the RVS methodology proposal evaluates the physical and socio-economic vulnerability, which generated result maps for damage scenarios which afford useful information for the planning of emergencies and evacuations, as well as the identification of critical objects vulnerable to seismic loads.In India, (Rajarathnam and Santhakumar, 2015) evaluate the seismic safety of buildings in Chennai using the RVS technique.They used aerial photographs in order to identify irregularities in the buildings with a Geographic Information System (GIS).(Ajay Kumar et al., 2017) propose a new format modified in order to carry out Rapid visual screening (RVS) in the state of Himachal, Pradesh.With the RVS scores, they obtained damage distribution curves for each typology of housing in order to understand the building distribution in the state.

Analytical Methods
Currently as we can observe in Table 2, there exists a great variety of methods which carry out an analytical evaluation.In this article, we focus on the program packet RMTK (Risk Modellers' Toolkit) by OpenQuake, developed by GEM (Global Earthquake Model), in which different cutting-edge methods for deriving solid analytical seismic vulnerability and fragility functions for individual structures or buildings (Figure 2).

Direct Nonlinear Static Procedures
The evaluation studies of the nonlinear structural response has been integrated into three direct nonlinear static procedures: (Ruiz-García and Miranda, 2007), (Vamvatsikos and Cornell, 2005) and (Dolšek and Fajfar, 2004) (Table 3).These are based on the use of capacity functions, resulting from nonlinear static Pushover analysis, in order to directly These parameters are used to represent a fragility curve as the probability of the limit state capacity (C) is exceeded by the demand (D), both expressed in terms of intensity levels (Sa,ds and Sa respectively), as shown in the equation 2 (Silva et al., 2015): The implemented methodologies allow the consideration of different shapes of the Pushover curve (multilinear and bilinear), entry to entry dispersion, and dispersion of the thresholds of damage state, in a systematic and coordinated way.

Static Nonlinear Procedures Based On Registries
Static nonlinear procedures described hereafter allow the calculation of the seismic response of various structures (Table 4) (in terms of the maximum displacement of the system equivalent to a single degree of freedom (SDOF)), considering a group of registries of ground movement.The development of these methods involves the numerical analysis of systems with particular dynamic and structural properties (for example, periods of vibration, viscous damping, hysteretic behaviour, among others) and accelerograms selected for specific regions of the world (for example, California, Southern Europe).For these reasons, their applicability to other types of structures and different ground movement registries requires adequate care (Silva et al., 2015).
The main results of each of these methodologies is a Probability Damage Matrix (which is to say, a section of assets by damage state for each ground movement registry, represented by the variable PDM), and the Spectral Displacement (which is to say, the maximum expected displacement of the equivalent SDOF system, represented by the variable Sd) for ground movement registries (Silva et al., 2015).

Nonlinear Time-History Analysis In Oscillators With A Degree Of Freedom
This methodology carries out a series of nonlinear time-history analyses (NLTHA) on one or various systems to a single degree of freedom (SDOF) (Table 5).In order to determine the structural capacity of the systems to multiple degrees of freedom (MDOF) under analysis, it is necessary to identify the relationship between the shear force and the roof displacement (which is to say, the PushOver curve).This curve should later convert into the capacity curve of an equivalent SDOF oscillator.
Ciencia Latina Revista Científica Multidisciplinar, Ciudad de México, México.ISN 2707-2207 / ISSN 2707-2215 (en línea),noviembre-diciembre,2022,Volumen 6,Número 6 p 14016 For buildings of low and medium height, it is typically assumed that the fundamental manner of vibration corresponds to the predominant response of the structure.Under this hypothesis, the SDOF oscillator represents the first response mode of the structure.This is usually valid for building with fundamental periods of vibration of up to approximately 1.0 s.On the contrary, they need to take into account the superior modes (Silva et al., 2015).
In this methodology, the demand is represented by a group of ground movement registries.The response of each structure comes from the solution of the movement equation 3 for an inelastic SDOF under seismic excitement: (https://github.com/GEMScienceTools/rmtk).
In the city of Antioquia, Colombia, they carried out a study of seismic risk on nonreinforced masonry buildings, for which they developed an exposition model based on the available land register information, survey data, and the opinion of experts.If the information was not accessible, they turned to a virtual survey carried out by Google Street View.They calculated the capacity functions through a pushover analysis, in order to subsequently develop the fragility functions for each building.The considered scenarios that could take place in the region of interest were carried out with the OpenQuake software.The results they obtained indicate that around 20% of the total property of the non-reinforced masonry structures would suffer big damages to the point of collapse, with the structures with four to six stories being the most vulnerable (Acevedo et al., 2017).
Ciencia Latina Revista Científica Multidisciplinar, Ciudad de México, México.ISN 2707-2207 / ISSN 2707-2215 (en línea),noviembre-diciembre,2022,Volumen 6,Número 6 p 14017 Through an analytical focus, fragility and vulnerability functions were developed for the majority of the types of common buildings in the world.Obtained from a global survey (50 countries), close to two hundred types of buildings were considered.The fragility models were developed for different tectonic environments.Two model groups were considered, one for the active surface crust, and the other for subduction.After having tested and calibrated the fragility models in order to provide viable seismic risk estimations, it is believed that the resulting database could be used as a standard for the comparison of existing models or as a starting point for the analysis of seismic risk for regions where there are other available models (Martins and Silva, 2018).
In the Andes region (Argentina, Bolivia, Chile, Colombia, Ecuador, Peru and Venezuela) (Villar-Vega et al., 2017) a uniform fragility model was carried out for 54 types of representative buildings for risk analysis on a grand scale.All the fragility functions are publicly available through the GEM vulnerability database on the OpenQuake platform (http://platform.openquake.org).Despite the usefulness of these models, it is important to recognise their limitations and their range of application.These fragility functions do not capture the specific characteristics of the building inventory on a local level.For the loss evaluation of earthquakes on a local scale, the models derived from using a more detailed methodology, and considering the local characteristics of the building inventory, should be considered.

Derivation Of Fragility And Vulnerability Functions
For the seismic vulnerability analyses of structures, the empirical method currently used are: Vulnerability Index Method (VIM), the macroseismic method by (Lagomarsino and Giovinazzi, 2006), and rapid visual screening (RVS) (FEMA P-155, 2015).
The main advantage of these methods is that they offer a more realistic vulnerability upon displaying the observed damages during the event.Some disadvantages could be the lack and weakness of data, not having a clear vision for investigating damage, depending mainly on the decisions of experts with differing opinions.In the diagram of Figure 3, the empirical methodology for the development of fragility functions is described (Kassem, Mohamed Nazri and Noroozinejad Farsangi, 2020).
The analytical method is the most accurate and can be used to consider all kinds of uncertainties.One of the main disadvantages of the analytical derivation of vulnerability functions is that the procedure is extremely intense from a computational perspective, and thus requires a lot of time.As such, vulnerability functions cannot be easily developed for different areas or countries with diverse building characteristics (Calvi et al., 2006) (Kassem, Mohamed Nazri and Noroozinejad Farsangi, 2020).
In the RMTK program OpenQuake, a probability damage matrix is used to derive a fragility function (which is to say, the probability of exceeding a number of damage states for a group of intensity level measurements), which can later be converted into a vulnerability function (which is to say, the loss index distribution for a group of intensity level measurements), using a consequence model.In this process, the portion of buildings in each damage state is multiplied by the associated damage index (from the consequence model) in order to obtain a loss index distribution for each type of intensity measurement (Silva et al., 2015).For the derivation of the vulnerability functions, the general process of the analytical methodologies of the RMTK program is described in (Figure 4).Table 2. Summary of analytical procedures for the evaluation of seismic vulnerability of masonry structures (D'Ayala, 2013).(Silva et al., 2015) Table 3. Summary of direct nonlinear static analytical methods from the RMTK program (Silva et al., 2015).

Method
Requirements Result (Vidic, Fajfar and Fischinger, 1994) 1.Capacity curves   (Silva et al., 2015) .In the previous descriptions, both empirical and analytical methods have been presented, along with their advantages and disadvantages, describing the inherent characteristics of each one.The choice of the most optimal or ideal vulnerability evaluation method will depend largely on the information available on the region in question.
If we wish to encompass the characteristics of each method, we could note that: The results of empirical methods are largely contributed by the judgement of experts, the definition of vulnerability is based on the characteristics of the building and the damage data of previous earthquakes.Damage data are used to calibrate the vulnerability functions, relating to the vulnerability index of global damages observed for buildings of the same typology, and extending the application in regions that have experienced the same level of macroseismic intensity or PGA.
Analytical methods, with the great variety of free access programs that currently exist appear to be a viable option to determine the vulnerability functions.Before choosing any of them, one must research their limitations and necessary characteristics so that they can be easily adapted to different types of buildings around the world.
The employment of two or more different methods which can complement and verify each-other will remain the choice of the investigator who wishes to obtain less uncertainty in the vulnerability evaluation.

Figure 1 .
Figure 1.Empirical and analytical methods for the evaluation of seismic vulnerability.

Figure 2 .
Figure 2. Analytical methods from the RMTK program for the evaluation of seismic vulnerabilities(Silva et al., 2015).

Figure 4 .
Figure 4. Steps to obtain vulnerability functions with analytical methods from the RMTK program (Adaptation by(Silva et al., 2014)).

Table 4 .
Summary of static nonlinear analytical methods based on registries from the RMTK program

Table 5 .
Summary of nonlinear time-history analysis methods in oscillators with a degree of freedom from the RMTK program