Probabilistic Loss Assessment for the Typology of Non-Ductile Reinforced Concrete Structures with Flat Slabs, Embedded Beams, and Unreinforced Infill Masonry

Abstract

Quito, the capital of Ecuador, a development pole, has experienced a population growth of 9% in the last five years. The structural system commonly chosen for housing is reinforced concrete frames with flat slabs, embedded beams, and masonry infill. This typology covers approximately 60% of the residential buildings in the city. Adding to the site’s seismic hazard, this fact results in a city with a high seismic risk. The research presented here is carried out within a probabilistic framework to determine the economic consequences of the main structural typology in the city. The methodology defines the seismic hazard by scaling a database of 200 records to the design spectrum. It models the typology to capture the variability between structures with a solid parametric study. Each capacity curve is analyzed through a nonlinear time history analysis using an equivalent one-degree-of-freedom system. The results show an average annual loss ratio of 0.16%. This metric indicates the vulnerability of the typology and the high repair costs of buildings that will be observed in case of an earthquake. The practical implications of these findings are significant as they contribute to urban planning and policy decisions. Finally, it is observed that the probabilistic method used efficiently generates fragility and vulnerability curves, saving computational time and obtaining expected results.

1. Introduction

Lately, seismic risk assessments have been carried out in many cities worldwide due to the technology and current information that some studies have provided. For instance, the Global Earthquake Model (GEM) Foundation has been collecting important data about seismic hazards from Quito, exposure models grouping different kinds of structures in some specific typologies, and fragility and vulnerability functions that represent consequence models for spectral accelerations. With this information and the OpenQuake Engine’s usage, seismic risk analyses and hazard assessments are developed. Then, economic losses, fatalities, and cost of replacement for structures are determined. Consequently, stakeholders develop mitigation strategies to reduce the abovementioned metrics [1].

In Asia, places such as the Philippines and Taiwan have developed risk assessment models. In the Philippines, ref. [2] updated the seismic hazard model considering active crustal shallow, subduction interface, and interslab ruptures to a peak ground acceleration (PGA) of 1.0 g with a 10% probability of exceedance in 50 years. As a result, hazard maps are presented as an option to determine the seismic demand for new buildings. In Taiwan, ref. [3] updated the seismogenic structure database, including new ground motion prediction equations and site amplification factors. So, non-experts without enough knowledge could use this information to establish alternatives to reduce an earthquake’s consequences.

Africa is another continent that has developed a seismic risk model using the OpenQuake Engine. For instance, ref. [4] uses previous seismic hazard maps of South Africa and improves them with seismicity and geological available data, systematically compiling and homogenizing an earthquake catalog using probabilistic seismic hazard analysis methodologies.

Moreover, ref. [5] describes the Euro-Mediterranean Seismic Hazard Model (ESHM20) update. In this study, the GEM Foundation and collaborators from different places use the OpenQuake Engine to generate quantitative performance metrics such as economic losses, fatalities, probable maximum losses, and risk maps that account for residential and non-residential buildings. The methodology used is cloud analysis, with several ground motion records and structures converted from MDOF to SDOF to use as nonlinear properties in the dynamics analysis, applying an extensive database of topologies developed in exposure models of Europe.

In America, ref. [6] developed a seismic vulnerability assessment of Montreal in Canada. Three attenuation functions were studied to consider the epistemic uncertainty in the hazard, and loss estimations were found to be very sensitive to the definition of the hazard function. Also, in the worst scenario, 5% percent of building stock could be damaged, and 500 people could be injured or die at a specific time of 2 p.m. In addition, ref. [7] developed a seismic risk assessment for Bogotá, Medellín, and Cali, all cities in Colombia. In this study, the seismic hazard, exposure models, and vulnerability functions are developed for each city. As a result, the present exceedance probability curves and average annual losses are shared openly with anybody and could be used to make decisions. Here, it is highlighted that the unreinforced infill masonry typology results in the worst economic losses because it is the most common building in the cities studied.

The studies mentioned above focus on the importance of seismic risk assessments worldwide, especially in countries such as Ecuador, which have a high seismic hazard due to the South American plate subducting the Nazca plate. A clear example of the devastating consequences is the 16 April 2016 earthquake in Pedernales, where 673 people died with USD 3 billion in economic losses [8]. With this premise, it is crucial to focus on the Metropolitan District of Quito (DMQ) as the capital of Ecuador. It is located on local faults prone to the VII intensity on the Medvedev scale associated with earthquake magnitudes of 6 to 6.5 [9]. Previous studies, such as [10], show the structural vulnerability of existing buildings in Ecuador. Indeed, ref. [10] highlights that 72% of the buildings analyzed register alarming levels, considering Ecuador’s constructive characteristics and typologies. In addition, ref. [11] evaluates the vulnerability of buildings in Quito, taking the city’s historic center as a sample. The evaluation considers 2606 buildings of different typologies, including the unreinforced masonry typology, corresponding to 63.28% of the buildings studied. The results show that the buildings will suffer significant damage. However, it mentions that the results can be improved for future work. Moreover, ref. [12] assessed the risk of DMQ with a reduction to 17 typologies, where the non-ductile cast-in-place reinforced concrete structures with infill masonry and flat slab (CR+CIP/LFLSINF+DNO) typology occupies the highest percentage of buildings and the one that concentrates the most significant losses in a seismic event. Here, the analysis exhibits that more than 80% of the buildings evaluated could suffer extensive damage and 45% complete damage for a 475-year return period earthquake.

These buildings generally report a lack of technical supervision during their construction. Therefore, unwanted structural responses could lead to flexural failure, shear failure, and failures corresponding to the infill masonry, which requires evaluations through numerical models.

As a reference, observations of the structural damage of the 2016 Pedernales earthquake showed flexural failures due to nonlinearities of the element in the areas where the plastic hinges are formed, shear failures due to the interaction between the frame and the infill masonry, and also failure modes related to diagonal tension shear slip in the mortar joint, crushing in the panel corners, and compression failure in the panel center. These failure modes have been extensively studied by [13,14,15,16], among others. For example, ref. [13] compiles experimental tests and presents a mathematical model that reflects the behavior of beam-column elements. The model includes the lack of adhesion due to insufficient overlaps or the failure due to a lack of transverse steel. This results in a hysteretic model with more marked puncture behavior than an element designed under current regulations. In addition, the results of the collected tests indicate that overlaps in areas of plastic hinges lead to unfavorable responses because they concentrate the damage at the ends and decrease the ability to dissipate energy.

This study aims to update the fragility and vulnerability functions considering studies after the current functions such as [17], which determines important nonlinear parameters to modeling the infill unreinforced masonry based on three real scale prototypes of an infill masonry frame with local materials, transverse sections, and constructive techniques typically used in Quito at first by [18]. Also, the local ground motions of the city are used to complement the database of the earthquakes to develop the cloud analysis using the VMTK software version 3 developed by [19] and used by the GEM Foundation. In addition, critical parameters, such as overlap length insufficiency, were analyzed for their influence on this typology response in this study. The final objective is to more accurately relate the seismic hazard to local variability construction methodologies, to determine representative loss estimates, and to characterize the CR+CIP/LFLSINF+DNO typology.

2. Materials and Methods

The city of Quito has two primary sources of seismicity: tectonic (cortical and subduction) and volcanic, with earthquakes of tectonic origin being those that reach greater magnitudes. Additionally, it is known that the city of Quito is built on a cortical fault system [20]. Therefore, it is necessary to define the seismic records with similar conditions to the site, so they are selected from the Pacific Earthquake Engineering Research database and the Geophysical Institute of the National Polytechnic School, Ecuador, database. The approach consists of determining the PGA to the different hazard levels as frequent (72 years), occasional (225 years), rare (475 years), and extreme (2500 years) based on the hazard curve described in the [21] in its chapter on seismic hazard. With these values, the NEC-15 spectrum is defined with the local characteristics of Quito. In addition, a PGA = 0.56 [g] is included to complement the set of earthquakes to cover a wide range of intensity measurements.

Once the spectra are defined, the previously mentioned database is used to select compatible records to scale them to the NEC-15 spectral form, as presented in Figure 1. The scaling procedure is carried out in the amplitude domain, considering that the average spectrum of all ground motions selected for each PGA has a minor fit error with the target spectrum [22].

It is worth mentioning that the average of the scaled signals reaches the target spectrum for the expected elastic period of the structure (T = 0.35 s). In total, 20 ground movements are considered with their two orthogonal components. This allows the creation of a database of 200 accelerograms that represent the seismicity of Quito with the wide range of intensities necessary to develop the fragility analysis. Once the database is formed, the intensity measures (IMs) are defined under the criteria of [23], where it is suggested to capture four IMs to take into account the dynamic properties of the large number of buildings that make up the studied structural typology. The intensity measurements are (i) PGA for rigid and low-rise buildings, (ii) spectral acceleration in 0.3 s (SA0.3 s) for low- and medium-rise buildings, (iii) spectral acceleration in 0.6 s (SA0.6 s) for medium-rise buildings, and finally, (iv) spectral acceleration in 1 s (SA1.0 s) for flexible and high-rise buildings.

Finally, 20 accelerograms are randomly chosen from the database for each intensity, summing up to 100. If there are not 20 records with sufficiently high intensities, the ground movements are scaled with a maximum factor of 2 to complete the records [23]. With these conditions, it is assumed that there is sufficient data to conduct an adequate statistical analysis and obtain satisfactory results in the structure’s response. The fragility and vulnerability curves are directly related to the seismic conditions of the analysis site. For this reason, accelerograms have been defined based on on-site conditions for Quito and local earthquakes; although they do not present large intensities, they have been used to determine low-intensity measurements.

The primary failure modes considered in the nonlinear model of the typology under study are diagonal tension shear slip in the mortar joint, crushing in the corner of the panel, and failure to compression in the center of the panel. Consequently, the model presented by [13] represents these failure modes through beam-column elements.

Plasticity is concentrated at the ends of the element to represent bending, and a decoupled spring captures shear. Nonlinear models were implemented in the OpenSees v3.2.2 software by [24].

The shear behavior in buildings with fill masonry is of utmost importance because it captures failure modes associated with the structure–masonry interaction. The influence of the masonry and its interaction with the surrounding frame is considered through equivalent strut models that represent the hysteretic behavior of this element, which can include the different failure modes and the influence of the openings, as in the studies developed by [13,14,25]. The hysteretic model used to represent the nonlinearity of the material is Pinching4 from OpenSees, together with the calibration parameters described in [16,17].

The openings in the infill walls represent a great uncertainty during the evaluation of the seismic behavior of a building due to their location within the panel, causing a concentration of damage in specific regions. For them, the [15] model was used to capture the openings’ influence on the panels’ behavior. It defines the $\rho$ factor as being responsible for reducing the resistance and rigidity due to the size of the holes present in the masonry frames. It is essential to recognize a limitation of the parametric model since how to capture the shear failure in the column due to the interaction with the masonry and openings was not resolved because these models are defined separately. It is not possible to know a priori if the opening will generate the “short column” effect without dividing the element. For this reason, it was decided to implement the equivalent diagonal and no other more complex models that capture the increase in the shear force transmitted from the panel to the columns.

To parameterize the typology and to model a wide variability in material geometry and mechanical properties, a three-story building is considered to start. A building survey is carried out to complete the technical information necessary to create the model. The details are presented in Figure 2, and it is assumed that the interior masonries do not have complete frames, so their contribution to the structure will be minor. It is known that there is a significant variability in the strength of concrete. Still, the typical value for the constructions trying to be achieved is an average compressive strength of 21 MPa. Likewise, the reinforcing steel available in the country has a minimum yield strength of 420 MPa and a minimum tensile strength of 550 MPa. The cross sections of both beams and columns typical of the typology are presented in Figure 2a. The figure also shows the typical materials used in masonry, with block being the most frequently used material according to [26]. The summary of the parameters used to model the nonlinear masonry is presented in

Table 1. Typical DMQ infill masonry parameters.

Variable	Description	Value	Units
tw	Thickness	150	mm
-	Strut	simple	--
fcb	Compressive strength of the block	0.97	MPa
ftb	Block tensile strength	0.22	MPa
d	Block length	400	mm
b	Block height	200	mm
fcj	Mortar compressive strength	20.19	MPa
j	Mortar joint height	10.11	mm
Uu	Friction coefficient	1.5	--
v	Poisson’s modulus of masonry	0.1524	--
fwu	Shear displacement resistance	0.1032	MPa
fws	Diagonal compressive strength	0.8162	MPa

; these values are defined by [17].

Variable values are included in the model of five parameters that directly influence the behavior of the structure. The geometric variation is considered by generating random values between 3 m and 6 m as distance between axes. This value is defined by considering the typical span in this typology. For the variation in the compressive strength of the concrete (f′c), the compressive strength of the joint mortar between masonry (fcj), and the compressive strength of the masonry block (fcb), variable values are generated that respond to a normal distribution considering a mean of 21 MPa, 20.19 MPa, and 0.97 MPa, respectively, and a standard deviation of 1.5, 3.65, and 0.16; these values are defined based on experimental tests of the DMQ materials by the authors of [17]. Finally, to consider the variability of masonry openings, random values for $\rho$ are generated between 0 and 1, where 0 means a vain opening and 1 means a wall without an opening. Although the openings respond to the architecture, it is not easy to adjust their variability in a model because the distribution of rooms and the location of doors or windows is unknown.

With this background, 275 nonlinear static “pushover” analyses are carried out on each mathematical model generated. Initially, 125 models were generated in five groups of 25, considering the variation of only one of its parameters and keeping the average value of the rest of the properties fixed to gain an idea of the incidence of each variable. Additionally, 150 models are made considering complete randomness, in which all the parameters vary without considering any fixed property. The code to generate the models is presented in the Supplementary Material section.

The analysis is only carried out in the “X” direction because it is the least resistant direction of the building and where the openings due to doors and windows in the fill masonry are generally located. The other direction is usually adjacent or attached to other buildings since the distribution of lots in the DMQ requires a minimum frontage of 10 m to divide the land. Therefore, the owner attaches the building to utilize the available space.

The methodology proposed by [23] is followed to develop the fragility and vulnerability curves, which requires transforming the pushover curves in a linearized manner to an Acceleration Displacement Response Spectrum (ADRS) system. First, it is required to transform from MDOF to SDOF using pushover curves previously obtained with Equations (1)–(3) of the 150 simulations, here $m_{eq}$ is the equivalent mass, ${\varnothing }_{1n}$ is the vibrations mode of the fundamental period, $m_n$ is the mass of each floor, $S_a$ is the spectral acceleration, $F$ is the base shear of the pushover curves, $S_d$ is the spectral displacement, ${\delta }_{roof}$ is the roof displacement in an MDOF system, and $\mathsf{\Gamma }$ is the participation factor of the first vibration mode.

$$m_{eq}={\displaystyle \frac{{\left({\sum }_{n=1}^N{\varnothing }_{1n}{m}_n\right)}^2}{{\sum }_{n=1}^N{\varnothing }_{1n}{m}_n}}$$

(1)

$$S_a={\displaystyle \frac{F}{m_{eq}}}$$

(2)

$$S_d={\displaystyle \frac{{\delta }_{roof}}{\mathsf{\Gamma }}}$$

(3)

Moreover, it is necessary to define the limits of linearization. Figure 3 presents a diagram of the transformation of an ADRS system to a quadrilinear model, in which you can see the four points used for the conversion. The first point is determined to be 75% of the maximum capacity of the structure, and this point corresponds to the creep of the building. Then, the second point is defined as its maximum capacity. The third point is the limit point of the first significant loss of capacity of the structure due to the failure of the infill masonry. Finally, point four is determined with a loss of 20% of the maximum capacity of the structure after the masonry has failed. The values corresponding to the spectral shift are taken from the ADRS curve for each established point.

Next, nonlinear response history analyses are carried out for the one-degree-of-freedom equivalent systems for the 150 linearized curves resulting from the simulations, considering the 100 ground movements chosen under the selection criterion previously discussed. The library used to perform the analyses is OpenSeesPy. Therefore, the Pinching4 material model, including degradation and an element of zero length, is used. The routines used are included in the work of [19] in their open-source Vulnerability Modelers Toolkit (VMTK) tool. Finally, in each simulation, the value of the maximum spectral acceleration is stored and defined as an IM, which will later be used to generate fragility and vulnerability curves. In the same way, the response of the maximum structure is stored in both accelerations and displacements.

The approach used to develop the fragility curves is defined by [27]. This methodology is known as Bayesian cloud analysis. It involves the dynamic nonlinear analysis of the typology subject to a database of earthquakes previously selected based on the characteristics of the Metropolitan District of Quito. Additionally, this method allows for the uncertainty of both ground movements and the structure’s capacity to be considered by explicitly modeling these two phenomena. Ref. [27] developed a comparison between both methods. The Incremental Dynamic Analysis usually represents a high computational cost since it needs to be repeated several times as a ground motion is scaled. In contrast, the Cloud Method is efficient since it develops a nonlinear analysis for an SDOF system using a set of unscaled ground motions. The great advantage of this analysis is the reduced computational time because, based on a simple linear regression, it allows us to determine a coefficient of determination that indicates the correlation of the curves obtained based on the periods of the structure.

The damage states represent limits of the engineering demand parameter (EDP) based on the level of damage that the structure will have. Ref. [28] is supported on the results of the experimental model of [29] and proposes four damage states of the infill masonry frame corresponding to cracking, yielding of the reinforcing bars, initiation of the compression failure of the concrete, and the buckling of rebars and failing of covering concrete, defining DS1 as 70% of the yield displacement, D2 as the yield displacement, DS3 as the yield displacement plus 25% of the difference in the ultimate displacement less the yield displacement, and DS4 as the ultimate displacement. Additionally, ref. [17] defines three damage states by observing the experimentation of the infilled masonry frames with characteristics of the city of Quito. The defined damage states are cracking appearance on the masonry panel, the system’s yielding, and the concrete frame’s cracking. However, the damage states initially defined by the GEM Foundation methodology are proposed by [30] with variation for the damage states DS2 and DS3 because they are distributed equidistantly [31]. Then, four levels of damage are defined: slight damage (DS1), moderate damage (DS2), extensive damage (DS3), and complete damage (DS4). The methodology discussed in [23] determines the damage limits, assuming DS1 occurs at 75% of the creep displacement. DS4, which represents complete damage, is defined as the ultimate displacement, and the intermediate damage levels DS2 and DS3 are obtained as equidistant between the two external limits. Figure 4 and

Table 2. Damage limit states.

Damage State		Threshold	Description
DS1	Slight	$0.7 \mathrm{S}\mathrm{d}\mathrm{y}$	Structural elements and contents remain functional. Small cracks appear in the walls.
DS2	Moderate	$0.75 \mathrm{S}\mathrm{d}\mathrm{y}+0.25 \mathrm{S}\mathrm{d}\mathrm{u}$	Damage occurs to non-structural elements, such as considerable cracks in the masonry. In addition, plastic hinges begin to appear on structural elements; however, the building is secure.
DS3	Extensive	$0.5 (\mathrm{S}\mathrm{d}\mathrm{y}+\mathrm{S}\mathrm{d}\mathrm{u})$	Severe damage occurs in structural and non-structural elements. Falling masonry could affect people’s safety.
DS4	Complete	$\mathrm{S}\mathrm{d}\mathrm{u}$	At this threshold, the structure collapses.

present the damage limit states.

In a probabilistic analysis, due to the large number of simulations carried out, responses of the structure can be obtained that generate a bias or poor fit of the linear regression. These points may be excessive deformations resulting from the nonlinear dynamic analysis concerning the ultimate deformations obtained from the capacity curves of the typology. Therefore, these points are treated following the work of [32], where a method of censored regression allows a better fit of the curve by treating these outlier points. For this reason, an upper limit (EDPc_upper) is established in the analysis, which delimits a maximum value and indicates that higher values represent the collapse or complete damage of the building. The recommendation of [23] is used to specify this limit, establishing a value of 1.5 times the EDP. Likewise, a lower limit (EDPc_low) of 10% of the creep displacement is considered; values lower than this limit are discarded. Once the censored points have been identified, the parameters are calculated using the maximum-likelihood method described in [33].

Vulnerability functions relate the IM caused by a ground motion with the percentage of economic losses given said intensity. The calculation of the vulnerability functions consists of the sum of the multiplication of the probability of occurrence of each damage state by the corresponding percentage of expected loss obtained from the consequence model. Several authors, such as [34,35,36], have defined consequence models; however, for this study, the consequence model defined in [23] is used.

3. Results

The results of the 150 models considering complete randomness described in Section 2, which corresponds to the capacity curves of the typology, are presented in Figure 5. A structure with non-reinforced infill masonry increases the capacity and initial rigidity of the structure until the point where the masonry fails; because of said failure, the first significant lateral load capacity loss is observed. Additionally, the wide range of capacity curves developed can be seen, highlighting that the peaks of the capacity curves fluctuate from a minimum value of 187.8 kN to a maximum value of 358 kN, and the average value and deviation of the maximum capacity are 271.23 kN and 31.49 kN, respectively. The displacements vary in a range of 0.33 m to 0.45 m, with a mean of 0.42 m and a standard deviation of 0.038 m, mainly capturing the variability of the typology.

Additionally, Figure 6 presents the different capacity curves obtained to appreciate the influence of span, concrete strength, mortar strength, masonry strength, and openings on the overall behavior of the building. Figure 6a presents the parameter corresponding to the distance variation between spans. The maximum capacity varies from 330 kN to 375 kN. That is, there is a variation of 14% at the peak of the curve. On the other hand, several curves did not reach the convergence of the nonlinear analysis, and others had a premature drop in capacity; this happens as the distance between spans increases because the greater the distance, the greater the stress on the structural elements that do not have adequate detailing to resist such efforts. Figure 6b shows the influence of the compressive strength of the concrete on the capacity curve. In it, we can see no more significant variation in the peak of the curves because the filling masonry determines it; the maximum points fluctuate between 350 kN and 362 kN, representing a 3.4% increase. However, in the second peak of the capacity curves, with low concrete resistance, the filling masonry begins to work until having a second significant loss of capacity. Concerning displacements, the variation is 10% (0.34 m to 0.38 m).

Similarly, Figure 6c shows the variation in the compressive strength of the mortar of the filling masonry joint; therefore unlike the compressive strength of concrete, there is a more significant variation in the maximum peaks, with values ranging from 300 kN to 360 kN, representing 20% variability. This result is expected because this parameter directly affects the resistance of the masonry. The ultimate displacement presents a 15% variability (0.39 m to 0.45 m). Figure 6d presents the parameter variation corresponding to the block’s compressive strength. This parameter has a more significant influence than those previously shown, capturing an increase of 38%. The values vary from 260 kN to 360 kN at the maximum points of the curves. This parameter, like the joint mortar, dramatically influences the capacity curve because it makes up the masonry, with the block or masonry used being the one that represents the most significant variability. The displacements remain at values similar to those previously obtained for the joint mortar parameter, which is 18% (0.38 m to 0.55 m).

Figure 6e presents the variability due to the $\rho$ factor, representing the loss of strength and stiffness due to openings in the masonry. This parameter captures a large variability in the capacity curves. The values fluctuate from 220 kN to 316 kN, covering an increase of 44% at the maximum peak of the curve. For displacements, values range from 0.23 m as a minimum value to 0.45 m as a maximum value (95% increase). Considering this parameter, capacity curves are captured that fall prematurely due to the influence of the openings and the pathologies that can affect the overall behavior of the building resulting from jointly modeling the masonry and the frames. In Figure 6f, capacity curves are presented considering the previously mentioned parameters, all of them random, which serves as a starting point to capture the variability of the 150 curves presented in Figure 5.

Figure 6 shows that each parameter contributes variability to the capacity curves, with the factor $\rho$ being the one that has the most significant influence on the response of the typology. Likewise, it can be noticed that the simulations are representative samples for the evaluation of the CR+CIP/LFLSINF+DNO typology and explicitly capture the uncertainty of the variation of the materials, the geometry, and the pathologies that we can find in the buildings corresponding to the typology studied in the DMQ.

Subsequently, it is necessary to transform the curves to linearize them to an ADRS system. It is important to emphasize that not all the proposed permutations reach numerical convergence in their randomly defined mathematical models; therefore, these simulations are discarded at this point in the methodology because they can generate bias in the results. Figure 7 presents the linearization of the capacity curves in a quadrilinear model. It shows the broad spectrum of capacity curves that cover the most significant number of buildings corresponding to the typology. These linearized curves allow us to capture the behavior of the pushover curves in a simplified manner while maintaining their characteristic values.

The best regression fit of the vulnerability and fragility functions is for a T = 1 s period, as shown in Figure 8. This result is comparable with [31], where the same result is reported for the risk assessment developed for the infill unreinforced masonry typology, and it is consistent with the inelastic period of the structure once the infill masonry fails. Besides, the fragility and vulnerability functions for the typology, considering determined vibration periods, are presented in Figure 9 and Figure 10, respectively. In addition,

Table 3. Result of fragility and vulnerability functions.

T [s]	Sa[g] NEC-15 Soil Type D	Probability of Exceedance Damage States [%]				Loss Ratio [%]
		DS1	DS2	DS3	DS4
PGA	0.54 g	100	26	2	0	10
0.3	1.19 g	100	98	74	40	68
0.5	1.19 g	100	98	74	40	68
0.8	1.04 g	100	94	56	22	50
1	0.83 g	100	84	28	8	30
1.5	0.55 g	100	26	2	0	10

presents a quantitative summary of the results.

To determine the probabilities of exceedance in Table 3, the fragility and vulnerability curves are considered for the period of the best adjustment, so the curves in Figure 9c are used. For example, a T = 0.5 s results in a spectral acceleration of 1.19 g for the 475-year return period according to the seismic hazard map of the Ecuadorian Construction Standard. Therefore, a 100% probability of exceedance for the damage state DS1 is established, meaning that the building will exhibit slight damage. Also, there is a 98% probability of exceedance for the damage state DS2, indicating that the building will suffer moderate damage. In addition, a 74% probability of exceedance for the damage state DS3 is reported, implying extensive damage in the building. Finally, there is a 40% probability of exceedance damage for the state DS4, representing complete damage. Moreover, Table 3 shows the loss ratio, namely the percentage of economic losses, concerning the initial cost of the building. For instance, the economic loss ratio for a spectral acceleration of 1.19 g is 68% of the original building cost. Finally, in this study, all the curves that can be used in a risk analysis are presented so that they can be chosen according to the needs of the study.

Considering all the building permutations, these curves represent the local characteristics of materials, construction methods, and the threat of DMQ. The fragility functions mean value and standard deviation are presented in an integrated manner in the graph. The authors of [23] validate the fragility and vulnerability analysis results. They present dispersion values reviewed in the literature by [37,38,39]; for regular buildings, the value for σ varies between the limits of 0.30 as a minimum and 0.80 as a maximum. Furthermore, ref. [23] presents a minimum σ value of 0.39 and a maximum of 0.61 in its curves. Considering this criterion, the obtained value for σ is 0.41, within the range of previous studies. Additionally, the seismic hazard curve in [40] found that DMQ could reach a PGA of 0.45 g. Considering the seismic hazard curve previously mentioned and the vulnerability curve for a period of T = 1 s as the curve to use because it has the best regression fit, the Annual Average Loss Ratio (AALR) = 0.16% is obtained for Quito. It is compared with the AALR = 0.20% obtained in [23] for the city of Oakland, having a PGA of 0.5 g, slightly higher than the DMQ threat. Therefore, the value is considered acceptable.

In addition, ref. [7] conducts a seismic risk study considering several typologies in three of the main cities of Colombia, Bogotá, Medellín, and Cali, using the GEM Foundation methodology. The study concludes that the CR+CIP/LFLSINF+DNO typology represents 35.3%, 71.5%, and 24.3% of the buildings studied for each of the cities, respectively, with Medellín being the city with a similar percentage to Quito. In addition, it states that the contribution of this typology in the calculation of the AALR for the mentioned cities is high, validating the high vulnerability of this type of building. The percentage of economic losses calculated for Bogotá ranges from 11% to 21%. In comparison, for Medellín, the losses are 14%, taking a PGA of 0.15 g for a 475-year return period for the two cities. The results of economic losses obtained for the typology in the present study are 30%, considering that the threat for the same return period is 0.45 g. These results can be compared considering the seismic variability at each site. However, it is essential to note that the studies conducted in Colombia include other structural typologies. Although infilled masonry frames are representative, different typologies could introduce variations in the results.

4. Conclusions

One hundred fifty models of the CR+CIP/LFLSINF+DNO typology have been presented in this study, varying parameters such as span, concrete strength, mortar strength, masonry strength, and openings in the infill masonry to cover an extensive range of buildings belonging to the analyzed typology. In addition, an equivalent strut model is used to represent the infill masonry. This simplification is made due to the time it takes to perform a large amount of non-linear analysis to form a robust database for cloud analysis. The simplification of an equivalent strut represents the structure’s global behavior and the main failure modes that occur in it. Still, the short column effects that typically form in the infill panel and frame interaction are neglected. This interaction could affect the behavior of the structure, causing brittle failures and premature collapses. This could cause the database of the vulnerability and fragility curves to change depending on the pathology that causes the failure of the building. Moreover, the models considering local material, transversal section geometry, and constructive techniques commonly used in the DMQ tend to reduce the uncertainty in the mathematical nonlinear model. Additionally, a database of two hundred ground motion records represents the seismic hazard in a Bayesian cloud analysis. Each model is converted from MDOF to SDOF, and a dynamic time history analysis is carried out for every record, considering the pushover curve as the constitutive of the SDOF system to obtain fragility and vulnerability functions. Representative findings of this study are presented below.

The fragility and vulnerability curves obtained by the Bayesian cloud analysis method are suitable due to the large number of simulations based on a robust database and the concentrated plasticity models. This allows for capturing the effects of collapse at a lower computational cost. Therefore, the proposed functions would improve the seismic risk assessment of DMQ.

The best fit of the linear regression for the typology studied corresponds to the T = 1 s period, which is congruent with the building’s inelastic period because the structure loses stiffness. The period is amplified when the structure is subjected to high-intensity demands. With this premise, the fragility and vulnerability curves that must be used correspond to the period of 1 s.

In the DMQ, an acceleration of 0.83 g is expected for a period of T = 1 s, which results in a 100% probability of exceeding the DS1 damage state, involving minor damage such as small cracks in the masonry. The DS2 damage state reports an 84% probability of exceedance, which indicates considerable damage to non-structural elements, such as large masonry cracks, coating detachment, and formation of plastic joints in the structural elements. There is a 28% probability of exceeding the DS3 damage state; the typology will report extensive damage to structural, non-structural, and content elements, affecting the inhabitants’ lives. There is an 8% probability of exceeding the DS4 damage state, which means total collapse. Finally, a 30% loss ratio will affect the typology. In other words, economic losses will amount to 30% of the total cost of each structure analyzed for that period and spectral acceleration.

The results obtained from the fragility curves with their corresponding exceedance probabilities for each damage state and the percentages of losses obtained from the vulnerability curves for the expected seismic threat in Quito provide valuable data to corroborate the high seismic risk and, at the same time, the economic consequences that this typology could cause. This study focuses on the typology covering 80% of the buildings in the DMQ; therefore, the associated economic losses will significantly impact the economy of the city and the country. Given the above, this research provides relevant data for governors since they can establish mitigation strategies to reduce these high percentages of consequences. For example, the publicization of these results alerts people of the consequences of a seismic event, in this way, fomenting work with the municipalities to generate massive reinforcement plans specific to this typology.

Future studies should focus on developing consequence models because these curves are made independently according to a database of construction costs for each site. Specific consequence models should be developed considering the typical characteristics of this typology. It should be considered that this type of building generally uses relatively inexpensive finishes and is often not finished enough to be inhabited, so an analysis of unit prices should be generated, and a database should be created considering the variability that this typology may have. With this database, a statistical analysis should be performed to determine an adequate consequence model that represents the typology and is related to each damage state. Finally, the vulnerability curves should be validated so that the economic losses presented should be verified.

One of the study’s limitations is using a one-equivalent strut to represent the infill masonry, so it is proposed as a future study to complement the analysis with a three-equivalent strut model because it allows us to capture the behavior of the short column. After all, this pathology usually occurs during the interaction of the infill masonry, the frame, and the openings. The one-equivalent strut model represents the masonry panel node to node to the frame. Hence, the equivalent strut model allows modeling a node-to-node strut and two additional struts for which we must create two extra nodes, one in the beam and one in the column, generating additional stresses at those points. Therefore, we can verify the amplification of forces that can occur due to this pathology and validate the results presented in the research.