Estimation of Uniform Risk Spectra Suitable for the Seismic Design of Structures

: The aim of this paper is to present a performance-based method to estimate uniform risk spectra (URS) for the seismic design and assessment of structures. These spectra, computed with the proposed methodology, provide the lateral capacity (In terms of spectral acceleration) that should be given to a structure, characterized by a reference single degree of freedom system, to achieve a predetermined exceedance rate of economic loss. This procedure involves the seismic hazard assessment necessary to deﬁne a seismic design level consistent with the accepted loss value, using a large enough number of synthetic seismic records of several magnitudes, which were obtained by means of an improved empirical Green function method. The statistics of the expected losses of a reference single degree of freedom system are obtained using Monte Carlo simulation, considering the seismic demand and the lateral strength of the structure as random variables. The method is divided into two main stages: (1) deﬁnition of the seismic hazard at the site of interest and (2) the probabilistic analysis of the seismic performance in terms of an economical loss ratio of nonlinear SDOF. To illustrate the proposed methodology and, subsequently, to validate it, a URS was computed for a site located in the Mexico City lake-bed zone, and its use in the design of three reinforced concrete frames is shown. The results show that the proposed spectra provide a sufﬁcient approximation between the seismic risk level considered in the seismic design and that of the designed structure. It is concluded that the proposed procedure is a signiﬁcant improvement over others considered in the literature and a useful research tool for the further development of risk-based earthquake engineering.


Introduction
A natural risk has several definitions, and from an economic point of view, it may be defined as the potential economic consequences derived from a natural phenomenon in a region, city, or country.Currently, seismic risk assessments have acquired importance because of the large losses presented due to recent earthquakes.The World Economic Forum [1] shows that even though the seismic codes are less permissive, earthquake-related losses keep increasing.This may be explained because, nowadays, there are more exposed assets (several new buildings are built) and most of the seismic building codes are oriented toward achieving only two main performance goals during the design process: (1) avoiding structural damage under frequent earthquakes and (2) avoiding building collapse under rare earthquakes, without considering the structural behavior and consequences of seismic events with intermediate intensities.This increases the uncertainty of damage occurrence and, consequently, the risk of uncontrolled losses, causing economic plans or emergency actions to lack a solid basis upon which to make decisions before and after a seismic event.Fortunately, there are very detailed methodologies [2][3][4][5] to assess structural performance, which employ different seismic response parameters, as well as to infer the corresponding time and reparation costs, such as that presented in FEMA P-58 [2].However, these techniques are usually carried out for academic research activities or for critical structures; on the other hand, common structures are usually designed employing traditional approaches, which are easily handled by civil engineers in professional practice.
Currently, most building codes around the world use response spectral analysis (RSA) [6][7][8][9] to estimate the design seismic demands in low-and mid-rise structures; therefore, to upgrade the current basic seismic design methodologies, an improvement of the provided design spectra are necessary.Unfortunately, the current design methods that use uniform hazard design spectra, as well as elastic approximations, may not allow for the design of structures that will exhibit the expected structural performance under highintensity seismic demands in which inelastic behavior may occur.This is evidenced by the damage patterns observed from recent earthquakes (Mexico 1985 [10,11], Loma Prieta 1989 [12], Northridge 1994 [13], Kobe 1995 [14], Ecuador 2016 [15], and Mexico 2017 [16]), where the expected safety levels provided by FEMA exhibited large uncertainties.
This paper proposes a new method to estimate the seismic demands to be employed in a risk-based design approach, the results of which may be easily implemented by civil engineers in practice, since RSA is still the predominant analysis method in many building codes.This approach would not require a subsequent seismic assessment once a structure is designed, as is required in other risk-based design methods.Such seismic demands are presented through uniform risk spectra (URS), which, through an RSA, define the lateral strength required by a structure so that it does not exceed an annual expected loss during each predefined time frame.The use of these type of spectra would allow the designer or building owner to decide on the acceptable risk level from the beginning of the design process.This could lead to, among other aspects, better financial protection plans for each designed building.The proposed URS represent inelastic spectra without reduction by overstrength for a given specific structural type.
The accuracy of the results obtained using these spectra was validated by computing the exceedance rate of the loss of three moment-resisting frames (MRFs), whose design demands were obtained using a URS.For this purpose, the methodology proposed by the Pacific Earthquake Engineering Institute Center (PEER) was used [4].
Different researchers have made efforts to improve the prediction of expected damage states, either from the design process through methodologies focused on reaching a given structural performance [17][18][19][20][21] or through seismic assessment methodologies, which are usually performed once a structure has already been designed or built [3][4][5]22,23].However, most seismic design codes have not included these methodologies or, in the best case, they have been adopted only for a small part of the building catalogue, such as tall towers or essential buildings.
One way to guarantee an accepted damage pattern in low-rise and mid-rise structures designed using RSA is by considering the key structural characteristics in the development of the seismic design spectra, where these spectra provide the necessary lateral strength (In terms of spectral pseudo-acceleration, Sa y ) so that a structure reaches a specific performance target.These spectra are obtained as a function of the dynamic characteristics of soil and the inelastic dynamic response of several single degree of freedom (SDOF) systems.One example are uniform ductility spectra, which are obtained by using reduction factors over elastic response or design spectra [24][25][26][27][28][29][30][31][32]; these factors are generally obtained from the statistical analysis and mathematical regressions of the response of instrumented buildings.These factors are easily implemented in building codes but, unfortunately, since only important buildings have been instrumented, only a few buildings are available to provide information.
Other types of spectra are oriented to achieve uniform rates of exceedance of engineering design parameters [33][34][35][36][37][38][39], such as damage indexes, damage states, or fragility approaches.These spectra are obtained from a statistical analysis of the SDOF response under seismic actions derived from seismic records, and they predict, with a good ap-proximation level, the seismic performance in terms of physical damage.Recently, several studies concerning risk-targeted seismic design demands have been developed [34].For instance, Kennedy and Short [40] and Cornell [41] developed a practical way to estimate the seismic risk of a structural system that, in addition, is also suitable for structural design, corresponding to a target reliability level where the limit states are defined in terms of the global ductility of the system.
Another important contribution is the approach proposed by Luco [42], where the main aim is to obtain the seismic hazard associated to a uniform collapse probability for structures located across the US.The approach using the means of inelastic GMPEs which, for instance, provides the distribution of an intensity measure associated with the capacity demand of SDOF systems, considers parameters as the yield strength [43][44][45].On the other hand, several approaches in which the main aim is to obtain the seismic demands, in terms of the spectral acceleration associated to a defined limit state, are estimated using parametric models, which depend on several factors associated to the main characteristics of a structure [46].However, they may not be suitable for the estimation of economic losses (i.e., risk), since they do not consider the reparation cost of the estimated damage.

Methodology
The proposed methodology is based on the total probability theorem, which, in this case, is used to quantify the annual exceedance rate of a pre-established loss value by considering both the seismic hazard and building vulnerability.Equation (1) shows the proposed model.
where v β M, R, T, α, Sa y is the annual exceedance rate of loss; dM is the seismic hazard represented as the occurrence probability of an earthquake with a defined magnitude from the i-th to the N-th seismic sources; and Pr B ≥ β M, R, T, α, Sa y is the structural vulnerability represented by the exceedance probability of loss given the magnitude, M, epicentral distance, R, vibration period, T, post-yielding stiffness, α, and lateral strength in terms of the yield spectral acceleration, Sa y .Moi and Mui are the minimum and maximum magnitudes of the seismic source.
To solve the proposed equation, the following steps are provided:

Stage 1
Definition of the seismic hazard accounting for the exceedance rate of the magnitudes of the sources considered and a set of seismic records (real or synthetics) representative of the seismic hazard of the site.

Stage 2
Computation of the nonlinear response in terms of loss for an SDOF whose vibration period and lateral strength values are predefined.The nonlinear response must be estimated for each seismic source and by using the corresponding seismic records obtained in the previous step.Loss associated to the nonlinear response is estimated by means of relating the expected damage patterns associated to a performance parameter (damage index) with the cost of their corresponding reparation actions.
For each seismic source, the exceedance probability of loss for a given magnitude, epicentral distance, vibration period, and lateral strength is obtained by analyzing statistically the results of the previous step.
The total exceedance rate of loss associated to the predefined dynamic and mechanical characteristics (T and Sa y ) of the SDOF is calculated using Equation (1).
If the previous steps are repeated for all magnitudes and considered values of lateral strengths, it is possible to obtain a 3D geometric surface that describes the relationship among the loss level, exceedance rate of loss, and lateral strength for a defined vibration period (Figure 1).The total exceedance rate of loss associated to the predefined dynamic and mechanical characteristics ( and  ) of the SDOF is calculated using Equation (1).
If the previous steps are repeated for all magnitudes and considered values of lateral strengths, it is possible to obtain a 3D geometric surface that describes the relationship among the loss level, exceedance rate of loss, and lateral strength for a defined vibration period (Figure 1).By computing several 3D surfaces (Figure 1) associated to a range of vibration period values and, subsequently, by selecting the lateral strength corresponding to the same loss value and loss exceedance rate in all of the computed 3D surfaces, the URS will be obtained.
The aforementioned process is graphically represented as follows (Figure 2).By computing several 3D surfaces (Figure 1) associated to a range of vibration period values and, subsequently, by selecting the lateral strength corresponding to the same loss value and loss exceedance rate in all of the computed 3D surfaces, the URS will be obtained.
The aforementioned process is graphically represented as follows (Figure 2).The total exceedance rate of loss associated to the predefined dynamic and mechanical characteristics ( and  ) of the SDOF is calculated using Equation (1).
If the previous steps are repeated for all magnitudes and considered values of lateral strengths, it is possible to obtain a 3D geometric surface that describes the relationship among the loss level, exceedance rate of loss, and lateral strength for a defined vibration period (Figure 1).By computing several 3D surfaces (Figure 1) associated to a range of vibration period values and, subsequently, by selecting the lateral strength corresponding to the same loss value and loss exceedance rate in all of the computed 3D surfaces, the URS will be obtained.
The aforementioned process is graphically represented as follows (Figure 2).

Application Example
For illustrating the implementation of the proposed methodology and its results, an example application is presented.The example consisted of the development of a URS in terms of the normalized repair cost and, subsequently, the seismic assessment of three structures designed with such spectrum.
The selected site for the development of the URS was the SCT seismic station, one of the most iconic places in the seismic history of Mexico City.This site represents the wave amplification of strong ground motions, which is a unique characteristic of the soil of the city [47][48][49][50][51][52][53][54][55][56].
For the assessment of the results provided by the proposed spectra, three structures of one, fifteen, and twenty-five stories, whose lateral strengths were obtained from URS, were assessed in terms of their seismic risk [3][4][5]57].It is expected that the risk level provided by the URS will be close enough to the risk calculated for such structures.

Defining the Seismic Hazard
In this example, for the sake of simplicity, only earthquakes generated from one seismic source were considered.Such seismic source is the Guerrero Gap, one of the most studied seismic zones by seismologist and civil engineers [58,59].The earthquake generation model for the subduction zone of Guerrero is described using a characteristic earthquake model [60], whose parameters are T00 = 80 years, Mo = 7.0, Mu = 8.4,D = 7.5, F = 0, σM = 0.027, and T o = 39.7 years [39,60].Figure 3 shows the estimated exceedance rate of magnitudes curve for the Guerrero Gap.

Application Example
For illustrating the implementation of the proposed methodology and its results, an example application is presented.The example consisted of the development of a URS in terms of the normalized repair cost and, subsequently, the seismic assessment of three structures designed with such spectrum.
The selected site for the development of the URS was the SCT seismic station, one of the most iconic places in the seismic history of Mexico City.This site represents the wave amplification of strong ground motions, which is a unique characteristic of the soil of the city [47][48][49][50][51][52][53][54][55][56].
For the assessment of the results provided by the proposed spectra, three structures of one, fifteen, and twenty-five stories, whose lateral strengths were obtained from URS, were assessed in terms of their seismic risk [3][4][5]57].It is expected that the risk level provided by the URS will be close enough to the risk calculated for such structures.

Defining the Seismic Hazard
In this example, for the sake of simplicity, only earthquakes generated from one seismic source were considered.Such seismic source is the Guerrero Gap, one of the most studied seismic zones by seismologist and civil engineers [58,59].The earthquake generation model for the subduction zone of Guerrero is described using a characteristic earthquake model [60], whose parameters are T00 = 80 years, Mo = 7.0, Mu = 8.4,D = 7.5, F = 0, σM = 0.027, and To = 39.7 years [39,60].Figure 3 shows the estimated exceedance rate of magnitudes curve for the Guerrero Gap.To characterize the seismic hazard at the site, synthetic seismic ground motions were used because there are not enough real earthquake records for the selected magnitude threshold at the selected site; however, if it is desired to use real ground motions, they must be able to represent the seismic hazard of the analyzed seismic source.The synthetic records employed were computed by Niño et al. [59], who used an improved empirical Green function approach in which a source spectrum is defined by two corner frequencies and a summation scheme divided in two stages.In this approach, the epicentral distance, R, and the site effects are considered implicitly by means of the selected earthquake seed; the EW component of the 25 April 1989 earthquake (M 6.9) recorded at the SCT station To characterize the seismic hazard at the site, synthetic seismic ground motions were used because there are not enough real earthquake records for the selected magnitude threshold at the selected site; however, if it is desired to use real ground motions, they must be able to represent the seismic hazard of the analyzed seismic source.The synthetic records employed were computed by Niño et al. [59], who used an improved empirical Green function approach in which a source spectrum is defined by two corner frequencies and a summation scheme divided in two stages.In this approach, the epicentral distance, R, and the site effects are considered implicitly by means of the selected earthquake seed; the EW component of the 25 April 1989 earthquake (M 6.9) recorded at the SCT station was used.A total of 550 soft-soil seismic record, fifty records for each magnitude considered (M 7.2 to M 8.2), were employed to define the seismic hazard at the site.These 550 seismic records were extracted as a representative sample of the seismic hazard of the city [61] from the eleven thousand seismic records obtained by Niño et al. [59].

Computing the Nonlinear Response in Terms of Loss
The nonlinear seismic responses were estimated in terms of the Terán and Jirsa (DI TJ ) damage index [62] and, subsequently, classified by magnitude.The computed nonlinear response of several SDOF systems, with vibration periods between 0.01 and 5 s, which cover the typical period range of response spectra, for the record set considered was used to define the damage spectra considering pre-established combinations of stiffness and strength (T and Sa y ) for a damping ratio of 0.05 and a post-yielding rate of α = 0.05 [63].
The hysteretic model used was the Takeda model [64], which is representative of reinforced concrete (RC) structures.It should be noted that URS may be built for any hysteretic model.To define the post-yield stiffness ratio, it was necessary to consider the residual displacements [65-67], since high values of the post-yield stiffness ratio led to small residual displacements [24,[68][69][70][71][72].
Figure 4 presents the estimated nonlinear responses as damage spectra using a postyielding rate of α = 0.05 [63], where, for each spectrum, all SDOFs have the same lateral strength.It is observed that the damage indexes have a large variability, mainly because of each seismic record's characteristics, such as intensities, frequency, and duration; this variability is considered in the computation of the probability of loss.
Buildings 2023, 13, x FOR PEER REVIEW 6 of 20 was used.A total of 550 soft-soil seismic record, fifty records for each magnitude considered (M 7.2 to M 8.2), were employed to define the seismic hazard at the site.These 550 seismic records were extracted as a representative sample of the seismic hazard of the city [61] from the eleven thousand seismic records obtained by Niño et al. [59].

Computing the Nonlinear Response in Terms of Loss
The nonlinear seismic responses were estimated in terms of the Terán and Jirsa (DITJ) damage index [62] and, subsequently, classified by magnitude.The computed nonlinear response of several SDOF systems, with vibration periods between 0.01 and 5 s, which cover the typical period range of response spectra, for the record set considered was used to define the damage spectra considering pre-established combinations of stiffness and strength (T and  ) for a damping ratio of 0.05 and a post-yielding rate of α = 0.05 [63].
The hysteretic model used was the Takeda model [64], which is representative of reinforced concrete (RC) structures.It should be noted that URS may be built for any hysteretic model.To define the post-yield stiffness ratio, it was necessary to consider the residual displacements [65][66][67], since high values of the post-yield stiffness ratio led to small residual displacements [24,[68][69][70][71][72].
Figure 4 presents the estimated nonlinear responses as damage spectra using a postyielding rate of α = 0.05 [63], where, for each spectrum, all SDOFs have the same lateral strength.It is observed that the damage indexes have a large variability, mainly because of each seismic record's characteristics, such as intensities, frequency, and duration; this variability is considered in the computation of the probability of loss.As it was mentioned previously, in this paper, the seismic risk was considered under economic terms so that the physical damage must be mapped to an economical value (repair cost ratio was considered).Specifically, some authors [73][74][75][76] have developed relationships between a damage index and the corresponding visible damage in the buildings; this allows for the definition of the necessary repair actions associated to each damage index limits and, subsequently, to estimate the economic cost.Table 1 shows the relationship proposed by Chacón and Paz [77] between the Park and Ang damage index (DIPA) [73] with the associated repair action and its repair cost ratio.As it was mentioned previously, in this paper, the seismic risk was considered under economic terms so that the physical damage must be mapped to an economical value (repair cost ratio was considered).Specifically, some authors [73][74][75][76] have developed relationships between a damage index and the corresponding visible damage in the buildings; this allows for the definition of the necessary repair actions associated to each damage index limits and, subsequently, to estimate the economic cost.Table 1 shows the relationship proposed by Chacón and Paz [77] between the Park and Ang damage index (DI PA ) [73] with the associated repair action and its repair cost ratio.To use the above relationship, the Park and Ang damage index values were mapped to the Teran and Jirsa [62] damage index by comparing the response of 50 inelastic SDOFs; the results are shown in Figure 5.By relating the data provided in Table 2, it is possible to fit a curve of the physical damage (DITJ)-structural loss ratio (β) for conventional RC structures.Figure 6 shows a fitted curve obtained through nonlinear regressions (Equation ( 2)); this curve was obtained by relating the midpoint of each DI interval and the corresponding loss ratio.In the same figure, it is possible to see that the physical characterization of damage does not have a linear relationship with the characterization of the economic loss.Reparation costs exhibit high variability; however, uncertainty between the repair cost and damage index was not considered in this study.
where β is the loss ratio, and DI is the selected damage index.By relating the data provided in Table 2, it is possible to fit a curve of the physical damage (DI TJ )-structural loss ratio (β) for conventional RC structures.Figure 6 shows a fitted curve obtained through nonlinear regressions (Equation (2)); this curve was obtained by relating the midpoint of each DI interval and the corresponding loss ratio.In the same figure, it is possible to see that the physical characterization of damage does not have a linear relationship with the characterization of the economic loss.Reparation costs exhibit high variability; however, uncertainty between the repair cost and damage index was not considered in this study.
where β is the loss ratio, and DI is the selected damage index.By using Equation ( 2), the nonlinear responses given by the damage spectra 4) were mapped to economic losses.

Estimating the Exceedance Probability of Loss
For estimating the exceedance probability of loss, the nonlinear responses in loss of the SDOFs were catalogued according to the M and R of the seismic recor sequently, a statistical analysis was performed with the purpose of fitting a pro distribution function (PDF).Kolmogorov-Smirnov tests were performed to as goodness of fit of several probability distributions.Accordingly, beta PDF was the most adequate.Furthermore, this PDF is delimited between zero and one, w consistent with the loss ratio values.Figure 7 shows three examples of the perform mogorov-Smirnov test, where it is possible to see a good fit of the beta PDF (i.e., ex (dashed line) to the statistical data (i.e., observed) (bars graphic) for a 5% sign level.
Thus, the loss PDF is defined as follows: where BETA is the beta function, and a and b are shape parameters of the PDF.S rameters are computed as follows: where c[] is the variation coefficient of loss, and E[] is the expected value of los were obtained from the aforementioned statistics analysis of the computed resp terms of loss for each family of inelastic SDOFs.By using Equation ( 2), the nonlinear responses given by the damage spectra (Figure 4) were mapped to economic losses.

Estimating the Exceedance Probability of Loss
For estimating the exceedance probability of loss, the nonlinear responses in terms of loss of the SDOFs were catalogued according to the M and R of the seismic records; subsequently, a statistical analysis was performed with the purpose of fitting a probability distribution function (PDF).Kolmogorov-Smirnov tests were performed to assess the goodness of fit of several probability distributions.Accordingly, beta PDF was deemed the most adequate.Furthermore, this PDF is delimited between zero and one, which is consistent with the loss ratio values.Figure 7 shows three examples of the performed Kolmogorov-Smirnov test, where it is possible to see a good fit of the beta PDF (i.e., expected) (dashed line) to the statistical data (i.e., observed) (bars graphic) for a 5% significance level.
Thus, the loss PDF is defined as follows: where BETA is the beta function, and a and b are shape parameters of the PDF.Such parameters are computed as follows: where c[β] is the variation coefficient of loss, and E[β] is the expected value of loss, which were obtained from the aforementioned statistics analysis of the computed response in terms of loss for each family of inelastic SDOFs.

Computing the Exceedance Rate of Loss
Upon defining the exceedance rate of the magnitudes of the analyzed seismic source and the exceedance probability of loss, it is possible to solve Equation (1). Figure 8 shows an example of an exceedance rate of loss computed with Equation (1) for an SDOF with T = 2.2 s and Sa y = 300 gals.

Computing the Exceedance Rate of Loss
Upon defining the exceedance rate of the magnitudes of the analyzed seismic source and the exceedance probability of loss, it is possible to solve Equation (1). Figure 8 shows an example of an exceedance rate of loss computed with Equation (1) for an SDOF with T = 2.2 s and  = 300 gals.

Computing the Exceedance Rate of Loss
Upon defining the exceedance rate of the magnitudes of the analyzed seismic source and the exceedance probability of loss, it is possible to solve Equation (1). Figure 8 shows an example of an exceedance rate of loss computed with Equation (1) for an SDOF with T = 2.2 s and  = 300 gals.By repeating the previous steps for all of the considered lateral strengths, Sa y , but keeping constant the vibration period, T, it is possible to obtain a 3D surface that represents the relationship among loss, lateral strength, and exceedance rate of loss.Figure 9 shows an example of the 3D surface β-ν(β)-Sa y for a family of SDOF with T = 2.2 s.By repeating the previous steps for all of the considered lateral strengths,  , but keeping constant the vibration period, T, it is possible to obtain a 3D surface that represents the relationship among loss, lateral strength, and exceedance rate of loss.Figure 9 shows an example of the 3D surface β-ν(β)- for a family of SDOF with T = 2.2s To build a uniform risk spectrum, it is necessary to compute a 3D surface (β-ν(β)- ) for each considered vibration period.Subsequently, couple of loss and an exceedance rate of loss values must be selected.According to the selected loss and exceedance rate of loss values, the corresponding lateral strength in each 3D surface (i.e., each vibration period) must be obtained.The URS is the spectra built with all of the lateral strengths of the considered vibration periods associated to the same loss and exceedance rate of loss.Figure 10 shows a comparison between the uniform hazard spectra (UHS) and URS for the analyzed site in terms of the return period (RP), i.e., the inverse of the exceedance rate.It can be readily seen that differences in the spectral ordinates for different values of loss and return period exist.For instance, for a constant value of β, the spectral ordinates were higher as the RP increased; on the other hand, if the RP was constant and β increased, the seismic demands decreased, i.e.,  the accepted risk was higher, and the required strength of the structure was lower.To build a uniform risk spectrum, it is necessary to compute a 3D surface (β-ν(β)-Sa y ) for each considered vibration period.Subsequently, a couple of loss and an exceedance rate of loss values must be selected.According to the selected loss and exceedance rate of loss values, the corresponding lateral strength in each 3D surface (i.e., each vibration period) must be obtained.The URS is the spectra built with all of the lateral strengths of the considered vibration periods associated to the same loss and exceedance rate of loss.Figure 10 shows a comparison between the uniform hazard spectra (UHS) and URS for the analyzed site in terms of the return period (RP), i.e., the inverse of the exceedance rate.It can be readily seen that differences in the spectral ordinates for different values of loss and return period exist.For instance, for a constant value of β, the spectral ordinates were higher as the RP increased; on the other hand, if the RP was constant and β increased, the seismic demands decreased, i.e., Sa y the accepted risk was higher, and the required strength of the structure was lower.

Accuracy of the Uniform Risk Spectra
In order to validate the URS' accuracy, three reinforced concrete moment-resisting frames (MRFs) were analyzed.The basic mechanical characteristics of the materials were defined according to the Mexico City Building Code [78], and the lateral demands were obtained using RSA with the proposed uniform risk spectrum.It is expected that the risk

Accuracy of the Uniform Risk Spectra
In order to validate the URS' accuracy, three reinforced concrete moment-resisting frames (MRFs) were analyzed.The basic mechanical characteristics of the materials were defined according to the Mexico City Building Code [78], and the lateral demands were obtained using RSA with the proposed uniform risk spectrum.It is expected that the risk level, β-ν(β), predicted using the URS is similar to that estimated via the Pacific Earthquake Engineering Research Institute (PEER) methodology [4,5].

Summary of Mechanical Characteristics
The concrete and steel reinforcement properties are Ec = 221,359 kg/cm 2 (21,707 MPa); f'c =250 kg/cm (24.5 MPa); Es = 2,010,000 kg/cm 2 (197,113 MPa); fy = 4200 kg/cm 2 (412 MPa).Table 3 summarizes the main structural characteristics for the analysis, and Figure 11 shows the structural typology for each structure.The design seismic demands of each structure were calculated with RSA using the URS.Thus, all structures were designed for a 10% loss every 475 years; hence, the corresponding URS was employed (Figure 12).The design seismic demands of each structure were calculated with RSA using the URS.Thus, all structures were designed for a 10% loss every 475 years; hence, the corresponding URS was employed (Figure 12).The design seismic demands of each structure were calculated with RSA using the URS.Thus, all structures were designed for a 10% loss every 475 years; hence, the corresponding URS was employed (Figure 12).For modeling the aforementioned MRFs, the DRAIN 2.0 [79] structural analysis program was used.To be in agreement with the used URS (Figure 11), the Takeda hysteretic model was adopted to represent their nonlinear behavior (beams and columns), considering a damping factor of 5% of the critical and a post-yielding stiffness ratio equal to 10%.For validation purposes, the strength of the structural elements was taken equal to the force demands of the structural elements obtained from the performed RSA.For modeling the aforementioned MRFs, the DRAIN 2.0 [79] structural analysis program was used.To be in agreement with the used URS (Figure 11), the Takeda hysteretic model was adopted to represent their nonlinear behavior (beams and columns), considering a damping factor of 5% of the critical and a post-yielding stiffness ratio equal to 10%.For validation purposes, the strength of the structural elements was taken equal to the force demands of the structural elements obtained from the performed RSA.
It has been observed that overstrength is an important factor in the design process.To consider its effects, reduction factors [31,80] have been proposed; however, the use of these reduction factors was not included in the URS shown here.

Risk Assessment Process
The equation proposed to carry out the risk assessment (Equation ( 6)) is provided by the Pacific Earthquake Engineering Research Centre [4,57]: where G (DV|DM) is the probability of a decision vector exceeding a specific parameter measure; G (DM|IM) is the damage exceedance probability given an intensity measure; and dλ(I M) is the exceedance rate of the magnitudes.Equation ( 6) considers both the seismic hazard and the building vulnerability under a probabilistic approach.It implies the expansion of the exceedance rate of the decision variables in terms of a loss measure and an intensity measure.Previous studies [3][4][5] have shown that this equation may be the best-known methodology for the seismic risk assessment of structures.
Since in this paper, the aim is to obtain the exceedance rate of loss, Equation ( 6) may be re-written as follows: where G (DM|IM) is the seismic vulnerability (vulnerability curve), and dλ(I M) is the seismic hazard (exceedance rate of intensities).
For estimating the seismic vulnerability of the analyzed structures, several incremental dynamic analyses (IDAs) [81] were carried out.For this purpose, the same 550 seismic records, as described previously, were used.Figure 13 shows the estimated IDA curves in terms of the maximum inter-story drift (γ max ) for each structure; the gray lines represent the IDA curves for each seismic record, and the black line represents the average curve.
On the other hand, to map the interstory drift obtained from the IDA curves to physical damage [82][83][84], the damage index vs.interstory drift curves were defined separately for three SDOF systems whose characteristics (T and Sa y ) were consistent with those of the analyzed structures.For this purpose, the values for T of the SDOFs were taken equal to the fundamental periods of the MRF structures, and the Sa y values were defined as the spectral ordinates Sa of the design URS for the corresponding periods.Later, the damage index was mapped to loss using Equation (2). Figure 14 shows the obtained relationships for each structure.The two types of curves computed previously (IDA and interstory drift-loss curves) (Figures 13 and 14) were related to estimate the corresponding vulnerability function for each structure; as a result, curves that relate the seismic intensity level and a loss value (Sa vs. β) were obtained.This process was performed for each computed IDA curve.The two types of curves computed previously (IDA and interstory drift-loss curves) (Figures 13 and 14) were related to estimate the corresponding vulnerability function for each structure; as a result, curves that relate the seismic intensity level and a loss value (Sa vs. β) were obtained.This process was performed for each computed IDA curve.The two types of curves computed previously (IDA and interstory drift-loss curves) (Figures 13 and 14) were related to estimate the corresponding vulnerability function for each structure; as a result, curves that relate the seismic intensity level and a loss value (Sa vs. β) were obtained.This process was performed for each computed IDA curve.
The vulnerability functions, VFs, are the result of the statistical analysis of the whole set of intensity measure vs. loss curves obtained for all of the considered seismic records; as a result, the expected value, E[β], and standard deviation, σ[β], associated to each intensity measure were obtained (Figure 15).The vulnerability functions, VFs, are the result of the statistical analysis of the whole set of intensity measure vs. loss curves obtained for all of the considered seismic records; as a result, the expected value, E [β], and standard deviation, σ [β], associated to each intensity measure were obtained (Figure 15).Generally, the exceedance rate of intensities is obtained through a probabilistic seismic hazard assessment (PSHA).In this paper, the exceedance rate of intensities associated to each structural vibration period corresponding to the three analyzed structures were obtained with an inverse process, i.e., from the UHS obtained for the studied seismic source obtained by Niño et al. [59], where the exceedance rate of the magnitudes (Figure 3) was considered.The annual exceedance rate of intensity associated with each structural vibration period of the analyzed structures are presented in Figure 16.Equation (7) was solved for several levels of seismic intensity to obtain the corresponding exceedance rate of loss associated to the analyzed structures.These curves represent the mean number of times that a loss value is exceeded annually.The computed curves of the exceedance rate of loss of the three designed frames (continuous line) are shown in Figure 17 in addition to the design exceedance rate of loss of the URS (square marker).Generally, the exceedance rate of intensities is obtained through a probabilistic seismic hazard assessment (PSHA).In this paper, the exceedance rate of intensities associated to each structural vibration period corresponding to the three analyzed structures were obtained with an inverse process, i.e., from the UHS obtained for the studied seismic source obtained by Niño et al. [59], where the exceedance rate of the magnitudes (Figure 3) was considered.The annual exceedance rate of intensity associated with each structural vibration period of the analyzed structures are presented in Figure 16.The vulnerability functions, VFs, are the result of the statistical analysis of the whole set of intensity measure vs. loss curves obtained for all of the considered seismic records; as a result, the expected value, E [β], and standard deviation, σ [β], associated to each intensity measure were obtained (Figure 15).Generally, the exceedance rate of intensities is obtained through a probabilistic seismic hazard assessment (PSHA).In this paper, the exceedance rate of intensities associated to each structural vibration period corresponding to the three analyzed structures were obtained with an inverse process, i.e., from the UHS obtained for the studied seismic source obtained by Niño et al. [59], where the exceedance rate of the magnitudes (Figure 3) was considered.The annual exceedance rate of intensity associated with each structural vibration period of the analyzed structures are presented in Figure 16.Equation (7) was solved for several levels of seismic intensity to obtain the corresponding exceedance rate of loss associated to the analyzed structures.These curves represent the mean number of times that a loss value is exceeded annually.The computed curves of the exceedance rate of loss of the three designed frames (continuous line) are shown in Figure 17 in addition to the design exceedance rate of loss of the URS (square marker).Equation (7) was solved for several levels of seismic intensity to obtain the corresponding exceedance rate of loss associated to the analyzed structures.These curves represent the mean number of times that a loss value is exceeded annually.The computed curves of the exceedance rate of loss of the three designed frames (continuous line) are shown in Figure 17 in addition to the design exceedance rate of loss of the URS (square marker).

Figure 17.
Comparison between the estimated exceedance rate of loss from the design (URS) and the estimated exceedance rate of loss after the design using the PEER approach.
Keeping in mind that the presented structures were designed to exceed (on average) a 10% loss every 475 years, for the 1-and 15-story frames, the proposed URS allowed for the design of structures that developed a risk level close to the one desired, since the risk levels calculated for the designed structures were similar to that of the URS used (error less than 20%).On the other hand, for the case of 25-story frame, the URS would seem to underestimate the necessary seismic forces associated with the accepted risk level.The results for the 25-story frame might be attributable to simplifications in the estimation of the exceedance rate of loss curve.In this paper the correlation between damage and structural seismic response is based on the dynamic response of a reference SDOF system subjected to the action of several seismic ground motion records, without taking into account the effects that are present in complex structures, such as the local and global damage definition.This generated inconsistences in the estimation of the associated VFs, where, in the same way, high loss values were presented under relatively low demands.For this reason, the risk obtained from the seismic assessment was larger than expected.
To avoid these inconsistences, a very detailed damage analysis should be carried out for the whole structure during the seismic assessment process, such as the local (element) to global (structure) damage and its correlation with a performance parameter including interstory drift; this will allow for the better estimation of the VF and, subsequently, the corresponding exceedance rate of loss.

Conclusions
The proposed formulation to compute the uniform risk spectra offers a rational way to provide a predetermined seismic risk level for structures in an explicit manner from the beginning of the design process without the need for later assessment.This provides valuable information for civil engineers on the economic losses that a structure will have in its lifetime, contrary to the current design spectra of most seismic codes, which do not provide, at least in an explicit way, the consequences of choosing a reduction factor in the seismic demands.It is necessary to start considering seismic risk in the planning of new cities; therefore, URS may be useful as a first tool for the development of new structures.
The proposed formulation considers the main aspects that define the structural response of any building, the structural characteristics that define the seismic response of a structural type, and the seismic hazard of the site where a given structure will be built.Furthermore, to consider the possible economic loss associated with the structural damage, reparation cost was considered as the damage variable in this study.Nonetheless, it is important to highlight that the reparation costs present high variability with time compared to the structural characteristics or the seismic hazard This represents a great challenge for future studies where this variability may be introduced in the computation.
For the sake of simplicity, the proposed formulation was exemplified with the calculation of a URS associated to a specific site, considering a single seismic source.However, Keeping in mind that the presented structures were designed to exceed (on average) a 10% loss every 475 years, for the 1-and 15-story frames, the proposed URS allowed for the design of structures that developed a risk level close to the one desired, since the risk levels calculated for the designed structures were similar to that of the URS used (error less than 20%).On the other hand, for the case of 25-story frame, the URS would seem to underestimate the necessary seismic forces associated with the accepted risk level.The results for the 25-story frame might be attributable to simplifications in the estimation of the exceedance rate of loss curve.In this paper the correlation between damage and structural seismic response is based on the dynamic response of a reference SDOF system subjected to the action of several seismic ground motion records, without taking into account the effects that are present in complex structures, such as the local and global damage definition.This generated inconsistences in the estimation of the associated VFs, where, in the same way, high loss values were presented under relatively low demands.For this reason, the risk obtained from the seismic assessment was larger than expected.
To avoid these inconsistences, a very detailed damage analysis should be carried out for the whole structure during the seismic assessment process, such as the local (element) to global (structure) damage and its correlation with a performance parameter including interstory drift; this will allow for the better estimation of the VF and, subsequently, the corresponding exceedance rate of loss.

Conclusions
The proposed formulation to compute the uniform risk spectra offers a rational way to provide a predetermined seismic risk level for structures in an explicit manner from the beginning of the design process without the need for later assessment.This provides valuable information for civil engineers on the economic losses that a structure will have in its lifetime, contrary to the current design spectra of most seismic codes, which do not provide, at least in an explicit way, the consequences of choosing a reduction factor in the seismic demands.It is necessary to start considering seismic risk in the planning of new cities; therefore, URS may be useful as a first tool for the development of new structures.
The proposed formulation considers the main aspects that define the structural response of any building, the structural characteristics that define the seismic response of a structural type, and the seismic hazard of the site where a given structure will be built.Furthermore, to consider the possible economic loss associated with the structural damage, reparation cost was considered as the damage variable in this study.Nonetheless, it is important to highlight that the reparation costs present high variability with time compared to the structural characteristics or the seismic hazard This represents a great challenge for future studies where this variability may be introduced in the computation.
For the sake of simplicity, the proposed formulation was exemplified with the calculation of a URS associated to a specific site, considering a single seismic source.However, for the actual design of applications, it is necessary to consider all seismic sources that affect the site of interest.Furthermore, a validation example of the seismic demands associated to a specific risk level was carried out through the seismic design of three different RC structures via RSA using one of the computed URS as the design spectra.The validation shows that the use of URS allows for the design of low-or medium-rise structures that approximate sufficiently the target seismic risk level.However, for the high-rise structures, the results were different to those expected, mainly due to simplifications in the calculation of the exceedance rate of loss curve using the PEER methodology.
It must be recognized that the main objective of the development of the URS shown herein is for them to be used in practical seismic design applications.Therefore, some simplifications that were considered in this work should be addressed in future studies, such as the effects of the overstrength or structural irregularities in the computation of the seismic demands using this type of spectra.

Figure 1 .
Figure 1.3D surface of the relationship among loss (), exceedance rate of loss ( ()), and lateral strength ( ) for a defined vibration period ().

Figure 1 .
Figure 1.3D surface of the relationship among loss (β), exceedance rate of loss (v (β)), and lateral strength (Sa y ) for a defined vibration period (T).

Figure 1 .
Figure 1.3D surface of the relationship among loss (), exceedance rate of loss ( ()), and lateral strength ( ) for a defined vibration period ().

Figure 2 .
Figure 2. Proposed flowchart for the estimation of URS.

Figure 2 .
Figure 2. Proposed flowchart for the estimation of URS.

Figure 3 .
Figure 3.Estimated exceedance rate of magnitudes for the Guerrero Gap.

Figure 3 .
Figure 3.Estimated exceedance rate of magnitudes for the Guerrero Gap.

Figure 4 .
Figure 4. Damage spectra for SDOF associated to Say = 100 gals and seismic records with magnitude of (a) 7.2 and (b) 7.3.

Figure 4 .
Figure 4. Damage spectra for SDOF associated to Sa y = 100 gals and seismic records with magnitude of (a) 7.2 and (b) 7.3.

Buildings 2023 ,
13, x FOR PEER REVIEW 7 of 20 0.10 < DI ≤ 0.25 Repair cracks including the use of epoxy to restore the elements and architectural features.0.30 0.25 < DI ≤ 0.40 Mortar covering should be replaced by specialized structural mortar and architectural features.0.62 0.40 < DI ≤ 1.0 Restore the detached concrete with structural mortar.Replacement of whole structural elements.1.00 To use the above relationship, the Park and Ang damage index values were mapped to the Teran and Jirsa [62] damage index by comparing the response of 50 inelastic SDOFs; the results are shown in Figure 5.

Figure 5 .
Figure 5.The Park and Ang damage index and Teran and Jirsa damage index relationship.(Dots are the computed damage indexes, and the black line is the fitted curve).

Figure 5 .
Figure 5.The Park and Ang damage index and Teran and Jirsa damage index relationship.(Dots are the computed damage indexes, and the black line is the fitted curve).

Figure 6 .
Figure 6.Relationship between the damage index and expected loss.

Figure 6 .
Figure 6.Relationship between the damage index and expected loss.

Figure 7 .
Figure 7. Kolmogorov-Smirnov fitted test applied to the statistical data of the SDOF with T = 2.2 s and lateral strength of (a) 200, (b) 500, and (c) 1000 gals under the action of M 8.0 seismic records.

Figure 7 .
Figure 7. Kolmogorov-Smirnov fitted test applied to the statistical data of the SDOF with T = 2.2 s and lateral strength of (a) 200, (b) 500, and (c) 1000 gals under the action of M 8.0 seismic records.

Figure 7 .
Figure 7. Kolmogorov-Smirnov fitted test applied to the statistical data of the SDOF with T = 2.2 s and lateral strength of (a) 200, (b) 500, and (c) 1000 gals under the action of M 8.0 seismic records.

Figure 8 .
Figure 8. Annual exceedance rate of the loss curve associated to Sa y = 300 gals and T = 2.2 s.

Figure 8 .
Figure 8. Annual exceedance rate of the loss curve associated to Say = 300 gals and T = 2.2 s.

Buildings 2023 , 20 Figure 10 .
Figure 10.Uniform risk spectra associated to several loss levels and return periods.

3 Figure 10 .
Figure 10.Uniform risk spectra associated to several loss levels and return periods.

Figure 16 .
Figure 16.Exceedance rate of intensities associated to the studied seismic source for structural vibration periods of 0.12, 1.25, and 1.80 s.

Figure 16 .
Figure 16.Exceedance rate of intensities associated to the studied seismic source for structural vibration periods of 0.12, 1.25, and 1.80 s.

3 Figure 16 .
Figure 16.Exceedance rate of intensities associated to the studied seismic source for structural vibration periods of 0.12, 1.25, and 1.80 s.

Figure 17 .
Figure 17.Comparison between the estimated exceedance rate of loss from the design (URS) and the estimated exceedance rate of loss after the design using the PEER approach.

Table 2 .
Relationship between the DIPA and DITJ.

Table 2 .
Relationship between the DI PA and DI TJ .

Table 3 .
Mechanical and structural characteristics of the models.