Issues Concerning the Seismic Design of Essential Mid-Rise MRF Buildings Exhibiting Linear Behavior

Abstract

This study evaluates the seismic performance and life-cycle economic implications of designing essential urban mid-rise reinforced concrete moment-resistant frame (MRF) buildings to maintain linear elastic behavior up to the Immediate Occupancy (IO) performance level. While most urban buildings are commonly designed to respond non-linearly in order to reduce initial construction costs, the current Mexico City Building Code (MCBC) permits that essential facilities, such as hospitals and schools, maintain linear behavior during moderate-to-strong earthquakes. This code establishes a maximum story drift ratio equal to 0.0075 for essential buildings constituted by MRF subjected to seismic events with a 250-year recurrence interval; in addition, it recommends ductile structural behavior to achieve Life Safety performance at a 450-year recurrence interval. Given the significant differences in occupancy, functionality, and contents of critical facilities, here it is analyzed whether the linear elastic design criterion is efficient for both secondary care hospitals and public schools. Two three-story and five-story MRF buildings, located on firm and transition soil, respectively, are analyzed. This study addresses the probability of brittle-type failure risk, the optimal allowable story drift at the IO performance level, the potential need for use-dependent drift limits, and the contribution of contents and nonstructural components to the total expected seismic losses. The seismic risk and economic performance are quantified through seismic hazard analysis, incremental dynamic analysis, fragility modeling, Monte Carlo simulation, and life-cycle cost evaluation.

1. Introduction

Some essential structures around the world are designed to present linear elastic behavior in some or in all of their structural components under a certain level of seismic intensity during their service life, for example, nuclear power plants, marine platforms, dams, etc. [1,2,3]; however, in general, worldwide, the seismic design of urban buildings assumes inelastic nonlinear behavior, leading to reduced structural demands and therefore lower initial costs compared to those presenting linear behavior [4,5]; however, it is possible to design essential urban structures (e.g., hospitals, schools, military facilities, transportation stations, etc.) exhibiting linear structural behavior up to an allowable maximum drift ratio corresponding to a specific performance level (limit state) when subjected to design earthquake motions from seismic events with a specified recurrence interval (RI). For example, the Mexican Seismic Design Guideline (Norma Técnica Complementaria para Diseño por Sismo-2023-NTC-DS-2023) [6], which is part of the Mexico City Building Code (MCBC) [7], establishes seismic design specifications for essential buildings constituted by moment-resisting frames, MRFs, to maintain linear behavior up to an allowable maximum story drift ratio ${\delta }_{IO}=$ 0.0075, associated with an Immediate Occupancy performance level (IO) during seismic events with an RI no less than 250 years. Furthermore, the guideline specifies that for the Life Safety performance level (LS), corresponding to seismic occurrences with RI ≥ 475 years, the buildings should be designed to develop moderate ductility demand.

Modern seismic design codes correspond with prevailing trends in performance-based seismic design [8,9,10], emphasizing the repercussions of earthquakes, and specifically addressing repair costs, business interruption losses, downtime, and, crucially, damage to nonstructural elements and building contents, which may constitute the predominant portion of total economic losses in essential facilities. This is especially pertinent for essential infrastructure, like hospitals and schools, where damage to assets and operational interruptions can result in significant social consequences, in addition to direct financial losses.

From the previous linear seismic design philosophy for essential building structures, included in NTC-DS-2023, several questions arise. These are discussed in the present study by means of a probabilistic risk analysis of two reinforced concrete three-story and five-story MRF buildings. Both structures are used as hospitals, and alternatively, as education institutions (school). The arising questions are:

(A) Considering that the IO performance level is expected to govern the seismic design, is it feasible to design the three-story and five-story MRF buildings such that they exhibit inelastic nonlinear behavior at the LS performance level? For which value of RI may a brittle-type structural failure occur?

(B) What is the optimal allowable maximum story drift ratio associated with IO (${\delta }_{IO}$) that minimizes the expected total cost during the buildings’ service lives? Is it ${\delta }_{IO}=$ 0.0075, or may it be another value?

(C) Is it necessary to specify a different design value of ${\delta }_{IO}$ for the buildings used as hospitals than for schools (given that they contain different types of content)? For example, hospitals typically contain high-value equipment compared to schools, even though both belong to the same category of essential facilities, which may lead to distinct cost–risk outcomes.

(D) What percentage of the total expected cost, for a given RI, corresponds to the expected content damage? And what percentage corresponds to the nonstructural component damage?

Answers to the above questions are discussed in the present study.

2. Theorical Framework

The methodology to address questions A, B, C, and D is described in Section 2.1, Section 2.2, Section 2.3 and Section 2.4, respectively. It is applied to two MRF reinforced concrete buildings with three and five stories; both structures have two different occupancies: (1) as hospital, and (2) as school. The three-story building is located in firm soil, and the five-story building in transition soil. The description and characteristics of the structures and of the soil are given in Section 3. Although the concepts of life-cycle cost and performance-based seismic design are well-developed in the literature, the present study offers specific contributions that distinguish it from existing studies. First, the present study focuses on essential buildings—hospitals and schools—to determine whether the linear elastic design criterion established by the MCBC is equally efficient for both types of use. Second, unlike other studies that use generic values, the present work presents detailed budgets for contents and specific nonstructural elements for both hospital and school facilities, allowing for a more precise and realistic estimation of the expected economic losses associated with seismic events. Given that the contents of these facilities—specialized medical equipment, diagnostic instruments, and school furniture, among others—represent a significant fraction of the total value of the building and, according to the findings of the present study, contribute the most to the total expected cost over the life-cycle, precise characterization is required for a realistic economic assessment.

2.1. Addressing Question A

The three-story and five-story MRF buildings are initially designed in accordance with the requirements of the current Mexican Seismic Design Guideline (NTC-DS-2023); each building structure is subjected to incremental dynamic analyses (IDA) [11]. The IDA results show that each building exhibits linear behavior up to the point that a structural brittle failure occurs, at specified seismic intensity, linked to a particular recurrence interval RI. Estimating the RI value at which the brittle failure occurs in each building and determining the potential risk of brittle failure are questions discussed in Section 4.

2.2. Addressing Question B

This section aims to calculate the optimum allowable maximum story drift (${\delta }_{IO}$), associated with the IO performance level, that minimizes the expected total cost $E\left[TC\right]$ during the buildings’ service lives. A probabilistic seismic risk analysis (PSRA) is conducted on the buildings, which are designed for different ${\delta }_{IO}$ values. The summarized methodology consists of first estimating the annual exceedance rate of the structural response—in terms of maximum interstory drift and, alternatively, floor acceleration—by combining the fragility functions specific to each component with the site’s seismic hazard function. This rate is transformed into a cumulative probability function, facilitating the simulation, via the Monte Carlo method, using a series of values for maximum story drift and, alternatively, floor acceleration, that are representative of the seismic events that may occur during the structure’s life-cycle. Thereafter, these simulated responses are transformed into economic losses utilizing the cost functions associated with each damage category: structural elements, nonstructural components, and contents. The losses incurred in each simulation are adjusted to the present value, considering the building’s lifespan and a suitable discount rate. This procedure is reiterated numerous times until statistical convergence is attained, thereby yielding a reliable estimate of the total expected cost throughout the structure’s life-cycle.

The PSRA consists of the following steps:

(I) Seismic hazard analysis, which provides the mean annual exceedance rate of seismic intensity, $\nu \left(y\right)$, at the site where each structure is located. In the present study, the seismic hazard functions at the sites are assumed to be known [12].

(II) Probabilistic evaluation of seismic structural demand, from which the mean annual exceedance rate of the structural response of interest (maximal story drift or peak floor acceleration), denoted as $\nu \left(d\right)$, is derived. For this purpose, fragility curves are computed to represent the probability that the structural response $D$ surpasses or equals a threshold $d$, conditioned on a seismic intensity $Y=y$. For its computation, Equation (1) is used:

$$P\left[D>d|Y=y\right]=1-\mathsf{\Phi }\left({\displaystyle \raisebox{1ex}{$(\ln d-{\mu }_{ln})$}\!\left/ \!\raisebox{-1ex}{${\sigma }_{ln}$}\right.}\right)$$

(1)

where ${\mu }_{ln}$ is the mean of the natural logarithm of the structural response at a specified intensity $y$, and ${\sigma }_{ln}$ is the standard deviation of the natural logarithm of the structural response at the same intensity $y$. Thereafter, the exceedance rate, $\nu \left(d\right)$, is computed for the response parameter d used in the preceding step, for the structure under analysis [13], using Equation (2).

$$\nu \left(d\right)={\int }_{y0}^{y1}P\left[D>d|Y=y\right]\left|{\nu }^{\prime }\left(y\right)\right|dy$$

(2)

where ${\nu }^{\prime }\left(y\right)$ is the derivative of the seismic intensity exceedance rate; $y_0$ and $y_1$ denote the minimum and maximum seismic intensity values for $\nu \left(y\right)$, respectively.

The floor accelerations as well as the maximum interstory drifts utilized in the loss estimation were derived from incremental dynamic analyses, which facilitated the characterization of the structural response at varying levels of seismic intensity while accounting for the structure’s nonlinear behavior. The dynamic responses were associated with economic damage to contents and nonstructural components through direct cost–damage correlations based on each response measure.

(III) Evaluation of the initial costs and the expected costs associated with structural, nonstructural, and content damage for the analyzed structures, which is carried out using a Monte Carlo simulation scheme [14], along with the estimation of the expected total cost, $E\left[TC\right]$. For this purpose, the following steps are performed:

• The cumulative distribution function $F_D\left(d\right)$ is derived from $\nu \left(d\right)$ using Equation (3).

  $$F_D\left(d\right)=1-{\displaystyle \frac{\nu \left(d\right)}{{\nu }_0}}$$

  (3)

  where $\nu \left(d\right)$ is obtained by integrating the product of the fragility function and the derivative seismic intensity exceedance rate, accounting for all potential seismic intensities that may occur at the site, and ${\nu }_0$ represents the minimal exceedance rate.
• Values of the parameter $d$ are simulated using $F_D\left(d\right)$, along with their related arrival times (Equation (4)), considering a time interval equivalent to the structure’s service life.

  $$T=-{\displaystyle \raisebox{1ex}{$\ln u$}\!\left/ \!\raisebox{-1ex}{${\nu }_0$}\right.}$$

  (4)

  where $u$ is a simulated value ranging from 0 to 1, characterized by a uniform probability density function.
• Using the time-history of the parameter $d$, the cost associated with damage to structural and nonstructural components, as well as the cost linked to damage in contents, is calculated for each value of $d$. Equation (5) is used to compute the present value of the cost for each simulated value of $d$.

  $$PV={\displaystyle \raisebox{1ex}{$FV$}\!\left/ \!\raisebox{-1ex}{${\left(1+\gamma \right)}^n$}\right.}$$

  (5)

  where $PV$ represents the present value, $FV$ the future value expected to occur at time $n$, and $\gamma$ the discount rate, which in the present study is taken as 5%, based on average macroeconomic indicators in Mexico, and is consistent with values commonly adopted in similar studies [15,16,17].
• The total cost, $TC$, is calculated as the sum of the initial cost and the present value of the cost associated with damage to structural and nonstructural components, as well as to cost linked to damage in contents. Steps 3 and 4 are repeated multiple times to estimate the expected value of the total cost.
• The expected total cost, $E\left[TC\right]$, is initially calculated for the hospital-use building; thereafter, $E\left[TC\right]$ is estimated for the school-use structure. The typical contents of hospital buildings and those of school buildings are examined. In both cases, representative content for each facility type is considered. The initial costs and costs related to damage to contents are expected to be greater for hospitals than for schools, owing to the availability of special equipment and systems not typically used in educational institutions.
• To estimate $E\left[TC\right]$ for each study case, the aforementioned methodology is applied to each structure that complies with the different allowable maximum story drift ratios, ${\delta }_{IO}$.
• As a result of the previous analysis, $E\left[TC\right]$ over the building’s life-cycle is obtained for each structure designed for a specified value of ${\delta }_{IO}$. The value of ${\delta }_{IO}$ that minimizes the expected total cost for a given RI is identified as the optimal allowable maximum drift value.

The methodological framework used in the present study is compatible with the probabilistic philosophy of the PEER (Pacific Earthquake Engineering Research Center) framework for loss assessment, which systematically integrates seismic hazard, structural response, component damage, and economic losses. The document, FEMA P-58 [9], provides a detailed procedure for estimating losses in buildings based on the seismic performance of structural and nonstructural components; however, a significant limitation of this document is that the inventories of contents and equipment are for general-use buildings located in the United States, with no detailed and specific inventories for hospital and school facilities in Mexico. In this sense, the present study adds to existing frameworks by developing detailed inventories with their respective initial costs of contents and nonstructural elements specific to these essential buildings, allowing for a more representative loss estimation that is consistent with the actual local conditions of the studied buildings.

2.3. Addressing Question C

To evaluate whether the hospital requires a design value ${\delta }_{IO}$ different compared to that for the school, the steps mentioned in Section 2.1 and Section 2.2 are implemented for the school and, alternatively, for the hospital structure. The optimal values ${\delta }_{IO}$ derived for both cases are then compared to determine whether the optimal ${\delta }_{IO}$ is identical (or not) for the school and the hospital, which is solved in Section 6.

2.4. Addressing Question D

Question D is solved in the present study by contrasting the expected total cost of damage to acceleration-sensitive contents with the expected total cost of the structural damage, in addition to nonstructural and content damage. In the present study the exceedance rates for each damage cost component (structural, nonstructural, and contents) are computed separately. The results are shown in Section 7.

3. Study Cases

Two three-story and two five-story reinforced concrete MRF buildings are analyzed following the approaches mentioned in Section 2. The buildings’ geometric characteristics are shown in Figure 1a,b. The five-story building is situated in the CO47 site, which corresponds to the transition soil zone of Mexico City, and the structure is named here as building B5, because it is located in zone B and it has five stories, while the three-story building is assumed to be located at the CU site, which corresponds to firm soil, and so is referred as building A3, because it is located in zone A. Two different occupancy types are assumed for each building: hospital and, alternatively, school.

Zone A is defined by high consistency and rocky conditions, situated in the elevated regions of the valley of Mexico. It comprises soils characterized by high strength and reduced compressibility. Conversely, Zone B displays transitional properties between firm and soft soil conditions; substantial deposits are located at depths of around 20 m or less, consisting of strata of sand and sandy silt. Figure 2 illustrates the elastic design spectra (EDS), the reduced design spectra (RDS) and the uniform hazard spectra (UHS) associated with an RI of 250 years for the two sites where the structures are located. The spectra were obtained from the SASID-2023 platform (Sistema de Acciones Sísmicas de Diseño) [18], as advised by NTC-DS-2023.

Both structures were designed in accordance with the NTC-DS-2023 guideline, which stipulates that MRF essential buildings should maintain linear behavior up to an allowable maximum story drift ratio equal to ${\delta }_{IO}$ = 0.0075, associated with the performance level IO, and RI ≥ 250 years, which correspond to the Design Base Earthquake.

The design of both structures was governed by the IO performance level. The expected fundamental vibration periods were 0.77 s and 0.66 s for buildings A3 and B5, respectively. The seismic hazard at a site is defined as the rate at which seismic intensities exceed a given threshold (seismic hazard function), and it represents the average annual number of events that equal or exceed that threshold. Rodríguez-Castellanos et al. [12] used a probabilistic seismic hazard analysis (PSHA) [13,19] to determine seismic intensity exceedance rates, which are presented in Figure 3. Seismic intensity is measured here using the spectral acceleration associated with the structure’s fundamental period, with a critical damping of 5%. This choice is based on the fact that the MCBC defines the seismic design spectrum in terms of this intensity measure, ensuring consistency between seismic hazard characterization and current Mexican guidelines. Although there are intensity measures with better efficiency and sufficiency properties [20,21,22], spectral acceleration is used here as the reference measure.

4. Solution to Question A

To solve Question A, buildings A3 and B5 were analyzed through IDA [11], using simulated accelerograms which were generated from the uniform hazard spectra (UHS) supplied by the SASID software (2023 version). The results, expressed in terms of maximum story drift ratio, are shown in Figure 4a and Figure 4b for buildings A3 and B5, respectively.

Figure 4a,b show that structural failure occurs at mean spectral accelerations of approximately 0.44 g and 0.8 g for buildings A3 and B5, respectively. According to Figure 3, it can be deduced that the associated expected recurrence intervals for these spectral acceleration levels are approximately 5200 and 3500 years, respectively. This means that in the case of a seismic event with a very high recurrence interval, B5 would be more susceptible to present brittle-type failure. In this study, building failure is identified when the structural response meets one or more of the following criteria: (i) shear demand exceeds the element’s shear capacity (brittle failure); (ii) confined concrete reaches its ultimate strain, typically associated with the fracture of transverse reinforcement; and (iii) tensile strain in the longitudinal reinforcement exceeds its fracture strain. These conditions are characterized by a sudden loss of load-carrying capacity with limited prior deformation.

5. Solution to Question B

5.1. Annual Rates of Exceedance of the Structural Response

To address Question B, the same buildings were redesigned considering different allowable values of ${\delta }_{IO}$, in addition to the previously used value ${\delta }_{IO}$ = 0.0075. The additional values used for the structural design were ${\delta }_{IO}$ = 0.005, 0.01, and 0.0125. The three-story structures situated in zone A and characterized by ${\delta }_{IO}$ values of 0.005, 0.0075, 0.01 and 0.0125 are designated here as A3-005, A3-0075, A3-010, and A3-0125, respectively.

Similarly, the five-story structures, located in zone B, are designated as B5-005, B5-0075, B5-010, and B5-0125, corresponding to ${\delta }_{IO}$ values of 0.005, 0.0075, 0.01, and 0.0125, respectively; however, designing this structure for a design distortion limit of 0.0125 is not feasible due to the high amount of longitudinal reinforcement needed to meet design demands. Although these amounts are within the Mexican Concrete Structures Design Guideline (Norma Técnica Complementaria para el Diseño de Estructuras de Concreto-2023- NTC-DCC-2023) [23] maximum limits, these reinforcements result in sections with significant congestion in the structural elements, making reinforcement installation and concrete casting difficult. This phenomenon is particularly critical in beam-column joints.

Fragility functions for the buildings were derived from Equation (1). Figure 5 and Figure 6 illustrate the outcomes for building B5, corresponding to the maximum story drift ratio and peak floor acceleration (PFA), respectively. The present study estimates fragility functions assuming that the structural response at a specific seismic intensity level follows a lognormal distribution. The variability in structural dynamic properties represents a smaller source of uncertainty compared to that associated with seismic hazard [24]. Although advanced techniques for the identification of structural dynamic properties are available [25], the coefficient of variation in the fundamental vibration period of medium-height reinforced concrete frames is on the order of 0.04 [26], whereas that of the empirical ground motion prediction models used in the calculation of seismic hazard is close to 1.0 [27]. As a result, it is assumed that the simplification used here in the characterization of dynamic properties does not compromise the accuracy of the results.

The subsequent results correspond to building B5, as it is subjected to a greater seismic risk compared to building A3; this means that building B5 experiences greater acceleration demands, as can be seen in Figure 2.

Next, the annual rate of exceedances for the response parameters, (a) maximum story drift ratio and (b) peak floor acceleration, are calculated using Equation (2). Results for building B5 are shown in Figure 7a and Figure 7b, respectively.

Figure 7 shows that the influence of ${\delta }_{IO}$ is greater for the maximum story drift ratio than for the peak floor acceleration, due to the fact that ${\delta }_{IO}$ directly governs the lateral stiffness and strength of the structural system, thereby primarily affecting displacement-related response quantities, while peak floor accelerations are more strongly controlled by the dynamic structural characteristics and the input ground motion.

5.2. Estimation of the Expected Total Cost

The expected total cost is evaluated following steps 1 through 7 described in Section 2.2 (subsection III). The total cost is expressed as the sum of the initial cost, $C_0$; the expected costs associated with damage to structural elements, $E\left[C_{EE}\right],$ and nonstructural elements, $E\left[C_{NE}\right]$; plus the expected cost of contents, $E\left[C_C\right],$ as defined in Equation (6).

$$E\left[TC\right]=C_0+E\left[C_{EE}\right]+E\left[C_{NE}\right]+E\left[C_C\right]$$

(6)

5.2.1. Initial Cost

In numerous studies about the optimization of seismic design parameters, life-cycle losses are typically assessed as a percentage of the initial construction expenditure [16,28,29,30]. This simplification is constrained by the fact that an erroneous assessment of the initial cost may considerably skew seismic design decision-making; for example, critical facilities like hospitals need specialized installations and advanced equipment, which markedly increase their value relative to standard buildings [31,32].

To estimate the initial costs of contents and nonstructural features, it is essential to first delineate a collection of common areas in both secondary care hospitals and public schools. This study uses the architectural design guidelines established by the Pan American Health Organization [33] and the National Institute for Educational Physical Infrastructure (INIFED) [34] as references, as they offer normative criteria and directives for the configuration and characterization of the spaces of interest. The following dimensions were assumed for hospitals: general consultation room: 17.28 m²; inpatient room: 25.92 m²; nursing station: 22.95 m²; emergency room: 62.36 m²; CT scan room: 40.32 m²; X-ray room: 60.48 m²; magnetic resonance imaging (MRI) room: 69.12 m²; operating room: 50.85 m²; and emergency laboratory: 20.16 m². For educational institutions, the specified dimensions were as follows: classroom: 77.76 m² and computer room: 90.72 m².

Table 1. Spatial distribution for building A3.

Level	Hospital Use	School Use
1	1 X-ray room 1 emergency room 1 CT scan room 1 MRI room 1 nursing station 30 general consultation rooms	2 computer rooms 8 classrooms
2	29 inpatient rooms 1 nursing station	2 computer rooms 8 classrooms
3	2 operating rooms 26 inpatient rooms	2 computer rooms 8 classrooms

and

Table 2. Spatial distribution for building B5.

Level	Hospital Use	School Use
1	1 X-ray room 1 emergency room 1 CT scan room 1 MRI room 1 nursing station 30 general consultation rooms	2 computer rooms 8 classrooms
2	29 inpatient rooms 1 nursing station	2 computer rooms 8 classrooms
3	29 inpatient rooms 1 nursing station	2 computer rooms 8 classrooms
4	29 inpatient rooms 1 nursing station	2 computer rooms 8 classrooms
5	2 operating rooms 26 inpatient rooms	2 computer rooms 8 classrooms

present the spatial distribution of the three-story and five-story buildings, respectively; the same spatial distributions are illustrated in Figure 8 and Figure 9.

For both buildings (A3 and B5) the initial cost of nonstructural parts and contents is determined based on the spatial distribution and the cost estimates for each location. It is noted that different structural designs corresponding to different ${\delta }_{IO}$ values have the same initial cost.

The initial cost distribution for structural and nonstructural components, contents, and installations for schools and hospitals, in building B5, is shown in Figure 10. The initial cost of contents in hospitals makes up 39% of the total, followed by structural elements (29%), nonstructural parts (22%) and installations (9%). These and nonstructural elements make up about 6% and 16% of the initial costs in schools, whereas structural elements account for 41% and contents for 36%. As mentioned in [5], it is not surprising that nonstructural components and contents have a significant impact on the initial cost.

5.2.2. Structural Damage Cost

Rehabilitation becomes extremely challenging when a structure’s damage index, $I_d$ (expressed in Equation (7)), exceeds 0.7; as a result, the repair cost is calculated as the initial cost plus the costs of demolition and debris removal [35]. The repair or reconstruction cost (CRR) in this study is determined as a function of the damage index, $I_d$, using Equations (8) and (9) [36]. Figure 11 illustrates the repair cost as a function of $I_d$ (Equations (8) and (9)).

$$I_d=\left({\delta }_i-{\delta }_y\right)/\left({\delta }_F-{\delta }_y\right)$$

(7)

$$C_{RR}=C_0·{I_d}^2; 0<I_d<0.7$$

(8)

$$C_{RR}=C_0·1.2; I_d\ge 0.7$$

(9)

5.2.3. Nonstructural Damage Cost

The functions mentioned in reference [37] were utilized here to estimate the costs related to damage to nonstructural elements. The functions were formulated from generic fragility functions based on empirical data and are utilized to assess damage in components. The present study utilized the functions correlating the maximum story drift ratio to the damage costs of nonstructural parts in reinforced concrete frames with restricted ductility. Figure 12 illustrates the corresponding function, normalized with respect to the total repair cost of nonstructural elements.

5.2.4. Content Damage Cost

Losses resulting from damage to building contents are essentially dependent upon the structure’s use and the type of items it contains. Numerous items are vulnerable to damage by floor acceleration at the locations in question. Appendix A, which is based on references [31,32,38], presents a methodology that initially facilitates the estimation of content damage losses for an inventory as a function of floor acceleration ($y_a$) and subsequently combines these individual estimates to formulate a comprehensive loss function applicable to multiple inventories within a building. Utilizing this version of floor acceleration as well as Equation (10), the cost functions for damage to contents were obtained. Figure 13 illustrates that losses resulting from damage to hospital structures are, as anticipated, considerably more than those associated with school buildings.

$$E\left[{\beta }_T\left|y_s\right.\right]=D_{max}{\sum }_{j=0}^{Npisos}\left(1-e^{\left({\theta }_j\ln 0.5\right)}\right)$$

(10)

5.3. Expected Total Cost

Using the methodology mentioned in Section 2, which uses the Monte Carlo simulation technique [14], the expected total cost is then estimated for the study cases, pertaining to both the health and the educational sector. Figure 14 shows the steps outlining the Monte Carlo simulation method.

Total Expected Loss of Buildings Located in Zone A and Zone B

Figure 15a,b show the total expected cost, normalized with respect to the initial cost, of buildings A3 and B5, respectively. The total expected costs for both hospital and school use have a comparable trend, failing to reveal an optimal ${\delta }_{IO}$ value that leads to a different reduction in the normalized total costs.

Figure 15a,b indicate that there is no clear optimal interstory drift value (${\delta }_{IO}$) that minimizes the expected total cost. This behavior can be explained by considering that the Immediate Occupancy performance level imposes high resistance and lateral stiffness on the structures, which significantly limits the demands for interstory drift and, consequently, reduces damage to structural elements and nonstructural components sensitive to drift, such as partitions and finishes, which contribute to the total cost and are not very sensitive to variations in ${\delta }_{IO}$. Structures designed with higher permissible ${\delta }_{IO}$ values have slightly longer fundamental periods and similar floor acceleration demands across different designs. Therefore, losses associated with nonstructural components and contents sensitive to acceleration, such as medical equipment, furniture, and ceilings, do not show significant variations depending on the adopted ${\delta }_{IO}$ value; so, the expected total cost is relatively uniform within the range of ${\delta }_{IO}$ values studied.

Based on Figure 15a,b, Question B is answered as follows: for the cases analyzed, there is no an allowable value of the maximum story drift ratio, associated with the OI, that minimizes the expected total cost over the buildings’ service lives.

6. Solution to Question C

Based on Figure 15a,b, it is unnecessary to specify a different design value of ${\delta }_{IO}$ for hospitals compared to schools.

7. Solution to Question D

The mean cost annual exceedance rates were calculated based on structural demand exceedance rates (see Figure 7) and on the cost functions described in Section 5.2.2 and Section 5.2.3, which establish the relationship between structural response and damage cost to determine the percentage contribution of the expected cost of damage to contents and nonstructural elements to the total expected cost of the building. The expected cost annual exceedance rates for buildings B-D005, B-D0075 and B-D010 are shown in Figure 16, Figure 17 and Figure 18, respectively. As illustrated in these figures, the contribution of acceleration-sensitive component damage costs to the total damage cost is predominant. This suggests that when the Immediate Occupancy performance level determines the linear elastic design of essential facilities, the permissible maximum story drift should be given by explicit design criteria for controlling floor accelerations rather than maximum interstory drifts.

Figure 16, Figure 17 and Figure 18 show that the expected content damage costs contribute approximately 60% of the total expected cost for an exceedance rate value of 1/475 years (2% of exceedance probability in 50 years).

8. Conclusions

An analysis was conducted on the significant consequences that may emerge during the service lifespan of mid-rise three- and five-story essential urban RC buildings constituted by moment-resisting frames when they are designed to develop linear elastic structural behavior. The following conclusions were obtained:

(A)

The IO performance level controlled the design of the buildings analyzed, ensuring that they stayed within the linear elastic structural behavior, even at acceleration levels linked to extremely high return periods. The RI for the structures analyzed was 3500 years for the five-story building and 5200 years for the three-story building.

(B)

In the investigated scenarios, no optimal allowable maximum story drift value, ${\delta }_{IO}$, linked to the OI performance level, was found for any of the structures analyzed.

(C)

As no optimal value of ${\delta }_{IO}$ was found for hospital nor school buildings, it is concluded that a different allowable maximum story drift value ${\delta }_{IO}$ is unnecessary for the analyzed hospital and school structures.

(D)

For the structure exposed to higher seismic intensity (building B5), the damage cost to contents was similar to the total expected cost; conversely, the impact of structural and nonstructural damage costs on the total expected cost was minimal. These findings indicate that design guidelines for safeguarding sensitive acceleration contents are necessary for buildings exposed to significant floor accelerations.

(E)

This study focused exclusively on the analysis of moment-resisting frames of low to moderate height, reflecting the characteristics of public educational schools and secondary care hospitals in Mexico. The results achieved and the established probabilistic framework enable the extension of the analysis to taller buildings, where the influence of higher modes and the impact of nonstructural components sensitive to acceleration may become important. In addition, it would be desirable to broaden the present study to include various structural systems (such as dual systems with concrete walls, braced frames, etc.) to assess the applicability and economic viability of the elastic-linear design provisions of NTC-DS-2023.

(F)

The results show that, while a reduction in ${\delta }_{IO}$ leads to smaller losses in structural elements and nonstructural components sensitive to interstory drift, the variable that dominates the expected total cost of the studied buildings is the floor acceleration, which does not show a clear reduction with variations in ${\delta }_{IO}$. This conclusion emphasizes the importance of developing criteria that include floor acceleration as an explicit design element in the seismic codes, in addition to the existing recognized tolerable value for the maximum interstory drift.

The present study has several limitations: (i) soil–structure interaction is excluded, a factor that may alter the seismic demand on the structural system, especially when this is located in soft soils; (ii) the substantial computational demands of the incremental dynamic analysis, added to Monte Carlo simulation, restrict the number of case studies to three- and five-story buildings; and (iii) the seismic hazard characterization is representative specifically of sites corresponding to firm and transition soils in Mexico City.

Due to the above limitations, the above conclusions should be taken with caution for other seismic contexts or soil types.