Loss and Downtime Assessment of RC Dual Wall–Frame Office Buildings Toward Resilient Seismic Performance

Abstract

This study quantitatively assesses the impact of seismic design strategies on the performance of reinforced concrete (RC) dual wall–frame office buildings by comparing direct and indirect economic losses and downtime in life-cycle terms. A high-rise archetype building located in Santiago, Chile, on stiff soil was evaluated as a benchmark case study. Three design strategies to potentially enhance the seismic performance of a building designed conventionally were explored: (i) incorporating fluid viscous dampers (FVDs) in the lateral load-resisting structure; (ii) replacing conventional non-structural components with enhanced ones (ENCs); and (iii) a combination of the previous two strategies. First, probabilistic structural responses were estimated through incremental dynamic analyses using three-dimensional nonlinear models of the archetypes subjected to a set of hazard-consistent Chilean ground motions. Second, FEMA P-58 time-based assessment was conducted to estimate expected annual losses (EALs) for economic loss estimation. Finally, for downtime assessment, a novel probabilistic framework, built on the FEMA P-58 methodology and the REDi guidelines, was employed to estimate the expected annual downtimes (EADs) to achieve specific target recovery states, such as reoccupancy (RO) and functional recovery (FR). Results revealed that seismically enhancing RC dual wall–frame buildings with FVDs significantly improves resilience by reducing loss and downtime. For example, the enhanced building with FVDs achieved an EAL of 0.093% and EAL of 8.6 days for FR, compared to the archetype base building without design improvements, which exhibited an EAL of 0.125% and an EAD of 9.5 days for FR. In contrast, the impact of ENCs alone was minor, compared to the effect of FVDs, with an EAL of 0.106% and an EAD of 9.1 days for FR. With this detailed recovery modeling, probabilistic methods, and a focus on intermediate recovery states, this framework represents a significant advancement in resilience-based seismic design and recovery planning.

1. Introduction

Prescriptive seismic codes for the design of buildings are primarily aimed at avoiding collapse under extreme earthquake loads. However, structural and non-structural damage is a latent possibility, usually not properly quantified [1]. Even without building collapse, if the damage is extensive, it could cause critical economic losses, long functional recovery times, and, consequently, significant environmental impacts due to reparation or demolition and reconstruction labor [2,3]. For example, recent earthquakes in Chile and New Zealand showed that the ‘life safety’ performance level has been successfully achieved with relatively few collapsed buildings [1], which is in agreement with recent numerical studies showing minimal probabilities of collapse even under extreme seismic events [4]. However, losses and downtime due to widespread damage in modern buildings could be substantial in earthquake-prone countries [5].

Specifically, due to the M_w 8.8 2010 Chile earthquake, total losses were around USD 30 billion (18% of the 2010 Chilean gross domestic product, GDP), and the international airports in Santiago and Concepcion were out of service for weeks due to non-structural damage [6]. Moreover, due to the M_w 6.3 2011 New Zealand earthquake, total losses were estimated to be over USD 15 billion (10% of the 2011 New Zealand GDP), about 50% of all the concrete and masonry buildings in the business district of Christchurch became unusable due to significant damage, and the district area was closed for months [7]. These numbers might not be acceptable for current societies that demand more resilient and sustainable urban built environments with limited structural damage and direct/indirect costs resulting from seismic demands, even in the case of large earthquakes.

To meet the expectations of current societies without compromising their future, buildings need to achieve enhanced and predictable performance levels beyond the minimum requirements of current seismic design codes [8]. Traditionally, such higher performance levels are reached by improving the lateral load-resisting system, which can be achieved by strengthening/stiffening the structural system and/or incorporating seismic protection systems. An alternative strategy is to replace non-structural components with new ones with enhanced performance levels that accommodate large displacement and acceleration demands with minimal damage [9]. In either case, more significant initial construction costs are expected, requiring deeper cost–benefit analyses to understand the trade-off between higher initial investments and future savings in earthquake-induced economic losses.

The performance-based earthquake engineering (PBEE) framework enables the evaluation of the probable seismic performance of structures over their operational life. The methodology proposed by the Pacific Earthquake Engineering Research (PEER) Center allows the assessment of new or existing buildings with a realistic and reliable understanding of the risks to life, occupancy, and economic loss that might occur in future earthquakes. In their FEMA P-58 [10] implementation, seismic performance is expressed in terms of decision variables (i.e., performance metrics) that can be understood by decision-makers, such as collapse probability, direct economic losses (i.e., building repair or replacement costs), repair time, and environmental impacts.

Recent investigations have proposed complementary methodologies for recovery time assessment, extending FEMA P-58 implementation and adopting a series of assumptions to connect building component damage to the functionality of key building systems (e.g., structural and non-structural components). For example, Terzic et al. [11] introduced a probabilistic framework that integrates models for evaluating post-earthquake functionality, mobilization time, and repair time. It employs fault tree analysis to identify critical subsystems affecting functionality and estimates the percent loss of building functionality for different damage scenarios. Cook et al. [12] built a framework which described building functionality at the tenant-unit level based on the performance of building systems, relating component damage to the operation of building systems and overall functionality through the use of fault trees. Molina Hutt et al. [13] estimate resilience metrics based on a four-component framework: (i) determining the post-earthquake usability, (ii) estimating impeding factor delays, (iii) evaluating the building repair time, and (iv) calculating the recovery time (downtime) as the combination of delays and repair time.

Despite the availability of these tools, little information is available on the expected collapse risk and earthquake-induced losses of RC buildings equipped with FVDs, especially in regions prone to subduction earthquakes, such as Chile. Consequently, the novelty of this study is in its quantitative assessment of the impact of different design strategies on the performance of modern code-conforming reinforced concrete (RC) buildings subjected to subduction earthquakes. Seismic performance is evaluated in terms of decision variables related to sustainability and resilience, such as expected economic losses and expected functional recovery times in a life-cycle framework. A high-rise dual wall–frame office building representative of the current design and construction practice in Chile was selected as a case study. Three strategies to potentially improve the performance of the benchmark building were explored (leading to a total of four cases, including the benchmark building designed per the current practice): (i) the incorporation of fluid viscous dampers (FVDs) into the lateral load-resisting system; (ii) the incorporation of enhanced non-structural components (ENCs); and (iii) a combination of the two previous strategies. To achieve the objectives, a fully probabilistic PBEE framework based on the FEMA P-58 [10] methodology was integrated with Molina Hutt et al.’s recovery time framework [13]. First, the probabilistic structural response was estimated by incremental dynamic analyses of 3D nonlinear models of the archetype subjected to hazard-consistent Chilean ground motions (GMs). For loss assessment, intensity-based assessments at several ground-shaking intensity levels were conducted to estimate the expected direct economic loss. For downtime assessment, the novel probabilistic framework, built on the FEMA P-58 methodology and the REDi guidelines, was utilized to predict the expected temporal recovery trajectory, robustness, and rapidity to reach specific target recovery states (e.g., reoccupancy and functional recovery). Finally, a time-based analysis was conducted, integrating the loss and downtime results with the respective hazard curve to estimate meaningful risk metrics, such as expected annual loss (EAL), expected annual downtime (EAD), and expected life-cycle cost (ELCC) to evaluate the effectiveness of each strategy.

To meet the above-mentioned objectives, this article is organized as follows. Section 2 presents the archetype buildings used in the study, including the seismic design strategies, as well as the nonlinear modeling. The FEMA P-58 methodology and adjustments adopted to evaluate the seismic performance of the archetypes are presented in Section 3, together with the selection of hazard-consistent ground motions, incremental dynamic analyses, and loss and downtime evaluation procedures. Section 4 provides detailed results of the risk estimations of each of the design strategies. Finally, conclusions and closing remarks are discussed in Section 5.

2. Archetype Buildings

2.1. Preliminary Definitions

The characteristics of the benchmark archetype building (BAB) were defined based on statistical data from the Chilean RC building inventory, gathered from the Chilean national database [14] as well as the state of design and local construction practice. According to the database, from buildings of at least 3 stories built between 2002 and 2020, the number of high-rise buildings (i.e., from 10 to 24 stories, 16 stories on average) has grown considerably in recent years and there is scarce information about their seismic performance. Therefore, a high-rise office building located in Santiago (moderate seismicity zone, classified as seismic zone 2 according to the Chilean seismic design code NCh 433 [15]) on soil type B (defined as fractured rock or very dense/firm soil according to DS 61 [16]) was considered as case study. The archetype selected has 3 underground levels surrounded by perimeter basement walls and 16 stories above the grade level (Figure 1a depicts the elevation of the base archetype). The structural system consists of a dual system, i.e., core shear walls and perimeter intermediate moment frames designed to take no more than 25% of the prescribed lateral seismic load (Figure 1b shows a plan layout of a typical floor). As commonly executed in Chilean practice, the thickness of the shear walls was set to be constant along the entire height. Regarding the frames, the beams had the same cross-section for all stories. Columns from the third underground story to the second story had the same cross-section, and columns from the third story to the top story had the same cross-section, as shown in

Table 1. Member cross-sections and reinforcements.

Archetype	Core Walls				Beams	Columns
	Flanges		Webs		(b × h)	(b × h)
	l	t	l	t		us3–s2°	s3°–s16°
BAB	4.2	0.7	16.0	0.5	0.7 × 0.6	1.0 × 1.0	0.8 × 0.8
ρlong. (%)	0.25		0.33		0.84	1.93	1.15

Note: l = length; t = thickness; b = width; h = height; us3 = third underground story; s2° = second story; s3° = third story; s16° = top story. ρ_long. = ratio of area of distributed longitudinal/vertical reinforcement to gross concrete area perpendicular to that reinforcement. Dimensions in meters.

. From Figure 1a,b, it is noted that the BAB has horizontal irregularity due to the orientation of the C-shaped core walls, and it has no vertical irregularity from the grade level for the upper 16 stories. These irregularities were considered in the design and analysis procedures. Indeed, torsional effects in the BAB due to horizontal irregularity were analyzed and controlled in the design procedures following the current Chilean seismic code for RC structures [15,16]. It is important to note that the linear elastic model used in the code-compliant design process was, as customary in Chilean seismic design practice, a full 3D model that included all the stories of the building, both below and above grade. Further details of the statistics of the national building database, archetype geometry, member cross-sections, and design issues can be found in Gallegos et al. [4].

Three improved design strategies are evaluated on the basis of the BAB: (i) the incorporation of FVDs into the lateral load-resisting system (BAB+FVD); (ii) the replacement of conventional non-structural components with ENCs (BAB+ENC); and (iii) a combination of both strategies (BAB+FVD+ENC). Regarding the archetypes equipped with dampers, based on the expert opinion of Chilean practitioners, it was decided to include this type of dampers without redesigning the BAB. Although not optimal (a more economical structure is possible), this approach is the most practical way to comply with the requirements of the current Chilean seismic design code for buildings with energy dissipation systems NCh 3411 [17] and is consistent with current Chilean practice for dual system buildings (e.g., ‘Las Condes Capital’ building).

Chilean RC buildings are characterized by a stiff structural system; therefore, FVDs were placed every three stories, starting from the grade level, in the mid span of the perimeter frames along the transverse direction of the building (i.e., the critical direction) to pursue the amplification of the small relative motion at the two ends of the dampers. The FVDs were arranged in a horizontal configuration attached with inverted chevron steel braces. A total number of 20 FVDs were required (two pairs every three stories, five pairs per perimeter side). The location of the FVDs in the transverse perimeter frames was considered to control possible torsional effects due to the horizontal irregularity of the building. In the design process, it was verified that the irregularities did not increase the demand for axial actions in the columns and flexural actions in the shear walls. This verification was performed following a performance-based design with the requirements of the current Chilean seismic design code [17]. Nonlinear dampers with a force capacity of 50 tonf and a velocity exponent ($\alpha$) of 0.35 were selected [18]. A single damper type was used in accordance with current Chilean practice. The force capacity was based on the maximum forces that the structure could take without strengthening (i.e., without additional construction costs). The behavior of the FVD was idealized as purely viscous. The relationship between force $F_d$ and velocity $\dot{x}$ is given by Equation (1), where $c_d$ is the damping constant.

$$F_d=sign\left(\dot{x}\right) c_d {\left|\dot{x}\right|}^{\alpha },$$

(1)

Regarding the archetypes that consider ENCs, recent studies [19,20] have highlighted the benefits of using the strategy of enhancing the non-structural components in the seismic performance of mid- and high-rise steel buildings for earthquake-prone regions. For these archetypes, the structural systems did not need to be redesigned because the non-structural components had no impact on the strength and stiffness of the archetypes. However, the fragility and consequence functions of the original non-structure components were replaced with those of the ENCs. More details of the non-structural components will be provided in Section 3.2.

2.2. Nonlinear Modeling

The members and dampers of the structural system were modeled as nonlinear elements. The 3D mathematical models of the BAB and BAB+FVD archetypes were developed in the structural analysis program Perform-3D [21]. This computational program was selected because it provides a good balance between accuracy and computational cost. In recent research studies [22], Perform-3D was used to implement nonlinear response history analyses of RC shear wall buildings, which indicates that is adequate for the seismic analysis of the type of archetypes considered in this study.

The slender shear walls of the archetypes were modeled with the ‘shear wall’ element, a fiber-based cross-section model with nonlinear concrete and steel fibers and a decoupled nonlinear shear layer. The material properties of the fibers were defined using the uniaxial constitutive relationship (i.e., stress vs. strain backbone curve) YULRX (Y: yielding, U: ultimate, L: loss, R: residual, X: maximum). Unloading and reloading stiffness degradation was accounted for through energy dissipation and stiffness reduction factors [21]. In addition, a shear layer was defined by a bilinear stress–strain backbone curve. Beams and columns were modeled as ‘frame type’ elements. A finite-length-plastic formulation was adopted to model plastic hinges, with fiber-based plasticity regions at both ends and a linear elastic region in between. YULRX backbone curves were also adopted to model the uniaxial stress–strain relationships of concrete and steel. Figure 1c shows a 3D view of the model of the benchmark archetype. Further details on the calibration of the model can be found in Gallegos et al. [4].

Moreover, the FVDs were modeled using the ‘viscous bar’ element, which includes a linear elastic bar component and a fluid damper component in series [21]. The bar component accounts for the in-series stiffness of the supporting brace element, the connections, and the intrinsic axial stiffness of the FVD. The fluid damper component represents the main characteristics of the FVD. As shown in Equation (1), two input values are required to define a FVD ($c_d$ and $\dot{x}$). The accuracy of the ‘viscous bar’ element was validated by comparisons with results given by the ‘viscousdamper’ element [23], implemented in OpenSees [24], which, in turn, has been previously validated by comparisons with experimental results [25]. Recent studies have demonstrated the suitability of the ‘viscous bar’ element of Perform-3D to model FVDs in mid- and high-rise buildings subjected to seismic loading [26]. Figure 1d shows a 3D view of the archetype equipped with FVDs, where the dampers (shown in red) are located at both transverse perimeter frames. The figure shows the arrangement of the set of 20 dampers in a horizontal configuration, fastened with inverted chevron steel braces.

3. Performance Assessment Methodology

A fully probabilistic PBEE framework with FEMA P-58 [10] implementation was adopted, considering some adjustments to adapt the methodology to the local engineering context. Figure 2 illustrates the research methodology to finally evaluate the direct and indirect economic loss of each design alternative. The performance of four archetypes was evaluated: (i) BAB; (ii) BAB+FVD; (iii) BAB+ENC; and (iv) BAB+FVD+ENC. The probabilistic structural responses were estimated by incremental dynamic analyses (IDAs) of the 3D nonlinear models of the archetypes subjected to sets of hazard-consistent Chilean GMs. For loss and downtime evaluations, a series of intensity-based assessments were conducted at several ground-shaking levels until collapse. The intensities (spectral acceleration at the fundamental period, S_a(T₁), was used in this study) also include distinctive hazard intensities with probabilities of exceedance equal to 50%, 10%, 2%, and 1% in 50 years, i.e., a service level earthquake (SLE), design basic earthquake (DBE), maximum considered earthquake (MCE), and very rare earthquake (VRE), respectively. Then, the intensity-based results were integrated with the respective hazard curve to complete the time-based analysis, providing meaningful risk metrics for comparison purposes.

3.1. GM Selection and IDA

As a first step within the framework, a probabilistic seismic hazard analysis (PSHA) was performed using the computational platform SeismicHazard [27], which integrates state-of-the-art seismic source models and GM models for the Chilean territory. A site-specific hazard curve was obtained for the fundamental period of the archetype (i.e., T₁ = 1.93 s), assuming that the archetype was located in Santiago on stiff soil. Then, a conditional spectrum (CS) was rigorously constructed and used as the target spectrum to select and scale GMs. The CS calculations considered the target Sa(T1) for the 2–50 years hazard level, and the mean causal magnitude M, mean causal distance R, and mean causal epsilon ε obtained from the PSHA deaggregation. From the Chilean strong ground motion database SiberRisk [28], a hazard-consistent set of 44 subduction GMs was selected and scaled to match the target mean, variance, and correlations of the S_a(T₁) values. The selected records incorporate actual shakings from past Chilean subduction earthquakes with M > 6.3 that occurred between 1985 and 2019, including GMs of the Mw 8.8 2010 megathrust seismic event. More details about the PSHA, CS, and the set of GMs can be found in Gallegos et al. [4].

The set of GMs was used to conduct the IDAs of the BAB and the BAB+FVD models. Numerous nonlinear response history (NLRH) analyses were performed. Each GM was systematically scaled until structural collapse. The adopted intensity measure (IM) was S_a(T₁), which has been shown to have a good correlation with nonlinear structural response [29]. The dynamic analyses were performed only along the direction of the shorter horizontal plan dimension, which is the critical direction (i.e., the direction along which the collapse fragility is larger). This decision was considered by pursuing an appropriate balance between accuracy in the results and computational burden; however, it is acknowledged that such an approach could restrict the capacity of the archetype model to capture the torsional effects in the building’s response due to its horizontal irregularity. Figure 3 presents the IDA results, where curves in Figure 3a,b show the relationship between the roof drift ratio (RDR) and S_a(T₁) for the BAB and BAB+FVD archetypes, respectively. The figures also present the 50th collapse percentile (median) as well as the 16th and 84th collapse percentiles (equal to one logarithmic standard deviation below and above the mean when a lognormal distribution is assumed). For each archetype, the S_a(T₁) values that triggered collapse were recorded for each GM. Assuming a lognormal distribution, the median collapse capacity, $\hat{\mathrm{\theta }}$, and the logarithmic standard deviation, $\hat{\mathrm{\beta }}$, were computed for each archetype. Figure 3c,d show the estimated lognormal collapse fragility curves, which are important inputs for loss assessment. The estimated parameters $\hat{\mathrm{\theta }}$ and $\hat{\mathrm{\beta }}$ for the BAB were 0.87 g and 0.27, respectively, whereas for the BAB+FVD they were 0.97 g and 0.26, respectively. It is evident that the archetype equipped with FVDs (i.e., BAB+FVD) performed better, i.e., it had a larger value of $\hat{\mathrm{\theta }}$ and, consequently, a smaller collapse probability at any given value of S_a(T₁).

It is worth noting that the evaluation of the buildings that include enhanced non-structural components (BAB+ENC and BAB+FVD+ENC) required neither additional structural design nor IDAs since the dynamic properties (and therefore the seismic demands on the BAB and BAB+FVD) did not change.

3.2. Economic Loss Assessment

Following the intensity-based assessment with FEMA P-58 implementation, economic losses were estimated for the four archetypes subjected to several S_a(T₁) intensities ranging from 0.05 g to 2.00 g, with increments of 0.05 g. At each shaking intensity, 2000 Monte Carlo simulations, called a realization, were performed to evaluate the mean earthquake-induced losses. It is worth noting that the number (2000) of Monte Carlo simulations was decided in order to balance accuracy (how close the results are to the true values) and efficiency (the computational cost of running simulations). The chosen number of simulations was based on a convergence. In our case, when the absolute difference between the results obtained using n simulations and using n + 1 simulations was below a target error, namely 1%, the process was finished. As this happened at approximately 2000 simulations, this number was taken as the target.

For each realization, the losses were calculated as follows: (i) engineering demand parameters (EDPs) were estimated from the results of the NLRH analyses; (ii) fragility functions were used in conjunction with the EDPs to determine the probability of reaching or exceeding associated damage states (DSs) for each structural and non-structural component; and (iii) consequence functions were then used to translate the DSs into repair/reposition costs.

This process required the estimation of the EDPs, which in this study were the following: (i) peak story drift ratio, PSDR; (ii) peak story tangential drift ratio, PSDRt; (iii) peak residual drift ratio, PRDR; and (iv) peak floor acceleration, PFA. The PSDRt, also called damageable story drift, eliminates rigid body rotation within wall panels and is considered a better proxy than PSDR for wall damage in mid- to high-rise shear wall buildings. Lognormal probability distributions, defined by the corresponding values of $\hat{\mathrm{\theta }}$ and $\hat{\mathrm{\beta }}$, were assumed to characterize the aleatory nature of the EDPs at each S_a(T₁) intensity. Figure 3a,b show an example of the variability of the structural response considering the RDR as the EDP.

The building performance models were then constructed by gathering detailed characteristics of the structural and non-structural components of the four archetypes. Initially, fragility and consequence functions of the components were taken from the FEMA P-58 database. However, some adjustments were implemented to represent the actual cost distribution of components in Chilean office buildings (i.e., 40% of the total initial cost is due to structural components and the remaining 60% is due to non-structural components, as obtained from actual RC office-building projects in Chile). This cost distribution is different from that reported for office buildings in the USA and, consequently, both the total cost and the cost of each structural and non-structural component were adjusted to reflect current Chilean RC office buildings. According to Taghavi and Miranda [30], non-structural components make up approximately 80% of the total economic investment in office buildings in the USA. Therefore, the Chilean buildings are less susceptible to presenting high damage costs due to the smaller relative cost of non-structural components. Moreover, even though this study assumes that the DSs of each structural and non-structural component are associated with the same values of $\hat{\mathrm{\theta }}$ and $\hat{\mathrm{\beta }}$ defined in the FEMA P-58 database, the repair cost of each structural and non-structural component associated with each DS followed the same relative costs defined in FEMA P-58, but normalized by the Chilean construction cost obtained from actual Chilean projects.

The replacement cost for the benchmark archetype (BAB) was estimated assuming a cost of USD 1660 per square meter on the basis of annual statistics for construction costs provided by Chile’s Ministry of Housing [31]. The cost of the FVDs (including the devices, testing, and additional steel braces) was assumed to be 3% of the BAB construction cost [18]. The cost of replacing the conventional non-structural components with enhanced ones was presumed to be 3% of the BAB construction cost. The latter was estimated based on the observation that the incorporation of non-structural components with improved seismic behavior might represent an increment of 5% of the total non-structural cost in a typical office building [19] and considering the component-cost distribution in Chilean office buildings (40–60% for structural and non-structural components, respectively) previously mentioned.

The quantities of structural components were calculated based on the structural design, and the quantities of non-structural components were estimated from typical quantities found in commercial buildings using the FEMA P-58 Normative Quantity Estimation Tool [10]. Though the NLRH analyses were conducted in the transverse direction, loss estimation requires inputs in both orthogonal directions. Therefore, quantities in the longitudinal direction were also considered. Non-structural quantities were consistent for all building archetypes. However, some non-structural component identifiers varied for the buildings with ENCs.

Table 2. Structural and non-structural components for BAB and BAB+FVD.

Fragility ID	Component	Category	Quant. d1	Quant. d2	Unit	Location	EDP
	Super Structure
B1041.021a	b/c connection	Structural	4	4	1 EA	stories	PSDR
B1041.021b	b/c connection	Structural	4	8	1 EA	stories	PSDR
B1044.093	Slender wall 12″	Structural	24.85	45.26	144 SF	undergrounds	PSDRt
B1044.103	Slender wall 18″	Structural		7.65	144 SF	stories	PSDRt
B1044.111	Slender wall 30″	Structural	4.11		144 SF	stories	PSDRt
B1049.032	s/c connection	Structural	24	24	1 EA	undergrounds	PSDR
B1049.032	s/c connection	Structural	8	4	1 EA	stories	PSDR
B2022.201	Curtain walls	Façade	103.33	103.33	30 SF	stories	PSDR
	Interiors
C1011.001a *	Wall partition (full)	Partitions	5.17	5.17	100 LF	stories	PSDR
C1011.001b	Wall partition (partial)	Partitions	5.17	5.17	100 LF	stories	PSDR
C2011.011b	Precast concrete stair	Egress	4.42		1 EA	undergrounds
C2011.011b	Precast concrete stair	Egress	1.03		1 EA	stories	PSDR
C3011.001a *	Gypsum+wallpaper	Partitions	0.78	0.78	100 LF	stories	PSDR
C3032.001d *	Suspended ceiling	Ceilings	0.83		2500 SF	stories	PFA
	Services
D1014.011	Traction elevator	MEP	8.06		1 EA	ground level	PFA
D2021.012a *	C/H potable (piping)	MEP	1.86		1000 LF	undergrounds	PFA
D2021.012a *	C/H potable (piping)	MEP	0.43		1000 LF	stories	PFA
D2021.012b *	C/H potable (bracing)	MEP	1.86		1000 LF	undergrounds	PFA
D2021.012b *	C/H potable (bracing)	MEP	0.43		1000 LF	stories	PFA
D2022.012a *	Heating hot water (p.)	MEP	1.86		1000 LF	undergrounds	PFA
D2022.012a *	Heating hot water (p.)	MEP	0.43		1000 LF	stories	PFA
D2022.012b *	Heating hot water (b.)	MEP	1.86		1000 LF	undergrounds	PFA
D2022.012b *	Heating hot water (b.)	MEP	0.43		1000 LF	stories	PFA
D2031.022a *	Sanitary waste (p.)	MEP	1.86		1000 LF	undergrounds	PFA
D2031.022a *	Sanitary waste (p.)	MEP	0.43		1000 LF	stories	PFA
D2031.022b *	Sanitary waste (b.)	MEP	1.86		1000 LF	undergrounds	PFA
D2031.022b *	Sanitary waste (b.)	MEP	0.43		1000 LF	stories	PFA
D3041.011b *	HVAC ducting	MEP	3.32		1000 LF	undergrounds	PFA
D3041.011b *	HVAC ducting	MEP	0.78		1000 LF	stories	PFA
D3041.031a *	HVAC diffusers	MEP	39.80		10 EA	undergrounds	PFA
D3041.031a *	HVAC diffusers	MEP	9.30		10 EA	stories	PFA
D3041.041a *	VAV box	MEP	8.85		10 EA	undergrounds	PFA
D3041.041a *	VAV box	MEP	2.07		10 EA	stories	PFA
D5012.013a *	Motor control center	MEP	11.51		1 EA	roof	PFA

Note: b/c: beam–column; s/c: slab–column; MEP: mechanical, electrical, plumbing. ^d1: transverse direction; ^d2: longitudinal direction. *: non-structural components that are susceptible to performance improvements.

provides a summary of the building performance model assumptions for the BAB and BAB+FVD archetypes, including the structural and non-structural components adopted in each component group (i.e., super structure, interiors, services), their fragility identifiers, component category, quantities in both directions (d1 and d2), unit, distribution of components throughout the building, and the EDPs the fragility functions were conditioned on. Moreover, the non-structural components that are susceptible to performance improvements have been identified with an asterisk (*).

Table 3. Structural and non-structural components for BAB+ENC and BAB+FVD+ENC.

Fragility ID	Component	Category	Quant. d1	Quant. d2	Unit	Location	EDP
	Super Structure
B1041.021a	b/c connection	Structural	4	4	1 EA	stories	PSDR
B1041.021b	b/c connection	Structural	4	8	1 EA	stories	PSDR
B1044.093	Slender wall 12″	Structural	24.85	45.26	144 SF	undergrounds	PSDRt
B1044.103	Slender wall 18″	Structural		7.65	144 SF	stories	PSDRt
B1044.111	Slender wall 30″	Structural	4.11		144 SF	stories	PSDRt
B1049.032	s/c connection	Structural	24	24	1 EA	undergrounds	PSDR
B1049.032	s/c connection	Structural	8	4	1 EA	stories	PSDR
B2022.201	Curtain walls	Façade	103.33	103.33	30 SF	stories	PSDR
	Interiors
C1011.001c *	Wall partition (full)	Partitions	5.17	5.17	100 LF	stories	PSDR
C1011.001b	Wall partition (partial)	Partitions	5.17	5.17	100 LF	stories	PSDR
C2011.011b	Precast concrete stair	Egress	4.42		1 EA	undergrounds
C2011.011b	Precast concrete stair	Egress	1.03		1 EA	stories	PSDR
C3011.001c *	Gypsum+wallpaper	Partitions	0.78	0.78	100 LF	stories	PSDR
C3032.003d *	Suspended ceiling	Ceilings	0.83		2500 SF	stories	PFA
	Services
D1014.011	Traction elevator	MEP	8.06		1 EA	ground level	PFA
D2021.013a *	C/H potable (piping)	MEP	1.86		1000 LF	undergrounds	PFA
D2021.013a *	C/H potable (piping)	MEP	0.43		1000 LF	stories	PFA
D2021.013b *	C/H potable (bracing)	MEP	1.86		1000 LF	undergrounds	PFA
D2021.013b *	C/H potable (bracing)	MEP	0.43		1000 LF	stories	PFA
D2022.013a *	Heating hot water (p.)	MEP	1.86		1000 LF	undergrounds	PFA
D2022.013a *	Heating hot water (p.)	MEP	0.43		1000 LF	stories	PFA
D2022.013b *	Heating hot water (b.)	MEP	1.86		1000 LF	undergrounds	PFA
D2022.013b *	Heating hot water (b.)	MEP	0.43		1000 LF	stories	PFA
D2031.023a *	Sanitary waste (p.)	MEP	1.86		1000 LF	undergrounds	PFA
D2031.023a *	Sanitary waste (p.)	MEP	0.43		1000 LF	stories	PFA
D2031.023b *	Sanitary waste (b.)	MEP	1.86		1000 LF	undergrounds	PFA
D2031.023b *	Sanitary waste (b.)	MEP	0.43		1000 LF	stories	PFA
D3041.011c *	HVAC ducting	MEP	3.32		1000 LF	undergrounds	PFA
D3041.011c *	HVAC ducting	MEP	0.78		1000 LF	stories	PFA
D3041.031b *	HVAC diffusers	MEP	39.80		10 EA	undergrounds	PFA
D3041.031b *	HVAC diffusers	MEP	9.30		10 EA	stories	PFA
D3041.041b *	VAV box	MEP	8.85		10 EA	undergrounds	PFA
D3041.041b *	VAV box	MEP	2.07		10 EA	stories	PFA
D5012.013c *	Motor control center	MEP	11.51		1 EA	roof	PFA

Note: b/c: beam–-column; s/c: slab–-column; MEP: mechanical, electrical, plumbing. ^d1: transverse direction; ^d2: longitudinal direction. *: improved non-structural components.

, on the other hand, provides a summary of the building performance model assumptions for the BAB+ENC and BAB+FVD+ENC archetypes. For these buildings, the non-structural components previously identified were replaced by those with enhanced seismic behavior. For example, conventional full-height wall partitions that are fixed below and above (C1011.001a) were substituted with wall partitions that include a slip track above with returns (C1011.001c). Suspended ceilings with vertical supports only (C3032.001d) were replaced by those with vertical and lateral supports (C3032.003d). An unanchored motor control center that is not vibration-isolated (D5012.013a) was substituted by equipment that is either hard-anchored or is vibration-isolated with seismic snubbers/restraints (D5012.013c).

Table 4. Damage state definitions.

Fragility ID	Component	DS1 (θ)	β	DS2 (θ)	β	DS3 (θ)	β
	Interiors
C1011.001a *	Wall partition (full)	0.005 rad	0.40	0.010	0.30	0.021	0.20
C1011.001c *		0.004	0.45	0.011	0.35	0.019	0.25
C3011.001a *	Gypsum+wallpaper	0.0021 rad	0.60
C3011.001c *		0.0020	0.70
C3032.001d *	Suspended ceiling	0.56 g	0.25	1.08	0.25	1.31	0.25
C3032.003d *		1.09	0.30	1.69	0.30	1.91	0.30
	Services
D2021.012a *	C/H potable (piping)	1.50 g	0.40	2.60	0.40
D2021.013a *		2.25	0.40	4.10	0.40
D2022.012a *	Heating hot water (p.)	0.55 g	0.50	1.10	0.50
D2022.013a *		0.55	0.50	1.10	0.50
D2031.022a *	Sanitary waste (p.)	1.20 g	0.50
D2031.023a *		3.00	0.50
D3041.011b *	HVAC ducting	1.50 g	0.40	2.25	0.40
D3041.011c *		1.50	0.40	2.25	0.40
D3041.041a *	VAV box	1.90 g	0.40
D3041.041b *		1.90	0.40
D5012.013a *	Motor control center	0.73 g	0.45
D5012.013c *		2.50	0.4

Note: Median (θ) and dispersion (β) values from FEMA P-58 database [10]. *: improved non-structural components. Units: g = gravity.

compares the median (θ) and dispersion (β) values of the damage state fragility curves for some conventional and enhanced non-structural components. These values are from the FEMA P-58 database.

PRDR values were also included in the analysis to account for cases where the buildings are assumed to be damaged beyond repair and need to be demolished. The distribution of PRDR is highly variable since it is the result of pure nonlinear structural behavior. Hence, the probability of demolition was estimated as a function of the PSDR by a cumulative lognormal distribution with a median value of 1% and a dispersion of 0.3, following the recommendations of FEMA P-58. However, after the Chilean M_w 8.8 2010 earthquake [6], many damaged RC buildings with residual drift ratios larger than 1% were stabilized and repaired [32]. Because of this observation, a median value of 1.5% was assumed, but further studies will be needed to refine this assumption. Building-specific collapse fragilities were adopted from the IDAs (Figure 3c,d) to determine varying probabilities of collapse at each intensity. Quantities in the transverse and longitudinal directions were considered. The EDPs were assumed to be the same in both directions as a conservative criterion.

Using this information, the methodology implemented in the software PELICUN v2.5 [33] accounted for the fragility functions to estimate the probability of damage for individual components (e.g., drift-sensitive or acceleration-sensitive components). Each damage state was associated with specific repair costs from the consequence functions for the economic loss, and these were aggregated to calculate the total direct economic loss, typically presented as a probabilistic distribution of repair costs, summarized by the expected (mean) value at each S_a(T₁) intensity.

Finally, for the FEMA P-58 time-based assessment, economic losses for the range of intensities (i.e., loss function), and their probabilities were weighted by the hazard curve developed from the PSHA. By integrating these weighted losses over all hazard levels, the EAL was computed, providing a measure of the annualized seismic risk. The EAL represents the average yearly economic loss due to earthquakes over the economic life of the building.

3.3. Downtime Assessment

The novel framework proposed by Molina Hutt et al. [13], which offers a probabilistic approach to model the post-earthquake recovery of buildings and to assess resilience metrics, was adopted. Downtime (i.e., recovery time) included the time for mobilizing resources after an earthquake and the time for conducting necessary repairs. The framework expands the FEMA P-58 [10] and the Resilience-based Earthquake Design initiative (REDi) [34] methodologies by modeling temporal buildings recovery timelines to target recovery states. Two target states to track the post-earthquake usability of the archetypes were considered: (i) reoccupancy (RO) and (ii) functional recovery (FR). RO was defined as a post-earthquake recovery state where the building has ensured pre-event habitability criteria. FR, on the other hand, was defined as a recovery state in which the building is sufficiently maintained or restored to support its pre-earthquake functionality. These states provide a strategic view of recovery trajectories, offering insights into intermediate milestones rather than focusing solely on the end state of full recovery.

According to the framework, it is necessary to probabilistically model the delays that hinder the initiation or progress of repairs (i.e., impeding factors). Key factors include the time required for inspection, stabilization, engineering and permitting, contractor mobilization, and securing financing. The delays, represented by lognormal distributions to account for variability and uncertainty, were calibrated with the scarce empirical information from the M_w 8.8 2010 Chile earthquake [6], where some buildings experienced significant damage and were repaired or demolished. Financing delays, which depend on factors such as the economic loss ratio and the type of funding mechanism, were adjusted to represent the local politics about insurance, public loans, and private loans.

Repair endeavors were organized into seven sequences (e.g., structural, interior, exterior, mechanical, electrical, elevator, and staircase repairs), which were grouped into parallel repair paths. It was assumed that repairs progress in phases, with multiple floors addressed simultaneously to reduce downtime. This strategy is supported by observations from the 1994 Northridge earthquake, where contractors typically repaired structural elements, elevators, and staircases concurrently, followed by interior repairs [35].

Using this information, the methodology, implemented in the software TREADS v2023 [13] and integrated seamlessly with PELICUN [33], provided the mean recovery trajectories to achieve both recovery states at each S_a(T₁) intensity. In particular, 2000 Monte Carlo simulations were performed to model potential recovery trajectories, integrating fragility data of damaged components and detailed repair sequencing. Each realization provided story-by-story downtime estimates (i.e., restoration/reconstruction over time), enabling a robust understanding of the uncertainties in recovery timelines. For collapsed or irreparable realizations, the downtime was equal to the building replacement time which consisted of the time for demolition and the time for reconstruction. Molina Hutt et al. [13] considered two weeks per floor to calculate the reconstruction time of US buildings. However, on the basis of the Chilean engineering construction practice, it was assumed as 3.5 weeks per floor.

In addition, two resilience-based performance metrics were calculated: (i) robustness and (ii) rapidity. The former measures the ability of a building to withstand seismic demands without significant degradation, while the latter evaluates the speed at which recovery milestones are achieved [36]. These resilience metrics enabled a comprehensive evaluation of the seismic performance of the archetype. Finally, for the time-based assessment, recovery times for the range of intensities (i.e., downtime function) and their probabilities were weighted by the hazard curve. By integrating these weighted times over the derivative of all hazard levels, the expected annual downtime (EAD) was computed, providing a measure of the annualized seismic risk of the building. The EAD reflects the average annual recovery time caused by earthquakes over the economic lifespan of a building.

4. Results and Analysis

4.1. Economic Loss Assessment Results

The expected loss ratio is one of the main outcomes of the intensity-based cost assessment. The ratio of the mean economic loss to the total construction cost defines the expected loss ratio for a specific shaking intensity. Figure 4 shows the loss functions, which represent the expected loss ratios for a range of S_a(T₁) intensities, for the benchmark archetype BAB (Figure 4a) and the archetype seismically enhanced with dampers BAB+FVD (Figure 4b). The figures also provide the loss breakdown in terms of the following: (i) the cost of building collapse; (ii) irreparable damage or demolition; and (iii) repairable damage, expressed in terms of a percentage (%) of the initial construction cost of the BAB. Similarly, Figure 5 shows the results for archetypes BAB+ENC (Figure 5a) and BAB+FVD+ENC (Figure 5b). According to the hazard curve from the PSHA, S_a(T₁) intensities could be defined for 72-, 475-, 2475-, and 4975-year return periods (i.e., SLE = 0.05 g, DBE = 0.15 g, MCE = 0.35 g, and VRE = 0.45 g, respectively).

It can be observed that the expected earthquake-induced repair costs are negligible and do not exceed 0.4% for the SLE intensity. Likewise, the expected losses are limited, with values of no more than 4.1%, for the DBE. Conversely, at MCE and VRE intensity levels, the expected costs are significantly larger. For example, the benchmark archetype BAB depicts loss ratios of 19.3% and 37.0% of the building replacement cost for the 2475 and 4975 return periods, respectively. Nonetheless, a comparison between these values and those reported in other studies on high-rise RC wall buildings under subduction seismicity [37] for the 475-year hazard level indicate smaller losses for Chilean RC buildings than for those located in Seattle (i.e., Cascadia subduction zone) which were designed to comply with the US codes. A common characteristic of Chilean RC building is the large lateral stiffness, due to stricter inter-story drift limits and base shear requirements, which yields small lateral displacement demands and, therefore, reduced damage and economic losses. Moreover, in Chilean buildings the cost contribution of non-structural components is significantly less important than in US buildings, which is another reason why economic losses are smaller in Chilean buildings.

Regarding the impact of FVDs, results indicate that the repair cost of the seismically improved Chilean RC dual wall–frame building with energy-dissipating devices (i.e., BAB+FVD) is 30–40% less than that of the conventional design (i.e., BAB). Certainly, this reduction is more meaningful at extreme intensities (MCE and VRE) than at frequent intensities (SLE and DBE). For instance, the repair cost drops from 19.3% to 13.1% of the building replacement cost at the MCE intensity, and from 37.0% to 21.1% at the VRE intensity.

Regarding the influence of the enhanced non-structural components, results show slight reductions in repair costs at all intensity levels, revealing that the cost–benefit of this strategy is limited (at least for the case study considered in this study). This observation must be regarded cautiously, since further studies will be needed to consider RC buildings with different heights, seismicity levels, and soil types. Previous studies on commercial buildings with more flexible structures, such as steel moment-resistant frame buildings [19], found more significant impacts due to the enhancement of the seismic performance of non-structural components such as partition walls.

From the loss breakdown, it is identified that the major contributor to the mean loss is repairable damage to structural and non-structural components, up to S_a(T₁) = 0.80 g for the BAB and BAB+ENC archetypes and up to S_a(T₁) = 0.90 g for the archetypes equipped with FVDs. For the benchmark archetype BAB, loss ratios due to irreparable damage become meaningful at very extreme intensities, between 0.45 g and 0.70 g, whereas collapse damage is meaningful at intensities greater than 1.0 g.

Despite the scarce information available related to ground motions, it is presumed that the 2010 Chile earthquake produced peak ground accelerations from 0.20 g to 0.25 g in the commercial district of Santiago [6]. RC high-rise buildings (T₁ ~ 2.0 s, similar to the fundamental period of the archetype buildings considered in this study) could experience S_a values in the order of 0.15 g to 0.20 g. Therefore, Figure 6 and

Table 5. Loss breakdown at S_a(T₁) = 0.20 g.

Components		BAB	BAB+ENC	BAB+FVD	BAB+FVD+ENC
Non-structural –acc. sensitive	MEP	1.3	1.1	1.1	0.9
	Elevator	0.5	0.5	0.5	0.5
	Ceilings	1.0	0.9	0.8	0.7
	Subtotal =	2.8%	2.5%	2.4%	2.1%
Non-structural – drift sensitive	Stairs	0.1	0.1	0.0	0.0
	Partitions	1.3	0.8	0.8	0.5
	Façade	0.2	0.2	0.1	0.1
	Subtotal =	1.6%	1.1%	0.9%	0.6%
NSC Subtotal =		4.4%	3.6%	3.3%	2.7%
Structural	s/c joints	2.0	2.0	0.8	0.8
	Walls	0.0	0.0	0.0	0.0
	b/c joints	1.1	1.1	0.3	0.3
SC Subtotal =		3.1%	3.1%	1.1%	1.1%
Irreparable		0.0	0.0	0.0	0.0
Collapse		0.0	0.0	0.0	0.0
TOTAL =		7.5%	6.7%	4.4%	3.8%

Note: MEP: mechanical, electrical, plumbing; s/c: slab–column; b/c: beam–column.

present the explicit metrics of the loss breakdown at the S_a(T₁) intensity of 0.20 g for the base building and for the seismically improved buildings. Some interesting observations can be pointed out, as follows. For the BAB, the major contributor to the repairable damage is non-structural damage, with a total value of 4.4% of the building replacement cost, whereas 3.1% is the value for structural damage (slab–column joints, shear walls, and beam–column joints). Regarding the non-structural damage, the acceleration-sensitive components (e.g., mechanical, electrical, plumbing, elevators, and ceilings) represent 2.8% of the losses, while the displacement-sensitive components (e.g., stairs, partitions, and façade) represent 1.6% of the losses. These results show a significant contribution of acceleration-sensitive non-structural components to losses in commercial buildings during the 2010 Chile earthquake, which is in agreement with studies that have evaluated in detail the economic losses actually generated by this event [1].

On the other hand, the building equipped with dampers (BAB+FVD) shows important loss reductions in the structural components (i.e., 60%), especially in the column joints, and smaller reductions for the non-structural components (i.e., reductions of 25%). The building with enhanced non-structural components (BAB+ENC) exhibits relatively small loss reductions in most of the non-structural components, but, as expected, important reductions in lightweight partitions (i.e., 40%, similar to that of the BAB+FVD). The last strategy (BAB+FVD+ENC) combines the beneficial reductions in structural and non-structural components, with significant reductions in partitions (i.e., 60%). Finally, costs related to demolition and collapse seem to be negligible at this specific S_a(T₁) intensity, and this observation agrees again with the empirical evidence observed after the 2010 Chile earthquake, where just one office building suffered a partial collapse in Concepcion [6].

Although comparisons at different intensity levels can be valuable, it should be considered that different S_a(T₁) intensities are associated with different annual exceedance rates (i.e., different probabilities of occurrence). To account for this issue, the economic losses were integrated with the absolute value of the derivative of the hazard curve to estimate the EAL, according to Equation (2). The EAL, which represents the average annual economic loss due to seismic damage, plays a crucial role in the evaluation of different enhancement strategies with FVDs and/or ENCs.

$$EAL={\int }_0^{\infty }E\left[L\left|IM=im\right.\right]\left|{\displaystyle \frac{d\lambda \left(IM\right)}{dIM}}\right| dIM,$$

(2)

where E[L|IM] represents the loss function and ${\displaystyle \frac{d\lambda \left(IM\right)}{dIM}}$ represents the derivative of the hazard curve.

Figure 7 and

Table 6. EAL breakdown in terms of percentage of BAB’s cost.

Contributor	BAB	BAB+ENC	BAB+FVD	BAB+FVD+ENC
Collapse	0.002	0.003	0.002	0.002
Irreparable	0.003	0.003	0.002	0.002
SC	0.020	0.021	0.010	0.010
NSC-DS	0.032	0.026	0.016	0.015
NSC-AS	0.067	0.054	0.064	0.054
TOTAL =	0.125%	0.106%	0.093%	0.082%
[US$/m2]	2.1	1.8	1.5	1.4

Note: SC: structural components; NSC-DS: displacement-sensitive non-structural components; NSC-AS: acceleration-sensitive non-structural components.

show the EAL as a ratio of the BAB construction cost (USD 1660 per square meter). Table 6 shows EAL values in the order of 0.125, 0.106, 0.093, and 0.082% for the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC archetypes, respectively. The strategy of incorporating FVDs into the base building gives a 25% reduction in the base EAL, while with the strategy of including ENCs a reduction of only 15% is achieved. For the strategy that combines FVDs and ENCs, the reduction is greater than 35% but at the expense of a higher initial construction cost (106% of the BAB cost, as previously assumed). Note that these results seem to be much smaller than those for steel office buildings in the USA, which have more flexible structural systems (and, consequently, a greater level of damage) as well as different building inventories, dominated by non-structural components that are more susceptive to earthquake damage than structural components [19]. For instance, for a special moment-resistant-frame building with T₁ = 1.86 s (similar to the fundamental period of the office buildings analyzed in this study), the EAL could be as large as 1.6%.

Table 6 also shows the disaggregated EAL information for the cost of collapse, irreparable damage, and repairable damage. In all cases, the contribution of collapse or demolition to losses is relatively low. Furthermore, a significant contribution of non-structural damage is evident, especially of those components sensitive to acceleration. This would indicate that Chilean stiff RC structures have important floor acceleration demands that are just slightly reduced with the incorporation of FVDs. Results revealed that the incorporation of ENCs, FVDs, and a combination of the previous two strategies achieved EAL values of 0.106%, 0.093%, and 0.082%, respectively, while the archetype base building without design improvements reached an EAL of 0.125%. The latter indicates EAL reductions (compared to the base case) of 15%, 26%, and 34% for the cases of BAB+ENC, BAB+FVD, and BAB+FVD+ENC, respectively, where the largest effect was provided by the incorporation of FVDs due to the reduced damage to both structural components and non-structural components sensitive to story drift ratios.

4.2. Downtime Assessment Results

For the intensity-based downtime assessment, the framework proposed by Molina Hutt et al. [13] was adopted to calculate the following outputs and resilience-based metrics: (i) the recovery trajectory, which shows the progress of building restoration or reconstruction over time to reach a target recovery state; (ii) the robustness; and (iii) the rapidity. As an example, Figure 8 shows the simulated recovery trajectories describing building usability at any time after an earthquake with S_a(T₁) = 0.20 g (associated with the 2010 Chile earthquake) for the four archetypes. Building usability ranges from 0% to 100%, where 100% indicates that all floors of the building have achieved the desired recovery state, and 0% indicates that none have. Specifically, Figure 8 displays the governing recovery trajectories to achieve the FR state for all 2000 realizations. The plots highlight the large uncertainty associated with downtime estimates (the 10th, 50th, and 90th percentiles are shown). From the figures, the median FR times are 212 (BAB in Figure 8a), 206 (BAB+ENC in Figure 8b), 166 (BAB+FVD in Figure 8c), and 161 (BAB+FVD+ENC in Figure 8d). Decreases in mean downtime values are noticeable when the building is equipped with FVDs (i.e., reductions of 20%). On the other hand, minimal reductions in the order of 3% are achieved when the building incorporates ENCs.

Regarding the buildings’ seismic performance in terms of robustness, which describes the probability of a building not achieving a target recovery state immediately after an earthquake, there are interesting behaviors. Results indicate that under the S_a(T₁) = 0.20 g intensity, the probabilities of not achieving RO and FR immediately after the earthquake are 100%, and 100%, respectively, for the BAB. Similar probabilities are reported for the other seismically enhanced buildings. The large outputs are mostly because of the damage to non-structural components such as mechanical–electrical–plumbing, ceilings, and partitions, which need to be repaired/replaced before occupants can reoccupy the office buildings.

Regarding the building performance in terms of rapidity, which describes the probability of not achieving a recovery state within a specified time, the results for the S_a(T₁) = 0.20 g intensity are summarized in Figure 9. Assuming as target recovery time that the buildings should be repaired within 4 months, the probabilities that the downtime required to achieve RO exceeds the target recovery time are 95% (BAB in Figure 9a), 90% (BAB+ENC in Figure 9b), 60% (BAB+FVD in Figure 9c), and 55% (BAB+FVD+ENC in Figure 9d). Moreover, the probabilities that the downtime required to achieve FR exceeds 4 months are 98% (BAB), 97% (BAB+ENC), 87% (BAB+FVD), and 85% (BAB+FVD+ENC). It is worth noting that these downtime assessments are related to the resilience of the built infrastructure, where more than 4 months to achieve FR implies large social impacts.

As mentioned before, although comparisons at different intensity levels can be valuable, it should be considered that each S_a(T₁) intensity is associated with a different probability of occurrence. Similarly to economic losses, the recovery times were integrated with the absolute value of the derivative of the hazard curve to estimate the EAD, according to Equation (3). The EAD, which represents the average annual downtime due to seismic damage, also plays a key part in the evaluation of different enhancement strategies with FVDs and/or ENCs.

$$EAD={\int }_0^{\infty }E\left[DT\left|IM=im\right.\right]\left|{\displaystyle \frac{d\lambda \left(IM\right)}{dIM}}\right| dIM,$$

(3)

where E[DT|IM] represents the downtime function and ${\displaystyle \frac{d\lambda \left(IM\right)}{dIM}}$ represents the derivative of the hazard curve.

Table 7. EAD in terms of percentage of building replacement time in absolute days.

	BAB	BAB+ENC	BAB+FVD	BAB+FVD+ENC
RO state [%]	0.65	0.39	0.49	0.26
[days]	6.6	4.0	5.0	2.6
FR state [%]	0.94	0.90	0.85	0.82
[days]	9.5	9.1	8.6	8.3

shows the EAD as a ratio of the BAB reconstruction time and also the time in units of days. In this study, the reconstruction time of the archetype building was calculated assuming that the construction of each story takes 24.5 days. This results in a total of 465.5 days for the 16-story archetypes. In addition, the median building demolition time was estimated as 550 days based on the data from the demolition times of the Chilean buildings that suffered severe damage in the 2010 earthquake. Therefore, the resultant total building replacement time is 1015.5 days. The table shows the EAD values to achieve reoccupancy RO state to be equal to 6.6, 4.0, 5.0, and 2.6 days for the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC archetypes, respectively. In addition, the EAD values to achieve FR state are equal to 9.5, 9.1, 8.6, and 8.3 days for the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC, respectively. A comparison between EAD values to achieve FR indicates that the strategy of incorporating FVDs gives a 10% reduction in the base EAD, while the strategy of including ENCs gives a reduction of only 4%. For the strategy that combines FVDs and ENCs the reduction is about 13% of the base EAD. These results, which correspond to median predictions of downtime, highlight how damage that requires only moderate repair costs could result in excessive downtimes, leading to the displacement of occupants with associated indirect costs.

4.3. Earthquake-Induced Life-Cycle Cost

Time-based assessments were performed to evaluate the probable performance of the four archetypes over a specified period of time (e.g., t = 50 years, assumed equal to the building economic life in this study) considering all earthquakes that could occur in that time period, and the probability of occurrence associated with each earthquake. Mean annual values such as EAL and EAD were taken from the analysis described in the former paragraphs. These annual values were useful for a benefit–cost study because they can be associated with reasonable insurance premiums. From the EAL, the annual direct economic costs were USD 2.1, USD 1.8, USD 1.5, and USD 1.4 per square meter for the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC archetypes, respectively. It was assumed that the cost associated with each day of downtime was USD 0.5 per square meter in lost rent. This results in annual indirect economic costs of USD 4.8, 4.6, 4.3, and 4.2 per square meter for the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC archetypes, respectively. Combining direct and indirect costs results in a total expected annual loss (EALt) of USD 6.9, 6.4, 5.8, and 5.6 per square meter for the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC archetypes, respectively.

The benefit–cost study involved a comparison between the net present value of average annual losses that are avoided through enhanced seismic performance and the initial construction costs associated with providing enhanced seismic performance. Using the discount rate (DR) concept, according to Equation (4), the EALt of the next 50 years can be brought to present value (PV) and combined with the initial construction cost (CC).

$$PV\left(EALt\right)={\sum }_{i=1}^t{\displaystyle \frac{EALt}{{\left(1+DR\right)}^i}},$$

(4)

The expected life-cycle cost (ELCC), a parameter with which decisions can be made, is calculated as shown in Equation (5). The ELCC facilitates the economic evaluation of the base archetype BAB and of each seismic performance improvement strategy.

$$ELCC=CC+PV\left(EALt\right),$$

(5)

Given the variations in interest rates worldwide as well as in the Chilean economy in recent years (post-pandemic era), a parametric study was carried out taking DRs around 6% ± 3%. The initial CCs of the BAB, BAB+ENC, BAB+FVD, and BAB+FVD+ENC archetypes were assumed equal to USD 1660, 1710, 1710, and 1755 per square meter, respectively. Figure 10 shows the variation in the ELCC at different DRs for the four archetype buildings. Moreover,

Table 8. Initial construction cost and expected life-cycle cost.

Cost [US$/m2]	BAB	BAB+ENC	BAB+FVD	BAB+FVD+ENC
CC	1660	1710	1710	1755
ELCC (DR = 9%)	1734	1778	1772	1818
(DR = 6%)	1767	1809	1799	1845
(DR = 3%)	1836	1873	1857	1901

Note: CC: construction cost; ELCC: expected life-cycle cost; DR: discount rate.

presents a summary of the costs. Regarding the outcomes, for all the range of DRs the BAB base building has the lowest ELCC, while the highest costs belong to the building that incorporates both FVDs and ENCs. For the BAB+FVD strategy, the ELCCs were slightly lower than those of the BAB+ENC strategy. For the largest DRs, costs were very similar for both the BAB+FVD and BAB+ENC strategies.

It is worth mentioning that even though the previous ELCCs have been analyzed based on varying levels of DRs (0% to 9%) and a reasonable number of ownership years (50 years, which is common for office building occupancy), the BAB base building always presented the lowest ELCC. This can be explained by the already-high seismic performance of code-conforming RC buildings in Chile (particularly due to their high stiffness, which reduces losses from damage to non-structural drift-sensitive components) and the lower cost fraction of non-structural components compared to other locations (e.g., California).

However, it is important to note that the decision to choose between these design strategies is not solely based on the ELCC; it might also involve stakeholders’ risk preferences and indirect costs (e.g., downtime) during the decision-making process. For example, a risk-averse stakeholder might still choose the strategy of BAB+FVDs to avoid low-probability/high-consequence losses and reduce the expected downtime, whereas a risk-neutral stakeholder would choose the BAB strategy. Further research is needed to incorporate these preferences.

Finally, since current Chilean seismic design provisions primarily focus on life safety and structural integrity during seismic events, with limited emphasis on recovery times, the resilience-based seismic design insights obtained in this study could play a crucial role in updating these provisions in Chile and other seismically active regions.

In the future, seismic design provision in Chile might include considerations for direct economic losses and downtime (it is worth noting that still Chilean codes do not include consideration for maximum probabilities of collapse, which are already included in ASCE 7-2022). As an initial step, quantifying economic losses and downtime for high-importance or high-occupancy buildings, such as hospitals, could become a requirement. This process would enable the establishment of future performance limits that balance enhanced resilience with reduced economic impacts and manageable initial construction costs.

By integrating these resilience-based principles into seismic design provisions, Chile and other seismically active regions can transition toward more comprehensive frameworks that address both immediate safety and long-term resilience, ensuring economic sustainability and faster recovery after seismic events.

5. Conclusions

This research evaluated the influence of three alternative design strategies to improve the seismic performance of code-conforming dual wall–frame buildings subjected to subduction seismicity. In particular, the impacts of the following strategies for the benchmark archetype building (BAB) were assessed: (i) fluid viscous dampers (FVDs); enhanced non-structural components (ENCs); and (iii) a combination of strategies (i) and (ii). The study estimated and compared the seismic performance of typical, representative Chilean RC dual wall–frame buildings in terms of economic losses and downtime, which are quantities that are meaningful to improve the resilience of the built environment towards sustainable development. As a case study, a reinforced concrete (RC) high-rise (16-story) office building was considered. Following the FEMA P-58 and REDi methodologies, loss and downtime assessments were conducted to calculate seismic performance metrics such as expected annual loss (EAL), expected annual downtime (EAD), recovery trajectories, robustness, rapidity, and expected life-cycle cost (ELCC) for each strategy. The summary of the outcomes is as follows:

• For all archetypes, expected direct economic losses are negligible at the SLE intensity level (≤0.4% of the building replacement cost) and are minimal (≤4.1%) at the DBE level. For the benchmark archetype, losses are considerable at the MCE and VRE levels (19.3% and 37.0%, respectively). For the archetype with FVDs, there are loss reductions of approximately 40% at most S_a(T₁) intensities.
• The archetype equipped with FVDs showed EAL reductions in the order of 25% with respect to a conventional design (from 0.125% to 0.093%). In contrast, slight decreases in the order of 12% were found for the archetypes that incorporate ENCs (from 0.125% to 0.106%, and from 0.093% to 0.082%). This reveals that there may be a limited cost–benefit trade-off when applying this latter strategy, at least for this type of RC building.
• Regarding the building performance in terms of robustness at the S_a(T₁) = 0.20 g intensity level (i.e., the 2010 Chile earthquake), the probabilities of not achieving RO and FR states immediately after the earthquake are equal to 100% for all the archetypes. The large outputs are mostly because of damage in non-structural components such as mechanical–electrical–plumbing, ceilings, and partitions, which need to be repaired/replaced before residents can reoccupy office buildings.
• Regarding the building performance in terms of rapidity at the same intensity, the probabilities that the downtime required to achieve FR exceeds 4 months are 98% (BAB), 97% (BAB+ENC), 87% (BAB+FVD), and 85% (BAB+FVD+ENC). It is worth noting that these downtime assessments are related to the resilience of the built infrastructure, where more than 4 months for FR implies large social impacts.
• The archetype equipped with FVDs showed EAD reductions in the order of 10% with respect to a conventional design (from 9.5 to 8.6 days) to achieve the functional recovery (FR) state. On the other hand, slight reductions in the order of 4% are reported for the archetypes that incorporate ENCs (from 9.5 to 9.1 and from 8.6 to 8.3 days). This reveals that there may be a limited cost–benefit trade-off when applying this latter strategy, at least for this type of RC building.
• Results indicate that in Chilean RC dual wall–frame buildings, incorporation of a seismic protection system such as FVDs could lead to higher levels of resilience. Nevertheless, the observations made in this study must be judged with caution since further studies will be needed to consider buildings with different heights, seismicity levels, soil types, and possibly a more refined review of the initial assumptions.
• Finally, the resilience-based insights from this study could guide updates to include considerations for direct economic losses and downtime in current Chilean seismic design provisions (that focus solely on life safety and structural integrity, lacking performance-based objectives, including resilience). As first step, quantification of losses and downtime for critical buildings, such as hospitals, could establish performance limits that balance resilience, economic impact, and construction costs. Incorporating these principles would enable Chile and other regions to adopt comprehensive frameworks that address both safety and long-term recovery, ensuring economic sustainability and faster post-earthquake recovery.

All these conclusions are based on results obtained from a large number of nonlinear response history analyses. In order to make possible such large amount of analyses, some simplifying assumptions had to be made, and some features of the seismic response (e.g., the torsional response) were not taken into account. The simplifying assumptions adopted in this study were carefully evaluated in order to minimize their impact on the results. However, although not expected to be significant, such an impact will nevertheless be evaluated in future studies.