Study on downtime calculation considering a series of seismic sequences in resilience performance assessment of RC buildings

Current seismic design primarily targets life safety under rare major earthquakes, while recent societal demands emphasize seismic resilience, particularly timely functional recovery and reduction of downtime. However, most existing downtime assessment frameworks implicitly assume that damage is governed by a single mainshock and do not explicitly track cumulative deterioration caused by repeated moderate earthquakes over the building’s service life. This omission can underestimate the downtime associated with the life-cycle maximum event. This study proposes a downtime evaluation method for RC office buildings that explicitly incorporates (1) delay times due to impeding factors, (2) repair times of structural components, (3) repair times of nonstructural components, and, as a distinctive feature, (4) cumulative deterioration induced by annual moderate earthquakes within a life-cycle seismic hazard framework. Five-, seven-, and ten-story shear-spring models with three ultimate ductility levels (strength-type, standard-type, and ductile-type) are analyzed under annual ground motion sequences consistent with the seismic hazard at Aobayama, Sendai, including both a single strong-motion record and a combined strong-motion sequence from the 2016 Kumamoto Earthquake. Structural damage is quantified using the Park and Ang damage index, nonstructural damage is evaluated via fragility curves, delay times are modeled with lognormal distributions for each impeding factor, and downtime is obtained from repair sequencing along multiple repair paths using Monte Carlo simulations. Results show that cumulative deterioration from moderate earthquakes, when propagated to the time of the life-cycle maximum event, systematically increases both damage indices and downtime compared with conventional mainshock-only assessments. Ductile-type models effectively limit this downtime growth in low-to mid-rise buildings. In particular, for the seven-story building subjected to the combined strong-motion sequence, converting the frame from the strength-type to the ductile-type reduces the estimated downtime by 17.3%. Although the average downtime is similar among the three building heights, the story-wise concentration of damage and the configuration of repair phases differ, indicating that explicitly modeling cumulative deterioration, together with optimized repair phasing and ductile design, is essential for reliably assessing and reducing downtime in resilience-oriented design of RC buildings.

1 Introduction

In recent years, the performance required of buildings post-earthquake has increased. Beyond ensuring a minimum level of safety during seismic events, attention has shifted toward resilience, with an emphasis on functionality recovery and restoration time. Bruneau and Reinhorn (2006) identified four key attributes of resilience: Robustness, the ability to withstand external forces without significant performance degradation; Redundancy, the availability of alternative load paths or functions; Resourcefulness, the capacity to identify problems and mobilize resources for recovery; and Rapidity, the ability to restore functionality in a timely manner. Figure 1 illustrates Bruneau’s resilience triangle, which depicts the time-varying functionality of a building from before an earthquake through to full recovery. Enhancing resilience involves both minimizing the extent of damage and disruption caused by the earthquake and reducing the time required for recovery. In terms of the resilience triangle, this corresponds to minimizing the shaded triangular area.

Figure 1

Figure 1. Conceptual representation of Bruneau’s resilience triangle.

However, the quantitative evaluation of building resilience remains challenging, and no universally applicable assessment methodology has yet been established. In particular, when estimating the recovery time (downtime) of buildings after earthquakes, recent studies have highlighted the importance of accounting for delay times, such as financing and mobilization of contractors (hereafter referred to as “impeding factors”), as well as the repair times of nonstructural components in addition to structural components (Gonzalez et al., 2023).

This study focuses on evaluating the seismic resilience of reinforced concrete (RC) buildings by developing a downtime-based assessment framework that explicitly integrates four critical aspects: (i) repair times of structural components, (ii) delay times caused by impeding factors, (iii) repair times of nonstructural components, and (iv) cumulative deterioration induced by moderate earthquakes during the service period. By applying this integrated framework to RC building models under various parameter conditions, the influence of these factors on resilience can be quantified in terms of total downtime. Specifically, the effects of cumulative deterioration on downtime are investigated by comparing two scenarios. One scenario is a major earthquake occurring immediately after construction, and the other is a major earthquake striking after years of service, when cumulative deterioration from moderate events has accumulated. In both scenarios, downtime is evaluated only for the maximum earthquake during its service-life, with the event reflecting the cumulative deterioration of the building. Furthermore, with a view toward long-term building use, the analysis assumes that the damaged building is repaired and remains in service, even in cases where demolition and replacement might be a more realistic option in practice, so that the influence of additional repair demand on the estimated downtime can be consistently quantified.

2 Current state of research on resilience performance evaluation

Carlos Molina Hutt et al. (2022) advanced the knowledge established in FEMA P-58 (FEMA, 2018) and the REDi methodology (Almufti and Willford, 2013) by proposing a downtime assessment framework that defines the recovery process of buildings in five stages. Their study, which analyzed a 12-story reinforced concrete (RC) residential building in Seattle, demonstrated that underground motions with a 475-year return period, the probability of requiring more than 4 months to achieve functional recovery reached 88%. The results highlighted that delay times due to impeding factors and the sequencing of repair activities contributed significantly to downtime.

In a preceding study by the same research group, a 42-story RC shear wall residential tower and a 40-story buckling-restrained braced frame (BRBF) office tower in San Francisco were examined (Molina Hutt et al., 2021). The median time to functional recovery was estimated to be 7.5 months for the residential building and 5.5 months for the office building. This research suggested that the integration of recovery planning at the design stage—such as establishing pre-event inspection frameworks and streamlining permitting processes—can substantially reduce functional recovery time.

At the urban scale, Hulsey et al. (2022) evaluated the resilience of downtown San Francisco by simulating a magnitude 7.2 earthquake on the San Andreas Fault. Their study focused on 87 of the 1,078 high-rise buildings in the area, quantifying the impact of post-earthquake safety cordons (restricted access zones) not only on individual buildings but also on the recovery of the city as a whole. The findings revealed that, within the first year of recovery, approximately one-third of lost office functionality days were attributable to access restrictions caused by collapse risks. This emphasized the necessity of strategic urban-scale measures in addition to improving individual building performance.

In another study, Gonzalez et al. (2023) evaluated RC school facilities in Puebla, Mexico, by comparing resilience values under three scenarios: (a) excluding both delay times from impeding factors and repair times of nonstructural components, (b) considering only delay times, and (c) considering both delay times and nonstructural repair times (Figures 2a–c). The results indicated that neglecting these factors leads to an overestimation of resilience, underscoring the importance of including them in downtime assessments.

Figure 2

Figure 2. Illustration of functionality recovery trajectories. (a–d).

In parallel, Akehashi and Takewaki (2022) proposed an availability-based resilience evaluation framework for elastic–plastic high-rise buildings subjected to resonant long-duration ground motions. Their model jointly considers structural frames, building services, and external factors such as lifelines and redundancy of facility systems, and formulates probabilistic indices including the expected total recovery time, the availability of structural and nonstructural systems immediately after an earthquake, and maximum-likelihood recovery curves. Through numerical analyses using pseudo-multi impulse inputs and passive viscous dampers, they demonstrated that enhancing structural control and providing redundant facility systems can significantly improve building resilience and reduce uncertainty related to repair manpower.

Field survey–based research in Japan by Nishino (2024) analyzed 250 cases of downtime related to building services (electrical, plumbing, HVAC, fire protection, etc.) damaged during the 2016 Kumamoto Earthquake. The study categorized recovery trends by service type and building use. For instance, medical and welfare facilities and hotels tended to recover relatively quickly, while among service types, electrical systems were restored the fastest, followed by plumbing, HVAC, and fire protection systems. The findings highlighted that even when structural damage is minor, prolonged functional disruption may occur due to service failures, pointing to the need for future consideration of nonstructural components in design and maintenance planning.

Additionally, Caruso et al. (2023) proposed a decision-making framework that integrates earthquake losses with energy consumption, contributing to the selection of optimal retrofit strategies by considering multiple performance indicators such as repair cost, CO₂ emissions, human impact, and payback period. Similarly, You et al. (2023) evaluated economic losses and downtime for a 20-story cross-laminated timber (CLT) shear wall residential building, demonstrating the relevance of resilience considerations even for emerging construction materials.

In summary, ongoing research in resilience performance evaluation is moving toward comprehensive frameworks that consider structural and nonstructural component performance, impeding-factor delays, environmental impacts, and city-scale effects, most of which build on the foundational methodologies of FEMA P-58 and REDi. However, these studies predominantly focus on single major earthquake events or sequences without explicitly quantifying the deterioration accumulated during the service life. Distinct from these approaches, the present study advances the resilience evaluation of RC buildings by extending the recovery trajectories considered in prior studies (Figures 2a–c) to explicitly incorporate the effects of cumulative deterioration induced by repeated moderate earthquakes—particularly those exceeding the short-term allowable stress limit—on downtime estimation after a major earthquake (Figure 2d). This enables a more realistic assessment of functional recovery for buildings that experience long-term use under frequent seismic excitation.

3 Downtime evaluation methods

3.1 Overview of the analytical model

The target structures are assumed to be five-, seven-, and ten-story reinforced concrete (RC) office buildings. Each structure is represented by a shear spring model, in which the story shear force coefficients follow the Ai distribution (Building Center of Japan, 2016). As analysis parameters, three types of building models with different ultimate story ductility factors, _uμ_i, were examined: the strength-type (_uμ_i = 1.5), the standard-type (_uμ_i = 2.0), and the ductile-type (_uμ_i = 4.0). For each building model, the story shear forces were determined using Newmark’s equal-energy rule so that the seismic performance related to structural safety would be approximately equivalent. Newmark’s equal-energy rule is expressed as:

$Q_{y1}=\frac{1}{\sqrt{2\mu -1}}\hspace{0.17em}{Q}_l=\gamma \hspace{0.17em}{Q}_l(1)$

where Q_l is the elastic response shear force, μ is the ductility factor, and Q_y1 is the yield shear force that provides equivalent energy dissipation for a given ductility factor. Here, the fractional term in Equation 1 is denoted as a positive coefficient γ. Accordingly, the yield shear force of the ith story, Q_yi, is given as Equation 2:

$Q_{yi}={f\left(\gamma \right)·C}_0·Rt·Z·Ai·\mathrm{\Sigma }Wi(2)$

where Q_yi is the yield shear force of story i, C₀ is the standard story shear coefficient (set to 1.0), R_t is the response modification factor related to the vibration characteristics, Z is the seismic zone factor, Ai is the Ai distribution factor, and ΣWi is the cumulative weight of the building above and including story i. In this formulation, the product of the structural characteristic factor D_s and the shape factor F_es from the calculation of ultimate lateral strength is regarded as a function proportional to γ, expressed as f(γ). The function f(γ) was calibrated such that the base shear coefficient of the standard-type model equals 0.3. The story shear coefficients (C_i), yield story shear forces (Q_yi), and yield linear stiffnesses (K_yi) for each building model are summarized in Table 1. For all building models, the floor weight, story height, and yield displacement were uniformly set to m_i = 500 t, h_i = 3 m, and δ_yi = 0.020 m for each story. The ultimate story ductility factor (_uμ_i) was assumed to be identical for all stories within each building model.

Table 1

Table 1. Parameters for each building model.

3.2 Downtime evaluation process

In this study, downtime is calculated through seismic response analyses under multiple ground motion scenarios expected to occur during the service life of the building. Note that downtime is evaluated only after the occurrence of the maximum earthquake during the service life; however, the cumulative deterioration caused by moderate earthquakes that occur during the service life is taken into account in this calculation. This section outlines the calculation procedure adopted in this paper, following the overall downtime evaluation process illustrated in Figure 3.

Step 1: Input ground motion model

Figure 3

Figure 3. Downtime evaluation process.

For the development of ground motion scenarios throughout the service life, the target site was set as the Aobayama Campus of Tohoku University, located in Aramaki-Aoba, Aoba Ward, Sendai City, Miyagi Prefecture. The seismic hazard data provided by the National Research Institute for Earth Science and Disaster Resilience (2024) (NIED) through the J-SHIS (Japan Seismic Hazard Information Station) were used. The seismic hazard curve employed in this study is shown in Figure 4. To establish a set of annual non-exceedance probabilities for earthquake ground motions during the service life (i.e., life-cycle ground motion scenarios), Hazen’s plotting position formula (A. Hazen, 1930) was adopted as shown in Equation 3:

$F\left(x{}_i\right)=1-\frac{i-\alpha }{N+1-2\alpha }(3)$

where N is the total number of observations, i is the rank of the observation when ordered in descending magnitude, x_i is the ith observation, F(x_i) is the non-exceedance probability, and α is a constant (0.5 in Hazen’s method). Assuming that the occurrence of earthquake ground motions follows a Poisson process as a stationary renewal process, the following relationship holds as shown in Equation 4:

$P\left(i\right)=1-\mathit{exp}\left[-\left(1-F\left(x{}_i\right)\right)T\right](4)$

where P(i) is the T-year exceedance probability of the ith largest observation. It should be noted that when Hazen’s method (α = 0.5) is applied, the return period of the maximum event among N years of annual maxima is fixed at 2N years. However, previous studies (Takahashi and Shiohara, 2004) have shown that the return period of the maximum ground motion during a building’s service life N is not necessarily constrained to 2N years, and alternative methods to handle this as a probability group have been proposed. Accordingly, in this study, α was determined such that the occurrence probability of the maximum ground motion during N years corresponds to the T-year exceedance probability P (1), leading to Equation 5:

$\alpha =\frac{\left(N+1\right)\mathit{ln}\left(1-P\left(1\right)\right)+T}{2\mathit{ln}\left(1-P\left(1\right)\right)+T}(5)$

Figure 4

Figure 4. Seismic hazard curve at Aobayama, Sendai.

By applying Equation 3, it is thus possible to obtain a set of N annual non-exceedance probabilities that includes the ground motion corresponding to the T-year exceedance probability 100P (1)%. This set represents the life-cycle ground motion intensity scenarios.

In this study, a service life of 100 years was assumed so as to represent a building intended to remain in use over a long period. Using Equation 3 in combination with the seismic hazard curve shown in Figure 4, groups of non-exceedance probabilities and peak ground velocities (at engineering bedrock) were computed. Here, the total number of events was set to N = 100, and the ground motion corresponding to the T-year exceedance probability of 100P (1)% was defined as the maximum ground motion during the period. Three representative scenarios were considered: 100-year exceedance probabilities of 75.1%, 19%, and 9.8%, corresponding to return periods of 72, 475, and 970 years, respectively.

For the time-history records, the strong ground motion observed in Kasuga, Nishi Ward, Kumamoto City, during the Kumamoto Earthquake on 14 April 2016, was employed as the reference record (Figure 5a). For each life-cycle scenario, this record was scaled so that the intensity of the input motion is consistent with the peak ground velocity (PGV) at engineering bedrock, x_w, obtained from the seismic hazard curve in Figure 4. Specifically, the unscaled acceleration record was first input to a single-degree-of-freedom oscillator with a natural period of 10 s and a high damping ratio, and the maximum pseudo-velocity response, v_max, was evaluated. The acceleration time history was then multiplied by a constant factor such that v_max becomes proportional to the target PGV x_w. In this way, the scaled record reproduces the hazard-consistent PGV while preserving the waveform characteristics of the observed motion.

Figure 5

Figure 5. Strong-motion acceleration records from the 2016 Kumamoto earthquake sequence. (a) Single earthquake. (b) Combined earthquakes.

In addition, to simulate the case where a building is struck by successive large earthquakes shortly after the annual maximum event, a “combined strong motion” was constructed by concatenating the ground motions observed in Kasuga on April 14 and 16, 2016, during the Kumamoto Earthquake sequence (Figure 5b). In Figures 5a,b, the effective duration of the strong ground motion (for the mainshock in (a) and for the aftershock in (b)) is also indicated, evaluated following Trifunac and Brady (1975) as the time interval between the instants when the cumulative integral of the squared acceleration reaches 5% and 95% of its final value. Because the subsequent downtime evaluation in this study utilizes only the peak response values (damage index D, inter-story drift ratio IDR, and peak floor acceleration PFA) in each year, this simplification is considered acceptable. Furthermore, to address the potential influence of the inter-event standstill period, a sensitivity analysis was conducted by inserting a sufficiently long zero-acceleration interval between the two records. Because the downtime calculation uses story-wise response quantities, the comparison was performed using all story-wise values of D, IDR, and PFA at the year when the maximum event occurs in each scenario. The results are presented in Supplementary Figures S1a–c (parity plots), showing that the story-wise responses are generally consistent between the no-gap concatenation and the gap-inserted input. Even in the cases with the largest deviations (maximum relative differences of 16.0151%, 12.0954%, and 10.4154%), the resulting downtime differences were only 5 days (467 vs. 462 days), 2 days (495 vs. 497 days), and 3 days (495 vs. 498 days), respectively (including the case where D crosses 1.0). To illustrate the frequency characteristics of the input motions, Figures 6a,b present 5%-damped pseudo-acceleration (pSA) and pseudo-velocity (pSV) response spectra of the scaled single and combined records. The elastic fundamental periods of the 5-, 7-, and 10-story models are also indicated in the figure for reference.

Figure 6

Figure 6. 5%-damped pseudo-acceleration and pseudo-velocity response spectra. (a) 5%-damped pseudo-acceleration spectra. (b) 5%-damped pseudo-velocity spectra.

In this study, to focus on the effect of continuous input of strong motions and cumulative deterioration under a controlled setting, an idealized repeated-sequence approach was adopted. The 2016 Kumamoto earthquake record was used to construct a “single” strong motion and a “combined” strong motion, in which the same record is applied sequentially according to the life-cycle ground motion scenarios. By using the same record repeatedly, the variability in the frequency content among different mainshock–aftershock pairs is eliminated, and the influence of sequential loading can be isolated more clearly.

It is acknowledged that this repeated approach does not fully represent the diversity of actual mainshock–aftershock or doublet-earthquake sequences, in which the spectral characteristics of successive events may differ significantly (Ruiz-García and Negrete-Manriquez, 2011; Yaghmaei-Sabegh and Ruiz-García, 2016). Therefore, in addition to this idealized setting, supplementary analyses using multiple strong-motion records are conducted and discussed in Section 4.4, in order to examine whether the main findings of this study hold when different ground motion characteristics are considered. To avoid ambiguity, the supplementary analyses in Section 4.4 are conducted in two complementary ways: (i) a record-by-record evaluation in which each observed waveform is repeatedly applied year-by-year within a life-cycle scenario and the results are averaged across input motions (Supplementary Figure S2), and (ii) an additional random-year record selection analysis in which the waveform is randomly sampled year-by-year from multiple observed records within a single 100-year simulation (Supplementary Figure S32).

Step 2: Structural seismic response model

The building was modeled as a multi-degree-of-freedom equivalent shear spring system. A tri-linear restoring force model was adopted for the story hysteresis characteristics, while the Takeda model (T. Takeda et al., 1970) was employed to represent the hysteretic rule. Structural damping was assumed to be proportional to instantaneous stiffness, with a damping ratio of 5% applied to the fundamental natural period. For each story, the cracking strength was taken as one-third of the yield strength. The post-yield stiffness degradation ratio was set to 0.3, and the stiffness after yielding was assumed to be 0.01 times the initial stiffness. Furthermore, when the story displacement exceeded the cracking point but did not reach the yield deformation before the next seismic excitation, it was assumed that the initial stiffness degraded toward the maximum experienced displacement. In cases where the yield point was exceeded, it was assumed that structural damage would be repaired, restoring the system to its original performance.

Step 3: Building response - Delay time model

Following previous studies (Gonzalez et al., 2023; Molina Hutt et al., 2022), the delay times caused by impeding factors were assumed to follow lognormal cumulative distribution functions. The median and variance of the lognormal distribution were set as shown in Table 2 (Gonzalez et al., 2023; Molina Hutt et al., 2022). Specifically, these parameter values were defined by combining the delay-time parameters proposed in the two studies, which were originally developed based on post-earthquake recovery data for Mexican school buildings and North American building stock. In the present study, they are used as a realistic numerical example to investigate downtime for RC office buildings in Sendai, rather than as Japan-specific calibrated values, and the development of delay-time models calibrated to Japanese buildings and institutional conditions is identified as an important topic for future research. The classification of delay times was based on the maximum inter-story drift ratio observed among all stories. The delay time curves for each impeding factor as a function of inter-story drift ratio are shown in Figures 7a–f, 8a–e. According to the specified lognormal cumulative distribution functions, the expected delay time was calculated for each type of impeding factors. For each repair sequence, the delay time was defined as the sum of the inspection time and the maximum value among the times required for expert judgment and approval, contractor mobilization, and financial arrangements.

Step 4: Building response - Damage model

Table 2

Table 2. Median and variance of delay time.

Figure 7

Figure 7. Delay-time curves for each impeding factor. (a) Inspection, Engineering, Permitting (≤0.005rad). (b) Inspection, Engineering, Permitting (≤0.01rad). (c) Inspection, Engineering, Permitting (≤0.015rad) (d) Inspection, Engineering, Permitting (≤0.02rad). (e) Inspection, Engineering, Permitting (>0.02rad) (f) Financing.

Figure 8

Figure 8. Delay-time curves for Contractor mobilization. (a) Contractor mobilization (≤0.005rad). (b) Contractor mobilization (≤0.01rad). (c) Contractor mobilization (≤0.015rad). (d) Contractor mobilization (≤0.02rad). (e) Contractor mobilization (>0.02rad).

The damage to structural components was evaluated using the Park and Ang damage model (Park and Ang, 1985):

$D=\frac{{\delta }_M}{{\delta }_u}+\frac{\beta }{Q_y{\delta }_u}\int dE(6)$

where D is the damage index, δ_M is the maximum deformation during an earthquake, δ_u is the ultimate deformation under monotonic loading, Q_y is the yield strength, β is a positive constant (0.05), and dE is the incremental hysteretic energy dissipation. For a practical interpretation of the correspondence between the damage index and actual damage, the first term of Equation 6, which mainly depends on the maximum deformation, was considered as component D_R, corresponding to immediate repair actions such as crack injection. The second term, which primarily depends on cumulative plastic deformation, was considered as component D_E, which reflects deterioration such as reinforcement fatigue and is associated with longer recovery times due to more extensive repair methods. It should be noted that the damage index D and its components, D_R and D_E, are calculated separately for each story of the building.

For nonstructural components, the amount of damage was evaluated using fragility curves following previous studies (Architectural Institute of JapanCommittee on Disaster, 1998). Unlike structural components, nonstructural components were assumed not to accumulate deterioration, as they are typically subject to planned maintenance and replacement. In general, fragility curves that follow a lognormal distribution are expressed as:

$F\left(X\right)=\mathrm{\Phi }\left(\frac{\mathit{ln}\left(X\right)-\lambda }{\zeta }\right)(7)$

where X represents an intensity measure of external demand (e.g., peak acceleration, peak velocity), F(X) is the probability of damage to the building (or element) under demand level X, Φ(x) is the standard normal cumulative distribution function, λ is the mean of ln(X), and ζ is the standard deviation of ln(X). For such a fragility curve, the mean value is given by exp (λ+ζ²/2), the variance by exp (2λ+2ζ²) −exp (2λ+ζ²), and the median by exp(λ).

According to the Seismic Design and Construction Guidelines for Nonstructural Components (Architectural Institute of Japan, 2003), damage to nonstructural components is classified into two main categories: damage primarily caused by inter-story drift ratio and damage primarily caused by peak floor acceleration. In addition, the progression of damage states differs among components, with each having different definitions of damage states (DS levels). Based on previous research (Beigi et al., 2015), the probability distribution medians and coefficients of variation used to define fragility curves for nonstructural components—classified according to the progression of their damage states—are summarized in Table 3. The parameters λ and ζ in Equation 7 were determined from Table 3, and the corresponding fragility curves were derived accordingly. Figures 9a–f show the fragility curves for each fragility group.

Table 3

Table 3. Fragility function parameters of nonstructural components.

Figure 9

Figure 9. Fragility curves for each fragility group. (a) Partitions. (b) Partition-like. (c) Windows. (d) IDR-Sensitive generic components. (e) Ceilings. (f) PFA-Sensitive generic components.

In this study, discrete damage states are explicitly defined for nonstructural components through these fragility curves. We consider four damage states (DS1–DS4) in addition to the undamaged state DS0. DS1 corresponds to slight damage with essentially full functionality (e.g., minor cracking or local misalignment that can be repaired without replacement), DS2 represents moderate damage requiring repair but with limited interruption of use, DS3 denotes extensive damage associated with partial loss of functionality and the need for replacement of significant portions of the component, and DS4 corresponds to near-collapse or complete damage, where the component is non-functional and must be fully replaced. For structural components, no discrete damage states are defined; instead, the continuous damage index D given by Equation 6 is directly used to quantify the severity of damage and to inform subsequent downtime calculations through the associated repair time models.

Step 5: Repair model

For structural components, when the maximum displacement of the structure exceeds the yield point, component D_R was assumed to be immediately repaired and reset to zero. Component D_E, on the other hand, was considered to accumulate as deterioration due to cumulative damage until the damage index D exceeds 1. Once D exceeds this threshold, major repair was assumed to be carried out, restoring D_E to zero.

For nonstructural components, since leaving damage unrepaired is considered rare, all earthquake-induced damages occurring during the service life were assumed to be repaired, and deterioration was assumed to be eliminated through scheduled maintenance.

Step 6: Damage - Repair time model

The repair time of structural components was calculated from the damage index D using the following logistic function, with reference to FEMA P-58 (FEMA, 2018):

$RT=T_{\mathit{min}}+\frac{T_{\mathit{max}}-T_{\mathit{min}}}{1+\mathit{exp}\left(-k\left(D-D_0\right)\right)}(8)$

where RT is the repair time of structural components, D is the damage index, T_min is the minimum repair time (7 days), T_max is the maximum repair time (120 days), D₀ is the midpoint (1.0), and k is the steepness parameter (5). When the damage index D exceeds 1, major repair is required. In this study, with an emphasis on the long-term use of the building, it is assumed that the structure is repaired and remains in service even in cases where D exceeds 1 and demolition or replacement might be a more realistic option in practice. Here, regardless of whether the damage index D exceeds 1, the repair time of the structural components on each floor is determined by Equation 8. Table 4 summarizes the relationship between the damage index D and the severity of damage to structural components (Park and Ang, 1987), while Figure 10 illustrates the relationship between the damage index D and the corresponding repair time.

Table 4

Table 4. Relationship between Damage index D and damage severity of structural components.

Figure 10

Figure 10. Relationship between Damage index D and repair time of structural components.

For nonstructural components, based on FEMA P-58 (FEMA, 2018), the repair time was assumed for each floor under the condition that all targeted nonstructural elements were in a given damage state (Table 5). The actual repair time was then obtained by multiplying this assumed repair time by the probability of damage occurrence derived from fragility curves, using either inter-story drift ratios or peak floor accelerations as demand parameters. These values were summed over all repair sequences to obtain the total repair time.

Step 7: Downtime

Table 5

Table 5. Repair time of nonstructural components.

The downtime estimation followed the methodology proposed by Carlos Molina Hutt et al. (2022). The repair sequencing was determined based on interviews with contractors and engineers who were involved in building repair projects after the 1994 Northridge Earthquake. According to Ref. (Molina Hutt et al., 2022), contractors often carried out the repair of structural, exterior, elevator, and stairway components simultaneously across groups of two to three stories, and immediately after completing the structural repair of a given floor, the repair of interior finishes, mechanical systems, and electrical systems was undertaken. For downtime evaluation, the repair process was modeled as progressing along four repair paths, as illustrated in Figure 11.

Figure 11

Figure 11. Repair sequencing and downtime estimation method.

Repair path A consists of the repair sequence of structural components on each floor, followed by the repair sequence of interior finishes, mechanical systems, and electrical systems. Repair path B corresponds to the repair sequence of exterior components, repair path C to elevator repair, and repair path D to stairway repair. Typically, repair path A is carried out within the building interior, repair path B around the building perimeter, and repair paths C and D at specific locations on each floor. This implies that the four repair paths can proceed in parallel.

It was assumed that repair proceeds in phases, each covering two to three consecutive stories starting from the lowest floor, with each group of floors referred to as a “repair phase.” For the five-story building, phase 1 included floors 1–3 and phase 2 included floors 4–5. For the seven-story building, phase 1 included floors 1–3, phase 2 included floors 4–5, and phase 3 included floors 6–7. For the ten-story building, phase 1 included floors 1–3, phase 2 included floors 4–6, phase 3 included floors 7–8, and phase 4 included floors 9–10. Let Φ_p denote the set of stories belonging to repair phase p, and let n(j) be the index of the repair phase that contains story j.

First, the maximum repair time of repair sequence i in repair phase p is defined as

${MRT}_{p\_i}={\max }_{j\in \mathrm{\Phi }p}{RT}_i^j(9)$

where ${RT}_i^j$ is the repair time of sequence i on story j.

Because all stories within a given repair phase are repaired simultaneously, the completion time $C_i^j$ of repair sequence i on story j is obtained by summing (1) the delay time associated with impeding factors for sequence i, ${IF}_i$; (2) the cumulative maximum repair times of sequence i over all preceding repair phases; and (3) the repair time of sequence i on story j:

${C_i^j=IF}_i+{\displaystyle \sum _{p=1}^{n\left(j\right)-1}}{MRT}_{p\_i}{+RT}_i^j(10)$

Next, the completion time of repair path A on story j, denoted by $C_A^j$, is defined. In path A, the interior finishes (sequence 2), mechanical systems (sequence 4), and electrical systems (sequence 5) start only after the structural repair (sequence 1) on each story is completed. Therefore, the completion time of repair path A is given by

$C_A^j=\max \left[C_1^j+\max \left({RT}_2^j,{RT}_4^j,{RT}_5^j\right),C_2^j,C_4^j,C_5^j\right](11)$

where $C_i^j$ is given by Equation 10 for i = 1,2,4,5.

For repair paths B, C, and D, the completion times on story j are equal to the completion times of repair sequences 3, 6, and 7, respectively:

$C_B^j=C_3^j,C_C^j=C_6^j,C_D^j=C_7^j(12)$

Finally, the downtime of the building, DT, is defined as the maximum completion time among all repair paths and all stories:

$DT={\max }_j\left\{C_A^j,C_B^j,C_C^j,C_D^j\right\}(13)$

Equations 9–13 correspond directly to the schematic representation of the repair paths shown in Figure 11.

4 Example of downtime estimation considering sequential earthquake events

4.1 Analysis cases and parameters

Downtime was evaluated for a total of 162 analysis cases, considering variations in building height (5-, 7-, and 10-story buildings) and structural performance (strength-type, standard-type, and ductile-type). The analyses employed the 2016 Kumamoto Earthquake records as both a single event and a combined strong-motion sequence, and the maximum ground motion was scaled to return periods of 72, 475, and 970 years. Furthermore, to account for cumulative deterioration over a 100-year service life, three scenarios were considered in which the maximum ground motion occurs in the 1st, 50th, or 100th year after construction (Figures 12a–c). To avoid an artificial re-initialization immediately prior to the prescribed maximum-event year in cases (b) and (c), the life-cycle ground-motion sequences were checked/screened so that D does not exceed 1 in the event immediately preceding the prescribed maximum-event year. In a limited number of cases, D > 1 can occur earlier than the prescribed maximum-event year; such cases were retained because they can represent a plausible life-cycle history where a major restoration occurs during the service period and deterioration is subsequently re-accumulated before the maximum event.

Figure 12

Figure 12. Year of occurrence of the maximum earthquake during the service period. (a) At 1st year. (b) at 50th year. (c) at 100th year.

In all cases, the financing method was assumed to be private loans (see Table 2). The combination of these parameters resulted in 162 distinct scenarios, for which downtime was computed and compared between the single and combined ground motion inputs.

4.2 Downtime calculation procedure

For each analysis case, nonlinear seismic response analyses were first conducted under the specified ground motion inputs to obtain, for each story, the maximum damage index D, inter-story drift ratio (IDR), and peak floor acceleration (PFA).

Based on the observed maximum IDR, the delay time associated with each impeding factor was modeled as a lognormal random variable, following the parameters summarized in Table 2. For each case, 100 Monte Carlo simulations were performed, and the mean value of the simulated total delay time was used in the subsequent downtime calculation.

The repair times for structural components were uniquely determined from the damage index D, whereas the repair times for nonstructural components were determined from the seismic response indices (primarily IDR and PFA). The total downtime for each case was then obtained by summing (i) the delay times due to impeding factors and (ii) the repair times of structural and nonstructural components over all repair phases and stories, according to the downtime evaluation procedure described in the previous section.

4.3 Results and discussion

4.3.1 Influence of building height

When comparing buildings with different numbers of stories (Figure 13), in the 5-story building both the damage index D and IDR reached their maximum at the 4th story, with particularly high peaks under the combined strong motion. This behavior is attributed to the shorter natural period and the dominance of the first vibration mode.

Figure 13

Figure 13. Results of downtime, Damage Index D, IDR, PFA (Influence of building height).

In the 7-story building, the natural period is longer and the contribution of higher modes slightly increases, leading to a more dispersed energy input. As a result, D and IDR become large in the middle stories (4th–6th) but do not exhibit a distinct peak at a single story, and their values are smaller than those of the 5-story building. In the 10-story building, the longer natural period and stronger higher-mode contributions further reduce and homogenize D and IDR over the height, and the influence of the continuous input becomes relatively small.

Consequently, the damage per story follows the trend 5-story > 7-story > 10-story. However, because the number of repair phases increases with building height (5-story < 7-story <10-story), the difference in average downtime among the three buildings becomes small (5-story ≈ 7-story ≈10-story). The increment of downtime due to continuous input of ground motions is largest in low-to mid-rise buildings, following the trend 5-story > 7-story > 10-story.

4.3.2 Influence of building model type

When comparing the building models (strength-type, standard-type, and ductile-type) (Supplementary Figures S4a–c), for the 5- and 7-story buildings, the average downtime is shortest for the ductile-type model. The increment of downtime caused by sequential ground motion input follows the order strength-type > standard-type > ductile-type, because the ductile-type model effectively suppresses the increase in the damage index D under both single and sequential ground motion inputs.

For the 10-story building, however, the differences in average downtime among the three model types (strength-type ≈ standard-type ≈ ductile-type) and the increase in downtime due to sequential input are both small. Similar tendencies are observed in the differences and increments of D and IDR among model types. These values decrease as the building height increases, which is consistent with the reduced impact of sequential input in taller buildings described in Section 4.3.1.

As a representative example, Figures 14a–f illustrate the recovery trajectories for each repair path in the case of a 7-story building subjected to a maximum ground motion with a 475-year return period occurring 100 years after construction. These trajectories clarify how the sequential strong motion input leads to extended downtime in the considered scenario.

Figure 14

Figure 14. Recovery trajectories for each repair path. (a) Strength/Single. (b) Strength/Combined. (c) Standard/Single. (d) Standard/Combined. (e) Ductile/Single. (f) Ductile/Combined.

4.3.3 Influence of return period of the maximum ground motion

When comparing different return periods of the maximum ground motion (Supplementary Figures S5a–c), for all building heights, longer return periods (i.e., stronger ground motions) lead to larger downtimes, as expected. However, the increase in downtime due to continuous ground motion input depends on the building height and return period.

Specifically, the increment of downtime caused by sequential input is largest for the 72-year case in the 5-story building, for the 475-year case in the 7-story building, and for the 970-year case in the 10-story building. This tendency is attributed to the height-dependent ground motion intensity at which the damage index D exceeds 1, beyond which the repair time increases sharply according to the repair-time model.

4.3.4 Influence of cumulative deterioration

When comparing the degree of cumulative deterioration (maximum ground motion occurring at the 1st, 50th, or 100th year of a 100-year service life; Supplementary Figures S6a–c), the average downtime follows the trend 100th year ≈ 50th year >1st year. The similarity between the 100th- and 50th-year results is attributed to the saturation of the cumulative damage component DE, which keeps the total damage index D at a high level in the later stages of the service life. In addition, the increments of D and IDR induced by moderate earthquakes during the service period, as well as those caused by the consecutive input of annual earthquake records, become smaller with increasing building height.

4.4 Supplementary analysis using multiple strong-motion records

To examine the robustness of the findings with respect to the characteristics of the input motions, an additional set of analyses was carried out using multiple strong-motion records. In this supplementary analysis, 5-, 7-, and 10-story buildings with three model types (strength-type, standard-type, and ductile-type) were considered. The return period of the maximum ground motion was set to 970 years, and a 100-year service life was assumed, with the maximum ground motion occurring in the 1st, 50th, or 100th year after construction.

First, following approach (i), a record-by-record evaluation was conducted in which the same waveform was repeatedly applied year-by-year within the 100-year life-cycle scenario (with annual intensity scaled to match the hazard-consistent PGV), and the downtime results were then averaged across input motions. In this approach, five input motions were considered: the 2016 Kumamoto earthquake (single motion and combined motion treated as two separate inputs), the 2005 Fukuoka-ken Seiho-oki earthquake, the 2011 Tohoku earthquake, and the 2018 Hokkaido Eastern Iburi earthquake. The results are presented in Supplementary Figure S1.

Second, to reflect year-to-year variability in waveform characteristics within a single life-cycle case, an additional analysis was performed following approach (ii): in each 100-year simulation, the waveform for each year was randomly sampled from four observed records (2016 Kumamoto earthquake—single motion only, 2005 Fukuoka-ken Seiho-oki earthquake, 2011 Tohoku earthquake, and 2018 Hokkaido Eastern Iburi earthquake), while keeping the same life-cycle conditions and hazard-consistent scaling as above. The results are presented in Supplementary Figure S2.

Overall, the strength-type model exhibits the longest downtime, followed by the standard-type model, while the ductile-type model yields the shortest downtime. In addition, the increase in downtime due to cumulative deterioration between the 1st year and the 50th or 100th year is largest for the strength-type model and smallest for the ductile-type model. These results indicate that the main qualitative findings of this study are robust, and remain consistent even when record-to-record variability and year-to-year variability in waveform characteristics are taken into account.

5 Concluding remarks

A comprehensive downtime evaluation method for reinforced concrete (RC) buildings was proposed, which explicitly integrates (i) the delay times associated with impeding factors, (ii) the repair times of structural components, (iii) the repair times of nonstructural components, and (iv) the cumulative deterioration caused by moderate earthquakes experienced during the service period. Using this integrated framework, downtime was calculated by varying the conditions of building height, building model type, strong ground motion record, return period of the maximum ground motion, and degree of cumulative deterioration. In this way, the contributions of structural repair, nonstructural repair, and delay times were synthesized into the total downtime, allowing the influence of these parameters on seismic resilience to be quantified. The findings obtained from the calculation results are summarized below.

1. When the effects of cumulative deterioration due to moderate earthquakes and the occurrence of successive large earthquakes shortly after the annual maximum event were considered, an increase in downtime was observed. This result highlights the importance of accounting for sequential earthquake effects from the perspective of seismic resilience assessment.

2. For low-to mid-rise buildings, the ductile-type models exhibited shorter downtimes, and even when consecutive large earthquakes were assumed to occur immediately after the annual maximum event, the increase in downtime was effectively suppressed in the ductile-type models. This finding suggests that ductile design can enhance the seismic resilience of low-to mid-rise buildings.

3. Although the difference in average downtime among buildings with different numbers of stories was small in this study, the story where damage was concentrated and the configuration of repair phases differed depending on the building height. Therefore, to further reduce downtime, it is essential to prioritize the rapid repair of damage-concentrated stories and to optimize the configuration of repair phases.

Future research should extend the proposed downtime-based framework to an integrated loss assessment that explicitly quantifies direct repair and replacement costs, life-cycle maintenance or retrofit expenditures, and business-interruption losses associated with functional downtime. Such an extension would require the careful formulation and validation of multiple economic assumptions (e.g., discount rates, inflation or deflation scenarios, and business continuity strategies), which we identify as a major challenge and therefore beyond the scope of the present engineering-focused study.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

TM: Conceptualization, Formal Analysis, Investigation, Methodology, Writing – original draft, Writing – review and editing. NT: Conceptualization, Formal Analysis, Methodology, Supervision, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work was supported by JSPS KAKENHI Grant-in-Aid for Challenging Research (Exploratory), Grant Number 23K17331, Principal Investigator: NT.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbuil.2025.1743082/full#supplementary-material

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