Sage Journals: Discover world-class research

Abstract

This study conducts the time-dependent seismic risk assessment of highway bridges in western Canada when subjected to Cascadia subduction earthquakes (CSE) and aftershocks (AS). The performance-based earthquake engineering framework is extended to assess the expected annual repair cost ratio and annual restoration time of a benchmark bridge class under CSE mainshock (MS) and AS events within a 5-day period. The epidemic-type aftershock sequence (ETAS) model is utilized to simulate the temporal evolution of AS events after the MS of CSE. Seismic hazard models for MS and AS are used to select CSE-consistent MS ground motions and build an AS ground motion database for selecting and pairing ETAS-matching MS-AS sequences. High-fidelity numerical bridge models are then developed and paired with these MS-AS motion series for non-linear dynamic response analyses. The Park and Ang damage index is modified into a demand-capacity ratio model to account for the long-duration-induced cumulative damage to bridge columns. Day-to-day time-dependent fragility models are then developed for the bridge class at both component and system levels under MS and time-changing AS events. Furthermore, these fragility models are integrated with the CSE hazard model and loss functions to estimate the risk metrics for the bridge class. The study highlights the significantly higher bridge risk when the AS hazard is considered, with the elevated risk being stabilized by the third day after the MS. However, this study also indicates that the commonly adopted one-MS-one-AS approach underestimates the bridge AS risk when facing CSE.

Keywords

Temporal epidemic-type aftershock sequence annual repair cost ratio annual restoration time fragility and risk assessment subduction earthquakes mainshock and aftershock highway bridges

Introduction

The province of British Columbia is located in a unique and complex seismic region (Hyndman and Rogers, 2010). Three earthquake sources contribute to the region’s seismic hazards: shallow crustal earthquakes within the crust of the North American plate, deep subcrustal earthquakes triggered within the Juan de Fuca plate, and the mega-thrust Cascadia subduction earthquakes (CSE) at the interface of these two plates (Atkinson and Goda, 2011; Cascadia Region Earthquake Workgroup, 2013; Rogers et al., 2015). The CSE has magnitudes between M_w-8 and M_w-9 (Cascadia Region Earthquake Workgroup, 2013; Mazzotti and Adams, 2004) and occurs at long intervals with an average return period of 432 years, according to the sixth-generation seismic hazard model of Canada (Adams et al., 2019). The odds of an M_w-9.0 CSE happening in the next 50 years in western Canada are estimated to range from 7% to 15% (Oregon State University, 2010).

The CSE has unique features when compared with crustal and subcrustal earthquakes, including large magnitude, long duration, and strong aftershocks (AS) (Cascadia Region Earthquake Workgroup, 2013; Clague, 1997). While CSE remains a significant concern for catastrophic damage, these unique features, particularly the effect of AS, have been rarely studied and considered. For instance, most seismic codes ignore the influence of time-dependent aftershock triggering and secondary earthquake clustering, which has resulted in unsatisfactory structural performance under AS during past major earthquake events, such as the 2011 Tohoku earthquake and Christchurch earthquake (Khan and Alam, 2019). The current Canadian Highway Bridge Design Code recognizes the need to consider AS effects yet provides very ambiguous statements regarding AS performance requirements (National Research Council of Canada, 2019). The code calls for the need to assess the aftershock capacity of damaged bridges. However, it does not provide specific performance limits or guidelines to enable such evaluations (Todorov and Muntasir Billah, 2022). Bridge evaluation and design under MS-AS sequences of CSE remain an open research question that demands immediate attention and comprehensive investigation.

Research efforts have been made to assess the seismic vulnerability and risk of building structures in British Columbia where CSE MS-AS events were taken into account. For instance, Salami and Goda (2014) employed 50 recorded and 50 artificial MS-AS sequences for simulating crustal and subcrustal earthquakes, and CSE events in proportion and estimated the seismic vulnerability of four residential wood-frame buildings in Vancouver. Their study found that AS would contribute an additional 5%–20% to the maximum structural response and seismic damage. Likewise, Tesfamariam and Goda (2015) carried out the seismic loss assessment for a four-story non-code-conforming RC building in Victoria, with each M_w-9 CSE MS followed with one major AS. This loss assessment was further utilized by Goda and Tesfamariam (2017) for enabling risk-informed management and decision-making. Historically, the 2012 M_w-7.8 Haida Gwaii earthquake in western Canada was identified as a megathrust event in the Cascadia subduction zone (Bird and Lamontagne, 2015; Hyndman, 2015). The MS of this earthquake was followed by numerous AS events (Zhang et al., 2020).

As crucial links in the transportation network, bridges are essential for emergency response, evacuation, and the delivery of critical supplies (Gidaris et al., 2022). The AS risk on bridges is of particular concern as often unlikely the MS-damaged bridges can be repaired before the occurrence of subsequent AS. This has prompted researchers worldwide to investigate the AS effect on the seismic fragility and risk of highway bridges. For example, Wang et al. (2022) performed the incremental dynamic analysis using 30 back-to-back one-MS-one-AS sequences and developed AS fragility of a multi-span reinforced concrete (RC) continuous girder bridge in China. Gidaris et al. (2022) utilized one-MS-one-AS sequences to estimate the expected functionality and resilience of a typical RC girder bridge in California.

However, no relevant studies have been conducted for highway bridges in British Columbia. Also, the commonly adopted one-MS-one-AS artificial ground motion sequence fails to represent realistic seismic hazards. AS events occur due to stress redistribution in the crust (Gee et al., 2022), with their occurrence rates generally decreasing with time according to a power-law decay law (Fusakichi, 1894). Enabling the time-dependent AS risk assessment is vital for post-earthquake decision-making regarding emergency response, transportation recovery, and community resilience (Jordan et al., 2011; Shokrabadi and Burton, 2019). Nevertheless, such temporal evolution and time dependency of the AS risk on buildings and bridges have been often ignored in the literature.

One key challenge lies in the development of a tailored risk assessment workflow specifically designed to evaluate the time-dependent MS-AS risk of bridges under CSE. These involve modeling CSE hazards and their associated temporal changes in AS hazards, selecting and sequencing hazard-matching MS-AS ground motions, and accounting for the proper timing of AS events. Moreover, response history analyses of bridge models, along with the corresponding probabilistic seismic demand modeling, fragility assessment, and risk evaluation, must incorporate a tailored mechanism that considers the ground motion duration effect, ensuring the impact of varying numbers of AS events at different times can be properly examined.

This study quantifies to what degree the MS and AS of CSE might affect the seismic fragility and risk of highway bridges in British Columbia. The two-span RC box-girder bridges with single columns are selected as the benchmark bridge class due to their widespread constructions. The performance-based earthquake engineering (PBEE) framework is enhanced by integrating analysis modules to deal with the MS-AS sequential events for CSE. These include the analysis of CSE hazards, the utilization of the epidemic-type aftershock sequence (ETAS) model for time-dependent simulation of triggered AS events, the selection and pairing of earthquake type- and ETAS-consistent MS-AS ground motions, the development of the Park and Ang damage index to quantify the cumulative damage of the bridge column. Subsequently, the Latin Hypercube sampling (McKay et al., 1979), Monte Carlo simulation (Choi et al., 2004), and bootstrap resampling technique (Efron, 1979) are integrated with the cloud analysis approach to develop component- and system-level seismic fragility models that capture various sources of uncertainties in bridge attributes, ground motions, and seismic demand models. Finally, the developed fragility models are convolved with bridge loss functions and the CSE hazard model to assess the seismic risk of the bridge class (i.e. the expected annual repair cost ratio (ARCR) and annual restoration time (ART)) on different days following the MS. The contribution of this study lies not only in being the first to examine the MS-AS risk of highway bridges in western Canada, but also in developing a tailored and enhanced methodological framework that enables reliable quantification of the temporal evolution of bridge risk under time-changing AS events. This framework bears the novelty to be applied more broadly for assessing the MS-AS risk of buildings and other critical infrastructure systems. This study highlights the significantly exacerbated risk due to AS events, pinpoints number of days after MS when risk metrics stabilize, and provides a comprehensive analysis framework that can be applied to assess the day-to-day seismic risk of highway bridges when facing MS-AS earthquake hazards.

The PBEE framework adapted for mainshock–aftershock seismic risk assessment

The PBEE proposed by the Pacific Earthquake Engineering Research Center (PEER) (Allin, 2000) has offered a viable route to assess the seismic risk of highway bridges (Muntasir Billah et al., 2013). As shown in Equation 1, a triple integral has been proposed by PEER based on the total probability theorem to incorporate different logical elements into the quantification of seismic risk.

$λ (DV) = \int \int \int P (DV | DM) | dP (DM | EDP) | | dP (EDP | IM) | | d λ (IM) |$ (1)

where λ(·) denotes the mean annual rate of exceedance, P(·) is the complementary cumulative distribution function (CDF), DV is the decision variable, DM stands for the damage measure, EDP denotes the engineering demand parameter, and IM denotes the ground motion intensity measure. Given the conditional independence assumption, the PBEE framework assesses the seismic risk through successive analysis modules (i.e. hazard, structural, damage, and loss), where the inherent uncertainty and variability from each module can be quantified and propagated. The PBEE framework has been extensively utilized to assess the seismic risk of bridges under earthquakes with MS events (Mackie and Stojadinović, 2005). To consider the AS effect, bridge fragility models (i.e. P(DM|EDP) in Equation 1) need to be developed under MS-AS ground motion sequences that are consistent with the CSE hazard in western Canada. As the number of AS events may decrease over time following the MS, the selected MS-AS sequences should account for this time dependency. In addition, the selected ground motions should match with historically recorded MS and AS events associated with subduction earthquakes. Moreover, one essential feature of integrating the MS-AS effect is to take into account the ground motion duration effect for capturing the cumulative damage (i.e. P(EDP|IM) in Equation 1) of bridge components. These considerations will be elaborated in the following sections.

Mainshock-aftershock seismic hazard model in western Canada

Seismic hazard model for different types of earthquakes

Crustal (CE) and subcrustal earthquakes (SCE), and CSE events would cause significant differences in seismic damage, fragility, and risk (Shao and Xie, 2024a). Event-specific seismic hazard models are needed to capture these differences. The probabilistic seismic hazard analysis (PSHA) based on the sixth seismic hazard model (SHM6) introduced in the National Building Code of Canada 2020 (NBCC2020) (Kolaj et al., 2020) is conducted using the platform of OpenQuake (Pagani et al., 2014) to derive seismic disaggregation data and generate seismic hazard curves for CE, SCE, and CSE in western Canada. The city of Victoria is considered the targeted location due to its high seismicity and significant socioeconomic impacts under seismic hazards. Figure 1a shows the seismic deaggregation results at the period of 1.0 s with a 2% probability of exceedance in 50 years at an equivalent site class C in Victoria. The results reveal the dominance of CSE events in Victoria, contributing 61% to the overall seismic hazard, while CE and SCE events contribute 31% and 6%, respectively. Figure 1b also presents the annual probabilities of surpassing different intensities of S_a(1.0) attributed to each earthquake type as well as the totalized regional hazard due to all earthquake types. The hazard contribution due to CSE significantly differs from those from CE and SCE. Specifically, CSE exhibits a lower exceedance probability at S_a(1.0) smaller than 0.2 g but a higher occurrence rate when S_a(1.0) ranges from 0.2 to 3 g.

Figure 1.

(a) Seismic deaggregation for the period of 1.0 s and 2% probability of exceedance in 50 years. (b) Component and total hazard curves for S_a(1.0), in Victoria, based on the SHM6 (Adams et al., 2019).

Seismic hazard model for AS under subduction earthquakes

An MS can trigger several AS events over a period of time and a range of distances. Studies have demonstrated that significant AS typically occur close to the MS’ epicenter (Das and Henry, 2003; Mendoza et al., 1988; Zhang et al., 2018). Therefore, this study concentrates on differentiating the temporal evolution of AS, not their spatial distribution. Among the various models for predicting AS occurrences, the temporal epidemic type aftershock sequence (ETAS) model (Ogata, 1988) is regarded as the state-of-the-art in statistical seismology. This model effectively captures the hierarchical nature of earthquakes by modeling secondary and subsequent generations of AS events (Gee et al., 2022; Zhuang, 2011). Therefore, this study utilizes the ETAS model to estimate the seismicity rate of AS events following CSE over the first 5 days, a period when most of the large-magnitude AS would occur.

The ETAS model describes the AS triggering rate, λ_t, as a non-stationary Poisson point process where every event can trigger a wave of AS events (Ogata, 1988):

$λ_{t} (t, H_{t}) = \sum_{j : t_{j} < t} g (t - t_{j}; M_{j})$ (2)

where H_t is the process history (time and magnitude) of the earthquake occurrence up to time t. The triggering function, g(t, M), consists of the productivity function, $K_{0} e^{α (M - M_{cut})}$ , and the temporal function, v(t):

$g (t, M) = K_{0} e^{α (M - M_{cut})} v (t)$ (3)

in which M_cut is the cutoff magnitude for selecting earthquakes larger than M_cut. K₀ (events per day) and α are the productivity parameters. K₀ measures the intensity of aftershock generation, defining the number of triggered events above M_cut, while α determines how the triggering productivity of an earthquake increases with magnitude. This function predicts the number of AS (children) generated by an earthquake of magnitude M. Given the large magnitudes of CSE, they are expected to generate a greater number of AS than other earthquake types.

The temporal function v(t) follows the normalized Omori-Utsu law:

$v (t) = c^{p - 1} {(t + c)}^{- p} (p - 1)$ (4)

where parameter c eliminates a singularity at t = 0, and p is associated with the decay rate of AS over time. This temporal function is the probability density function (PDF) of the time interval between a parent event and its AS events. The outputs of the ETAS consist of synthetic catalogs of AS earthquakes, including the number of AS, their occurrence times, and their magnitudes.

In addition to the triggered AS, the ETAS model includes a background component that accounts for spontaneous seismicity, namely those not related to the MS. As the probability of another strong spontaneous earthquake occurring within the short post-MS simulation period is much lower than that of triggered events (Papadopoulos et al., 2020), also forecasting overall seismicity is not the attention, the component of spontaneous seismicity (i.e. the background rate in the ETAS model) is excluded in this study.

The ETAS model has been extensively applied and developed for CE hazards due to the abundance of recorded data (Seif et al., 2017). The challenge exists to deal with the CSE in western Canada, as very few events have been observed or recorded in the region (Wang and Tréhu, 2016). In this regard, Zhang et al. (2020) developed a set of global M_w-9.0 class subduction-zone ETAS parameters, which are compared against recent well-recorded subduction events worldwide, including the 2004 Aceh-Andaman earthquake, the 2010 Maule earthquake, and the 2011 Tohoku earthquake. The proposed ETAS parameters used in their study for subduction earthquakes are K₀ = 0.04 ± 0.02, α = 2.3, c = 0.03 ± 0.01, and p = 1.21 ± 0.08. Their model parameters do not show significant variability across different subduction-zone regions.

The ETAS model parameters have embedded uncertainties. Following a single MS event, one can simulate numerous AS scenarios through a Monte Carlo simulation. 320 AS simulations following a M_w-9.0 MS event are illustrated in Figure 2a by assuming a normal distribution of each uncertain parameter and considering a M_cut value of 5.0. As shown in the figure, AS events with M_w ≥ 5.0 are decreased from 20 to 55 events on Day 1 to 2 to 8 events on Day 5. Results from a single simulation are also illustrated in Figure 2b by showing the number of events each day with M_w stays above 5.0. This simulation indicates 43, 19, 12, 9, and 7 AS events across the first 5 days, with the majority having M_w < 6.0. Note that only the AS with M_w ≥ 6.0 are considered influential to causing additional damage and risk to the bridges in British Columbia.

Figure 2.

(a) Frequency distribution of AS events over 5 days for a M_w-9.0 CSE MS based on 320 simulations of the ETAS model. (b) Magnitude distribution of AS events from one representative simulation.

Ground motion selection for mainshock–aftershock sequences

Mainshock ground motions

The input ground motions should reflect distinct features associated with the CSE. This is achieved by selecting ground motions that match the conditional mean spectrum (CMS). Unlike the uniform hazard spectrum (UHS), which conservatively implies that the enveloping spectral values will occur at all periods within a single ground motion of any earthquake type (Bommer et al., 2000; Naeim and Lew, 1996; Reiter, 1992), the CMS provides the mean response spectral ordinates conditioned on the occurrence of a specific spectrum value at a single period of interest. The CMS provides more realistic spectral shapes (Baker, 2011; Baker and Cornell, 2006) to preserve the characteristics of contributing earthquakes. This study develops the CSE-specific CMS consistent with the seismic hazard of SHM6 using S_a(1.0) at an equivalent site class C in Victoria with a return period of 2475 years. The CMS is developed based on the distinct earthquake characteristics of CSE derived from seismic deaggregation analysis (Figure 1a) and the relevant ground motion prediction equation (GMPE) model proposed by Parker et al. (2022).

There is a scarcity of available ground motion records for CSE events. To address this, recent large thrust earthquakes are used as proxies for future CSE events. These earthquakes include the 2003 M_w-8.3 Tokachi Oki earthquake and the 2011 M_w-9.0 Tohoku earthquake (Tesfamariam and Goda, 2015; Tirca et al., 2015), both having durations of around 300 s and showing macro-level similarity against the expected CSE (Goda and Tesfamariam, 2015; Tirca et al., 2015). Ground motions from these earthquake events are extracted from the Center for Engineering Strong Motion Data (CESMD) (The Consortium of Organizations for Strong-Motion Observation Systems, 2012), PEER Next-Generation Attenuation for subduction zone regions (NGA-Sub) (Kishida et al., 2018), and three strong-motion seismograph networks in Japan (i.e. K-NET, KiK-net, and SK-net) (The National Research Institute for Earth Science and Disaster Resilience, 1996).

A suite of 320 bi-directional ground motions is selected by comparing the geometric mean response spectra of the candidate records with the corresponding target CMS. As displayed in Figure 3a, ground motion records are selected to have their mean spectra and statistical bounds match the target CMS with plus/minus two standard deviations of the hazard model. The period range for the spectral matching is from 0.5 T₁ to 2.0 T₁, where T₁ = 0.98 s refers to the median value of the first periods for the 320 bridge samples considered in this study. To have CMS-matching spectra, the ground motions are scaled, with the scaling factor constrained within 0.8–3 to preserve the motions’ physical characteristics.

Figure 3.

(a) Response spectra of a suite of 320 selected MS ground motions for CSE, and frequency distributions of (b) PGA, (c) S_a(1.0), and (d) D_5-95.

Figure 3b to d also provides additional intensity and duration characteristics of the selected CSE ground motions. Figure 3b shows the frequency distribution of the peak ground acceleration (PGA). Figure 3c shows that the S_a(1.0) values of the selected ground motions mostly stay between 0.2 and 0.8 g, a range consistently indicated in the CMS at the period of 1.0 s. Figure 3d presents the distribution of ground motions’ significant duration, D_5-₉₅, computed as the time interval for 5%–95% of the Arias Intensity [47]. The D_5-95 values for many of the selected CSE ground motions fall between 50 and 120 s, which is significantly longer than CE events (i.e. with D_5-95 typically remain below 20 s (Afshari and Stewart, 2016)).

Aftershock ground motion database

Even fewer ground motions have been recorded from historical AS events. Most ground motion databases do not provide as-recorded MS-AS sequences, especially for large-magnitude events (Papadopoulos et al., 2020). Instead, past studies have frequently used the same sets of records for both MS and AS through random permutation. Others created AS records by scaling and cloning MS ground motions. These approaches fail to realistically capture distinctions in ground motion features (e.g. duration, peak amplitude, and frequency content) between MS and AS earthquakes (Goda and Taylor, 2012; Ruiz-García, 2012). To improve this, this study generates occurrence times and magnitudes of AS earthquakes based on the ETAS model, and selects suitable AS records by developing an AS database for subduction earthquakes.

The K-NET, KiK-net, and SK-net ground motion databases in Japan consist of more systematic recordings for AS earthquakes (Goda and Taylor, 2012). These databases are used to compile the AS database, which includes AS ground motions from the 2003 M_w-8.3 Tokachi Oki earthquake and the 2011 M_w-9.0 Tohoku earthquake. AS events are identified as those that occur within a typical 100-day time window and a 30-km difference in the focal depth from the MS. Using this criterion, 764 AS records are compiled, with their magnitude distribution shown in Figure 4a. The figure reveals noticeable peaks in the numbers of records for AS with M_w 6.0–6.5, and M_w-7.0, yet fewer records are available for M_w ≤ 5.5 and M_w ≥ 7.5. Moderate to strong AS (i.e. M_w 6.0–7.0) occur more often following a subduction earthquake; they are significantly larger in magnitudes than AS for CE—the latter with M_w close to 5.0 (Toker, 2013) and rarely exceed 6.0 (Kim and Shin, 2017; Toker, 2013). Subduction earthquake AS that have M_w ≥ 6.0 are expected to cause extra damage to bridge structures. The response spectra of these ground motions (i.e. 453 AS records with M_w ≥ 6.0) are provided in Figure 4b. The figure shows that AS motions exhibit shorter predominant periods compared to MS motions, which is consistent with previous research findings (Goda and Taylor, 2012; Kim and Shin, 2017).

Figure 4.

(a) Magnitude frequency distributions of the AS motion database. (b) Response spectra of M_w ≥ 6.0 AS records.

Mainshock–aftershock generation, and ground motion selection and sequencing

Figure 3 shows that 320 MS motions are first selected. The ETAS model is then applied against each MS motion. With M_cut = 6.0, the ETAS model provides the daily occurrence number of AS events that have M_w ≥ 6.0 for the first 5 days. Figure 5 illustrates the frequency distribution of AS events following these 320 MS events. For MS earthquakes with M_w < 9, most cases resulted in one AS on Day 1, with few scenarios generating more than one AS on Day 1 and one event on Day 2, Day 3, or Day 4. For MS events with M_w ≥ 9.0, the number of AS events on Day 1 ranges from 1 to 7, approximately following a normal distribution, with some cases further having 1-3 AS events on Day 2. From Day 3 to Day 4, about 150 of the 320 MS motions were followed by one AS event per day, with very few cases generating 2 AS events on Day 3. On Day 5, approximately 120 cases still generated one AS. Figure 5 generally indicates that AS activity with M_w ≥ 6.0 is most pronounced on Days 1 and 2, with frequent multiple occurrences, while the number of AS events with M_w ≥ 6.0 likely reduces to one or none after Day 3.

Figure 5.

Frequency distribution of the daily number of M_w ≥ 6.0 AS events over 5 days following 320 MS events.

The development of a target spectrum for selecting AS motions remains an open question. For instance, no specific GMPE is available for AS of CSE. Most of the GMPE either excluded AS (Boore et al., 2014; Campbell and Bozorgnia, 2014; Chiou and Youngs, 2014) or did not differentiate against MS motions (Idrissa, 2014). While recent GMPE include AS by correlating it with MS through scaling factors, they were developed based on CE events (Kim and Shin, 2017). In a related context, positive correlations have been identified between MS and AS for various IM, such as peak amplitudes (Fayaz and Galasso, 2022; Xu and Lu, 2024; Yu et al., 2024) and spectral accelerations (Fayaz and Galasso, 2022; Xu and Lu, 2024). Spectral accelerations of MS and AS motions are correlated at closely spaced periods (Papadopoulos et al., 2019); this correlation is also observed for subduction-type earthquakes (Fayaz and Galasso, 2022; Ruiz-García, 2012). This study leverages the observed MS-AS correlation in spectral accelerations to select AS motions. In particular, the statistical distance from the CMS when selecting MS in Figure 3a is preserved by multiplying a scaling factor that is randomly sampled from a normal distribution with a mean of 1.0 and an assumed dispersion of 0.2. The scaled distance is utilized as the one from the median spectrum to select each AS motion from the AS dataset shown in Figure 4b. The parent MS and its following various numbers of AS on different days are then paired to develop 320 MS-AS motion sequences.

Figure 6 presents three examples of MS-AS acceleration sequences, each with an MS followed by several AS motions over the 5 days. The figure illustrates the decaying rate of AS events over time, with a noticeable reduction in AS motion numbers as post-MS days progress. The AS also indicate smaller magnitudes, lower intensities, and shorter durations when compared with the MS for CSE (Fayaz and Galasso, 2022; Xu and Lu, 2024; Yu et al., 2024). The constructed MS-AS ground motions are used as inputs for conducting non-linear response history analyses, developing probabilistic seismic demand models (PSDMs), and assessing the seismic fragility and risk of the bridge class.

Figure 6.

Examples of acceleration time-history sequences of an MS followed by subsequent ASs over 5 days.

The benchmark bridge class and its numerical modeling

The two-span continuous concrete bridges with single circular columns are selected as the benchmark bridge class. These bridges take a significant portion of the bridge inventory in British Columbia (Shao and Xie, 2024a). Bridges constructed in the post-1990 era are considered; their design is compliant with the modern seismic design code of Canada (National Research Council of Canada, 2019). Table 1 tabulates statistical distributions of bridge modeling parameters to capture intraclass variances in geometry, material properties, and design detailing (Mangalathu, 2017; Shao and Xie, 2024a).

Table 1.

Input parameters and their distributions considered in the bridge model (Mangalathu, 2017; Shao and Xie, 2024a)

Components	Variable	Distribution (µ, σ)^a
Geometry	Span length, L (m)	Normal (28.9, 6.0)
	Deck width, B_d (m)	Normal (11.9, 0.6)
	Column height, H_c (m)	Lognormal (7.6, 1.4)
	Column diameter, D_c (m)^b
Column	Concrete compressive strength, f’_c (MPa)	Normal (31.4, 3.9)
	Rebar yielding strength, f_y (MPa)	Normal (475.7, 37.9)
	Longitudinal reinforcement ratio, ρ_l (%)	Uniform (2.0, 0.3)
	Transverse reinforcement ratio, ρ_t (%)	Uniform (0.9, 0.07)
Pile group	Translational stiffness, K_ft (kN/mm)	Normal (270.7, 122.6)
	Transverse rotational stiffness, K_fr (GNm/rad)	Normal (5.7, 1.1)
	Transverse/longitudinal rotational stiffness ratio, k_r	Lognormal (1.5, 1.5)
Abutment on piles	Abutment backwall height, H_a (m)	Lognormal (3.5, 0.6)
	Pile stiffness, K_p (N/mm)	Lognormal (116.9, 44.8)
Elastomeric bearing	Stiffness per deck width, K_b (N/mm/m)	Lognormal (574.6, 301.2)
	Coefficient of friction, μ_f	Normal (0.3, 0.1)
Gap	Longitudinal gap (pounding), Δ_l (mm)	Lognormal (20.8, 12.6)
	Transverse gap (shear key), Δ_t (mm)	Uniform (19.5, 11.0)
Shear key	Acceleration for shear key capacity, a_sk (g)	Lognormal (1.0, 0.2)
Mass	Mass factor, m_f	Uniform (1.05, 0.003)
Damping	Damping ratio, ξ (%)	Normal (4.5, 1.25)

Median (µ) and standard deviation (σ) are shown in the parentheses following each distribution.

Circular columns feature a mix of diameters: 1.5 m (17%), 1.7 m (41%), 1.8 m (25%), and 2.1 m (17%).

High-fidelity numerical models are developed in OpenSees (McKenna, 2011) for the benchmark bridge class. As displayed in Figure 7, the bridge deck is modeled using elastic beam elements with mass lumped along the centerline. The connections between the bridge deck and end abutments are preserved through rigid transverse beams. The columns are simulated using fiber-discretized force-based nonlinear beam-column elements to specify distinct nonlinear material properties at different locations across the cross section (Shao and Xie, 2024a). A nonlinear spring system is established to simulate the dynamic interactions among various abutment components. For example, the trilinear material model is employed to model the abutment pile foundation. The contact element developed by Muthukumar and DesRoches (2006) is adopted to simulate the pounding effect between the deck and the abutment wall. The Steel01 material is used to model the elastomeric bearing. The force-displacement response of shear keys is modeled using a trilinear curve based on the Caltrans-UCSD field experiments (Megally et al., 2002). The passive resistance of abutment backfills (i.e. without considering at-rest or active pressure conditions) is simulated using a hyperbolic material (Shamsabadi and Yan, 2008). The connection details of these abutment components, and their associated constitutive envelop curves, are also illustrated in Figure 7.

Figure 7.

Numerical modeling of various bridge components for the case study bridge class.

Parameter distributions listed in Table 1 are utilized for a Latin Hypercube Sampling process to generate 320 bridge samples, with each bridge sample utilized to build the numerical model and randomly paired with one bi-directional MS-AS ground motion sequence. Numerical models of these bridge samples show that their natural periods range from 0.57 to 2.17 s, with a fitted lognormal median of 0.98 s and a dispersion of 0.2. Nonlinear time response analyses are then conducted on these motion-bridge samples for subsequent seismic demand, fragility, and risk assessment.

Seismic fragility assessment

Park and Ang damage index

Seismic fragility assessment is conducted by convolving the PSDM for each bridge component with the corresponding capacity limit state (LS) model. Both seismic demand and capacity models are defined against a shared engineering demand parameter (EDP). For bridge columns, this study employs the damage index proposed by Park and Ang (1985), an EDP that combines the peak deformation and absorbed energy under dynamic loading.

$D_{PA} = \frac{Δ_{m}}{Δ_{u}} + ψ_{PA} \frac{E_{h}}{F_{ey} Δ_{u}}$ (5)

where D_PA is the Park and Ang damage index; Δ_m is the maximum deformation in demand during the loading; Δ_u is the ultimate deformation capacity determined experimentally; ψ_PA is a parameter that reflects the effect of cyclic loading on the cumulative damage of the column and is taken as 0.15 for RC bridge columns (Fajfar, 1992; Mohammed, 2016; Todorov and Muntasir Billah, 2021); F_ey is the equivalent yield force; and E_h is the total dissipated hysteretic energy. D_PA has shown good agreement with experimental test data for buildings (Chai et al., 1994; Ellingwood, 2001; Ghosh et al., 2011), and it has been further utilized in developing seismic fragility models for bridge columns under various types of earthquakes (Ghosh et al., 2015; Shao and Xie, 2024a). Under dynamic loading, E_h is the parameter that increases with the ground motion duration, whereas Δ_m is the response under the most dominant seismic pulse of the earthquake event. As a result, D_PA monotonically increases with earthquake duration, making it particularly suitable for assessing column fragility under MS-AS events. The capacity LSs will be defined for the column using the Park and Ange damage index in the next section.

Capacity limit state models of bridge components

The capacity LS models quantify observable behaviors of bridge components when reaching different damage states (DS). The Park and Ang (1985) damage index was initially defined against the complete collapse of the structure (i.e. Δ_u is considered in Equation 5). To adapt it to assess the column fragility under other DS, such as slight, moderate, extensive, and complete DS defined in HAZUS (FEMA, 1999), the ultimate deformation, Δ_u in Equation 5, is substituted by the corresponding displacement LS threshold value at other DS. By synthesizing various existing capacity models for RC bridge columns, a uniform capacity model (with lognormal median and dispersion) for column drift LS is proposed for different DS (Shao and Xie, 2024b). Under each DS, the capacity drift is sampled and combined with the seismic demand Δ_m to compute the modified Park and Ang damage index (i.e. S_{PA, i} for the ith DS) (Shao and Xie, 2024a).

In general, bridge components are classified into (1) primary components whose failure would severely affect the stability of the entire bridge and (2) secondary components that are less likely to render a complete bridge failure (Mangalathu, 2017). Bridge column and span seating are considered two primary components with four DS (FEMA, 1999), whereas the remaining components are deemed secondary with only slight and moderate damage (Mangalathu, 2017). As listed in Table 2, the lognormally distributed median and dispersion values of bridge capacity LS models are determined based on recommendations from previous studies (Mangalathu, 2017; Zheng, 2021). Since the modified Park and Ang damage index combines column demand and capacity into a normalized ratio, the damage occurs at ith DS when S_{PA, i} is larger than 1.0.

Table 2.

EDPs and capacity limit state models of various bridge components (Mangalathu, 2017; Zheng, 2021)

Component	EDP (unit)	Median				Dispersion
		Slight (DS₁)	Moderate (DS₂)	Extensive (DS₃)	Complete (DS₄)	All DSs
Abutment seat	Unseating (mm)	38.1	114.3	355.6	533.4	0.35
Abutment	Displacement (mm)	76.2	254.0	—	—
Foundation	Displacement (mm)	25.4	101.6	—	—
Bearing	Shear strain (–)	1.5	3.0	—	—
Shear key	Displacement (mm)	83.8	327.7	—	—
Joint seal	Movement rating (–)	1.5	3.0	—	—

Probabilistic seismic demand model

The PSDM is commonly formulated as a linear regression model for EDP-IM pairs in the logarithmic space.

$\ln (S_{d}) = \ln (a) + b \ln (IM)$ (6)

where IM is the intensity measure of ground motions, a and b are the regression coefficients, and S_d is the median estimate of the demand in terms of an EDP. Dispersion, β_d/IM, is evaluated based on a statistical analysis of EDP-IM pairs:

$β_{d | IM} = \sqrt{\frac{\sum_{i = 1}^{N} {(\ln (d_{j}) - \ln (S_{d}))}^{2}}{N - 2}}$ (7)

where d_j is the jth demand out of N simulations. This study employs S_a(1.0) of the MS as the IM due to its effectiveness in establishing the PSDM for the bridge class (Shao and Xie, 2024a). It is noted that the PSDM is subject to the limitations of the cloud analysis method, including assumptions such as a linear EDP-IM regression function in the logarithmic space, constant dispersion across the entire IM range, and a lognormal EDP distribution at each IM level. It only predicts the combined MS-AS bridge fragility for each day. Addressing other types of fragility and risk assessments, such as post-MS, state-dependent AS fragility, would necessitate modifications to the cloud method or alternative approaches, such as back-to-back incremental dynamic analyses. For the bridge column, S_d becomes S_PA, which is the ratio between demand and capacity, and the β_PA/IM accounts for uncertainties in bridge parameters, ground motions, and capacity models. Figure 8 presents the S_PA,₄-S_a(1.0) pairs under MS and AS events within the first 5 days, along with color-highlighted 20 data where S_PA,₄ show substantial increases after the MS. On each day, more AS motions are added to the MS-AS sequences for nonlinear response history analyses to obtain the new seismic demand, which is further integrated into the modified Park and Ang damage index for a logarithmic linear regression to develop the day-to-day PSDM. Evolutions are observed, particularly in the highlighted 20 demand data, transitioning from MS to MS-AS events on Day 1 and Day 2, which contribute to the increase of the PSDM. Also, AS motions amplify the dispersion of the PSDM, where the β_PA,4/IM is increased from 0.70 for MS to a range of 0.76–0.79 that considers AS events.

Figure 8.

S _a(1.0)-based demand-capacity ratio model for D_PA,4 under MS and daily AS over the first 5 days.

Component- and system-level fragility curves of the bridge class

The PSDM is combined with the corresponding capacity LS model to generate component-level fragility curves, which compute the damage probability P_i,k for component k under DS i. For components other than the column, P_i,k is estimated as:

$P_{i, k} = Φ [\frac{\ln (S_{d, k} / S_{c, i})}{\sqrt{β_{d | IM, k}^{2} + β_{c, i}^{2}}}]$ (8)

where S_c,i and β_c,i represent the median estimate and dispersion of the capacity at ith DS, respectively; S_d.k and β_d|IM,k refer to the median and dispersion of the PSDM, respectively; and Ф(·) is the standard normal CDF. The modified Park and Ang damage index directly computes column demand and capacity as a ratio, which can be directly utilized to compute the column damage probability, P_i,col.

$P_{i, col} = Φ (\frac{\ln S_{PA, i}}{β_{PA, i | IM}})$ (9)

where S_PA,i and β_PA,i|IM refer to the median and dispersion of the regressed demand-capacity ratio model for the column (e.g. Figure 8).

Figure 9 presents column fragility curves developed for both MS and AS events on different days. These curves are developed based on column response data from the entire sequence of MS-AS motions up to each day and, therefore, represent MS-AS combined fragilities. A common approach in the literature considers the MS followed by only one AS motion (Tesfamariam and Goda, 2015), where the corresponding fragility curve is also provided. The fragility curves reveal a progressive increase in the probabilities of exceeding extensive and complete DSs (DS3 and DS4) of bridge columns due to daily AS following the MS. For DS1 and DS2, the fragility remains relatively stable from the MS, due to the limited additional effect of AS on already slightly and moderately damaged columns. The most substantial rise in fragility for DS3 and DS4 occurs from the MS to Day 1, indicating an immediate increase in vulnerability due to AS. Subsequent days show a continued, though diminishing, fragility increase, with the curves being stabilized by Day 3. Specifically, the median S_a(1.0) values (i.e. the S_a(1.0) at 50% damage probability) for DS3 are decreased by about 15% on Day 1, 2% on Day 2, and 1% each subsequent day, totaling a reduction of approximately 20% compared to the MS alone. For DS4, the median S_a(1.0) decreases by around 25%, with reductions of 20% on Day 1, 3% on Day 2, and 1% on each following day. Under the one-MS-one-AS case, the fragility curves stay between those of the MS alone and Day 1. The common one-MS-one-AS approach underestimates the column fragilities, especially at higher DS. These findings underscore the impact of AS in elevating the probability of severe damage or collapse in bridge columns, highlighting the necessity of incorporating time-dependent AS effects in seismic fragility and risk assessments.

Figure 9.

Bridge column fragility curves under MS and AS on different days.

The bridge system fragility is further developed using the joint probabilistic seismic demand models (JPSDM) (Nielson, 2005) to capture inter-component correlations in seismic demands. The JPSDM consists of marginal distributions of component seismic demands and a computed covariance matrix. These component demands are re-sampled with the embedded correlation and compared with the corresponding capacity LS values. The bridge system is considered to reach a certain DS should it be exceeded by any component. The frequency ratio of DS exceedance is further regressed as a lognormal distribution function to constitute the system-level fragility curve.

Figure 10 presents component- and system-level fragility curves of the bridge class under MS and AS after Day 1. Among different bridge components, span unseating exhibits high fragility at DS1 and DS2, while the bridge column dominates DS3 and DS4. Compared with MS, additional AS motions on Day 1 lead to substantially increased column fragilities, particularly at extensive and complete DS. The increased column fragility also contributes to higher bridge fragilities at the system level when incorporating AS events.

Figure 10.

Bridge fragility curves for MS and the first day after MS (Day 1) across all DS.

One PSDM was utilized to develop the fragility model. To further propagate the associated uncertainty, the bootstrap resampling technique (Efron, 1979) is utilized to generate 1000 sets of seismic demands each day for developing a stochastic set of PSDM for each bridge component. Each set of PSDM is then utilized to develop one realization of component- and system-level seismic fragility models. Figure 11 shows 1000 bootstrapped system-level fragility curves for obtaining the median curves and [5%, 95%] confidence bands. These curves show a similar trend in time evolution where (1) AS motions on subsequent days would increase the bridge system fragility, particularly against extensive and complete DS, and (2) the increase of system fragility is substantial on Day 1 and becomes somewhat stable after Day 3. These stochastic sets of bootstrap fragility curves are employed in the next section for seismic risk assessment.

Figure 11.

Fragility curves and their confidence bands for MS and AS over the first 5 days (each gray curve represents the fragility from one realization of the bootstrapped PSDM).

Seismic risk assessment

Expected annual repair cost ratio

The expected annual repair cost ratio (ARCR) is deemed a reliable estimate of the seismic risk in a short period (Der Kiureghian, 2005). It is computed by coupling the seismic hazard model, fragility model, and loss functions. First, the repair cost ratio (RCR) of the bridge class is computed by considering the repairing costs of different bridge components under different DS.

$RCR = \sum_{k = 1} \sum_{i = 1} p_{i, k} d_{i, k} c_{k}$ (10)

where subscripts i and k indicate the ith and kth DS and bridge component, respectively, p_i,k is the damage probability within each DS, d_i,k refers to the damage ratio that defines the percentage damaged for component k in ith DS, and c_k is the replacement cost ratio of the kth bridge component relative to the whole bridge. The system-level RCR connects to component-level fragility functions through the parameter p_i,k, which computes the differentiated conditional probability within the ith DS:

$p_{i, k} = {\begin{matrix} P_{i, k} - P_{i + 1, k}, for 1 \leq i \leq 3 \\ P_{i, k}, for i = 4 \end{matrix}$ (11)

Table 3 summarizes the loss functions for the benchmark bridge class. They feature lognormal distributions to quantify probabilistic damage ratios and replacement cost percentages by considering possible repair actions at each DS, the required quantities, and the unit costs of associated materials (Ghosh and Padgett, 2011; Ning and Xie, 2022; Padgett, 2007; Sobanjo and Thompson, 2001; Xie and Zhang, 2017). Second, the expected ARCR is computed by convolving the IM-based RCR with the corresponding seismic hazard model for CSE shown in Figure 1b, which essentially eliminates the dependence on earthquake IM.

$ARCR = \int_{IM} RCR (IM) | \frac{d λ}{d (IM)} | d (IM)$ (12)

Table 3.

Damage ratio and replacement cost of various bridge components (Ghosh and Padgett, 2011; Ning and Xie, 2022; Padgett, 2007; Sobanjo and Thompson, 2001; Xie and Zhang, 2017)

Component	Damage ratio				Replacement cost (%)	Dispersion
Component	Slight (DS₁)	Moderate (DS₂)	Extensive (DS₃)	Complete (DS₄)	Replacement cost (%)	Dispersion
Column	0.03	0.08	0.25	1.00	42.0	0.35
Unseating	0.03	0.08	0.25	1.00	24.6
Abutment	0.25	1.0	—	—	16.5
Foundation	0.13	1.0	—	—	7.2
Bearing	0.50	1.0	—	—	7.9
Shear key	0.50	1.0	—	—	0.5
Joint seal	0.50	1.0	—	—	1.3

Monte Carlo simulation is employed to combine the 1000 sets of bootstrap component-level fragility curves with the sampled loss metrics to compute the time-dependent ARCR of the bridge class for each day. Figure 12 displays the computed histograms of the ARCR, along with their fitted lognormal CDFs. The ARCRs of the bridge class range from 2.0% to 12% under MS alone and increase to 2.7% to 20% 5 days after the MS. Their lognormal CDF feature median ARCR of 4.6%, 6.3%, 6.9%, 7.0%, 7.0%, and 7.0%, with standard deviations of 0.26, 0.29, 0.30, 0.31, 0.31, and 0.31, respectively, when subjected to the MS and AS occurring in 5 days. In addition, the one-MS-one-AS approach results in a median ARCR of 5.7% with a standard deviation of 0.28, positioning it between the MS alone and Day 1 results. Figure 12 shows that the benchmark bridge class in Victoria will experience higher ARCR under CSE when subjected to AS motions, yet the ARCR will become stable by Day 3. Specifically, the bridges’ median ARCR are increased by 35% and 50%, on Day 1 and Day 3, respectively, when compared with the MS alone. Conversely, the one-MS-one-AS approach falls short of fully capturing the cumulative impact on bridges’ ARCR posed by consecutive AS events after the MS of CSE. This underscores the importance of considering the impact of seismic loss due to AS when subjected to CSE, particularly given that the potential influence of AS events has not been considered in Canada’s national seismic hazard model and bridge design code (Adams et al., 2019).

Figure 12.

(a) Histograms of ARCRs. (b) Their lognormal CDF fits.

Expected annual restoration time

As one major metric to assess seismic resilience, the expected annual restoration time (ART) provides insights into the post-earthquake serviceability and functionality of the bridge network. Its computation involves the seismic hazard model, system-level bridge fragility, and the associated repair time estimation.

$RT = \int_{IM} \sum_{i = 1} p_{s, i} t_{i} | \frac{d λ}{d (IM)} | d (IM)$ (13)

where subscripts s and i indicate the bridge system and DS, respectively, p_s,i is the damage probability of the bridge system within each DS, t_i is the post-earthquake repair time of the bridge against the ith DS; other variables have been defined previously. The repair time, t_i, in Equation 13 depends on the repair measures considered for the bridge under each DS. This study adopts the normally distributed repair time models recommended by HAZUS (FEMA, 2003), as listed in Table 4. As such, the 1000 sets of bootstrap system-level fragility models are randomly paired with the repair time models through Monte Carlo simulation, which is further utilized to integrate the seismic hazard model for CSE to compute the expected time-dependent ART of the bridge class for each day after the MS.

Table 4.

Post-seismic restoration times for highway bridges (FEMA, 2003)

	Slight (DS₁)	Moderate (DS₂)	Extensive (DS₃)	Complete (DS₄)
Mean	0.6	2.5	75	230
Standard deviation	0.6	2.7	42	110

The ART histograms of the bridge class under MS and the following AS within 5 days, along with their fitted lognormal CDFs, are presented in Figure 13. As shown in the figure, CSE would cause expected recovery ranging from 1.4 to 25.3 days/year under MS alone, and 2.5 to 48.4 days/year when considering AS after 5 days. The fitted CDF show medians of 11.1, 15.7, 17.3, 17.7, 18.0, and 18.3 days/year, and standard deviations of 0.40, 0.43, 0.44, 0.45, 0.45, and 0.45, respectively, in time recovery when the bridge class is exposed to the MS and subsequent AS hazards within the first 5 days. Moreover, the corresponding histogram and CDF for the one-MS-one-AS approach are also presented in Figure 13. In this case, the CDF curve falls between those for the MS alone and Day 1, with a median ART of 13.8 days and a standard deviation of 0.48, which underestimates the expected ART caused by CSE AS events. The expected ART caused by MS-AS sequences is considerably higher than that caused by the MS alone, while the rate of increase somewhat stabilizes by Day 3. The AS occurring the first day after the MS causes a rapid median ART growth of around 40%, whereas all AS over 5 days results in an approximately 60% increase in ART. It is evident to observe the impact of CSE AS in hindering the functionality recovery of the bridge class and post-earthquake mobility of the transportation network. This study evaluates the ART in a combined AS-MS environment, accounting for temporally increased bridge fragility and extended restoration time due to the occurrence of AS events on each day. A more detailed seismic resilience assessment and informed decision-making regarding bridge repair and restoration can be achieved by developing both state and time-dependent AS bridge fragility models, which is the focus of ongoing research by the authors.

Figure 13.

(a) Histograms of ART. (b) Lognormal CDF fits of ART annual exceedance probability.

Conclusion

This study extends the PBEE methodology to assess the time-dependent seismic risk of two-span RC box-girder bridges in western Canada subjected to CSE MS and AS hazards. The CSE hazard model is employed to select ground motions for the MS, whereas the temporal ETAS model is applied to simulate the temporal occurrence of AS events. These are further used to pair CSE-consistent MS-AS ground motion sequences. The Park and Ang damage index is modified to account for the cumulative MS-AS-induced damage to bridge columns. Time-dependent component- and system-level fragility models are then developed for the bridge class, with the expected ARCR and ART estimated as key metrics for evaluating the bridge risk under CSE MS and AS hazards over a 5-day period. The main findings of the study are as follows:

The constructed MS-AS ground motion sequences indicate a consistent temporal evolution of AS occurrences following the CSE MS, with the ETAS model simulating the changes in AS events over 5 days.

Bridge columns and systems exhibit higher seismic fragility under MS-AS sequences compared to MS alone, particularly against extensive and complete DSs.

The expected ARCR and ART increase as more AS events occur, with the most rapid increase happening within the first day. The estimated bridge risk stabilizes by the third day. Considering AS risk would increase the median ARCR by about 50% and the median ART by approximately 60%.

The one-MS-one-AS approach underestimates the impact of AS on bridge fragility and risk, with results falling between the MS-only scenario and the MS-AS outcomes after the first day.

Although destructive CSE has not occurred in decades, its large magnitude and the triggering of numerous strong AS would cause significant damage and loss to civil engineering structures in western Canada. The elevated risk from CSE MS and AS must be considered in the seismic analysis, design, and retrofit of bridge infrastructure. The findings of this study also provide valuable insights to aid in emergency response planning for the region’s transportation network following a CSE.

Supplemental Material

sj-xlsx-1-eqs-10.1177_87552930251335209 – Supplemental material for Time-dependent seismic risk assessment of highway bridges in western Canada under mainshock-aftershock subduction earthquakes

Supplemental material, sj-xlsx-1-eqs-10.1177_87552930251335209 for Time-dependent seismic risk assessment of highway bridges in western Canada under mainshock-aftershock subduction earthquakes by Yihan Shao and Yazhou Xie in Earthquake Spectra

Footnotes

The authors also thank Chunxiao Ning from McGill HMIR Lab for his help in conducting PSHA and developing the conditional mean spectra for ground motion selection. Any opinions,findings,conclusions,or recommendations expressed in this paper are those of the authors,do not necessarily reflect the official views or policies of the funding agencies,and do not constitute a standard,specification,or regulation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This research is supported by the NSERC Discovery Grant of Canada under Funding No. RGPIN-2020-04156. The first author would like to thank FRQNT for providing the B2X Doctoral Scholarship.

ORCID iD

Yazhou Xie

Data and resources

The data of fragility and risk results are provided in the supplementary material.

Supplemental material

Supplemental material for this article is available online.

References

Adams

Allen

Halchuk

Kolaj

(2019) Canada s 6th generation seismic hazard model, as prepared for the 2020 National Building Code of Canada. In: The 12th Canadian Conference on Earthquake Engineering, Quebec City, QC, Canada, 12–20 June.

Afshari

Stewart

(2016) Physically parameterized prediction equations for significant duration in active crustal regions. Earthquake Spectra 32(4): 2057–2081.

Allin

(2000) Progress and Challenges in Seismic Performance Assessment. Luxembourg: PEER Newsletter.

Atkinson

Goda

(2011) Effects of seismicity models and new ground-motion prediction equations on seismic hazard assessment for four Canadian cities. Bulletin of the Seismological Society of America 101(1): 176–189.

Baker

(2011) The conditional mean spectrum: A tool for ground motion selection. Journal of Structural Engineering 137(3): 322–331.

Baker

Cornell

(2006) Spectral shape, epsilon and record selection. Earthquake Engineering & Structural Dynamics 35(9): 1077–1095.

Bird

Lamontagne

(2015) Impacts of the October 2012 Magnitude 7.8 Earthquake near Haida Gwaii, Canada. Bulletin of the Seismological Society of America 105(2B): 1178–1192.

Bommer

Scott

Sarma

(2000) Hazard-consistent earthquake scenarios. Soil Dynamics and Earthquake Engineering 19: 219–231.

Boore

Stewart

Seyhan

Atkinson

(2014) NGA-West2 equations for predicting PGA, PGV, and 5% damped PSA for shallow crustal earthquakes. Earthquake Spectra 30(3): 1057–1085.

10.

Campbell

Bozorgnia

(2014) NGA-West2 ground motion model for the average horizontal components of PGA, PGV, and 5% damped linear acceleration response spectra. Earthquake Spectra 30(3): 1087–1114.

11.

Cascadia Region Earthquake Workgroup (2013) Cascadia Subduction Zone Earthquakes: A Magnitude 9.0 Earthquake Scenario. Washington, DC: FEMA.

12.

Chai

Priestley

MJN

Seible

(1994) Analytical model for steel-jacketed RC circular bridge columns. Journal of Structural Engineering 120(8): 2358–2376.

13.

Chiou

BSJ

Youngs

(2014) Update of the Chiou and Youngs NGA model for the average horizontal component of peak ground motion and response spectra. Earthquake Spectra 30(3): 1117–1153.

14.

Choi

DesRoches

Nielson

(2004) Seismic fragility of typical bridges in moderate seismic zones. Engineering Structures 26: 187–199.

15.

Clague

(1997) Evidence for large earthquakes at the Cascadia subduction zone. Reviews of Geophysics 35(4): 439–460.

16.

Das

Henry

(2003) Spatial relation between main earthquake slip and its aftershock distribution. Reviews of Geophysics 41(3): 1013.

17.

Der Kiureghian

(2005) Non-ergodicity and PEER’s framework formula. Earthquake Engineering & Structural Dynamics 34(13): 1643–1652.

18.

Efron

(1979) Bootstrap methods: Another look at the jackknife. In: Kotz

Johnson

(eds) Breakthroughs in Statistics (Springer Series in Statistics). New York: Springer, pp. 569–593.

19.

Ellingwood

(2001) Earthquake risk assessment of building structures. Reliability Engineering & System Safety 74(3): 251–262.

20.

Fajfar

(1992) Equivalent ductility factors, taking into account low-cycle fatigue. Earthquake Engineering & Structural Dynamics 21(10): 837–848.

21.

Fayaz

Galasso

(2022) A generalized ground-motion model for consistent mainshock–aftershock intensity measures using successive recurrent neural networks. Bulletin of Earthquake Engineering 20(12): 6467–6486.

22.

FEMA (1999) Earthquake Loss Estimation Methodology: User’s Manual. Washington, DC: Federal Emergency Management Agency.

23.

FEMA (2003) Multi-hazard Loss Estimation Methodology: Earthquake Model. Washington, DC: Federal Emergency Management Agency.

24.

Fusakichi

(1894) On the after-shocks of earthquakes. The Journal of the College of Science, Imperial University Oftokyo 7: 111–200.

25.

Gee

Peruzza

Pagani

(2022) The power of the little ones: Computed and observed aftershock hazard in Central Italy. Earthquake Spectra 38(1): 702–724.

26.

Ghosh

Padgett

(2011) Probabilistic seismic loss assessment of aging bridges using a component-level cost estimation approach. Earthquake Engineering & Structural Dynamics 40(15): 1743–1761.

27.

Ghosh

Padgett

Sánchez-Silva

(2015) Seismic damage accumulation in highway bridges in earthquake-prone regions. Earthquake Spectra 31(1): 115–135.

28.

Ghosh

Datta

Katakdhond

(2011) Estimation of the Park–Ang damage index for planar multi-storey frames using equivalent single-degree systems. Engineering Structures 33(9): 2509–2524.

29.

Gidaris

Padgett

Misra

(2022) Probabilistic fragility and resilience assessment and sensitivity analysis of bridges incorporating aftershock effects. Sustainable and Resilient Infrastructure 7(1): 17–39.

30.

Goda

Taylor

(2012) Effects of aftershocks on peak ductility demand due to strong ground motion records from shallow crustal earthquakes. Earthquake Engineering & Structural Dynamics 41(15): 2311–2330.

31.

Goda

Tesfamariam

(2015) Multi-variate seismic demand modelling using copulas: Application to non-ductile reinforced concrete frame in Victoria, Canada. Structural Safety 56: 39–51.

32.

Goda

Tesfamariam

(2017) Seismic risk management of existing reinforced concrete buildings in the Cascadia Subduction Zone. Natural Hazards Review 18(1): 1–10.

33.

Hyndman

(2015) Tectonics and structure of the Queen Charlotte Fault Zone, Haida Gwaii, and Large Thrust Earthquakes. Bulletin of the Seismological Society of America 105(2B): 1058–1075.

34.

Hyndman

Rogers

(2010) Great earthquakes on Canada’s west coast: A review. Canadian Journal of Earth Sciences 47(5): 801–820.

35.

Idrissa

(2014) An NGA-West2 empirical model for estimating the horizontal spectral values generated by shallow crustal earthquakes. Earthquake Spectra 30(3): 1155–1177.

36.

Jordan

Chen

Y-T

Gasparini

Madariaga

Main

Marzocchi

Papadopoulos

Sobolev

Yamaoka

Zschau

(2011) Operational earthquake forecasting: State of knowledge and guidelines for utilization. Annals of Geophysics 54(4): 315–391.

37.

Khan

Alam

(2019) Reinforced concrete substructure bridge seismic performance for mainshock-aftershock scenarios–review of findings and state of the art. In: 12th Canadian Conference on Earthquake Engineering, Quebec City, QC, Canada, 12–20 June, pp. 1–8.

38.

Kim

Shin

(2017) A model for estimating horizontal aftershock ground motions for active crustal regions. Soil Dynamics and Earthquake Engineering 92: 165–175.

39.

Kishida

Contreras

Bozorgnia

Abrahamson

Ahdi

Ancheta

Boore

Campbell

Chiou

Darragh

Gregor

Kuehn

Kwak

Kwok

Lin

Magistrale

Mazzoni

Muin

Midorikawa

Silva

Stewart

Wooddell

Youngs

(2018) NGA-sub ground motion database. In: 11th U.S. National Conference on Earthquake Engineering, Los Angeles, CA, 25–29 June.

40.

Kolaj

Adams

Halchuk

(2020) The 6th generation seismic hazard model of Canada. In: 117th World Conference on Earthquake Engineering, Sendai, Japan, 13–18 September.

41.

Mackie

Stojadinović

(2005) Fragility basis for California highway overpass bridge seismic decision making. Pacific Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley.

42.

Mangalathu

(2017) Performance based grouping and fragility analysis of box-girder bridges in California. PhD Thesis, Georgia Institute of Technology, Atlanta, GA.

43.

Mazzotti

Adams

(2004) Variability of near-term probability for the next great earthquake on the Cascadia subduction zone. Bulletin of the Seismological Society of America 94(5): 1954–1959.

44.

McKay

Beckman

Conover

(1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2): 239–245.

45.

McKenna

(2011) OpenSees: A framework for earthquake engineering simulation. Computing in Science & Engineering 13: 58–66.

46.

Megally

Silva

Seible

(2002) Seismic response of sacrificial shear keys in bridge abutments, Report no. SSRP-2001/23. San Diego, CA: Department of Structural Engineering, University of California.

47.

Mendoza

Seismological

(1988) Aftershock patterns and main shock faulting. Bulletin of the Seismological Society of America 78(4): 1438–1449.

48.

Mohammed

(2016) Effect of earthquake duration on reinforced concrete bridge columns. https://scholarwolf.unr.edu/items/7546e2b8-2bce-4b7a-a505-806587164262

49.

Muntasir Billah

AHM

Shahria Alam

Rahman Bhuiyan

(2013) Fragility analysis of retrofitted multicolumn bridge bent subjected to near-fault and far-field ground motion. Journal of Bridge Engineering 18(10): 992–1004.

50.

Muthukumar

DesRoches

(2006) A Hertz contact model with non-linear damping for pounding simulation. Earthquake Engineering & Structural Dynamics 35(7): 811–828.

51.

Naeim

Lew

(1996) On the use of design spectrum compatible time histories. Earthquake Spectra 11(1): 111–127.

52.

National Research Council of Canada (2019) Canadian Highway Bridge Design Code S6-19. Ottawa, ON, Canada: National Research Council of Canada.

53.

Nielson

(2005) Analytical fragility curves for highway bridges in moderate seismic zones. PhD Thesis, Georgia Institute of Technology, Atlanta, GA.

54.

Ning

Xie

(2022) Risk-based optimal design of seismic protective devices for a multicomponent bridge system using parameterized annual repair cost ratio. Journal of Structural Engineering 148(5): 04022044.

55.

Ogata

(1988) Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association 83(401): 9–27.

56.

Oregon State University (2010) Odds are 1-in-3 That a Huge Quake Will Hit Northwest in Next 50 Years. Corvallis, OR: Oregon State University.

57.

Padgett

(2007) Seismic vulnerability assessment of retrofitted bridges using probabilistic methods. PhD Thesis, Georgia Institute of Technology, Atlanta, GA.

58.

Pagani

Monelli

Weatherill

Danciu

Crowley

Silva

Henshaw

Butler

Nastasi

Panzeri

Simionato

Vigano

(2014) OpenQuake engine: An open hazard (and risk) software for the global earthquake model. Seismological Research Letters 85(3): 692–702.

59.

Papadopoulos

Kohrangi

Bazzurro

(2019) Correlation of spectral acceleration values of mainshock-aftershock ground motion pairs. Earthquake Spectra 35(1): 39–60.

60.

Papadopoulos

Kohrangi

Bazzurro

(2020) Mainshock-consistent ground motion record selection for aftershock sequences. Earthquake Engineering & Structural Dynamics 49(8): 754–771.

61.

Park

Ang

AHS

(1985) Mechanistic seismic damage model for reinforced concrete. Journal of Structural Engineering 111(4): 722–739.

62.

Parker

Stewart

Boore

Atkinson

Hassani

(2022) NGA-subduction global ground motion models with regional adjustment factors. Earthquake Spectra 38: 456–493.

63.

Reiter

(1992) Earthquake hazard analysis: Issues and insights. Surveys in Geophysics 13(3): 297–298.

64.

Rogers

Halchuk

Adams

Allen

(2015) 5th generation (2015) seismic hazard model for southwest British Columbia. In: 11th Canadian Conference on Earthquake Engineering, Victoria, BC, Canada, 21–24 July, pp. 21–24.

65.

Ruiz-García

(2012) Mainshock-aftershock ground motion features and their influence in building’s seismic response. Journal of Earthquake Engineering 16(5): 719–737.

66.

Salami

Goda

(2014) Seismic loss estimation of residential wood-frame buildings in Southwestern British Columbia considering mainshock-aftershock sequences. Journal of Performance of Constructed Facilities 28(6): 1–10.

67.

Seif

Mignan

Douglas Zechar

Werner

Wiemer

(2017) Estimating ETAS: The effects of truncation, missing data, and model assumptions. Journal of Geophysical Research: Solid Earth 122(1): 449–469.

68.

Shamsabadi

Yan

(2008) Closed-form force-displacement backbone curves for bridge abutment-backfill systems. In: Hiltunen

Manzari

(eds) Geotechnical Earthquake Engineering and Soil Dynamics IV. Reston, VA: American Society of Civil Engineers, pp. 1–10.

69.

Shao

Xie

(2024a) Seismic risk assessment of highway bridges in western Canada under crustal, subcrustal, and subduction earthquakes. Structural Safety 108: 102441.

70.

Shao

Xie

(2024b) Unified seismic capacity limit state models of reinforced concrete bridge columns. In: Gupta

Sun

Brzev

, et al. (eds) Canadian Society of Civil Engineering Annual Conference. Cham: Springer, pp. 435–452.

71.

Shokrabadi

Burton

(2019) Regional short-term and long-term risk and loss assessment under sequential seismic events. Engineering Structures 185: 366–376.

72.

Sobanjo

Thompson

(2001) Development of agency Maintenance, Repair & Rehabilitation (MR&R) cost data for Florida’s bridge management system. Final Report, no. Contract no. BB-879. Tallahassee, FL: Florida Department of Transportation.

73.

Tesfamariam

Goda

(2015) Loss estimation for non-ductile reinforced concrete building in Victoria, British Columbia, Canada: Effects of mega-thrust Mw9-class subduction earthquakes and aftershocks. Earthquake Engineering & Structural Dynamics 44(13): 2303–2320.

74.

The Consortium of Organizations for Strong-Motion Observation Systems (2012) Center for Engineering Strong Motion Data. San Francisco, CA: The Consortium of Organizations for Strong-Motion Observation Systems.

75.

The National Research Institute for Earth Science and Disaster Resilience (1996) Strong-Motion Seismograph Networks (K-NET. Kik-net). Tsukuba, Japan: The National Research Institute for Earth Science and Disaster Resilience.

76.

Tirca

Chen

Tremblay

(2015) Assessing collapse safety of CBF buildings subjected to crustal and subduction earthquakes. Journal of Constructional Steel Research 115: 47–61.

77.

Todorov

Muntasir Billah

AHM

(2021) Seismic fragility and damage assessment of reinforced concrete bridge pier under long-duration, near-fault, and far-field ground motions. Structures 31: 671–685.

78.

Todorov

Muntasir Billah

AHM

(2022) Post-earthquake seismic capacity estimation of reinforced concrete bridge piers using machine learning techniques. Structures 41: 1190–1206.

79.

Toker

(2013) Time-dependent analysis of aftershock events and structural impacts on intraplate crustal seismicity of the Van earthquake (Mw 7.1, 23 October 2011), E-Anatolia. Central European Journal of Geosciences 5(3): 423–434.

80.

Wang

Tréhu

(2016) Invited review paper: Some outstanding issues in the study of great megathrust earthquakes—The Cascadia example. Journal of Geodynamics 98: 1–18.

81.

Wang

Deng

Luo

Dang

Guo

(2022) Aftershock fragility assessment of continuous RC girder bridges using a modified damage index. Buildings 12(10): 1675.

82.

Xie

Zhang

(2017) Optimal design of seismic protective devices for highway bridges using performance-based methodology and multiobjective genetic optimization. Journal of Bridge Engineering 22(3): 04016129.

83.

(2024) An extended generalized conditional intensity measure method for aftershock ground motion selection. Earthquake Engineering & Structural Dynamics 53(4): 1509–1536.

84.

Zhou

(2024) A practical approach of probabilistic seismic hazard analysis for vector IMs Regarding mainshock with potentially largest aftershock. Journal of Earthquake Engineering 28(3): 637–658.

85.

Zhang

Werner

Goda

(2018) Spatiotemporal seismic hazard and risk assessment of aftershocks of M 9 megathrust earthquakes. Bulletin of the Seismological Society of America 108(6): 3313–3335.

86.

Zhang

Werner

Goda

(2020) Variability of etas parameters in global subduction zones and applications to mainshock–aftershock hazard assessment. Bulletin of the Seismological Society of America 110(1): 191–212.

87.

Zheng

(2021) Advanced seismic risk assessment of California box-girder bridges using emerging modeling techniques and innovative risk models. PhD Thesis, Georgia Institute of Technology, Atlanta, GA.

88.

Zhuang

(2011) Next-day earthquake forecasts for the Japan region generated by the ETAS model. Earth, Planets and Space 63(3): 207–216.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.02 MB

0.00 MB