Resilience assessment of bridges considering multi-hazard of earthquakes and blasts

Abstract

Natural and human-made hazards pose significant risks to bridges, disrupting transportation systems and causing severe economic and social impacts. Earthquakes and blasts are particularly critical in evaluating long-term bridge resilience. Current resilience assessment methods, however, focus primarily on single and deterministic hazards, neglecting the uncertainty associated with hazard randomness, hazard interrelationship, structural robustness, and variability in restoration. This can underestimate risks and lead to structural failures, highlighting a critical knowledge gap. This paper proposes a novel approach to assess bridge resilience under multi-hazards, specifically earthquakes and blasts. The approach incorporates underexplored uncertainties, accounts for damage accumulation through state-dependent fragility, and introduces the resilience quantification probabilistically. An illustrative case study demonstrates its application, showing that hazard randomness, particularly the sequence and timing of sequential hazards during restoration, significantly influences bridge resilience. The findings emphasize the importance of detailed and probabilistic consideration of hazard randomness and interrelationship in the multi-hazard context. The proposed approach has the potential for broader application to other hazard types and structural systems, addressing an urgent need for resilience assessment in infrastructure systems subjected to multiple hazards.

Graphical abstract

Keywords

multi-hazards / bridge / resilience assessment / earthquake and blast / fragility analysis

1 Introduction

Bridges might be subjected to multiple natural or man-made hazards during their service life such as earthquake, wind, blast, and fire, etc. [1]. These hazards overlap in space and time, cause damage, disrupt transportation systems, and hinder rescue and recovery efforts [2]. Of the many natural threats, earthquake is recognized as one of the most destructive ones known to human beings. Although it only corresponds to 15% of hazard occurrence with consequences, it causes the greatest fatality and economic losses with the global percentage of 78% and 45%, respectively [3]. It can cause serious effects on bridges (i.e., site and structural damage) and the regional transportation networks [4,5]. For example, from 2011 to 2018, the earthquakes of Tohoku (2011, M9, 15,000 victims) and Nepal (2015, M7.8, 8000 victims) were among the most catastrophic events by destroying production sites and disrupting transport lines [6,7]. Furthermore, bridges have been severely affected by blast attacks in recent years due to their critical role in the supply chain and emergency relief, particularly through transport vehicles carrying flammable materials and explosives [8]. According to statistics from the US National Consortium for the Study and Responses to Terrorism, the number of bombing attacks against bridges increased from fewer than 70 in 1998 to nearly 400 in 2015, making them increasingly frequent targets [9]. The 2022 Crimean Bridge explosion exemplifies the severe consequences of such attacks, where a bomb-laden truck led to structural failure and multiple fatalities [10]. Given these escalating threats, ensuring the resilience of bridges against multiple hazards, particularly sequential earthquake and blast events, is of paramount importance for maintaining transportation functionality during the structural lifecycle and safeguarding lives and property.

Addressing multi-hazard risks has become a global priority due to factors such as industrial expansion, geopolitical instability, regional conflicts, and climate change [11–13]. International policies, including the European Commission’s Risk Assessment and Mapping Guidelines for Disaster Risk Management [14], the Sendai Framework for Disaster Risk Reduction [15,16], and the Union Civil Protection Mechanism [17], underscore the necessity of multi-hazard resilience assessment. Recent research has increasingly shifted toward this domain, yet significant gaps persist in assessing infrastructure resilience under multi-hazard conditions [18,19]. There have been a few studies proposing methodologies for hazard combinations, such as scour-earthquake [20], earthquake-flood [21,22], and hurricane-earthquake [23], however, consideration of a wider variety of hazard portfolios is still lacking [24].

A key challenge in multi-hazard resilience assessment is the insufficient understanding of inter-hazard relationships. While existing research categorizes earthquake and blast events as non-concurrent hazards due to their independent occurrence probabilities, their sequential effects on structural integrity remain largely unexplored [25,26]. In these studies, earthquakes and blasts exhibit stochastic dynamic behavior, triggering different structural responses and failure mechanisms. Current approach for hazard analysis using superposition of single-hazard effects to produce a maximum effect for various types of hazards, such as in design codes AASHTO [27]. This is not applicable for multi-hazards since it could reduce the effectiveness of risk reduction strategies or increase vulnerability due to asynergistic effects [28]. A more refined methodology that explicitly accounts for inter-hazard relationships is necessary for accurate resilience assessment.

In sequential hazard scenarios, the sequence and timing of events critically influence structural resilience [29,30]. For instance, a bridge compromised by an initial earthquake may experience exacerbated damage when subjected to a subsequent blast due to reduced structural capacity [31,32]. The structural fragility in this context is known as state-dependent fragility, referring to the influence of the prior hazard on structures subjected to the subsequent hazard, conditional on a given damage level from the first one, which is more complex than for individual hazards. Some studies have explored fragility modeling for successive hazards such as earthquake-tsunami, earthquake-wind, earthquake-hurricane, and scour-earthquake [33–35]. However, to the best of the authors’ knowledge, limited studies have investigated state-dependent fragility for bridges under sequential earthquake and blast loads, highlighting the necessity of more research efforts in this area.

Beyond fragility modeling, a comprehensive resilience assessment must also incorporate post-hazard restoration. Mapping the bridge vulnerability to appropriate recovery patterns is essential for achieving targeted resilience outcomes [36]. However, existing literature on restoration models remain limited due to inherent difficulties related to hazard variability, damage mechanisms, structural significance, resource availability and local practice, etc [37,38]. Some studies have introduced linear, stochastic stepwise, or six-parameter sinusoidal models, demonstrating that probabilistic methods are most effective for capturing uncertainties in the restoration process [39]. Most of these models focus on single hazard scenarios, whereas limited models had incorporated the interrelationship of hazardous events including uncertainty in hazard timing and the sequential effects of various hazards, affecting their applicability in multi-hazard contexts [40]. Thus, adapting current models to account for hazard interrelationship during restoration presents another key challenge for multi-hazard resilience assessment.

This study aims to bridge these gaps by providing an approach for resilience assessment of bridges subjected to multi-hazard of earthquakes and blasts. Although earthquakes and blasts originate from independent sources with various impacts, their sequential effects and random occurrence can significantly impact bridges in the perspective of structural lifecycle. This study explores these effects and emphasizes the associated influence on bridge resilience assessment. Novel contributions include: 1) modeling sequential earthquake and blast hazards while considering hazard randomness (i.e., intensity, sequence, and occurrence time); 2) quantifying state-dependent fragility for resistance reduction across consecutive hazards; 3) refining restoration models to integrate hazard uncertainty and interdependency. The application of this method is illustrated using a case study consisting of a typical two-span concrete highway bridge exposed to sequential earthquake and blast loads. In Section 2, steps of the approach used to conduct this state-of-the-art research is described. Section 3 presents the case study to illustrate the implementation. Results of the study are also presented and discussed in this section, with conclusions summarized in Section 4. The findings contribute to a broader understanding of multi-hazard resilience assessment by incorporating probabilistic hazard interactions beyond traditionally interdependent hazardous events. It lays foundation for facilitating decision-making process for stakeholders and infrastructure owners, to enhance safety, sustainability and resilience of critical infrastructure.

2 Resilience assessment approach

This section outlines the approach for resilience assessment of bridges subjected to multi-hazard of earthquakes and blasts. The approach is scalable and can be extended to accommodate multiple sequential hazards. To simplify discussion in the study, the paper focuses on two sequential hazards: an earthquake and a blast, referred to as Haz-1 (the first hazard) and Haz-2 (the second hazard). As mentioned in the Introduction, earthquakes and blasts are recognized as non-concurrent, independent hazards with varying impacts. The intensity, sequence, and timing of the two hazards can vary considerably. In this context, bridge functionality-defined by its traffic carrying capacity, with damage states corresponding to different levels of functionality from 0 (complete loss) to 1.0 (full capacity)-is time-dependent, and the following three scenarios are possible: 1) Haz-2 (e.g., an earthquake) may occur immediately after Haz-1 (e.g., a blast) (Fig. 1(a)), without any intervening repair; 2) Haz-2 may occur during the restoration, at a point when functionality loss from Haz-1 has not been fully recovered (Fig. 1(b)); 3) restoration begin immediately after Haz-1 and is completed before the initiation of Haz-2 (Fig. 1(c)). While bridge owners may consider the scenario in Fig. 1(c) as the ideal condition, scenarios shown in Figs. 1(a)–1(b) are also realistic, especially when factoring in limited resources and the unpredictable nature of hazards. Indeed, Figs. 1(a) and 1(c) can be considered as the boundary conditions of Fig. 1(b), where Haz-2 occurs either at the start or end of the restoration from Haz-1. Efforts were given to the approach that can describe various hazard scenarios, analyzing the hazard interrelationship while encapsulating structural robustness, redundancy and resourcefulness. Details of the approach is outlined in the four steps shown in Fig. 2. Each step is further described and explained in the subsections below.

2.1 Hazard analysis

The first step is to define the magnitude of hazards to be considered. Each hazard can be described through the hazard intensity indicator, which quantifies the significance of hazard to influence bridges. In the case of seismic hazard, intensity can be described by indicators such as peak ground acceleration (PGA), peak ground velocity (PGV) or spectral acceleration (SA) at the fundamental period of bridges [41]. In the case of blasts, such as those caused by vehicle explosions, hazard intensity indicators include overpressure (P), impulse (I), scaled distance (Z), or hybrid variables made of P and I (e.g., X=P^{{\alpha }_1}{I}^{{\alpha }_2}, X={\left(PI\right)}^{1/2}). This is different from the seismic hazard, which is often characterized by a single variable such as PGA. Blast loads require two variables, P and I, to adequately represent hazard intensity. This distinction arises because earthquake-induced structural response is primarily governed by inertial forces, whereas blast effects depend on both the overpressure and the duration of force application due to the involvement of shock waves and momentum transfer. In this study, a hybrid variable made of P and I was employed to allow for the derivation of empirical blast fragility under the assumption of a bivariate lognormal probability distribution [42,43]. In this research, a scenario-based approach is adopted to consider the impact of multi-hazards, where hazards with predetermined intensities are assumed to occur at specific times during the service life of bridges, without special emphasis on exceedance probabilities (Fig. 2(a)).

2.2 Fragility analysis

Structural damage is typically represented through fragility functions, which describe the probability of assets exceeding various limit states at a given hazard intensity. Among the methods for deriving fragility functions, the analytical curve is generally adopted due to its flexibility in accommodating a broader range of hazard scenarios and structural types. Fragility curves are developed using numerical methods, typically through finite element analysis of bridges subjected to various hazard intensity levels. This process requires the development of advanced numerical models that capture both the structural geometry and mechanical behavior. Besides, Engineering Demand Parameters (EDPs) are employed to describe the measured performance of structural components correlating with various damage states. For seismic fragility analysis, commonly used EDPs include the drift ratio and curvature of piers, displacement and shear deformation of bearings, residual displacement and deformation of abutments [44]. For blast fragility analysis, relevant EDPs encompass response time, stress and strain in structural elements, vertical bearing capacity of piers, and residual displacement and deformation of structural components [45]. In the case of multi-hazard scenarios as depicted in Fig. 1(b), state-dependent fragility should be made available to quantify the damage resulting from Haz-2, conditional on the damage level incurred from Haz-1. This can be achieved by introducing a variable for structural resistance reduction termed as reduction factor (\epsilon) using Eq. (1), to account for the degradation of structural capacity due to damage accumulation caused by the prior hazard.

(1)\epsilon =\left(1-\frac{S_i}{S_{\mathrm{c}\mathrm{o}\mathrm{m}}}\right)\cdot \eta ,

where S_i denotes the corresponding value of EDPs under hazard intensity I=i; S_{\mathrm{c}\mathrm{o}\mathrm{m}} represents the value of EDPs corresponding to the complete damage of the structure. The correction coefficient, denoted as \eta, is determined according to engineering practice [46,47]. The reduction factor can be quantified using various EDPs for each hazard type.

Next, Eq. (2) is adopted to express the probability of exceeding a particular damage state {DS}_i for a given hazard intensity I=i [24]:

(2)P\left(DS\ge {DS}_i|I=i\right)=\Phi \left(\frac{\mathrm{l}\mathrm{n}\left({\displaystyle \frac{I}{{\mu }_i}}\right)}{{\sigma }_i}\right),

where {\mu }_i denotes the median value of hazard intensity required for the structure to reach its i\mathrm{t}\mathrm{h} damage state; {\sigma }_i denotes the logarithmic standard deviation of hazard intensity; \Phi is the standard normal cumulative distribution function. It is important to note that values of {\mu }_i and {\sigma }_i are different for each damage state. The reduction factor from Eq. (1) can be directly applied to adjust the structural resistance while calculating state-dependent fragility using Eq. (2). For varying occurrence of Haz-2 under sequential hazards, the fragility parameters under Haz-1 is same as the individual hazard effect, and the fragility parameters under Haz-2 are interpolated linearly based on occurrence time between two extremes: the first bound, where the two hazards occur consecutively without intermediate restoration, and the second bound, where each hazard occurs with complete restoration, to finalize the fragility analysis (Fig. 2(b)).

2.3 Restoration models

The restoration model refers to characterizing the recovery process of a structure’s functionality over time following hazardous events [48]. It plays a crucial role in quantifying structural resilience, which is defined as the structure’s ability to absorb and resist loading impacts, maintain functionality, and recover quickly to adapt to future scenarios [49]. The restoration models are often represented by recovery curves, that typically begin at the moment of the hazardous event with a residual functionality, and gradually rises over time as repairs are made, eventually reaching a target functionality, either full or near full recovery [50]. The process is quantified by key elements including downtime and recovery path. During the downtime, essential preparatory tasks are undertaken before the actual restoration starts, such as emergency response, inspection and condition assessment [37]. The recovery path serves as a conceptual and graphical tool that illustrates the various stages of restoration, the duration and speed it takes, and the completeness of recovery. It is influenced by various factors, such as the hazard itself, the severity of damage, resource availability, construction techniques, and scheduling, etc. These factors, in turn, affect variables, i.e., residual functionality, downtime, recovery speed and duration, that quantify the restoration process [51].

As discussed in the Introduction, several methods have been developed to estimate the restoration, including expert opinions, empirical and analytical methods [24,39]. To model the recovery path with a high level of uncertainty, a generalized six-parameter sinusoidal-based continuous function proposed by Bocchini et al. [39] is employed. This function can capture various types of restoration patterns, including stepwise, linear, positive and negative exponential, and sinusoidal curves, making it an ideal candidate due to its flexibility. The evolution of functionality over time using this method is depicted in Fig. 2(c). Specifically, the shape and position of the curve are determined by six parameters: residual functionality Q_{\mathrm{r}}, downtime {\delta }_i (months), recovery duration {\delta }_{\mathrm{r}} (months), target functionality Q_{\mathrm{t}}, and two additional shape parameters, s and A. The residual functionality Q_{\mathrm{r}} corresponds to the level of damage sustained by the bridge under various hazards, while the position of the inflection point and amplitude of the curve are controlled by the shape parameters (s and A). Recovery duration {\delta }_{\mathrm{r}} denotes the time required for restoration to complete, while target functionality Q_{\mathrm{t}} represents the fully restored structural capability. Each of these parameters can be modeled as independent random variables, with ranges for different damage states. Three parameters, i.e., Q_{\mathrm{r}}, Q_{\mathrm{t}}, and {\delta }_{\mathrm{r}}, are assumed to follow triangular distribution, and parameter {\delta }_i is assumed to follow uniform distribution. s and A are determined based on engineering experience with specification for different damage states. Monte Carlo approach is used to sample from these distributions, thus providing a probabilistic quantification of restoration for various damage states.

Once obtained the restoration model, the bridge functionality can be quantified accordingly considering all the damage states. If no hazard occurred during restoration, the bridge functionality under each individual hazard can be calculated using Eq. (3) [39]:

(3)Func=\sum _{i=1}^{n}F{R}_{D{S}_i}\cdot P_{D{S}_i|I},

where {FR}_{{DS}_i} represents the structural functionality at the ith damage state {DS}_i, while P_{{DS}_i|I} is the probability of being in the i-th damage state at a hazard intensity I. n refers to the total number of damage states considered for the specified hazard. P_{{DS}_i|I} can be calculated based on the results of fragility analysis using Eq. (4) [52]:

(4)

where P\left({DS}_{ij}|EDP\right) represents structural fragility at the jth damage state {DS}_{ij} of the ith structural component, conditioned on the EDPs of interest; F_{D{S}_{ij}}\left(EDP\right) is the fragility curve, computing the probability of being or exceeding the jth damage state of the ith structural component.

When considering the interrelationship between hazards, e.g., Haz-2 occurs during restoration after Haz-1 before complete recovery is achieved, the functionality calculated from Eq. (3) needs to be adjusted using Eq. (5) [20]:

(5)Func=\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}{FR}_{{DS}_{i,1},{DS}_{j,2}}\cdot P_{{DS}_{j,2}|{DS}_{i,1},I},

where {FR}_{{DS}_{i,1},{DS}_{j,2}} represents the structural functionality that needs to recover from damage state {DS}_{i,1} due to Haz-1 and damage state {DS}_{j,2} due to Haz-2, and P_{{DS}_{j,2}|{DS}_{i,1},I} is the probability of being in damage state {DS}_{j,2} for Haz-2, conditional on being in damage state {DS}_{i,1} at intensity I for Haz-1. n_1\mathrm{a}\mathrm{n}\mathrm{d}{n}_2 represent the number of damage states for Haz-1 and Haz-2, respectively. {FR}_{{DS}_{i,1},{DS}_{j,2}}covers the restored functionality following Haz-1 with reduction subsequently due to damage incurred from Haz-2. This modification applies specifically to the restoration after the occurrence of Haz-2, while the functionality in the earlier stage prior to Haz-2, is assumed to follow the same trajectory as that of individual hazards. For varying occurrence of Haz-2, the parameters for restoration are interpolated linearly based on occurrence time between two extremes: the first bound, where the two hazards occur consecutively without intermediate restoration, and the second bound, where each hazard occurs with complete restoration, to finalize the functionality (Fig. 2(c)).

2.4 Resilience quantification

Quantification of resilience is performed through established functionality curves. The commonly used way is through a resilience index using Eq. (6) [53]:

(6)R=\frac{{\int }_{t_0}^{t_{\mathrm{h}}}Func\left(t\right)\mathrm{d}t}{t_{\mathrm{h}}-t_0},

where Func\left(t\right) denotes the functionality curves derived in the previous step, which is time-dependent relying on the restoration; t_0 is the time of occurrence for hazards; t_{\mathrm{h}} is the investigated time horizon, i.e., the time that structural functionality fully recovered.

It is recognized that under multi-hazard scenarios, the time intervals between hazards are stochastic, either very short representing concurrent events, or refer to much longer periods. As per sequential hazards (i.e., Haz-1, Haz-2) considering the random occurrence of Haz-2 during restoration, the resilience index calculated from Eq. (6) is treated as a random variable. To model the occurrence of Haz-2, Monte Carlo approach is employed, assuming its occurrence follows a uniform distribution between the onset of Haz-1 and the completion of restoration, creating a large number of sampled scenarios as outlined in Fig. 2(d) [24,54]. This helps model the occurrence of second hazard in a manner that reflects maximum uncertainty without introducing unwarranted biases in the absence of detailed recorded data indicating a varying likelihood of hazard occurrence over time. The resulting resilience index and its statistical characteristics, encapsulating the entire loss and restoration, can help facilitate the efficient allocation of resources, planning, and interventions to streamline the decision-making process (Fig. 2(d)).

3 Application to a reinforced concrete bridge

3.1 Description of the case study

This section illustrates the application of the multi-hazard resilience assessment approach to a realistic case study involving a reinforced concrete (RC) bridge, as shown in Fig. 3. This is a two-span bridge exposed to sequential hazards including an earthquake and a blast caused by vehicle explosion passing underneath. It represents a common class of bridges, with two equal spans of 35 m, totaling 70 m in length. The bridge features a single pier and two bearings located on integral abutments at each end, supported by well-compacted backfill materials. The deck consists of an 8-m-wide and 1.2-m-high box girder. The pier has a square cross-section measuring 1.5 m × 1.5 m and a height of 6.5 m. The longitudinal and transverse reinforcement ratios for the pier are 2% and 0.67%, respectively. The concrete grade for both the girder and pier is C40, while the reinforcement has a yield strength of 400 MPa. The structural damping ratio is 5%. The bridge resilience has been assessed under two scenarios: 1) an earthquake followed by a blast (noted as Earthquake & Blast); 2) a blast followed by an earthquake (noted as Blast & Earthquake), both accounting for the temporal variability of occurrence for the second hazard. For earthquake events, this study selected 100 ground motions from the Pacific Earthquake Engineering Research Center (PEER) strong-motion database based on shear wave velocity, earthquake magnitude, and source-to-site distance. Each ground motion was scaled in terms of PGA to achieve varying PGA levels for conducting nonlinear time-history analyses. Detailed information on the selected ground motions can be found in the Appendix.

3.2 Fragility analysis

For sequential hazards, fragility analysis involves determining the state-dependent fragility considering damage accumulation, as well as random occurrence of the second hazard. To achieve this, fragility parameters have been studied for two boundary conditions under each scenario: 1) the first bound, where both hazards occur consecutively without intermediate restoration; 2) the second bound, where each hazard occurs with complete restoration. The fragility parameters presented in this paper were primarily derived from numerical analysis without taking into account the 3D effects to cover the particular needs of this study.

(1) Fragility for Earthquake & Blast

The seismic fragility of bridge was analyzed using a nonlinear dynamic model in OpenSees (Fig. 3). As the superstructure remains undamaged under seismic loading, elastic beam elements were used to model the main girder and deck slabs. Zero-length elements were used to simulate the abutments, with HyperbolicGapmaterial and Hysteretic material models representing the active and passive earth pressures, respectively [55,56]. The behavior of elastic bearings was simulated using a bilinear material model, and the dispBeamColumn element was used to capture the nonlinearity of pier performance during earthquake [57]. The longitudinal reinforcement was modeled using the uniaxial bilinear steel material [58], while the core concrete was represented using the Popovics concrete material [59,60], and the outer concrete represented using the Kent-Scott-Park concrete material [61]. The pier base was constrained in all degrees of freedom, without considering pile-soil interaction effects.

Four potential damage states-slight, moderate, serious, and complete damage-were considered, utilizing PGA as the hazard intensity indicator. EDPs including pier base curvature, deformation of bearing and abutment were analyzed. Probabilistic seismic fragility analysis was conducted for each component as well as for the entire structure, following the method established by Nielson and DesRoches [62]. The analysis identified potential seismic failure modes for each damage state as illustrated in Table 1. The bridge, characterized as a high-capacity frame structure, demonstrated low vulnerability to seismic actions. The damage was predominantly of a geotechnical nature under earthquakes, affecting the backfill-abutment system rather than the pier and deck. Significant structural failure is anticipated only under high levels of seismic intensity. Therefore, when considering earthquake as the first hazard in sequential hazard scenarios, the seismic damage was concentrated on abutments and bearings. Parameters for the individual seismic fragility corresponding to various damage states following Eq. (2) were summarized in Table 2 (see Earthquake in Table 2).

Three levels of seismic input were considered as the initial hazards, specifically PGA of 0.15g, 0.5g, and 1.5g, corresponding to slight, moderate, and serious damage of the bridge. Given that the pier is the primary component affected by the subsequent blast load caused by vehicles explosion, assessing resistance reduction for pier due to prior earthquake was necessary. In this context, the residual plastic energy that the pier can absorb following an earthquake was employed to calculate the reduction factor using Eq. (1). It was quantified as the area under the Moment–Curvature (M–C) curve from pushover analysis. Specifically, S_i denoted the area under the M−C curve corresponding to certain seismic intensity, while S_{\mathrm{c}\mathrm{o}\mathrm{m}} represented the area at the maximum curvature, which indicated the complete damage. The correction factor \eta of 0.8 was applied for safety considerations based on engineering practice. Calculated reduction factors for the three seismic events were 0.79, 0.78, and 0.75, respectively. These were applied to adjust the pier capacity while deriving the fragility parameters for consecutive blast load without considering any intermediate restoration (see Earthquake & Blast in Table 3).

(2) Fragility for Blast & Earthquake

The pier was the main damaged component as per the blast load. Accordingly, reduction factor was introduced for the pier to account for damage accumulation for subsequent earthquake. Where the pier base was fixed and the pier top was constrained in terms of horizontal and rotational displacement, three failure modes may exist for the pier: shear, bending, and combined bending-shear failure. The occurrence of different failure modes depends on the characteristics of blasts. Specifically, under impulsive blast, the failure mode of pier is primarily governed by shear failure, whereas under quasi-static loading, the pier exhibits bending failure. In the transitional status between the two, the pier experiences a combined bending-shear failure. Potential failure modes induced by blasts for each damage state in the case study were summarized in Table 1. The overpressure–impulse (P–I) diagram is a widely used method for analytically characterizing and predicting blast damage to structures, as per Eq. (7) [63,64].

(7)\left(P-P_0\right)\left(I-I_0\right)=12{\left(P_0/2+I_0/2\right)}^{1.5},

where P denotes the overpressure, and I denotes the impulse. P_0 and I_0 refer to the critical values of overpressure and impulse for different damage states of the RC columns, respectively. The diagram divides the entire (P,I) space into two parts (Fig. 4): (1) (P,I) located in the upper-right space of the curve indicates that structural damage exceeds the specified damage state, and (2) while (P,I) located in the lower-left indicates that structure has not yet reached the specified damage state [63]. Different P−I curves serve as boundaries that delineate various damage states, dividing the entire (P,I) space into distinct regions. Since RC columns primarily serve as vertical load-bearing components, the reduction in the vertical load-bearing capacity was recognized as the damage criterion as calculated in Eq. (8), to quantify various damage states with thresholds of D = 0.2, 0.5, and 0.8 for slight, moderate and serious damage as listed in Table 1 [65,66].

(8)D=1-\frac{P_{{\mathrm{N}}_{-}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}}^{\prime }}{P_{{\mathrm{N}}_{-}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{\prime }},

where P_{{\mathrm{N}}_{-}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{\prime } denotes the vertical load-bearing capacity of the intact pier, while P_{{\mathrm{N}}_{-}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}}^{\prime } denotes the residual load-bearing capacity of the pier after blast load.

P−I diagrams directly obtained from experimental methods are usually labor-intensive, resource-intensive, and costly. Therefore, analytical formulae established by Shi et al. [65] based on numerical modeling results to predict the overpressure and impulsive asymptotes (i.e., P_0,I_0) for each damage state was utilized in this study. The computed (P_0,I_0) considered the distributions of random variables including material and geometric parameters using their statistical characteristics (i.e., mean µ and standard deviation σ) corresponding to each damage state. Blast fragility analysis was conducted to derive fragility surface for piers following the procedure proposed by Parisi [67] and a discrete number of (P,I) levels was obtained for various damage states. Assuming that the blast fragility was a bivariate lognormal probability distribution, (P,I) can be fitted to empirical fragility values by means of the Ordinary Least Squares (OLS) or the Maximum Likelihood Estimation (MLE) method through a hybrid EDP X as per below [67]:

(9)X=P^{{\alpha }_1}{I}^{{\alpha }_2},{\alpha }_1+{\alpha }_2=1,

where P denotes the peak overpressure in kPa and I denotes the impulse in kPa·ms. {\alpha }_1 and {\alpha }_2 are best-fit coefficients using OLS or MLE, creating units for X as {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}.This study utilized the OLS method to derive the empirical fragility in the function of hybrid variable X through minimisation of the sum squared errors. The estimated parameters {\alpha }_1,{\alpha }_2 can be obtained through solving the following optimisation equation:

(10)\begin{array}{rl}& \frac{min}{{\epsilon }_0,{\epsilon }_1,{\epsilon }_2}\sum _{j=1}^m{\left\{p_j-\varnothing \left[{\epsilon }_0+{\epsilon }_1\mathrm{l}\mathrm{n}{P}_j+{\epsilon }_2\mathrm{l}\mathrm{n}{I}_j\right]\right\}}^2,\\ & {\epsilon }_0=-\frac{\mu }{\beta },{\epsilon }_1=-\frac{{\alpha }_1}{\beta },{\epsilon }_2=-\frac{{\alpha }_2}{\beta },\end{array}

where \varnothing (\cdot ) is the normal cumulative distribution function; \mu is the mean of lnX; \beta is the standard deviation of lnX. p_j is the empirical fragility value. Accordingly, blast fragility parameters corresponding to various damage states derived through this process were summarized in Table 3 (see Blast in Table 3).

Three levels of input were considered for the initial occurrence of blast loadings, specifically X = 4000, 7000, 10000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}, corresponding to slight, moderate, and serious damage of the bridge. Resistance reduction factor was introduced to consider the damage accumulation of pier for subsequent seismic fragility analysis. In this context, the hybrid variable X was employed to reflect damage levels to pier and calculate the reduction factor using Eq. (1). Specifically, S_i denoted the value of X corresponding to certain blast intensity, while S_{\mathrm{c}\mathrm{o}\mathrm{m}} represented the value of X indicating the serious damage state. The correction factor \eta of 0.9 was applied for safety considerations based on engineering practice. Thus, reduction factors for three blast hazards were 0.58, 0.33, and 0.090, respectively. These factors were applied to adjust the pier resistance while deriving the fragility parameters for sequential earthquake without considering any intermediate restoration (see Blast & Earthquake in Table 2).

The fragility parameters listed in Table 2-3 were used as the boundary conditions to estimate fragility parameters under the sequential hazards of an earthquake and a blast considering the random occurrence of the second hazard during restoration after the first one. Linear interpolation based on the timing of occurrence for the hazard was employed to achieve it. For Earthquake & Blast, the upper bound represented the individual blast effect, while for Blast & Earthquake, it represented the individual seismic effect. The lower bound in both scenarios represented the consecutive hazards scenarios.

3.3 Structural restoration

As introduced in Subsection 2.3, the development of restoration models representing bridge functionality changing over time was influenced by various factors with high level of uncertainty. A generalized six-parameter sinusoidal-based continuous function was employed in this study to model a wide range of recovery patterns. The six parameters were treated as independent random variables with ranges for different damage states. A summary of the damage states and corresponding restoration tasks under seismic and blast loads were listed in Table 1. As per seismic loads, the expected most severe damage was concentrated on the backfill-abutment system of geotechnical nature, which is relatively easier and faster to repair. For blast loads, the damage was mainly concentrated on pier, the structural component providing vertical support for girder. If the pier was completely destroyed, the collapse of the bridge would occur and result in the reconstruction, which required a longer time for restoration. In this paper, it is assumed that functionality of the fully recovered bridge is the same as that of the intact one. The effect of hazards on bridge is only affected by the lost functionality itself but not by the hazard nature. For example, a 20% loss of functionality due to an earthquake followed by a 10% loss due to the blast means a total loss of 30% if no restoration occurs. The parameters for restoration models under individual and sequential hazards of an earthquake and a blast were summarized in Tables 4–11, which were referenced from Argyroudis et al. [24], Bocchini et al. [39], Wilson et al. [68] and FEMA [69]. The parameters listed in Tables 4–11 were used as the boundary conditions to estimate the parameters under sequential hazards considering the random occurrence of the second hazard during restoration. Linear interpolation based on the timing of occurrence for the hazard was employed.

3.4 Resilience quantification and results discussion

This section contains the results of analyses performed to the case study to highlight the impact of multi-hazards considering a sequential earthquake and blast on a typical RC bridge. Various hazard sequences, intensities and the random occurrence during restoration were considered for assessing the bridge resilience following the approach introduced in Section 2. To ensure the precision of simulation, 0.3 d was selected as the sampling interval for simulating the occurrence of the second hazard after the first hazard. Details of results are listed as follows.

1) Earthquake & Blast

Figure 5 shows the results corresponding to the sequential occurrence of earthquake (Haz-1) and blast (Haz-2). Five blast levels (X = 4000, 6000, 8000, 10000, 12000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}) were considered, with their occurrences distributed during restoration for three seismic intensity levels (PGA = 0.15g, 0.5g, 1.5g). The black dashed lines represent scenarios where a blast occurs at the onset of restoration, while the red dashed lines indicate the blast occurrence at the end. Grey lines represent the random occurrence of blasts at different moments throughout the entire process. The green lines represent the expected resilience of all sampled scenarios. As indicated in Fig. 5(a), when a blast of lower intensity (X = 4000, 6000, 8000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}) occurs shortly after an earthquake with a low PGA (0.15g), the bridge retains some functionality despite the damage. However, with higher blast intensities (X = 10000, 12000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}), bridge functionality can be entirely lost. In contrast, when an earthquake (PGA = 0.2g) follows a blast (X = 4000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}) shortly without restoration, the structural functionality drops to zero (Fig. 6(a)). This difference is due to the varying vulnerabilities of components under different hazards. When earthquake occurs first, it primarily damages the abutments, highlighting the seismic vulnerability of the whole structure, while the following blast mainly affects the pier. As the hazard effects are distributed across different components, functionality remains when both hazards are of low intensity. However, when blast occurs first, pier damage caused by the blast reduces the seismic resistance, making the pier the most vulnerable component under subsequent earthquake. The combination of hazard effects on the same component results in a more significant reduction in functionality. Figure 5(b) presents the second case, where a blast occurs following an earthquake with intensity PGA = 0.5g. The difference in the expected resilience between this scenario and the one with a PGA of 0.15g is small. This is because seismic damage is concentrated in the abutments, while blast primarily affects the pier. For higher blast intensities (X > 6000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}) combined with seismic events, the subsequent blast significantly impairs bridge functionality, even after most of the restoration tasks. Figure 5(c) illustrates the third case, in which a blast occurs after an earthquake with a PGA of 1.5g. The high-intensity earthquake causes severe damage to the bridge, substantially reducing its functionality. In this case, complete loss of structural functionality occurs even with a low-intensity blast following.

2) Blast & Earthquake

Figure 6 shows the results corresponding to the sequential occurrence of blast (Haz-1) and earthquake (Haz-2). Five seismic intensity levels (PGA = 0.2g, 0.4g, 0.6g, 1.0g, 1.5g) were considered, with the occurrences randomly distributed during restoration for three levels of blasts (X = 4000, 7000, 10000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}). The black dashed lines represent scenarios where an earthquake occurs at the onset of restoration, while the red dashed lines indicate earthquake occurrence at the end. Grey lines represent the occurrence of earthquakes at different moments throughout the entire process. The green lines represent the expected resilience of all sampled scenarios. As indicated in Fig. 6(a), even a lower level of PGA (0.2g), soon after the blast, is sufficient to reduce functionality to zero. The combination of reduced functionality after the initial hazard and high seismic fragility due to pier damage makes it highly vulnerable to the second hazard in a short period. In all cases, the effect of earthquake on resilience diminishes when it occurs long after blast, as illustrated by the gray and red curves. However, for higher PGA levels (1.0g, 1.5g), structural functionality is significantly reduced even when earthquake occurs near complete restoration. Figure 6(b) shows the second case, where an earthquake occurs following a blast with intensity X = 7000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}. A PGA of 0.2g striking when 50% of the lost functionality is restored, can cause a complete loss of bridge functionality. In contrast, for the scenario involving the blast (X = 4000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}) and earthquake (PGA = 0.2g), the earthquake occurring at the same stage does not reduce the functionality to zero. Thus, higher PGA levels (0.6g,1.0g, 1.5g) can severely compromise functionality. Figure 6(c) shows the case in which an earthquake occurs following a blast with intensity X = 10000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}. In this case, the first hazard significantly reduces the structural resistance and functionality, making it highly vulnerable to even low-intensity seismic events following. This represents the worst-case scenario, where Haz-1 leads to a substantial drop in functionality, together with increased structural vulnerability, and prolonged restoration time. Figure 6 shows that the expected resilience decreases with increasing X and PGA, and the occurrence of Haz-2 plays a crucial role in resilience assessment. Two hazards occurring at shorter intervals cause greater damage, resulting in low resilience, where rapid succession leads to the worst case. When two hazards occur at longer intervals, especially when Haz-2 occurs at the end of Haz-1 restoration, the damage is less severe, leading to higher resilience. The introduction of coefficient of variation for resilience quantifies the uncertainty associated with the occurrence of Haz-2. The variability enables a reliability-based assessment of bridge resilience, highlighting the importance of using probabilistic approach for resilience assessment, particularly for multi-hazards of which the occurrence time can play a role.

Comparing Fig. 5 to Fig. 6, it is noted that even with the same hazard intensity, different hazard sequences produce distinct functionality patterns and resilience estimates (Fig. 5(c)): PGA = 1.5g & X = 4000, 10000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2}; Fig. 6(a): X = 4000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2} & PGA = 1.5g and Fig. 6(c): X = 10000 {\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)ּ}^{{\alpha }_1}{(\mathrm{k}\mathrm{P}\mathrm{a}\cdot \mathrm{m}\mathrm{s})}^{{\alpha }_2} & PGA = 1.5g). Besides, the functionality curves, in this case, exhibit a more dispersed distribution over time, with a less discrete resilience estimate. This is because when the occurrence of Haz-2 is much closer to the end of restoration, the restoration time is closer to the individual Haz-2. The time required for individual blast is significantly longer than that for individual earthquake (see Tables 4-5). As a result, functionality curves for the scenario where blast occurs after earthquake span a wider time range and show more pronounced discontinuities. However, due to the coarse classification of damage states for blast relatively to earthquake, the bridge failure probability for each damage state is not sensitive to blast intensity as that for earthquake. Consequently, the variation in resilience across simulations remains relatively small. In other words, the occurrence of blasts has little effect on bridge resilience in this hazard sequence. This finding highlights the importance of considering hazard sequence in resilience assessments, especially in a multi-hazard context.

4 Conclusions

This paper provides an approach for resilience assessment of bridges under multi-hazards of earthquakes and blasts. The approach considers hazard randomness, structural robustness and variability of restoration to quantify resilience against multiple hazards. It can aid stakeholders and infrastructure owners in decision-making, helping maximize structural resilience through risk mitigation and restoration during service life. The approach was demonstrated through a case study of a two-span highway bridge subjected to sequential hazards: an earthquake and a blast. Two hazard sequence with various intensity were considered and analyzed: Earthquake & Blast, and Blast & Earthquake. For each case, random occurrence of the second hazard was considered to develop functionality curves and estimate the resilience. The results lead to the following conclusions.

1) Bridge resilience decreases as hazard intensity increases in all cases. The occurrence of the second hazard during restoration significantly affects resilience, making it lower than when hazards occur and recover independently. Shorter intervals between hazard occurrence cause heavier damage and lower resilience, where rapid succession leads to the worst case. While longer intervals, especially when the second hazard occurs after complete restoration for the first hazard, result in less damage and higher resilience. Damage from the previous hazard can reduce the structural resistance subjected to subsequent hazard, potentially causing complete loss of structural functionality. Multi-hazards cannot be treated independently, as this may overestimate bridge resilience. To ensure the accurate and safety-oriented resilience assessment, it is essential to consider hazard interrelationships, including uncertainty in hazard timing and the sequential effects of multiple hazards. These findings demonstrate the need for accounting the hazard randomness, interrelationship and structural robustness to assess structural resilience subjected to multiple hazards of earthquakes and blasts.

2) Bridge resilience under multi-hazards of earthquakes and blasts depends on the hazard nature, failure mechanisms, and practical restoration. Even subjected to the same hazard intensity, different hazard sequences produce distinct restoration patterns and resilience estimates. For example, unlike the Blast & Earthquake scenario, the Earthquake & Blast scenario shows restoration curves that span a wider time range with noticeable disruption. This highlights the impact of hazard randomness and variability of restoration on bridge resilience, emphasizing the need to consider it in a detailed manner using probabilistic approach within the multi-hazard context.

This paper demonstrates the approach for assessing bridge resilience under multi-hazards of earthquakes and blasts. It can be adapted globally to quantify the resilience of various critical infrastructures, such as buildings, power transmission towers, and dams against diverse hazard combinations, e.g., winds, tsunamis, floods, and scour.

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