Sage Journals: Discover world-class research

Abstract

Although fragility function development for structures is a mature field, it has recently thrived on new algorithms propelled by machine learning (ML) methods along with heightened emphasis on functions tailored for community- to regional-scale application. This article seeks to critically assess the implications of adopting alternative traditional and emerging fragility modeling practices within seismic risk and resilience quantification to guide future analyses that span from the structure to infrastructure network scale. For example, this article probes the similarities and differences in traditional and ML techniques for demand modeling, discusses the shift from one-parameter to multiparameter fragility models, and assesses the variations in fragility outcomes via statistical distance concepts. Moreover, the previously unexplored influence of these practices on a range of performance measures (e.g. conditional probability of damage, risk of losses to individual structures, portfolio risks, and network recovery trajectories) is systematically evaluated via the posed statistical distance metrics. To this end, case studies using bridges and transportation networks are leveraged to systematically test the implications of alternative seismic fragility modeling practices. The results show that, contrary to the classically adopted archetype fragilities, parameterized ML-based models achieve similar results on individual risk metrics compared to structure-specific fragilities, promising to improve portfolio fragility definitions, deliver satisfactory risk and resilience outcomes at different scales, and pinpoint structures whose poor performance extends to the global network resilience estimates. Using flexible fragility models to depict heterogeneous portfolios is expected to support dynamic decisions that may take place at different scales, space, and time, throughout infrastructure systems.

Keywords

Fragility functions surrogate models risk metrics network efficiency infrastructure resilience

Introduction

Measuring and modeling resilience has become imperative for establishing community resilience baselines, determining actions to improve it, and monitoring its changes (National Academies of Sciences, Engineering, and Medicine, 2019; National Research Council, 2012). A key component of any seismic risk and resilience modeling framework is damage modeling of physical assets, including structures and infrastructure systems (e.g. Cimellaro et al., 2010; Deierlein et al., 2003; van de Lindt et al., 2023). These performance estimates are a precursor for subsequent loss analyses, repair and recovery modeling, and interdependent systems analyses. The common practice is to use fragility functions to predict systems’ performance while recognizing the influence of inherent sources of uncertainty. A fragility function is often defined as the conditional probability of the response of a system exceeding its capacity, expressed as:

P_{f | X} = P [Y \geq C | X] = P [ℓ (X) \leq 0 | X]

(1)

where $Y$ is a measurable structural response (also called engineering demand parameter), $C$ is the capacity of the system measured in the same units as $Y$ , and $ℓ (X) = C - Y$ is the limit state function in terms of a basic variable $X$ (Melchers and Beck, 2018). In classical approaches, $P_{f | X}$ is assumed to be conditioned only on the seismic intensity measure, considered to be the main descriptor of the damage probability, and denoted here as one-parameter fragilities. For regional analysis, past studies have proposed the use of archetype fragility functions to reflect the vulnerability of portfolios of structures (Federal Emergency Management Agency [FEMA], 2020). That is, the performance of similar structures is lumped into single fragility models as shown in Figure 1a. However, important differences in predicted performance and risk estimates have been observed when structures assumed to be similar turn out to have a very different seismic response (Heresi and Miranda, 2022; Mangalathu et al., 2017). To somehow address this, archetypes fragilities must be defined for classes subdivided considering several combinations of system parameters (Jaiswal et al., 2011; Martins and Silva, 2021). These models are widely used given they are a practical tool for regional-scale probabilistic analysis.

Figure 1.

Strategies for assigning fragility functions for regional-scale analysis: (a) archetype fragilities, (b) parameterized fragilities, and (c) structure-specific fragilities.

The limitations of archetype fragilities have propelled a growing body of researchers to propose models that capture variations in the fragility estimate given differences in structural parameters. Modeling each structure separately, as graphically depicted in Figure 1c, may not be feasible; besides, the benefit of such an endeavor is not yet clear. This approach may be hampered by the lack of data to characterize individual structures or by the computational costs associated with nonlinear simulations of thousands of structures across a network. Even if achieved, the derived structure-specific fragilities could become outdated due to aging processes or other situations that can provoke changes in the portfolio (Choe et al., 2008; Ghosh and Padgett, 2011; Rao et al., 2017). Hence, recent studies on analytical fragility derivations have focused on reducing the cost of nonlinear dynamic simulations by either lowering the resolution of the models, utilizing data-driven approaches or a combination of these (e.g. Ghosh et al., 2013; Muntasir Billah and Shahria Alam, 2015; Patsialis and Taflanidis, 2020; Rincon and Padgett, 2022; Seo et al., 2012). The faster computation of the system response, typically achieved through surrogate models, allows more refined analysis at the regional scale. Besides computational efficiency for response prediction, some researchers have exploited such methods to propose more expressive fragility models that capture the portfolio heterogeneity and system conditions. These models assume the failure probability, $P_{f | X}$ , to be conditioned on a vector of basic variables $X = (X_{1}, \dots, X_{j})$ —for example, the intensity measure, structural, material, and geometrical features—thus are known as multiparameter or parameterized fragilities (illustrated in Figure 1b). With this model, one can approximate the failure probability of each structure $i$ in a portfolio as a function of the hazard intensity and structural characteristics $x_{i}$ (Ghosh et al., 2013; Seo et al., 2012).

This article addresses the unanswered question of how and if different fragility modeling practices (including those that leverage emerging surrogate modeling, machine learning (ML) techniques, and parameterized approaches) actually influence the outcomes of seismic risk and resilience quantification. Along the way we propose a systematic approach for comparing fragility methods that influence a wide range of performance measures that span from the structure to infrastructure network scale. To this end, the next section discusses common steps for the derivation of seismic fragility functions and presents the details of performance measures for comparison of assessment results ranging from the structure level to the network scale (e.g. conditional probability of damage, risk of losses to individual structures, portfolio risks, network recovery trajectories). The theoretical backgrounds, similarities, and differences among fragility modeling approaches are described and applied to case studies that consist of multiple spans simply supported bridges subjected to bidirectional ground motion records. The subsequent section presents the implications of adopting alternative traditional and emerging fragility modeling practices within seismic risk quantification of structures and portfolios, and in the final section, to resilience estimation of infrastructure networks.

Methods

The proposed comparisons are presented in three main steps: (1) derivation and comparison of fragility functions for multiresponse systems, (2) evaluation of implications on risk metrics at the structure and portfolio scale, and (3) measurement of influence on network resilience estimates. This section summarizes the overall steps used for fragility function derivation, while the details of the compared modeling practices are presented in the next section. The following “Methods” subsections present the performance measures proposed in this study to quantify the influence of such practices in risk and resilience outcomes across scales.

Our study evaluates multiresponse systems, that is, structures whose performance (or damage condition) is defined by $k > 1$ components or failure modes (Aslani and Miranda, 2004; Bichon et al., 2011; Choe et al., 2008; Du and Padgett, 2020; Muntasir Billah and Alam, 2015; Nielson and DesRoches, 2007b). This entails that the system responses, and the capacity, must be represented by multivariate random vectors $Y = (Y_{1}, \dots, Y_{k})$ and $C = (C_{1}, \dots, C_{k})$ . In this case, the failure region is computed by considering the components’ limit state functions $ℓ_{i} (x) = C_{i} - Y_{i}$ , for $i = 1, \dots, k$ and the arrangement of the components into the system (Bichon et al., 2011; Kang et al., 2008; Nielson and DesRoches, 2007b). Hence, the fragility function is defined in terms of the composite limit state function $L (X)$ . Equations 2 and 3 present the failure probability of structures whose system failure is described by series and parallel arrangement, respectively. Other intermediate configurations can be defined following similar expressions (Dueñas-Osorio and Padgett, 2011):

P [⋃ ℓ_{k} (X) \leq 0] = P [min (ℓ_{1} (X), \dots, ℓ_{K} (X)) \leq 0]

(2)

P [⋂ ℓ_{K} (X) \leq 0] = P [max (ℓ_{1} (X), \dots, ℓ_{K} (X)) \leq 0]

(3)

Derivation of fragility functions for systems with multiple failure modes

Common fragility derivation techniques share the steps depicted in Figure 2. As shown in the figure, the main difference lies in the procedure used for the estimation of the seismic response (Step 5); hence, we focus our attention on this task. While more details on the investigated techniques are presented in the next section, the general steps are described as follows:

Records selection: a set of representative ground motion records are selected. These are expected to cover a wide range of intensity measures values (e.g. peak ground acceleration). In fact, this suite should be representative of the joint distribution of high-dimensional intensity measure descriptions (i.e. multivariate hazard consistency) (Du and Padgett, 2021). Only unscaled ground motions are used in this study to avoid scale-induced bias (Mai et al., 2017; Sudret et al., 2015).

Model definition and structural variables: the computational model, modeling assumptions and modeling parameters are defined to capture the uncertain and complex system failure modes. For forward uncertainty propagation, modeling parameters that may be significant in the system response may be treated as random variables, represented by the vector $X = (X_{1}, \dots, X_{j})$ (Padgett and DesRoches, 2007).

Definition of the experimental design: considering exploration of the entire input space $X$ infeasible, a set of $n$ samples ${x_{1}, \dots, x_{n}}$ , with $x_{i} \in R^{j}$ , are obtained from the joint probability density function (pdf) $f_{X} (x)$ . Different sampling methods such as Monte Carlo, factorial designs, or Latin Hypercube Sampling can be used (Burton et al., 2021; Simpson et al., 2001). The samples, arranged in an $n \times j$ matrix, are called the experimental design $X$ . Two experimental designs are created for most of the derivation methods: (1) a first design $X_{1}$ with feasible size for computing expensive nonlinear simulations and (2) a second (larger) design $X$ to be evaluated through demand prediction models (Step 5).

Nonlinear dynamic analyses: computer algorithms $h (\cdot)$ are used to obtain the response of the system $y = h (x)$ . This is the most time-consuming step and may require using high-performance computers (Rathje et al., 2017). The responses $Y_{1}$ of the initial experimental design $X_{1}$ are obtained from nonlinear dynamic analyses.

Construction of demand prediction model and response estimation: the data set ${X_{1}, Y_{1}}$ is used to construct the demand prediction models. These are subsequently leveraged to estimate the response of the larger experimental design $X$ , extending the input parameter uncertainty propagation. The details of the examined practices are presented in the next section.

Capacity definition and computation of failure events: the components’ capacities $C$ must be sampled from probabilistic descriptions or obtained as a function of the system parameters $X$ for all $n_{ds}$ damage states. Following the recommendations of previous studies, a total of $n$ capacities $c$ are paired with the $n$ response predictions $\hat{y}$ obtained from Step 5 (Ghosh et al., 2013). Using the failure indicator function $1_{L (x) \leq 0}$ , each system’ response-capacity pair is translated into a binary (survive-failure) state, mapping to 1 the system failure events ( $L (x) \leq 0$ ) and 0 otherwise. This results in a vector of Bernoulli trials $B$ . This procedure is repeated for all $n_{ds}$ possible capacities, and the $B$ vectors are summed to obtain a single damage classification vector, that is, $\sum_{b} 1_{L_{b} (x) \leq 0}$ , for $b = 1, \dots, n_{ds}$ . For brevity, it still termed $B$ , but now it comprises the realizations of system damage state, $b \in D = {0, 1, \dots, n_{ds}}$ .

Fitting a fragility model: while a common approach is to fit lognormal and logistic distributions to the conditional failure probabilities, non-parametric methods can also be used (Lallemant et al., 2015). Multinomial logistic distribution has also been deemed particularly suitable for sequential damage fragility functions (Andriotis and Papakonstantinou, 2018). In this model, the probability of a damage state $b$ is computed using Equation 4:

P (DS = b | X = x) = Λ_{b} (x) = \frac{e^{(x β_{b})}}{\sum_{i \in D} e^{(x β_{i})}}

(4)

Figure 2.

Common steps of fragility derivation practices.

Using multiclass logistic regression, one can obtain the set of coefficients $β_{i}$ , $i \in D$ , that better predict a damage state $b$ from the predictors $X$ . Note that it can be applied to either one-parameter or parameterized fragilities, where in the case of the former, the descriptor is only, usually, the intensity measure in the logarithmic space. Using this approach, the fragility function of a damage state $b$ is just the summation of probabilities up to $b$ , that is, $P (DS \geq b | X = x) = \sum_{i = 1}^{b} P (DS = i | X = x)$ .

Implications of fragility derivation methods across risk estimates

Although (visual) comparisons of a “benchmark” fragility model with derived fragilities are commonplace, this strategy offers little information about the expected implications on risk and resilience estimates. Hence, the following subsections explain how we propose to compare the derived fragilities at different scales and across performance measures, considering that outcomes at each stage can be of interest for different stakeholders. The proposed metrics for comparing methods are based on the cumulative distribution function $F (x)$ , the probability distribution function $dF (x) / dx = f (x)$ , or aggregated metrics (such as the expected value $E [\cdot]$ ) obtained from the outcomes.

Statistical comparison between fragility functions

Independent of the type of derivation method, a primary outcome of such fragility analyses is often a single fragility curve that is derived in the form of a univariate function; parameterized fragilities require additional steps to arrive at this form as explained later within the context of the case study. We use statistical concepts to measure the “differences” between the evaluated techniques and a benchmark model. Given that a fragility function resembles a cumulative distribution function (cdf), the Kolmogorov–Smirnov statistic $D_{KS}$ (Equation 5) is a good and practical metric to compare fragilities (Miller, 1956). Noting that this metric only focuses on the maximum difference across cdfs, we adapted additional metrics such as the Hellinger distance $H$ (Equation 6) and the Wasserstein distance $W_{1}$ (Equation 7) (Balakrishnan, 2019). These give information in relation to differences between probability distribution functions (pdf) $f$ and $\hat{f}$ , and cdfs, $F$ and $\hat{F},$ respectively, across the whole conditional parameter $x$ . All the selected distances are symmetric metrics, allowing comparisons between distributions without the need of defining a “reference” distribution.

The $D_{KS}$ and the Hellinger distance $H$ are bounded between zero and one as they measure the distances in terms of probability units. However, these metrics differ on how they weight differences in the statistical distributions across $x$ . For example, while the $D_{KS}$ highlights the maximum difference on the cdfs that occurs at a single $x$ value, the Hellinger distance evaluates the cumulative distances across $x$ (i.e. the Euclidean norm), focusing on the entire range of overlap between pdfs. On the contrary, the Wasserstein metric measures the “amount of work” required to transform one distribution function into the other. That is, it measures the “work” associated with how long and how much the mass of probability has to be “moved” from one distribution to represent a second distribution (Balakrishnan, 2019). In the one-dimensional case, $W_{1}$ in Equation 5 results in a metric of distance in units of “work” measured in units of the sample space (thus not comparable to the first two metrics). That is, if the distribution function measures the stochastic time to recover functionality after an event, then $W_{1}$ is in units of time:

D_{KS} (F, \hat{F}) = sup_{x} | F (x) - \hat{F} (x) |

(5)

H (F, \hat{F}) = \frac{1}{\sqrt{2}} {[\int {(\sqrt{f (x)} - \sqrt{\hat{f} (x)})}^{2} dx]}^{1 / 2}

(6)

W_{1} (F, \hat{F}) = \int_{R} | F (x) - \hat{F} (x) | dx

(7)

Measurements of differences in structural and regional risk

Practical comparisons should be made based on the differences found in reliability, risk, or resilience metrics. These explicitly include the exposure of the assets to site-specific hazards and potential consequences of this exposure (Mahsuli and Haukaas, 2013). As one example, this article considers losses at the structure and portfolio scale when comparing the implication of adopting different fragility methods. Given a defined seismic hazard model, the expected relative loss of a particular structure conditional to the hazard intensity $(IM = im)$ can be obtained as follows:

E {[L | IM = im]}^{(s)} = \sum_{i}^{n_{ds}} L_{i} \cdot P (DS = i | IM = im)

(8)

where $L_{i}$ represents the expected repair costs (relative to the structure replacement cost) for damage state $i = 1, \dots n_{ds}$ , associated with structure $s$ . The expected relative annual loss of structure $s$ can be obtained as follows:

E {[L]}^{(s)} = EA L^{(s)} = \int E {[L | IM = im]}^{(s)} | \frac{d ν {(IM)}^{(s)}}{dIM} | dIM

(9)

Where $| d ν {(IM)}^{(s)} / dIM |$ represents the slope of the hazard curve computed at the structure’s location.

The previous risk metrics can be translated to a portfolio of structures. In this case, other considerations such as spatially correlated hazard intensities (Goda and Hong, 2008; Jayaram and Baker, 2009) must be included, and the analysis may require the use of simulation-based approaches. The computation of relative losses at the structure level is presented in Equation 10, where $(Z = z)$ represents an earthquake event with known location and magnitude $M$ . Note the loss for a single event is computed as the expectation over $Q$ spatially correlated hazard realizations:

E {[L | Z = z]}^{(s)} = \frac{1}{Q} \sum_{q = 1}^{Q} \sum_{i}^{n_{ds}} L_{i} \cdot P (DS = i | IM = i m^{(s)})

(10)

To aggregate structural losses across the portfolio, the replacement value $W_{s}$ must be considered. The expected portfolio loss $TL$ for a particular event $z$ is then obtained as follows:

E [TL | Z = z] = \frac{1}{Q} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} W_{s} \cdot E {[L | IM = im]}^{(s)}

(11)

Finally, the expected portfolio annual loss, for a particular fault, is computed as follows:

E [TL] = EA L_{portfolio} = λ_{M \geq m_{\min}} \int E [TL | E = e] f_{M} (m) dm

(12)

where $λ_{M \geq m_{\min}}$ represents the rate of events $z$ with magnitude $M \geq m_{\min}$ , and $f_{M}$ describes the probability distribution of those events’ magnitude.

Influence of fragility on network performance measures

An infrastructure system consists of a set of structures connected (or somehow related) to others that work together to serve a system function. In fact, the way system components (i.e. structures in this case) interact with each other may hold more significance when assessing network performance than their individual performance alone, enabling a large body of analyses beyond direct risk metrics (Bocchini and Frangopol, 2011, 2012; Dueñas-Osorio et al., 2007). To evaluate how fragility function derivation practices impacts network functionality, we adopt a select set of performance measures that fall under the general class of time-dependent connectivity-based evaluations. Using a graph $G (V, A)$ , the network structure is defined by a node set $V$ and a set of arches (or edges) $A$ . Fragility functions can be used to inform the state of nodes or links, whichever is selected to represent the structure. The links are chosen as the elements that can suffer loss of functionality in this work.

We define network functionality in terms of connectivity-based metrics. For this, we use the concept of network efficiency $η$ , which depicts the global communication effectiveness between nodes in the graph $G$ (Dueñas-Osorio et al., 2007; Cui et al., 2010):

η (G) = \frac{1}{{| V (G) |}^{2} - | V (G) |} \sum_{i \neq j \in G} \frac{1}{d_{ij}}

(13)

In Equation 13, $| V (G) |$ is the number of nodes in the network and $d_{ij}$ represents the shortest path distance (in number of links) between nodes $(i, j)$ . For infrastructure systems (e.g. transportation, power, and water networks), recovering the original efficiency $η_{0}$ after an event is critical. Therefore, we propose the time-dependent normalized global efficiency $η_{r}$ to measure the (relative) network functionality at time $t$ as follows:

η_{r} (G, t) = \frac{η (G, t)}{η (G, t = 0)} = \frac{\sum_{i \neq j \in G (t)} \frac{1}{d_{ij}}}{\sum_{i \neq j \in G (t = 0)} \frac{1}{d_{ij}}}

(14)

where $G (t)$ depicts the modified network $G$ at time $t$ , which links have been removed (after the hazard event) and restored (during the recovery phase). In our definition, $d_{ij}$ represents the cost of communicating nodes $i$ and $j$ , for example, the path length in distance units, thanks to the elimination of the normalizing term. Recovery trajectories are retrieved by plotting $η_{r} (G, t)$ against time. Comparison between trajectories produced by different fragility approaches can be performed using the statistical distances explained in previous sections. Besides, the Wasserstein distance—given its definition in Equation 7—could further relate to similarities between resilience indexes, provided the latter are computed by considering the area under the functionality curve.

Overview of traditional and recent demand prediction models for fragility derivation of multicomponent response systems

This section probes the differences and similarities in traditional and more recent alternative practices for demand modeling (Step 5 in Figure 2). The approaches studied here differ in the way they can relate system and/or hazard features with the multiple system responses, in the extent they propagate input uncertainty into the demand estimation, and in the overall assumptions needed by each one. Each of the methods presented below are further evaluated through illustrative examples in the next sections.

Monte Carlo analysis

Monte Carlo analysis is a straightforward method to propagate uncertainties in the hazard intensity and structure features. A large number of samples $n$ are used to ensure the design of experiments $X$ approaches the input space $X$ . The output of the analysis is assumed to be representative of the “true” system response distribution $f_{Y} (y)$ (Karamlou and Bocchini, 2015; Mai et al., 2017). Convergence of the statistics of $f_{Y} (y)$ should be used as a stopping rule; however, in the literature of fragility methods, typically tens of thousands of simulations are used as a rule of thumb. Accordingly, applying Monte Carlo analysis for fragility derivation of each structure within a region is typically prohibitive (particularly when using unique numerical models for each structure); hence, this approach is only feasible for conducting structure-specific analyses.

Linear regression model

Foundational work in seismic fragility development often considered the relationship between the intensity measure $IM$ and individual system responses $Y$ to follow a power law model, which is linear in the log-scale (Cornell et al., 2002; Ellingwood et al., 2007):

\ln Y = \ln \bar{Y} + R = (β_{0} + β_{1} X_{1}) + R

(15)

where parameters $X_{1} = \ln IM$ , $β_{0}$ and $β_{1}$ are obtained from regression analysis, and $R$ represents the residuals, that is, the difference between the mean estimate $\ln \bar{Y}$ and the observations (Cornell et al., 2002; Ellingwood et al., 2007; Sudret et al., 2015). In the log-transformed space, the residuals are assumed to follow a normal distribution, centered in zero with variance $σ_{R}$ . For systems with multiple failure modes or component responses, parameters $β_{0}$ and $β_{1}$ are computed for each component and the residuals are assumed to follow a multivariate normal distribution $N_{k} (0, Σ_{R})$ , with $k$ -dimensional mean vector $0$ and $k \times k$ covariance matrix $Σ_{R}$ . The latter is obtained using the $n \times k$ matrix of regression residuals $R$ as follows (Du and Padgett, 2020; Ghosh et al., 2013):

{\hat{Σ}}_{R} = \frac{R^{T} R}{n - 1}

(16)

Besides the linear assumption, the model also assumes a constant variance (homoscedasticity) of the residuals in the log-log scale. Only tens to hundreds of nonlinear analyses are required, which makes it attractive to derive structure-specific fragilities; however, its application to portfolios is hampered by the use of strong assumptions and the need to still perform nonlinear analyses.

Multivariate pdfs at multiple intensity levels

To relax the homoscedastic and linear relationship assumptions, past authors have proposed fitting multivariate pdfs to the logarithm of the responses at discrete intensity values (or bins) (Karamlou and Bocchini, 2015). Although non-parametric versions have been proposed, the multivariate normal distribution is used here given that satisfactory results have been empirically observed. The pdf takes the form $f_{\ln Y | bi n_{i}} ~ N_{k} (μ, Σ)$ , where $μ$ is the vector of mean component responses (in the log-scale) at $bi n_{i}$ , and $Σ$ is the covariance matrix of the responses at bin $i$ . Since this method estimates a demand model $f_{\ln Y | bi n_{i}}$ for each bin, a larger number of nonlinear simulations are required if many bins are of interest for damage analysis. Like Monte Carlo-based practices, its application is restricted to the analysis of a few (archetype) structures.

Surrogate-based demand prediction

Using different techniques from the ML field, it is possible to develop more flexible seismic demand models that capture the variability in system configurations (Ghosh et al., 2013; Seo et al., 2012; Xie et al., 2020). ML methods enable more complex structures and are deemed to capture nonlinearities in the functional relationship between $X$ and $Y$ . Similar to the linear regression model, the surrogate models predict the system responses by adding the multivariate mean response prediction, from a surrogate model, and the correlated model errors $R$ (Du and Padgett, 2020):

Y = \bar{Y} + R = \hat{h} (X) + R

(17)

where $\bar{Y} = \hat{h} (\cdot)$ represents the multivariate mean estimate of the surrogate model. As with the two previous techniques, it is common to work with the responses (along with other parameters, including the intensity measure) in the logarithmic space (i.e. using $\ln Y = \ln \bar{Y} + R = \hat{h} (X) + R$ ). The goal is to build surrogate models that can accurately predict the response of any system configuration, enabling their application to derivation of fragility functions for entire portfolios. In this regard, developed surrogate models can be used in two ways: (1) to estimate demands for specific system configurations, next used to fit one-parameter structure-specific fragilities or (2) to estimate demands across a suite of portfolio configurations first used to derive parameterized fragility functions, which are then used to predict structure-specific fragilities. Given their objectives, they are termed here as “demand-driven” and “fragility-driven,” respectively. Although the latter introduces a second approximation through ML models, it results in a straightforward path for model sharing and practical application, therefore receiving more attention in past studies.

Fragility functions derived with different techniques

Bridge structures (and their role in transportation network functionality) are adopted to explore the influence of fragility modeling practices, including the demand modeling approach adopted as well as the strategy for mapping fragility model to structure within the portfolio. Non-seismically designed multispan simply supported (MSSS) concrete girder bridges, typical of the central and eastern US (CEUS), are considered to illustrate the systematic comparative assessment across scales and implement the proposed statistical distance measures at distinct stages of analysis. Scripts for three-dimensional nonlinear analytical models, developed in Opensees (McKenna and Feneves, 2023), and a set of 348 unscaled ground motions records (representative of the CEUS region) are retrieved from DesignSafe repositories (Kameshwar et al., 2019). The models retrieved were developed in a parameterized fashion (including geometrical, material, and aging parameters) which facilitates the use of experimental designs for fragility assessment and uncertainty propagation. Slight modifications of these have been performed for the purpose of this study. The different fragility derivation techniques presented above will be illustrated using four different bridge configurations (refer to Table 1). The configurations correspond to existing bridges in the National Bridge Inventory database found in the CEUS region (National Bridge Inventory (NBI), 2022); the locations have been arbitrarily modified in this study.

Table 1.

Representative bridges used for analysis of multispan simply supported (MSSS) concrete girder bridges in the central and eastern US (CEUS) region

Bridge ID	Latitude (°)	Longitude (°)	Number of spans	Mid-span length (m)	Column height (m)	Deck width (m)
Bridge 1	35.0906	−90.0256	5	22.60	5.03	21.30
Bridge 2	34.8680	−90.2385	4	8.70	6.50	19.10
Bridge 3	35.0469	−89.6156	3	8.80	6.35	10.50
Bridge 4	35.2797	−89.6803	2	27.40	4.97	12.70

Experimental designs

The varying approaches for fragility derivation are contingent on whether the technique aims to generalize to diverse structural configurations or not (i.e. whether the fragility represents a single structure, an entire portfolio or generalizes to any system configuration), which thereby influences the design of experiments employed (Step 3 in Figure 2). Two types of designs of experiments are defined to construct the fragility functions. The first experimental design, $X_{1}$ , consists of the number of actual finite element models analyzed to build the seismic demand model. The second, $X$ , is used to propagate the uncertainty in the input parameters $X$ by exploiting the uncovered relationship between $X$ and system responses $Y$ .

In this study, the traditional methods based on Monte Carlo, multivariate pdfs at multiple intensity levels and linear regression analyses are used to develop structure-specific fragilities for the four case study bridges. The known parameters of these bridges are fixed across samples; the rest of parameters (geometrical, material, and aging parameters) are sampled from probabilistic models defined in previous studies (Ghosh and Padgett, 2011; Kameshwar et al., 2019; Nielson and DesRoches, 2007a). Unrealistic combinations of parameters are avoided by considering design rules and appropriate sampling strategies. A similar strategy is used to create designs representative of the general class of bridges (portfolio fragilities). For this case, the experimental designs include the geometrical parameters shown in Table 1 as random variables. The probabilistic distributions of such parameters are obtained from a statistical analysis of the NBI database and from previous studies (NBI, 2022; Nielson and DesRoches, 2007a). This design is used for building seismic demand models of the archetype and surrogate-based models. Latin hypercube sampling is used to obtain all the above-mentioned experimental designs, except for the case of Monte Carlo analysis.

One seismic ground motion (consisting of a pair of orthogonal unscaled ground motion records) is paired with each bridge sample. First, the descriptor selected to represent the record intensity in the design of experiments is defined as the geometric mean of the peak ground accelerations in the two horizontal components $PG A_{gm} = \sqrt{PG A_{x} * PG A_{y}}$ (in the log-scale) (Nielson and DesRoches, 2007a). Then, we divide the original set of records into bins according to their $PG A_{gm}$ ; the same number of samples are drawn at each bin and paired to the random bridges. Note we propose this step to have a regular distribution of bridge samples across the different seismic intensity levels, but it is not mandatory. The records per sample are randomly rotated in the range $[0, 2 π]$ to create variations of the seismic inputs and intensity measures.

Demand estimation from nonlinear simulations

The Texas Advance Computer Center (TACC) capacities are leveraged through the DesignSafe cyberinfrastructure (Rathje et al., 2017) to have access to computing resources needed to conduct the nonlinear dynamic analyses (approximately 33,000 simulations in total). The vector of system responses $Y$ corresponds to eight bridge components: column curvature ductility, abutment displacements (active, passive, and transverse), and fixed and expansion bearing displacements (longitudinal and transverse directions). These components have been revealed to contribute to the vulnerability of MSSS concrete girder bridges (Nielson and DesRoches, 2007a). Portfolio components’ responses are presented in Figure 3a and b as a function of the $PG A_{gm}$ . The observed relationship is not linear nor homoscedastic, confirming the need of more complex functional forms than linear regression models in the logarithmic scale. The joint distribution of the two components, presented in Figure 3c, portrays the large interaction between component responses during seismic events. The three plots use a gradient color to depict the influence of additional parameters, in this case of the span length, highlighting the importance of investigating seismic response and fragility sensitivity to parameters beyond ground motion intensity measures.

Figure 3.

Relationship between components responses, $PG A_{gm}$ , and span length for the portfolio responses: (a) column responses, (b) abutment passive responses, and (c) joint component responses.

Seismic demand models using traditional methods

As indicated in previous subsections, traditional methods rely on lognormality assumption of the response distribution conditional on the intensity measure. The linear regression model assumes a linear relationship between responses and intensity measure plus a Gaussian noise (in the log-scale). The expected value of the expression in Equation 15 corresponds to the dashed line presented in Figure 4a; the coefficients of the linear regression model are obtained via ordinary least squares solution. The residuals between components’ prediction and analytical model responses are used to obtain the $k \times k$ covariance matrix $Σ_{R}$ (see Equation 16). These are later used for prediction purposes. Gray dots presented in Figure 4 plots correspond to the Monte Carlo simulations.

Figure 4.

Seismic demands estimated with different approaches for Bridge 3: (a) linear regression, multivariate pdfs and Monte Carlo estimates, (b) univariate surrogate demands based on lasso regression, and (c) multivariate surrogate demands based on kernel ridge regression.

On the contrary, multivariate lognormal distributions fitted at multiple intensity levels allow capturing the variation in the descriptive statistics of the responses at discrete intensity values (shown as red squares in Figure 4a). To obtain such pdfs, the responses are first grouped in bins of 0.1 g, then the within bin component’s responses and associated intensity metric are adjusted following recommendations in (Sudret et al., 2015). That is, the responses of each sample $i$ are scaled making to match the representative seismic intensity $i m_{t}$ of the bin $t$ with the original $i m_{i}$ as follows:

{\hat{Y}}_{i} (i m_{t}) = Y_{i} (i m_{i}) \frac{i m_{t}}{i m_{i}}, for i = 1, \dots, n

(18)

where $Y_{i} (\cdot)$ refers to the (log-scale) component responses, and ${\hat{Y}}_{i} (\cdot)$ refers to the (log-scale) adjusted responses in bridge sample $i$ . Acceptable adjusting factors are obtained using small bin sizes. At each bin, the mean value of each component responses is obtained to construct the sample mean vector $μ = (μ_{1}, \dots, μ_{j})$ (see Figure 4a). Subsequently, the sample covariance at bin $i$ can be obtained using Equation 16; in this case, $R = (\ln Y - μ)$ . Figure 4a shows the relationship of the response with large intensity values seems to be nonlinear (in the log-log scale); however, it also depicts the importance of having enough samples at each bin to avoid biasing the statistics toward few sample responses when working with this method.

Seismic demand models using ML-based methods

Building the surrogate model entails some steps to assure the model is not trained with ill-conditioned data, does not suffer data leakages, is not overfitting, does not produce inaccurate predictions, among other potential problems. Common good practices include holding out validation sets before any other step is performed, selecting models able to handle collinear predictors, performing standardizing of the data before performing the training process, performing feature selection, among others. A thorough explanation of these steps and common recommendations can be found elsewhere (Du and Padgett, 2020; Ghosh et al., 2013).

For the case study bridges, we used lasso regression and kernel ridge regression methods to develop surrogate demand models. While the former was performed for individual components responses, the latter used simultaneously the eight components’ responses, termed as univariate and multivariate surrogates, respectively. In contrast to the lasso model, the kernel ridge regression model maps the original data set into a high-dimensional space where the interaction between observations depends on the (kernelized) distance between them. Hence, this model is considered more complex than the lasso regression and prone to overfitting. However, both models use regularization seeking to avoid overfitting and for handling collinearities (Vu et al., 2015). The model parameters (or hyperparameters) of these models correspond to a regularization term $λ$ (respectively, the $l_{1}$ and $l_{2}$ norms), the degree of the polynomial transformation, and the kernel hyperparameters (for the kernel ridge regression case). We used a grid-search approach to explore and select the “best” combination of hyperparameters for each surrogate. Cross-validation was implemented within the grid-search method, and the model with lower validation error was selected as the seismic demand surrogate model.

The mean predictions ( $\hat{h} (X)$ ) of the univariate and multivariate surrogate models are presented in Figure 4b and c, respectively. In contrast to traditional methods, the expected values are found to be spread which occurs due to the multiple parameters (beyond the intensity measure) introduced for demand estimation. A gradient color used in Figure 4 further highlights the importance of structural parameters in the seismic response variation (the columns diameter in this case). Besides the mean prediction, correlated residuals are added to the demands obtained via surrogate models to obtain the total demand prediction (illustrated with dark red dots). Once the model is trained and validated, Equation 17 can be used to estimate the components responses of a particular structure or structural portfolios with parameters described by $X = x$ .

One-parameter and parameterized fragility models

The demand estimation modeling performed above produces the vector of responses $y = (y_{1}, \dots, y_{k})$ for each bridge sample $x \in X$ . These are compared to sampled vectors of component capacities $c^{(j)} = (c_{1}, \dots, c_{k})$ , per damage state $j = 1, \dots, 4$ , to determine the damage classification vector $B$ . Lognormal pdfs found elsewhere (e.g. Ghosh, 2021; Ghosh and Padgett, 2011; Ramanathan et al., 2012) are used in this study to represent such component capacity limit states. If the experimental design consisted of structure-specific configurations, then one-parameter fragility functions are obtained by applying the multinomial logistic regression using only the $\ln PG A_{gm}$ as the descriptor parameter. Contrarily, to compute the parameterized fragility functions first, the demands for portfolio-level experimental designs are obtained via demand-driven surrogate models (the lasso regression model in this case given the reduced complexity it offers). The multinomial logistic regression is applied in this case using $X$ as the predictors. Note $X$ can be reduced to only those features that could be observed, measured, or sensed from the structures and the uncertainty from the rest of parameters results embedded in the fragility function. To develop the parameterized fragility functions, we use elastic net regularized logistic regression, proven in previous studies to produce suitable fragility models for damage prediction in the presence of collinear features (Misra and Padgett, 2019). Unlike the other methods, a set of coefficients is obtained to represent any system configuration fragility.

Typical one-parameter fragility expressions can be obtained from the parameterized fragilities if one plugs the set of parameters $x$ directly in Equation 4 and aggregates the probabilities of damage states up to the one of interest. Considering some parameters may be unknown, one may require computing the multidimensional integration as presented in Equation 19 (Ghosh et al., 2013) or to numerically approximate it using Monte Carlo integration, as shown in Equation 20. In both cases, these probabilities are summed up to the damaged state of interest:

P_{ds = b | x} = \int \dots \int Λ_{b} (X) f_{X} (x) d x

(19)

{\hat{P}}_{ds = b | x} = \frac{1}{M} \sum_{m = 1}^{M} Λ_{b} (x_{m})

(20)

Figure 5 presents the obtained fragilities for Bridges 1 and 3, for all the above-mentioned methods (listed in Table 2), depicting the variation in model performances. Differences among the obtained fragilities are exacerbated as the damage state increases. While Bridge 1 seems to be fairly represented by most of the models, Bridge 3 shows large differences between the benchmark Monte Carlo model and the rest of the approaches (especially the archetype fragility).

Figure 5.

Fragility functions obtained for Bridges 1 (upper) and 3 (lower) for damage states slight to complete.

Table 2.

Design of experiments for different fragility types

Fragility type nomenclature	Description	Design for demand model $\| X_{1} \|$	Design for fragility fitting $\| X \|$	Application	Steps in Figure 2^a
Monte Carlo	Monte Carlo analysis to directly derive one-parameter fragility of each structure	$-$	5000^b	One structure	5a, 6, and 7a
Linear regression	Power law demand model for structure response estimation, followed by one-parameter fragility fitting	200^b	100,000^b		5b, 6, and 7a
Multivariate pdfs at multiple intensities	Multivariate pdfs for demand model of structure using multiple stripes, followed by one-parameter fragility fitting	2000^b	100,000^b		5b, 6, and 7a
Archetype	Demand model to represent a suite of portfolio configurations (multivariate pdfs adopted), followed by one-parameter fragility fitting	5000^c	100,000	Bridge class	5b, 6, and 7a
Univariate_lasso (lasso regression)	Univariate surrogate demand models across portfolio parameter space, applied to each structure for one-parameter fragility fitting	5000^c	100,000^b	Each structure at the regional scale	5c, 6, and 7a
Multivariate_kernel (kernel ridge regression)	Multivariate surrogate demand models across portfolio parameter space, applied to each structure for one-parameter fragility fitting	5000^c	100,000^b		5c, 6, and 7a
Parameterized	Surrogate demand model (lasso adopted) followed by parameterized fragility function (elastic net logistic regression) fitting, both across portfolio parameter space	5000^c	100,000^d		5c, 6, and 7b

All the methods require Steps 1–4.

Number of samples corresponds to the experimental design of each structure $i, i = 1, \dots, 4$ .

Samples in these sets are equivalent.

This design of experiments represents system configurations in a portfolio, not a particular structure.

Statistical differences among derived functions

Visual comparison of fragility functions lacks objectivity and does not allow conclusions to be drawn about the appropriateness of a derivation technique. Hence, the methods to measure differences between cdfs or pdfs explained before are adopted herein. The three different statistical distances are computed separately for each method in Table 2, the four damage states and the four bridges under analysis, using the Monte Carlo fragilities as the benchmark model. Figure 6 shows such statistical distances and reveals the fragility models that are closer to the benchmark models, as is the case for the multivariate surrogate (kernel ridge regression) and the multivariate pdfs at multiple intensities (traditional) methods. Linear regression, univariate surrogate (lasso regression), and parameterized fragilities presented slightly larger distances compared to the two previous methods. Finally, the archetype fragility diverges the most from the benchmark model, especially for the fourth damage state.

Figure 6.

Statistical distances for each fragility derivation method per damage state: 1 slight, 2 moderate, 3 extensive, and 4 complete. Each dot represents one of the four bridges. The boxes in the x-axis represent each of the demand modeling methods evaluated.

The Kolmogorov–Smirnov statistic $D_{KS}$ shows the range of maximum differences between conditional exceeding probabilities. While the closest fragilities result in average values ranging between 8% and 9%, the second-best models’ $D_{KS}$ average range between 15% and 17%, and the “farthest” fragility (the archetype model) shows averaged maximum $D_{KS}$ distances of 24%. The Hellinger distance shows similar tendencies to $D_{KS}$ , although the distance has a different meaning (explained earlier), confirming the same “order” of favorability. For the Wasserstein distance, it was found that the larger “costs” independent of the model occurs at the larger damage states. This interesting result suggests that probability measures of damage states of highest interest are more impacted by the approximations of demand estimation models. W₁ values diminish the differences between methods, except for the archetype case that remains as the fragility that diverges the most with respect to the benchmark model.

Risk estimates at the structural and regional scale

The small-to-large differences found in fragilities derived through different practices should be positioned in terms of its influence on subsequent risk and resilience analyses. In this section, we focus on performance measures related to risks at the structure and regional or portfolio scale.

Risk estimates at the bridge level

The four individual structures previously analyzed are assumed to be part of the highway bridge network presented in Figure 7a; the location of the four bridges is portrayed with blue dots. The bridge network consists of 130 bridges, hypothetically placed in the Memphis Metropolitan and Statistical Area (MMSA) highway system. For illustrative purposes, the area is subjected to earthquakes produced by a point-source located at 35.927 N, 89.919 W. The event magnitude is assumed to follow a truncated Gutenberg–Richter model with $M_{\min} = 5$ , $M_{\max} = 8$ , slope parameter equal to 1, and annual rupture frequency of the fault $λ_{M \geq m_{\min}} = 0.05$ . The Atkinson and Boore (1995) ground motion model is used to compute the mean estimate of the intensity at each location. The mean hazard curves for the four bridge locations are presented in Figure 7b.

Figure 7.

Illustrative example of a highway bridge network subjected to seismic events: (a) network of bridges and (b) mean hazard curves for Bridges 1–4.

The first comparison can be performed in terms of the expected relative loss conditional to the hazard intensity, $E [L | IM = im]$ . The relative repair costs $L_{i}$ associated with damage states $i = 1, \dots, 4$ are assumed to be 3%, 8%, 25%, and 100%, respectively (FEMA, 2020). Using Equation 8, $E [L | IM = im]$ is computed for varying intensity measures to obtain the so-called vulnerability (or mean damage ratio) curves. The results of such curves obtained from the different fragility derivation practices are presented in Figure 8a; differences between archetype vulnerabilities and the rest of models are noticeable for this performance measure. Note the minor differences for Bridge 1 are consistent with the slight differences in fragility functions observed before (refer to Figure 5). These differences between vulnerability curves become over- or under-estimated economic impacts, with larger errors expected when scenario-based seismic risk assessments are conducted. For decision-making purposes based on economic losses (or benefit cost analysis), such differences can be misleading in intervention prioritization schemes.

Figure 8.

Mean damage ratio curves (a) and relative expected annual losses at the structure level (b).

Another risk metric of interest is the relative expected annual losses (EALs), obtained from Equation 9. The differences that arise in these values between methods (presented in Figure 8b) may be somewhat counterintuitive. For example, in Bridge 4, larger differences are observed between the vulnerability curves of traditional methods (yellow and pink curves) with respect to the benchmark model (black line), compared to those observed for the surrogate-based models (blue curves). However, the relative EAL in the traditional methods resulted closer to the benchmark estimation. This is explained by the fact that lower intensity values are the actual contributors toward the EAL, diminishing the effect of observable differences that occur in higher intensity measures. Overall a similar order of magnitude is obtained between the different approaches, possibly attributed to using probability weighting to compute the EALs.

Risk estimates at the portfolio scale

The seismic hazard at the regional scale is modeled using simulation-based approaches to consider spatially correlated intensities; the correlation model for PGA values used in this study corresponds to the proposed by Jayaram and Baker (2009). The same set of residual samples was kept during the comparisons to guarantee that differences measured are only a product of the differences in fragility models. An example of a simulated scenario for a magnitude $M = 6.0$ is presented in Figure 7, depicting the wide range of possible intensities that the bridges face. Pertaining to exposure characteristics, the replacement value of each bridge $W_{s}$ is approximated as the deck area times a unitary cost of US$100/m². The geometrical parameters presented in Table 1 are also obtained for the 130 bridges from existing bridges within the region (NBI, 2022; Nielson and DesRoches, 2007a), and the unknown parameters are assumed to follow the same pdfs used for building the experimental designs.

Given the infeasibility of traditional methods to scale up to big portfolios, the three traditional approaches (Monte Carlo, multivariate pdfs for different intensity levels, and linear regression) are disregarded from the subsequent analysis. Archetype fragilities are kept as they are commonly used in regional-scale analysis, and fragility functions for each system configuration are computed using the surrogate-based models (both, demand-driven and fragility-driven approaches). The lasso regression model is termed in the subsequent analyses as surrogate-based model. Thus, the remainder of this article presents a comparison between archetype, surrogate-based (lasso model), and parameterized models.

To estimate the influence of fragility modeling practices at the regional scale, earthquakes with varying magnitude are simulated to estimate the conditional portfolio loss $E [TL | Z = z]$ . In this study, only the magnitude is variable, allowing to compute a curve of “portfolio vulnerability.” Such curves are presented in Figure 9a; the average values are presented with solid lines, and percentiles 5th and 95th are depicted with dashed lines. For this study, the discrepancies found on this performance measure among approaches are attributed to large differences present in particular bridge fragility functions. While traditional regional-scale models use the same archetype fragility, emergent methods based on ML may sometimes pinpoint unusually highly vulnerable bridges. Then, these fragile structures easily reach larger damage states when a seismic event occurs, exacerbating (and dominating) the losses in the portfolio as shown in Figure 9b. That is, expected losses of some bridges estimated with ML-based fragilities (for $M = 6$ ) show large discrepancies, sometimes an order of magnitude larger than archetype-based estimates. This finding highlights the importance of capturing granularities within the portfolio fragility definition.

Figure 9.

Portfolio risk outcomes: (a) conditional portfolio total losses and (b) average bridge losses for an earthquake $M$ = 6.

From Figure 9a, the surrogate-based models present slightly closer portfolio loss estimates (solid lines) between them than compared against archetype-based estimations. This similarity, however, does not yield very similar expected annual portfolio losses $EA L_{portfolio}$ (Equation 12), resulting in US$2.63, US$4.93, and US$8.13 million, respectively, for archetype, surrogate-based, and parameterized models. Differences in the $EA L_{portfolio}$ show that particular performance measures may increase (or reduce) the differences between fragility methods given these consider the events likelihood. The overall results at the portfolio scale suggest that the ML-based methods achieve an acceptable level of accuracy, making them suitable for introducing approximate individual fragility functions when analyzing portfolios of structures. In this way, these surrogate-based methods could contribute to capturing granularities in the risk and resilience outcomes, much needed for objective decision-making.

Estimation of implications at the network scale

We further investigate the impact of differences in fragility estimates by considering the interaction of bridge functionality within the transportation network. In this study, a bridge that reaches or exceeds moderate damage state, $P (DS \geq 2 | X = x)$ , is assumed to lose its functionality (Padgett and DesRoches, 2007); hence, the communication between the two adjacent nodes is lost while inspections and repair actions take place. That is, just a binary state is considered for the links: these can be either functional or not functional. The total time of repair is computed as the sum of an idle time and a repair time, both obtained from the probabilistic model proposed by Decò et al. (2013). The total repair time of the network is defined as the instant when all the links return to their original condition and depends on the scheduling of repair crews. To this end, a simple back-to-back scheduling is used, the number of crews is arbitrarily fixed in ten, and the order of crew’s deployment (i.e. “bridge priority”) is defined based on the betweenness centrality of the edges (Brandes and Fleischer, 2005; Newman, 2010). Figure 7a illustrates the “bridge priority” using larger circle sizes for those bridges that will be repaired first during the network recovery process. A simulation-based scheme is used here to propagate uncertainties that cut across scales, such as the ones represented by correlated hazard samples, idle and repair simulated times, and network event conditions. The sampling strategy is defined in such manner that these samples are shared across analyses of fragility modeling practices, narrowing attention to the implications of fragility models on network functionality trajectories.

The fragility functions are used in this stage to estimate the probability of the edge’s failures (i.e. reaching or exceeding moderate damage state), then the edge state (functional or not) is sampled. Therefore, having different fragility derivations primarily impacts the different sampled network conditions. After computing the network event conditions, the idle and recovery sampled times are assigned only to those bridges that lost functionality. Subsequently, the scheduling is defined according to the number of crew members and order of priority predefined for repair purposes (if any). The network time-dependent functionality trajectories are constructed using the scheduled repair times. Every instant $t$ that one link recovers its functionality, the relative global efficiency $η_{r} (G, t)$ is calculated again, until all the bridges restore operation. The functionality recovery trajectories shown in Figure 10 depict the temporal evolution of $η_{r}$ relative to the original global efficiency. Noting that $η_{r}$ only depends on the system state, efficient strategies can be developed to avoid re-computing the temporal global efficiency in a brute force manner.

Figure 10.

Time-dependent network functionality trajectories using three fragility input models for events with $M = 7$ and $M = 8$ .

Comparison of all the recovery trajectories simulated for earthquakes of magnitude $M = 7$ and $M = 8$ is presented in Figure 10. This figure confirms the significant impact of the event’s magnitude in the system resilience, which causes larger variability in the recovery trajectories as the magnitude increases. Surprisingly, the 5th, 50th, and 95th percentiles computed over the trajectories and the maximum time to recover the entire functionality look quite similar across the different fragility approaches. To quantify such similarities, statistical distances are used to measure the difference among trajectories (shown in Figure 11). The use of statistical distance is valid in this setting given that functionality trajectories, as the ones proposed here based on time-dependent normalized global efficiency $η_{r} (G, t)$ , reassemble empirical distribution functions (Sharma et al., 2018).

Figure 11.

Statistical distances between network recovery trajectories. The boxes arranged in vertical columns represent pair-wise comparison between (a) archetype versus surrogate univariate model, (b) archetype versus parameterized fragility, and (c) surrogate univariate model versus parameterized fragility.

The distances are measured for each pair of methods as a function of the magnitude of the event (shown in Figure 11). The distances measured confirm the relationship between the magnitude and the statistical distances observed previously. As the magnitude increases, the statistical distance also increases between any pair of methods, especially for the archetype-based methods. For instance, $D_{KS}$ distances show from negligible maximum differences in the time-dependent normalized global efficiency $η_{r} (G, t)$ , for lower magnitudes, to differences that can go up to ∼20% (for events of $M = 8$ ) when any method is compared to the archetype-based case. The two methods based on ML algorithms differ at most in ∼10% for the higher moment magnitudes and are found to be the less divergent pair of methods, also in the case of $H$ and $W_{1}$ distances. The distribution of $W_{1}$ metrics presented in Figure 11c highlights the pronounced similarities between trajectories produced by these two methods. This small $W_{1}$ distance suggests that, if measured, resilience indexes obtained by computing the area under the functionality curve (e.g. metrics proposed by Bocchini and Frangopol, 2012; or Cimellaro et al., 2010) would differ less between ML-based methods than expected compared to differences with archetype-based analysis.

In general, differences at this scale are quite small compared to the vast differences found in risk outcomes at a regional scale. We attribute this to three major factors: (1) the differences between failure probabilities (among derivation methods) only matters at this scale if they produce different network events states and the “distance” itself is not as important as it is when it informs the likelihood of the performance measure (e.g. the conditional relative loss at the structure level); (2) the routes covered by the critical pinpointed bridges in previous stages can be easily replaced by other routes, as it happens on networks with redundant connectivity; and (3) the use of recovery models that do not consider system parameters to predict the individual time to repair (i.e. that act as “archetype” recovery models) hamper the gained granularity of the ML-based fragility models.

Conclusion

This study has targeted the unanswered question of how and if different fragility modeling practices (including traditional and those that leverage emerging surrogate modeling) actually influence seismic risk and resilience quantification. The discussion of fragility derivation approaches centers on (1) the procedures and assumptions of various techniques for demand modeling and (2) the agility of these to yield either one-parameter or parameterized fragility functions, that is, fragilities conditioned on system and/or hazard features. In contrast to existing studies, the implications of fragility techniques are evaluated over performance measures that cut across scales, including from conditional damage probabilities, to expected losses (at the individual and portfolio scale), to network functionality metrics (e.g. recovery trajectories). Case studies using bridges and transportation networks are leveraged to test such implications and systematically contrast their outcomes using posed statistical distance metrics.

Small statistical distances were found between proven traditional methods (for structure-specific fragilities) and ML-based approaches, confirming the ability of the latter to be used for structure-specific analysis with acceptable accuracy. This conclusion extended to estimated risk metrics at the individual structure scale, for example, conditional expected relative losses, where emerging techniques captured particularities on the performance of heterogeneous system configurations. To further support these findings, a hypothetical highway network consisting of 130 bridges was studied. Outcomes showed that capturing granularities in the individual performance is not only important for capturing subtleties in losses predictions but also it may pinpoint fragile structures within the region that could exacerbate, and dominate, the losses in portfolio analyses. These observations showed that resolution in modeling portfolios could impact the risk outcomes and lead to totally different conclusions, possibly affecting objective investment plans (e.g. retrofitting programs). Regarding network functionality analyses, results showed that divergencies in recovery trajectories are not as dramatically impacted by differences in fragility description as they were for the portfolio losses. At this scale, other modeling choices, such as the estimation of individual recovery time without consideration of system parameters, may reduce the impact of fragility model choice and provides an opportunity for further investigation.

While the systematic comparisons are presented using highway bridge networks, the findings of this study can be generalized to other structures embedded in networked systems. The approach proposed in this study to methodically compare fragility model implications can serve as a basis for future fragility method evaluation, propelling objective quantification of their impacts on risk and resilience pipelines. The use of the proposed statistical distance metrics also promotes comparison of regional-scale performance and resilience measures that keep the heterogeneity of the individual performances (i.e. those that are not based on aggregation), as was shown by comparing individual network recovery trajectories. Overall, the findings of this study suggests that emerging techniques are equipped to go beyond efficient estimations to enable the construction of models of heterogeneous portfolios, better supporting decisions processes that may occur at different scales, space, and time, throughout infrastructure systems.

Footnotes

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: The authors acknowledge the support provided by the National Science Foundation under Award Number CMMI-2227467. The first author acknowledges the grant support from the Fulbright-Minciencias scholarship program. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or the Fulbright-Minciencias program.

ORCID iDs

Raul Rincon

Jamie Ellen Padgett

References

Andriotis

Papakonstantinou

(2018) Extended and generalized fragility functions. Journal of Engineering Mechanics 144(9): 04018087.

Aslani

Miranda

(2004) Component-level and system-level sensitivity study for eEarthquake loss estimation. In: 13th world conference on earthquake engineering, Vancouver, B. C., Canada: Earthquake Engineering Research Institute. Paper No. 1070.

Atkinson

Boore

(1995) Ground-motion relations for eastern North America. Bulletin of the Seismological Society of America, 85(1): 17–30. DOI: 10.1785/BSSA0850010017

Balakrishnan

(2019) The Fundamental Statistical Distances. 36-705: Intermediate Statistics, Lecture 35. Carnegie Mellon University. Available at: https://www.stat.cmu.edu/∼siva/teaching/705/lec35.pdf (accessed 14 August 2023).

Bichon

McFarland

Mahadevan

(2011) Efficient surrogate models for reliability analysis of systems with multiple failure modes. Reliability Engineering & System Safety 96(10): 1386–1395.

Bocchini

Frangopol

(2011) A stochastic computational framework for the joint transportation network fragility analysis and traffic flow distribution under extreme events. Probabilistic Engineering Mechanics 26(2): 182–193.

Bocchini

Frangopol

(2012) Restoration of bridge networks after an earthquake: Multicriteria intervention optimization. Earthquake Spectra 28(2): 427–455.

Brandes

Fleischer

(2005) Centrality measures based on current flow. In: Diekert

Durand

(eds) STACS 2005, Lecture Notes in Computer Science (Vol. 3404). Berlin and Heidelberg: Springer, pp. 533–544.

Burton

(2021) Design of computer experiments for developing seismic surrogate models. Earthquake Spectra 38: 384–406.

10.

Choe

D-E

Gardoni

Rosowsky

Haukaas

(2008) Probabilistic capacity models and seismic fragility estimates for RC columns subject to corrosion. Reliability Engineering & System Safety, Probabilistic Modelling of Structural Degradation 93(3): 383–393.

11.

Cimellaro

Reinhorn

Bruneau

(2010) Framework for analytical quantification of disaster resilience. Engineering Structures 32(11): 3639–3649.

12.

Cornell

Jalayer

Hamburger

Foutch

(2002) Probabilistic basis for 2000 SAC federal emergency management agency steel moment frame guidelines. Journal of Structural Engineering 128(4): 526–533.

13.

Cui

Kumara

Albert

(2010) Complex networks: An engineering view. IEEE Circuits and Systems Magazine 10(3): 10–25.

14.

Decò

Bocchini

Frangopol

(2013) A probabilistic approach for the prediction of seismic resilience of bridges. Earthquake Engineering & Structural Dynamics 42(10): 1469–1487.

15.

Deierlein

Krawinkler

Cornell

(2003) A framework for performance-based earthquake engineering. In: Proceedings of the 7th pacific conference on earthquake engineering, Christchurch, New Zealand, 13–15 February.

16.

Padgett

(2020) Investigation of multivariate seismic surrogate demand modeling for multi-response structural systems. Engineering Structures 207: 110210.

17.

Padgett

(2021) Multivariate return period-based ground motion selection for improved hazard consistency over a vector of intensity measures. Earthquake Engineering & Structural Dynamics 50(2): 415–435.

18.

Dueñas-Osorio

Padgett

(2011) Seismic reliability assessment of bridges with user-defined system failure events. Journal of Engineering Mechanics 137(10): 680–690.

19.

Dueñas-Osorio

Craig

Goodno

(2007) Seismic response of critical interdependent networks. Earthquake Engineering & Structural Dynamics 36(2): 285–306.

20.

Ellingwood

Celik

Kinali

(2007) Fragility assessment of building structural systems in Mid-America. Earthquake Engineering & Structural Dynamics 36(13): 1935–1952.

21.

Ghosh

(2021) Next generation fragility functions for seismically designed highway bridges in moderate seismic zones. Natural Hazards Review 22(1): 04020051.

22.

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.

23.

Ghosh

Padgett

Dueñas-Osorio

(2013) Surrogate modeling and failure surface visualization for efficient seismic vulnerability assessment of highway bridges. Probabilistic Engineering Mechanics 34: 189–199.

24.

Goda

Hong

(2008) Spatial correlation of peak ground motions and response spectra. Bulletin of the Seismological Society of America 98(1): 354–365.

25.

Federal Emergency Management Agency (FEMA) (2020) Hazus earthquake model technical manual. Available at: https://www.fema.gov/sites/default/files/2020-10/fema_hazus_earthquake_technical_manual_4-2.pdf

26.

Heresi

Miranda

(2022) Structure-to-structure damage correlation for scenario-based regional seismic risk assessment. Structural Safety 95: 102155.

27.

Jaiswal

Wald

D’Ayala

(2011) Developing empirical collapse fragility functions for global building types. Earthquake Spectra 27(3): 775–795.

28.

Jayaram

Baker

(2009) Correlation model for spatially distributed ground-motion intensities. Earthquake Engineering & Structural Dynamics 38(15): 1687–1708.

29.

Kameshwar

Navya

Jamie

(2019) Response and Fragility Modeling of Aging Bridges Subjected to Earthquakes and Truck Loads. DesignSafe-CI. DOI: 10.17603/ds2-5tzv-qz91.

30.

Kang

W-H

Song

Gardoni

(2008) Matrix-based system reliability method and applications to bridge networks. Reliability Engineering & System Safety 93(11): 1584–1593.

31.

Karamlou

Bocchini

(2015) Computation of bridge seismic fragility by large-scale simulation for probabilistic resilience analysis. Earthquake Engineering & Structural Dynamics 44(12): 1959–1978.

32.

Lallemant

Kiremidjian

Burton

(2015) Statistical procedures for developing earthquake damage fragility curves. Earthquake Engineering & Structural Dynamics 44(9): 1373–1389.

33.

McKenna

Feneves

(2023) Open system for earthquake engineering simulation: Home page. Available at: https://opensees.berkeley.edu/index.php (accessed 9 August 2023).

34.

Mahsuli

Haukaas

(2013) Seismic risk analysis with reliability methods, part I: Models. Structural Safety 42: 54–62.

35.

Mai

Konakli

Sudret

(2017) Seismic fragility curves for structures using non-parametric representations. Frontiers of Structural and Civil Engineering 11(2): 169–186.

36.

Mangalathu

Jeon

J-S

Padgett

DesRoches

(2017) Performance-based grouping methods of bridge classes for regional seismic risk assessment: Application of ANOVA, ANCOVA, and non-parametric approaches: Performance-based Grouping Methods of Bridge Classes. Earthquake Engineering & Structural Dynamics 46(14): 2587–2602.

37.

Martins

Silva

(2021) Development of a fragility and vulnerability model for global seismic risk analyses. Bulletin of Earthquake Engineering 19(15): 6719–6745.

38.

Melchers

Beck

(2018) Structural Reliability Analysis and Prediction. 3rd ed. Hoboken, NJ: Wiley.

39.

Miller

(1956) Table of percentage points of Kolmogorov statistics. Journal of the American Statistical Association 51(273): 111–121.

40.

Misra

Padgett

(2019) Seismic fragility of railway bridge classes: Methods, models, and comparison with the state of the art. Journal of Bridge Engineering 24(12): 04019116.

41.

Muntasir Billah

AHM

Alam

(2015) Seismic fragility assessment of concrete bridge pier reinforced with superelastic shape memory alloy. Earthquake Spectra 31(3): 1515–1541.

42.

Muntasir Billah

AHM

Shahria Alam

(2015) Seismic fragility assessment of highway bridges: A state-of-the-art review. Structure and Infrastructure Engineering 11(6): 804–832.

43.

National Academies of Sciences, Engineering, and Medicine (2019) Building and Measuring Community Resilience: Actions for Communities and the Gulf Research Program. Washington, DC: National Academies Press, p. 25383.

44.

National Bridge Inventory (NBI) (2022) National Bridge Inventory—Management and preservation—Bridges & structures—Federal Highway Administration. Available at: https://www.fhwa.dot.gov/bridge/nbi.cfm (accessed 4 October 2022).

45.

National Research Council (2012) Disaster Resilience: A National Imperative. Washington, DC: National Academies Press, p. 13457.

46.

Newman

(2010) Networks: An Introduction. Oxford and New York: Oxford University Press.

47.

Nielson

DesRoches

(2007a) Analytical seismic fragility curves for typical bridges in the central and southeastern United States. Earthquake Spectra 23(3): 615–633.

48.

Nielson

DesRoches

(2007b) Seismic fragility methodology for highway bridges using a component level approach. Earthquake Engineering & Structural Dynamics 36(6): 823–839.

49.

Padgett

DesRoches

(2007) Bridge functionality relationships for improved seismic risk assessment of transportation networks. Earthquake Spectra, 23(1): 115–130. DOI: 10.1193/1.2431209

50.

Padgett

DesRoches

(2007) Sensitivity of seismic response and fragility to parameter uncertainty. Journal of Structural Engineering 133(12): 1710–1718.

51.

Patsialis

Taflanidis

(2020) Reduced order modeling of hysteretic structural response and applications to seismic risk assessment. Engineering Structures 209: 110135.

52.

Ramanathan

DesRoches

Padgett

(2012) A comparison of pre- and post-seismic design considerations in moderate seismic zones through the fragility assessment of multispan bridge classes. Engineering Structures 45: 559–573.

53.

Rao

Lepech

Kiremidjian

(2017) Development of time-dependent fragility functions for deteriorating reinforced concrete bridge piers. Structure and Infrastructure Engineering 13(1): 67–83.

54.

Rathje

Dawson

Padgett

Pinelli

J-P

Stanzione

Adair

Arduino

Brandenberg

Cockerill

Dey

Esteva

Haan

Hanlon

Kareem

Lowes

Mock

Mosqueda

(2017) DesignSafe: New cyberinfrastructure for natural hazards engineering. Natural Hazards Review 18(3): 06017001.

55.

Rincon

Padgett

(2022) Seismic reliability analysis of complex structural systems using system-level surrogate models. In: Proceedings of the 12th national conference in earthquake engineering, p. 5. Salt Lake City, UT: Earthquake Engineering Research Institute.

56.

Seo

Dueñas-Osorio

Craig

Goodno

(2012) Metamodel-based regional vulnerability estimate of irregular steel moment-frame structures subjected to earthquake events. Engineering Structures 45: 585–597.

57.

Sharma

Tabandeh

Gardoni

(2018) Resilience analysis: A mathematical formulation to model resilience of engineering systems. Sustainable and Resilient Infrastructure 3(2): 49–67.

58.

Simpson

Poplinski

Koch

Allen

(2001) Metamodels for computer-based engineering design: Survey and recommendations. Engineering with Computers 17(2): 129–150.

59.

Sudret

Mai

Konakli

(2015) Assessment of the lognormality assumption of seismic fragility curves using non-parametric representations. arXiv. DOI: 10.48550/arXiv.1403.5481.

60.

van de Lindt

Kruse

Cox

Gardoni

Lee

Padgett

McAllister

Barbosa

Cutler

Van Zandt

(2023) The interdependent networked community resilience modeling environment (IN-CORE). Resilient Cities and Structures, Integrated Modeling of Cities to Improve Natural Hazards Resilience 2(2): 57–66.

61.

Snyder

Rupp

Chen

Khelif

Müller

K-R

Burke

(2015) Understanding kernel ridge regression: Common behaviors from simple functions to density functionals. International Journal of Quantum Chemistry 115(16): 1115–1128.

62.

Xie

Ebad Sichani

Padgett

DesRoches

(2020) The promise of implementing machine learning in earthquake engineering: A state-of-the-art review. Earthquake Spectra 36(4): 1769–1801.

Fragility modeling practices and their implications on risk and resilience analysis: From the structure to the network scale

Abstract

Keywords

Introduction

Methods

Derivation of fragility functions for systems with multiple failure modes

Implications of fragility derivation methods across risk estimates

Statistical comparison between fragility functions

Measurements of differences in structural and regional risk

Influence of fragility on network performance measures

Overview of traditional and recent demand prediction models for fragility derivation of multicomponent response systems

Monte Carlo analysis

Linear regression model

Multivariate pdfs at multiple intensity levels

Surrogate-based demand prediction

Fragility functions derived with different techniques

Experimental designs

Demand estimation from nonlinear simulations

Seismic demand models using traditional methods

Seismic demand models using ML-based methods

One-parameter and parameterized fragility models

Statistical differences among derived functions

Risk estimates at the structural and regional scale

Risk estimates at the bridge level

Risk estimates at the portfolio scale

Estimation of implications at the network scale

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References