A stochastic computational framework for the joint transportation network fragility analysis and traffic flow distribution under extreme events

Transportation networks are critical lifelines during and after extreme events (earthquakes, floods, hurricanes, explosions, etc.), as their post-event performance governs emergency response and regional recovery. Prior research has addressed reliability, maintenance, resilience, and socio-economic impacts for lifelines and networks. However, network performance under extreme events depends not only on individual component fragility but also on spatially correlated damage across bridges. This paper presents a stochastic computational framework that integrates structural fragility analysis, random field simulation of bridge damage, and combined traffic distribution/assignment to quantify how spatial correlation of bridge damage affects network performance. The central research question is how the correlation distance of bridge damage influences network connectivity and efficiency metrics (e.g., total travel time), and how sensitive those metrics are to assumptions about correlation versus independence. The framework enables parametric control of correlation via random fields, addressing the inadequacy of assuming independent bridge damages.

The paper situates the work among multiple strands: (i) reliability analysis, life-cycle cost and maintenance optimization for bridge networks (Frangopol and co-workers); (ii) seismic retrofit benefits including traffic redistribution and resilience (Shinozuka and co-workers); (iii) reliability and cascading failures in interdependent lifelines (Dueñas-Osorio and co-workers); (iv) infrastructure network vulnerability (Murray and co-workers); (v) probabilistic, monitoring, and maintenance optimization for infrastructure networks (van Noortwijk and co-workers); (vi) community and facility resilience (Bruneau and MCEER team); (vii) matrix-based system reliability for networks (Song and others); and (viii) pre-disaster network strengthening decisions (Peeta and co-workers). Random fields have been widely used in seismic engineering to model spatially varying demands/effects. The present work builds on these by using random fields to model bridge damage directly, enabling explicit control of spatial correlation and integrating with traffic assignment to assess network performance sensitivity.

The framework comprises five stages. 1) Structural fragility analysis: Bridges are the vulnerable components. For natural hazards (e.g., earthquakes), the HAZUS-MH tool provides class-based and bridge-specific fragility curves combining ground shaking and ground failure effects. For each scenario, the probability of damage states (no, minor, moderate, major, collapse) is computed and summarized into an expected damage index per bridge Ib in [0,4] by weighting states 0–4 by their probabilities. 2) Spatial correlation estimation: Bridge geographic coordinates (lat/long) are mapped to 2D Cartesian coordinates. Using the multi-scenario damage dataset, for each bridge pair the correlation coefficient of damage is estimated across scenarios. Discrete correlations versus inter-bridge distance are then fit (least squares or maximum likelihood) to a decaying model to estimate the correlation length (lambda) and functional form. The study adopts an exponential-type model and reduces to an isotropic, homogeneous 1D distance dependence to maximize statistical use of limited scenarios. 3) Random field simulation: The bridge damage level is simulated as a 2D non-Gaussian random field with (i) marginal distribution assumed Uniform[0,4] (neutral choice to avoid biasing correlation effects) and (ii) the estimated correlation structure. Corresponding autocorrelation and spectral density functions for a homogeneous, isotropic field are used. Strongly non-Gaussian simulation algorithms generate samples over the network region; values at bridge locations are extracted. This enables direct parametric variation of lambda without repeated fragility analyses and scales efficiently with bridge count. 4) Network performance analysis: The transportation network is modeled as a directed graph. Inputs include edge free-flow time and practical capacity (from segment length, speed, and lanes), node trip generation/attraction O and D, and a congestion function that relates time to flow. Bridge damage reduces practical capacity using an exponential residual-capacity function based on the maximum damage on each edge (bottleneck assumption). A combined traffic distribution and assignment algorithm is used: initialize shortest paths under free-flow, estimate an origin-destination matrix via a gravitational model with balancing factors, assign flows to shortest paths, then iteratively update edge times, shortest paths, OD matrix, and flows using convex combination and line search to minimize a standard objective that enforces user equilibrium and OD consistency. 5) Statistical result analysis: For each lambda, many random field samples are generated. Two performance metrics are computed: Fully Connected Ratio (FCR), the percentage of samples where all nodes are mutually reachable; and total travel time P, the sum over edges of time times flow. Samples with disconnected networks have divergent P and are excluded uniformly across lambdas by using, for all lambdas, the same minimum count of connected samples equal to the smallest N_conn(lambda) across lambdas. Statistics (percentiles, standard deviation) of P are then computed on the retained sets. Implementation details: Example analyses used 500 samples per lambda, lambdas spanning near-uncorrelated (0.1 mi) to highly correlated (20 mi), grid discretizations sufficient to resolve bridge-level damage, and HAZUS-derived network attributes. OD values were calibrated for the example via optimization to match flows; for sensitivity analyses, post-event trip reductions were set to zero to avoid biasing performance indicators.

- Correlation length estimation from scenario-based fragility outputs yielded markedly different lambdas depending on scenario design: using same epicenter with varying magnitudes produced an estimated lambda of about 15.3 miles; using same magnitude with varying epicenters produced about 9.8 miles. This highlights the difficulty of representing correlation via limited scenarios and motivates random-field-based parametric control. - Parallel-type network example (one bridge per edge): • Lambda was varied from 0.1 to 20 miles; 500 random samples per lambda were generated with Uniform[0,4] marginal damage and exponential residual capacity mapping. Approximately 45% of bridges had residual capacity below 50% due to the uniform damage and exponential mapping. • FCR(lambda) exhibited strong sensitivity to correlation distance. The minimum FCR occurred at lambda ≈ 8 miles and was 40% (i.e., 200 of 500 samples remained fully connected). For very small lambda (weak correlation), FCR approached about 60%; for very large lambda (nearly fully correlated), FCR tended toward about 50% (network either largely intact or largely disrupted depending on the single field realization). • Total travel time P had best (lowest) values for small lambda, degraded substantially for intermediate lambda (roughly 4–10 miles), and improved again for large lambda. The variability (standard deviation) of P was also largest in the intermediate lambda range, indicating higher uncertainty in performance. - Series-type network example (multiple bridges per edge with bottleneck assumption): • Both FCR and P indicated worse performance than the parallel case, especially for small lambda where at least one bridge per edge tends to be severely damaged. For large lambda, metrics approached those of the parallel case because nearly uniform damage made the number of bridges per edge less influential. - Overall, assuming independence among bridge damages was shown to be unrealistic and can lead to large, unconservative errors in network performance assessment. Correlation length strongly influences both connectivity and efficiency metrics, with the most detrimental and variable outcomes at intermediate correlation distances that align with network spatial scales.

The framework directly addresses the research question by decoupling marginal damage severity from spatial dependence, enabling controlled variation of correlation length and quantifying its impact on network-level metrics. Findings demonstrate that spatial correlation of bridge damage is a critical driver of post-event connectivity and travel efficiency. Small correlation lengths allow damage to average out across nearby alternatives, preserving connectivity and yielding lower total travel time. Intermediate correlation lengths, comparable to node-to-bridge spacing, tend to co-localize high damages around specific nodes or corridors, producing isolation and congestion hotspots, which minimize FCR and maximize P with high variability. At large correlation lengths, damage is nearly uniform across the network, producing bimodal outcomes (either broadly functional or broadly impaired), hence FCR approaching 50% and reduced variability of P relative to the intermediate peak. These insights underscore the importance of modeling spatial dependence in fragility-informed network analyses, for both emergency access planning (FCR) and longer-term mobility and economic recovery (P).

The paper contributes: (1) a computational framework integrating bridge fragility analysis, non-Gaussian random field simulation of damage, and combined traffic distribution/assignment to evaluate transportation network performance under extreme events; (2) a sensitivity analysis showing strong dependence of connectivity and travel-time performance on the spatial correlation of bridge damage; and (3) a practical approach to estimate correlation length from limited fragility scenarios. Results indicate that neglecting correlation (assuming independence) can substantially overestimate network performance, particularly at correlation distances comparable to network spatial scales. For practice, correlation should be explicitly incorporated in network performance, maintenance planning, monitoring, and vulnerability assessments. Potential future extensions include multi-variate random fields to represent bridge classes with different fragilities and marginals, incorporation of more realistic non-homogeneous and anisotropic correlation structures, inclusion of other damaged components (e.g., roads, tunnels), time-variant reliability and life-cycle considerations, and refined modeling of post-event trip generation/attraction changes.

- Damage modeled only for bridges; other network components were assumed undamaged. - Random fields assumed homogeneous, isotropic, and quadrant with a single correlation length in both spatial directions; non-homogeneity or anisotropy was not modeled. - Marginal damage distribution was assumed Uniform[0,4] to isolate correlation effects; actual damage marginals may differ and could affect results. - The residual capacity mapping from damage used an exponential function with parameters selected by engineering judgment; different mappings would alter performance metrics. - Bottleneck assumption (edge capacity governed by the most damaged bridge) is conservative, especially in urban settings where local detours may exist. - Post-event reductions in trip generation/attraction were set to zero for sensitivity isolation; real events would alter O and D. - OD estimation used a gravitational model and assumed vertices as both origins and destinations; more detailed demand models and temporal variation were not considered. - Scenario-based correlation estimation relied on limited HAZUS scenarios and an assumed functional form; estimated lambdas may be sensitive to scenario selection. - The approach aggregates bridge structural differences in an average sense for the uni-variate field; using class-specific or bridge-specific fields would better reflect heterogeneity.