Transportation networks are critical for post-event recovery activities following extreme events (natural or man-made). Significant research has focused on lifeline analysis, including integrated frameworks for reliability analysis, life-cycle cost assessment, and maintenance optimization of bridge networks (Frangopol et al.), seismic bridge retrofit analysis incorporating traffic flow redistribution and network resilience (Shinozuka et al.), utility lifeline reliability considering flow redistribution and cascading failures (Dueñas-Osorio et al.), vulnerability assessment of infrastructure networks (Murray et al.), life-cycle management and maintenance optimization (van Noortwijk et al.), and resilience concepts for infrastructure facilities (Bruneau et al.). This paper introduces a stochastic computational framework that combines structural fragility analysis with traffic flow assessment. Network performance depends on the damage level of all its components; therefore, this framework is used to investigate the influence of correlation among individual bridge damage levels on network performance indicators. Rather than analyzing numerous extreme event scenarios, random field theory is employed to directly control and analyze the correlation structure of the damage.

The paper reviews existing research on lifeline and transportation network analysis under extreme events. It cites works by Frangopol and colleagues on bridge network reliability and optimization, Shinozuka's group on seismic retrofit and socio-economic impacts, Dueñas-Osorio's work on cascading failures in utility networks, Murray's research on network vulnerability, van Noortwijk's contributions to life-cycle management, and Bruneau's framework on community resilience. The authors also mention the matrix-based system reliability method used by Song and others, and Peeta's work on network interactions. The review highlights the need for a method that considers both individual component fragility and the overall network response, incorporating the spatial correlation of damage.

The proposed framework (Fig. 1) combines several analysis steps: 1. **Structural Fragility Analysis:** Bridges are considered the most vulnerable components. Damage is assessed using HAZUS-MH MR4, considering ground motion and failure. An average damage level (Ib) is computed for each bridge for each scenario (Eq. 1). 2. **Spatial Correlation Estimation:** Bridge coordinates are transformed into a Cartesian system (Eq. 2). Damage statistics (Eqs. 4 and 5) and the correlation coefficient (Eq. 3) are calculated. The correlation coefficient is interpolated (Eq. 6) to estimate the spatial correlation length (λ). 3. **Random Field Simulation:** The bridge damage level is modeled as a two-dimensional random field. A uniform distribution [0,4] is assumed for the marginal distribution. The correlation structure (Eqs. 8-10) is derived from the estimated correlation coefficient. Damage level samples are generated for various correlation lengths. 4. **Network Analysis:** Network connectivity is represented using graph theory. Edge characteristics (Eqs. 12-13) are computed. Post-event bridge characteristics are updated (Eq. 14). An iterative optimization algorithm (based on Evans' method) combines traffic distribution and assignment, incorporating congestion functions (Eq. 11) and computing network performance indicators (Eqs. 36-37). 5. **Statistical Result Analysis:** The minimum FCR(λ) is computed. The best P results are selected, discarding samples with disconnected nodes. Statistics of P(λ) are then calculated.

Two numerical examples are presented: "bridges in parallel" and "bridges in series". **Bridges in Parallel:** This example uses a simplified network with six vertices and eight highway segments. Nine earthquake scenarios were used to estimate the correlation coefficient. A sensitivity analysis was performed for λ ranging from 0.1 mi to 20 mi. Results show a strong dependence of the Fully Connected Ratio (FCR) and total travel time (P) on the correlation length. For small λ (weak correlation), FCR is high (~60%), indicating good network connectivity. As λ increases, FCR decreases significantly (below 50%), increasing again for large λ. The behavior of P mirrors that of FCR. The highest variability in network performance is observed when λ is comparable to the average distance between bridges. **Bridges in Series:** This example includes additional bridges, creating a "series" configuration. The results show that network performance is considerably worse for low λ, than in the "parallel" case, especially for FCR. However, for fully correlated damage (large λ), both FCR and P approach the values from the "parallel" example. The "bottleneck" assumption is noted as over-conservative. The findings demonstrate the importance of considering spatial correlation in bridge damage when assessing network performance. Ignoring correlation leads to overly optimistic estimations of network performance.

The findings highlight the significant impact of spatial correlation in bridge damage on transportation network performance. The study demonstrates that assuming independence between bridge damage levels can lead to substantial errors in assessing network resilience and functionality. The methodology presented provides a way to quantitatively assess this correlation using a realistic model of bridge damage and traffic flow. The results are relevant for a variety of applications, including risk assessment, planning, design and maintenance decisions. Both examples emphasize the importance of correlation, particularly for low to moderate correlation lengths. The results also underscore the importance of considering network topology when assessing the consequences of spatially correlated bridge damage.

This paper presents a novel framework for integrating structural fragility analysis and traffic flow distribution under extreme events. The sensitivity analysis demonstrates the importance of considering damage correlation. The methodology allows for a more realistic assessment of transportation network vulnerability and facilitates better-informed decision-making regarding infrastructure management and emergency preparedness. Future research could explore more complex correlation models, different damage states, and the incorporation of other infrastructure components.

The study focuses on bridge damage and assumes a simplified model of traffic flow. The use of a uniform distribution for bridge damage levels might not fully represent real-world scenarios. The “bottleneck” assumption used for bridges in series is over-conservative. More complex models of traffic flow, incorporation of alternative routes and other infrastructure components beyond bridges, and refined damage models could further enhance the accuracy and applicability of the framework.