System Seismic Vulnerability Analysis of Steel Tube-Concrete Arch Bridges Using Copula Function

Literature Overview

This research presents a comprehensive system-level seismic vulnerability analysis of steel tube-concrete (SRC) arch bridges using the Copula function approach. Unlike traditional component-based vulnerability assessments, this study considers the spatial correlation between multiple bridge components and their combined influence on overall structural performance. SRC arch bridges are increasingly employed for medium-to-long span bridges (100–500 m) due to their favourable strength-to-weight ratio, aesthetic appeal, and efficient use of materials. The system-level analysis provides a more realistic assessment of seismic performance by accounting for the interaction between the arch ribs, deck system, hangers, and foundation components.

Structural System and Component Characteristics

SRC arch bridges typically comprise the following structural components, each with distinct seismic vulnerability characteristics:

The steel tube in the arch rib is typically a circular or elliptical section with an outer diameter of 600–1500 mm and wall thickness of 12–30 mm, fabricated from Q345 or Q420 steel conforming to GB/T 8163 or API 5L standards. The concrete fill is high-strength (C50–C80) with appropriate admixtures for workability in the confined tube geometry.

Copula Function Methodology

The Copula function approach addresses the critical limitation of traditional vulnerability analysis, which assumes independence between component failures. In reality, seismic demand is spatially correlated, and component responses are strongly dependent on the global structural behaviour. The Copula function provides a mathematical framework for modelling the joint probability distribution of multiple correlated random variables.

The key steps in the Copula-based vulnerability analysis are:

1. Component vulnerability curves: Each component (arch rib, deck, hangers, bearings, foundation) is characterised by a fragility curve expressing the probability of exceeding a damage state given a seismic intensity measure (e.g., PGA, SA(T1)).
2. Correlation matrix: The spatial and temporal correlation between seismic demands at different components is quantified using the conditional mean measure (CMM) or rank correlation coefficient.
3. Copula selection: The appropriate Copula family (Gaussian, Clayton, Gumbel, Frank) is selected based on the observed dependence structure. The Gaussian Copula is commonly used for seismic vulnerability analysis due to its symmetric dependence structure and mathematical tractability.
4. System vulnerability derivation: The system-level vulnerability is computed by integrating the component vulnerabilities through the selected Copula function, accounting for the failure logic (series, parallel, or k-out-of-n systems).

Key Technical Parameters and Results

The study typically evaluates vulnerability across multiple damage states:

The system vulnerability is typically 15–30% higher than the component-based vulnerability when positive correlation is accounted for, demonstrating the importance of the Copula approach. The dominant failure mode for SRC arch bridges is typically the flexural yielding of the arch rib at the springing or crown, followed by concrete crushing in the compression zone.

Welding and Fabrication Quality Implications

The seismic performance of SRC arch ribs is directly influenced by the quality of steel tube fabrication and welding:

1. Longitudinal weld quality: The longitudinal weld in the steel tube creates a continuous stress concentration path that can initiate fatigue cracking under cyclic seismic loading. Full-penetration welds with smooth transition profiles are essential.
2. Transverse weld continuity: If the arch rib is fabricated from multiple tube segments, the transverse butt welds must achieve full fusion with minimum residual stress. Magnetic particle testing (MT) and ultrasonic testing (UT) are mandatory for all structural welds.
3. Concrete-steel interface: The bond between the steel tube and concrete fill is critical for composite action. The steel tube should be cleaned to Sa 2.5 grade, and the concrete should be placed using a self-compacting mix with a slump flow of 280–320 mm to ensure void-free filling.
4. Residual stress management: Welding residual stresses in the arch rib can reduce the effective seismic capacity by 5–15%. Stress relief treatment (thermal or mechanical) should be considered for critical connections.

Engineering Practice Considerations

For engineers designing SRC arch bridges in seismic regions, the Copula-based vulnerability analysis provides several actionable insights:

• Component redundancy: The system vulnerability analysis identifies critical components whose failure leads to disproportionate system damage. These components should be designed with higher performance objectives (e.g., collapse prevention rather than damage limitation).
• Foundation-structure interaction: The correlation between foundation response and superstructure demand is often underestimated in component-based analysis. The Copula approach captures this interaction and may reveal the foundation as a more critical component than anticipated.
• Seismic isolation: The analysis can evaluate the effectiveness of seismic isolation bearings in reducing system vulnerability. Isolated SRC arch bridges typically show a 40–60% reduction in DS3 and DS4 probabilities compared to fixed-base designs.
• Performance-based design: The vulnerability curves can be used to calibrate performance-based seismic design objectives, ensuring that the bridge meets specific functionality requirements after design-level earthquakes.

Key Questions and Reflections

The Copula-based approach, while more realistic than independent-component analysis, still relies on simplified assumptions about the dependence structure. The actual correlation between component responses may vary with seismic intensity, ground motion duration, and structural nonlinearity. Future research should investigate non-stationary Copula functions that evolve with increasing seismic demand.

Additionally, the vulnerability analysis should be complemented with fragility analysis of the bridge's functional components (e.g., deck integrity, hanger tension capacity) to provide a comprehensive assessment of post-earthquake usability. The transition from structural vulnerability to functional vulnerability is essential for transportation network resilience planning.

Study Insights and Outlook

The Copula-based system vulnerability analysis represents a methodological advancement in seismic assessment of SRC arch bridges. By accounting for spatial correlation and system-level interaction, the approach provides more realistic vulnerability estimates that can directly inform risk-informed design and retrofit decisions. Future work should integrate this methodology with probabilistic seismic hazard analysis to provide full risk-based assessments, and should extend to consider multi-hazard scenarios including earthquakes combined with flooding, landslides, or traffic loading. The technology also has direct applications for existing bridge inventory assessment, where rapid vulnerability screening can prioritise retrofit investments.