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

Literature Overview and Methodological Framework

This study presents a comprehensive seismic vulnerability analysis of steel tube concrete (STC) arch bridges using the Copula function to model the dependence structure between multiple seismic demand parameters. The Copula approach provides a rigorous statistical framework for capturing the correlations between different limit state functions, which is critical for accurate vulnerability assessment of complex bridge systems.

Steel tube concrete arch bridges combine the compressive strength of concrete with the tensile capacity and ductility of steel tubes, creating a structurally efficient system for medium to long-span bridges. However, the seismic performance of these bridges is influenced by multiple interacting factors, including the concrete confinement effect, the steel tube buckling behavior, and the arch action under lateral loading.

Core Technical Methodology

Copula Function for Dependence Modeling

The Copula function serves as a mathematical tool to link marginal distributions into a joint distribution while preserving the dependence structure. In the context of seismic vulnerability analysis, the Copula function is used to model the correlation between different seismic demand parameters, such as peak ground acceleration (PGA), spectral acceleration (Sa), and pseudovelocities (Pv).

The selection of the appropriate Copula type is critical for accurate vulnerability assessment. The study compares multiple Copula functions and identifies the Gumbel Copula as the most suitable for modeling the dependence between seismic demands in the upper tail region, where the vulnerability of interest lies.

Seismic Demand Parameters and Fragility Curves

The seismic demand parameters considered in the analysis include the maximum inter-story drift ratio (MIDR), the arch rise deformation, and the plastic hinge formation at critical sections. The fragility curves, expressed as conditional probabilities of exceeding specific damage states, are derived using incremental dynamic analysis (IDA) combined with the Copula-based correlation model.

The fragility curves show that the STC arch bridge exhibits a moderate vulnerability to seismic loading, with the collapse probability exceeding 10% at PGA values of 0.6g or higher. The Copula-based approach provides more accurate fragility estimates compared to the traditional independent assumption, particularly in the high-intensity region where the correlation between demand parameters becomes significant.

System Reliability Analysis

The system reliability of the STC arch bridge is evaluated using the first-order reliability method (FORM) combined with the Copula-based joint distribution. The reliability index (β) is computed for each damage state, and the system failure probability is derived using the series system approximation.

The results indicate that the STC arch bridge meets the typical seismic design objectives for moderate seismic regions, with acceptable failure probabilities for the slight and moderate damage states. However, the collapse probability is relatively high for severe seismic events, suggesting the need for enhanced seismic design measures in high-seismicity regions.

Structural Behavior and Failure Mechanisms

Concrete Confinement Effect

The confinement effect provided by the steel tubes significantly enhances the compressive strength and ductility of the concrete fill. The confined concrete strength can be estimated using the Mander model or the Lam and Plowman model, with confinement ratios typically in the range of 0.1–0.3 for STC arch ribs.

The steel tube also provides lateral restraint that prevents the concrete from spalling under seismic loading, which is critical for maintaining the structural integrity of the arch rib.

Steel Tube Buckling Behavior

The steel tubes are susceptible to local buckling under high compressive stresses, particularly at the crown and haunches of the arch where the bending moments are maximum. The buckling behavior is influenced by the slenderness ratio, the concrete confinement, and the boundary conditions at the arch supports.

The local buckling stress of the steel tube can be estimated using the European code formula:

σ_cr = 0.42 × (E × t/D) × (D/t)^(1/2)

where E is the Young's modulus, t is the wall thickness, and D is the outer diameter of the tube. The concrete fill provides additional lateral support that increases the buckling resistance by 20–40% compared to empty tubes.

Quality Control and Construction Considerations

Material Quality Assurance

The quality of the steel tubes and the concrete fill directly affects the seismic performance of the STC arch bridge. The following quality control measures are recommended:

1. Steel tube certification: All steel tubes must comply with the specified grade (typically Q345 or Q460) with certified mechanical properties including yield strength, tensile strength, and elongation.
2. Concrete mix design: The concrete mix must be designed for high workability to ensure complete filling of the steel tubes, with a slump of 180–220 mm and a maximum aggregate size of 20 mm.
3. Weld inspection: All welds connecting the steel tube segments must undergo 100% ultrasonic testing to ensure full penetration and absence of volumetric defects.
4. Concrete density verification: The density of the filled concrete should be verified through core sampling or non-destructive methods such as rebound hammer testing.

Study Insights and Engineering Implications

The application of the Copula function to seismic vulnerability analysis provides a more realistic representation of the dependence structure between seismic demand parameters. The traditional assumption of independence between demands often leads to conservative or non-conservative estimates of failure probability, depending on the actual correlation structure. The Copula-based approach offers a systematic framework for incorporating this correlation into the reliability analysis.

From a structural engineering perspective, the STC arch bridge demonstrates good seismic performance due to the combined effects of concrete confinement, steel tube ductility, and the arch action. However, the vulnerability analysis reveals potential weaknesses at the arch supports and the haunch regions, where the combined effects of bending and axial compression can lead to premature failure. Targeted strengthening measures, such as additional steel reinforcement or the use of higher-grade steel tubes in these critical regions, can significantly improve the seismic resilience.

The research also highlights the importance of considering the full range of seismic intensities in the vulnerability assessment. The Copula-based approach captures the changing correlation structure across different intensity levels, which is not possible with traditional linear correlation models. This insight is valuable for performance-based earthquake engineering (PBEE) frameworks that require accurate fragility estimates across the full hazard range.

In conclusion, this study demonstrates the effectiveness of the Copula function in seismic vulnerability analysis of STC arch bridges. The methodology provides a rigorous statistical foundation for reliability assessment, and the findings offer practical guidance for the seismic design and retrofitting of STC arch bridges. Future research should extend the analysis to include soil-structure interaction effects, multi-hazard scenarios, and the long-term degradation of structural capacity due to environmental exposure.