Keywords

1 Introduction

With the continuous development of technology, various aesthetically pleasing new types of bridges have emerged. Among these, spatial special-shaped arch bridges have garnered significant attention in existing literature, primarily focusing on their design and construction [1, 2]. However, there has been relatively limited research on the seismic fragility of spatial special-shaped steel box arch bridges. Analyzing a bridge’s response to lateral seismic forces allows us to better assess and improve its seismic performance, ensuring that the bridge remains safe and stable during earthquakes. This is of critical importance for enhancing a bridge’s earthquake resistance, as it helps reduce potential losses and risks associated with earthquakes.

Kim et al. [3] utilized the Monte Carlo simulation method and response spectrum analysis to establish fragility curves for bridges. The findings revealed that bridges considering multiple excitation factors are more susceptible to seismic damage compared to those considering uniform excitation. The fragility curves were represented using different seismic indicators such as PGA, PGV, and SA. It was concluded that PGA and SA are more suitable and efficient seismic indicators. Choi et al. [4] simultaneously considered the correlation between the failure modes of bearings and pier columns under seismic actions. They introduced the first-order boundary method to establish the upper and lower bounds of systemic fragility curves for bridges. Li et al. [5] considered the correlation between the failure modes of pier, bearing, and abutment components and established systemic fragility curves for medium-span reinforced concrete continuous beam bridges using a comprehensive second-order boundary method. Pan [6] conducted a seismic fragility analysis on a steel bridge in New York State and found that nonlinear regression outperforms traditional linear regression in fitting the bridge’s response data.

Currently, most research on seismic fragility focuses on conventional bridges, with limited studies on the seismic fragility of irregular arch bridges. This paper takes a specific spatial irregular steel box arch bridge as an example and establishes a finite element model. By applying fragility theory and conducting incremental dynamic analysis (IDA), the fragility curves of bridge components are developed to evaluate their vulnerability. Additionally, the paper explores the systemic fragility curve of such bridge structures using the weight method. The findings of this study can serve as a reference for similar vulnerability research on bridges.

2 Bridge Analytical Model

To ensure the accuracy and reliability of the finite element numerical model, this paper used the finite element analysis software Midas/Civil to establish a complete spatial finite element model of the bridge. In this model, the main beam, main arch, pile foundation, and abutment were simulated using beam elements, while the cables were simulated using truss elements. The model consisted of 1,209 beam elements and 12 truss elements. The main beam is elastically connected to the abutments, whereas the main arch is rigidly connected to the foundation of the arch seat. The bearings are simulated using a combination of rigid and elastic connections. The connections between the main beam pile foundations and the abutments, as well as between the arch seat pile foundations and the foundation of the arch seat, are treated with master-slave constraints. The finite element model of the bridge is shown in Fig. 1.

Fig. 1.
figure 1

Bridge Finite Element Model

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There are No. 1 to No. 4 bearings on the main beam, and the specific arrangement is shown in Fig. 2. Among them, bearing 1 is a fixed bearing, bearing 2 is a longitudinal moving bearing, bearing 3 is a transverse moving bearing, and bearing 4 is a bidirectional moving bearing. The maximum allowable displacement of the bearing is 200 mm.

Fig. 2.
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Bearings Arrangement

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3 Seismic Fragility Analysis of Bridge

3.1 Selection and Division of Damage Index

The damage indicators for the bearing components are calculated based on the maximum allowable displacement of the bridge’s movable bearings and coefficients corresponding to different damage states. The damage indicators for the bridge abutments and arch bearings are determined based on the damage states proposed by the scholar Zheng Kaifeng [7]. The damage state and damage index of the bearing, abutment, and arch support platform are shown in Table 1.

Table.1 Component failure index
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3.2 Seismic Fragility Analysis of Components

The ‘frequency statistics method’ is used for the statistics of the failure state of the component under different PGA. When the ‘ frequency statistics method ‘ is used to draw the fragility curve, because the probability points corresponding to the failure state of each level under each level of PGA are directly connected, the obtained will be a set of broken lines. The current fitting methods for probabilistic points assume that the fragility curves of components conform to a lognormal distribution. These methods employ the probability density function of the lognormal distribution to perform regression analysis on the probabilistic points. This analysis enables the determination of the mean and standard deviation of the failure probability density function for each component at various levels. Subsequently, these computed mean and standard deviation values are inserted into the cumulative distribution function of the standard normal distribution. By doing so, the fragility curves for the different damage states of each component can be derive [8]. This article utilizes the lognormal distribution function and employs maximum likelihood estimation in the Matlab software to fit the lognormal distribution for structural fragility.

Assuming that the seismic fragility curves conform to a two-parameter lognormal distribution:

$${F}_{j}(IM)=\varPhi \left[\frac{ln(IM/{c}_{j})}{{\zeta }_{j}}\right]$$
(1)

where, \({\text{F}}_{\text{j}}(\cdot )\) represents the fragility function of the damage state, whereas \({\text{c}}_{\text{j}}\) and \({\upzeta }_{\text{j}}\) represent the median and logarithmic standard deviation, respectively, corresponding to that damage state. The fragility curve of components is shown in Fig. 3.

Fig. 3.
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Bridge Abutment and Bridge Arch Support Platform and Bridge Bearing Fragility Curve

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The probability of failure states for the four bearings of the irregular arch bridge under different PGA values was analyzed statistically, as shown in Figs. 4 and 5.

Significant differences exist in the probability of complete failure among the four bearings, At a PGA of 1.0g, the probabilities of complete failure for bearings 2 and 3 are 70% and 90%, respectively. Bearing 4 exhibits the lowest probability of complete failure among the four bearings at a PGA of 1.0g, with a probability of 35%. This demonstrates its strong seismic resistance and higher safety performance.

Fig. 4.
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Extensive Damage Fragility Curve of Bearing

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Fig. 5.
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Complete Damage Fragility Curve of Bearing

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3.3 Seismic Fragility Analysis of System

The bridge structure is a complex system composed of a large number of different components. If only the fragility analysis of a certain component is carried out, the seismic performance of the whole bridge system cannot be fully reflected. The calculation results show that the damage level and failure probability of the components are quite different, so it is necessary to carry out the fragility analysis of the bridge system.

The weighting method takes into account the importance of each component in the bridge system. It categorizes the major components, which are the ones that, if damaged, would lead to the failure of the entire bridge system, as one group, and the remaining components as another group, considering their respective weights. The weighting method has been proven to be suitable for the analysis of bridge system fragility [9].

In this paper, the bridge abutment and arch support platform are divided into main components, and the bearing is divided into other components [10]. The weight of each component is calculated as shown in Table 2. The corresponding calculation weights are computed based on the provided weights of the components, as shown in Eq. (2):

$${\omega }_{\upeta }=\frac{{\omega }_{q}}{{\omega }_{mq}}$$
(2)

where, \({\upomega }_{\upeta }\) represents the calculated weight of the component, \({\upomega }_{\text{q}}\) represents the weight of the component, and \({\upomega }_{\text{mq}}\) represents the weight of the main component.

Table 2. Member weight calculation
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The weight method [9] is used to calculate the failure probability of the bridge system. The calculation formula of the weight method is shown in Eq. (3).

$$ P_{{{\text{sys}}}} = \mathop {{\text{max}}}\limits_{{{\text{i}} = {1}}}^{{\text{m}}} P_{{{\text{fi}}}} + \eta \mathop \Sigma \limits_{{{\text{j}} = 1}}^{{\text{n}}} P_{{{\text{fj}}}} \le 1 $$
(3)

where, \({\text{P}}_{\text{sys}}\) is the failure probability of the bridge system. \({\text{P}}_{\text{fi}}\) is the probability of failure of the i th main member.m is the number of main components. \({\text{P}}_{\text{fj}}\) is the probability of failure of the j th other component.n is the number of other components. η is the corresponding weight.

It is evident from Fig. 6 that the maximum bandwidths for the system and its main components, ranging from slight failure to complete failure, are 0.111, 0.121, 0.113, and 0.085, respectively. This indicates that the probability of failure at each level on the system’s fragility curve is higher than that of the main components, aligning with the actual earthquake-induced damage observed in the project.

Fig. 6.
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Fragility Curve of System and Main Components in Failure State

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4 Conclusions

The conclusions derived from this study are as follows:

A comparative fragility analysis of multiple components reveals that the maximum probability of complete failure for bearings is 70%, for the bridge abutment is 40%, and for the arch support platform is 10%. This indicates that the bridge abutment and bearings are susceptible to damage under seismic actions, while the arch support platform exhibits better seismic performance. It suggests that the bridge abutment and bearings are the vulnerable components of the bridge, and their seismic protection should be given priority.