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Comprehensive assessment of cablestayed bridge based on Pagerank algorithm
Advances in Bridge Engineering volume 4, Article number: 32 (2023)
Abstract
This paper develops an improved structural health assessment method for cablestayed bridge to address the issue of neglecting component correlations in existing assessment standards. Firstly, the directed graph of fault transmission between components in the cablestayed bridge system was constructed. The Pagerank algorithm was used to analyze the degree of correlation between these components, and then the influencing degree of and the influenced degree of each component were determined. Secondly, considering the failure rate of individual components and the influenced degree of other component faults, a condition evaluation method with component correlation for cablestayed bridge was proposed. Finally, the improved assessment method was applied to a super largespan steel cablestayed bridge as a case study and compared with the relevant assessment specifications. The results show that main girder alignment, cable force and main tower alignment have a greater degree of correlation with other components and are important indicators for bridge health monitoring. Visual inspection of main girder and bridge bearing are the fault appearance components and should be paid attention to in preventive maintenance. The drainage system and electromechanical facilities are the fault source components and must be kept in good condition in daily inspections. The proposed method considers the interrelationships among components more comprehensively and can provide more reliable bridge health assessment results to support bridge maintenance decisions.
1 Introduction
The cablestayed bridge represents a significant form of modern bridge engineering. Currently, China holds the highest number of cablestayed bridge (Pang et al. 2022). However, with the increasing service life of cablestayed bridge, environmental erosion and repeated loading can easily lead to aging and damage of materials and components, potentially resulting in catastrophic bridge accidents (Li and Feng 2023). Therefore, conducting timely and accurate assessments and maintenance for cablestayed bridge is of utmost importance. Currently, in engineering practice, the assessment of cablestayed bridge status largely relies on normative methods (JTG/T 3365012020, 2020; JTG/T H212011, 2011). This method is based on the Analytic Hierarchy Process (AHP), known for its welldefined metrics and ease of application, and has therefore found widespread use. However, this approach has its shortcomings, including subjectivity and a lack of consideration for interrelationships among metrics (Yang et al. 2019). To solve these problems, scholars have proposed various methods aimed at mitigating subjective biases.
Some scholars have introduced methods such as Fuzzy Analytic Hierarchy Process (Jakiel and Fabianowski 2015), neural networks (Gai et al. 2023), and reliability theory assessment (Raizer 2004) to reduce subjective biases. Others have employed evidence reasoning theory (Bolar et al. 2013) to handle redundant or conflicting information between manual inspection and health monitoring data.
Zong et al. introduced a fuzzy evaluation method that constructs a threelevel logical evaluation structure with 21 basic influencing factors using the Analytic Hierarchy Process. They used the Delphi method to determine the weights of each judgment matrix, providing theoretical support for the assessment of largespan steel bridge (Zong et al. 2021). Yang et al. proposed a novel comprehensive condition assessment method based on Improved Interval Evidence Theory (IIET), Fuzzy Analytic Hierarchy Process (AHP), and natural fusion. They applied this method to evaluate a threespan PSC continuous box girder bridge, demonstrating its advantages over existing AHP assessment methods and traditional combination methods (Yang et al. 2019). Moufti et al. put forth a Fuzzy Evidence Hierarchy Reasoning method, systematically processing and aggregating detected bridge defect measurements, establishing an enhanced reliable bridge assessment platform for detailed condition assessment of concrete bridge under uncertain conditions (Moufti et al. 2014). Wang et al. introduced the use of Evidence Reasoning (ER) methods to model the inherent uncertainty in bridge subjective assessments and aggregate assessments of bridge components, offering a comprehensive evaluation of bridge condition (Wang and Elhag 2008).
However, cablestayed bridge represents highorder hyperstatic composite structure. It possesses characteristics such as complex stress distribution (Kildashti et al. 2020) and strong intercomponent correlation. For example, damage to stay cables can affect cable tension and main girder deflection (Tian et al. 2021). The cable forces in cablestayed bridge control internal force distribution and bridge deformations (Huang et al. 2018). Moreover, auxiliary piers have an impact on the structural response of cablestayed bridge (Wei et al. 2020). Currently, existing assessment methods do not account for the intricate correlations among components in cablestayed bridge. These methods have also failed to demonstrate the interdependencies among individual components.
In the field of engineering fault diagnosis, extensive research has delved into component correlations. Typically, the Pagerank algorithm is employed for analysis (Shen et al. 2019; Zhang et al. 2019) (Deng et al. 2023, Huang et al. 2023) to gain a deeper understanding of these associations. This algorithm is known for its simplicity and computational efficiency, enabling the ranking of component importance.
Wang et al. proposed a timeseries data mining and forecasting strategy based on information causality and the Pagerank algorithm. This strategy was applied to study the interaction between hydraulic, mechanical, and electromagnetic factors in a hydraulic power generation system with frequent changes in operating conditions (Wang et al. 2023). Su et al. introduced a method based on the Pagerank algorithm to rapidly and accurately identify critical nodes in power systems, and they validated this method using the IEEE 39node system (Su et al. 2021). Zhu et al. relying on the Pagerank algorithm and resource iteration concept, proposed the Improved Pagerank Algorithm (IPRA) to identify influential process nodes. Comparative analyses with existing algorithms demonstrated that this method excels in identifying influential nodes in complex production networks (Zhu et al. 2023).
Therefore, the objective of this paper is to apply the Pagerank algorithm to propose an improved method for assessing the condition of a cablestayed bridge. Based on the Pagerank algorithm to calculate the correlation between the components of a cablestayed bridge, the degree of influence and the degree of being influenced between the indexes are derived and verified with an actual cablestayed bridge as a case study. This method intends to remedy the problem of component correlation, which is neglected in the existing assessment methods, so as to assess the condition of a cablestayed bridge more accurately.
2 Correlation analysis method for components
The Pagerank algorithm is a search engine algorithm established by Sergey Brin and Larry Page, the two founders of Google. The algorithm relates the ranking of web pages to a number of factors related to the web page (Brin and Page 1998). The main principle of the PageRank algorithm is that if page A is able to link to page B, then it is assumed that page A passes an importance value (PR value) to page B. Based on the linking relationship, this value is calculated iteratively until it is smooth and is calculated as shown in Eq. (1).
where D_{PR}(P) is the PageRank value of a webpage, d is the damping factor, taking the value of 0 < d < 1, there is a damping factor because not all users will browse along the hyperlinks of the webpage, generally take d = 0.85; N is the number of all pages; T_{1}, T_{2}, ⋯T_{n} is the webpage pointing to the webpage P; D_{PR}(P) is the webpage T_{i} pointing to the webpage P has a PageRank value; \({N}_{T_i}\) is the webpage T_{i} has the number of export links.
Based on this principle, the interactions between the components of a cablestayed bridge can be viewed as the interlinking of pages in the Internet, with failures passing along the relational path. By analyzing the correlation degree of fault degradation among the components of the cablestayed bridge, it can provide reliable data for the subsequent scoring. The PR value represents the degree of influence of one component on another component. The higher the value, the greater the impact and correlation of the component on other components.
Based on the above idea, a directed graph of fault transmission among the components of the cablestayed bridge is established. Based on the Pagerank algorithm, the intercomponent correlation is solved by iterative calculation of the probability transfer matrix, as in Eq. (2) (Brin and Page 1998). Figure 1 shows the flow of the algorithm implementation.
Where E is the (n × 1) order matrix whose elements are all 1, n is the number of system components, d is the damping factor, which is selected as 0.3 with reference to the related literature (Long et al. 2017), C′ is the equal probability transfer matrix, and the initial PR value of each node is W′, and W^{x +1} denotes the (n × 1) order matrix composed of the importance degree of each node obtained from the (x + 1) iteration.
The correlation between components of cablestayed bridge is divided into the degree of influence C_{I} and the degree of being influenced C_{K}. C_{K} represents the degree of probability that a component is affected by the transmission of faults from other components and fails, and C_{I} represents the probability that a component will have an influence on other components. In this paper, with the help of the Pagerank algorithm and with reference to Eq. (2), we obtain iterative formulas for the calculation of the value of C_{K} and C_{I}.
In this paper, the power method is used to solve the C_{K} and C_{I} values. Generally, the initial value of PageRank (PR) does not significantly impact the convergence of the final PR value and the ultimate ranking. As a common practice, the initial values for C_{K} and C_{I} are usually set as a matrix with (n × 1) elements, all equal to 1.
3 Comprehensive assessment method for cablestayed bridge based on Pagerank algorithm
Based on the foundation of previous research (Liu et al. 2017a), this paper proposes a common method of dividing firstlevel indicators, which is categorized according to the perspective of the main components. Therefore, this paper integrates the manual inspection and health monitoring data, constructs the corresponding evaluation index system, and assigns the weights, which is shown in Fig. 2.
There is a correlation between the components in the overall cablestayed bridge system. This leads to a combined failure rate of the components of the cablestayed bridge that consists of two aspects. First, the components themselves may fail, generating a corresponding failure rate. Secondly, due to the correlation of failures between components, this may lead to the propagation of failures, which may result in failures of other components. As a result, the overall cablestayed bridge failure rate may increase when component correlations are taken into account compared to the case of individual failures, and therefore the scoring results obtained may be lower. We introduce the concept of a method for calculating the independent failure rate of equipment components into the condition assessment of cablestayed bridge (Long et al. 2017). The component scoring result considering correlation can be expressed as the componentindependent failure scoring result minus the failure effects of other components on it, as described below.
Assuming that the considered relevance score of each component V_{i} is S_{ii}, and the result of the separate failure score is S_{i}, the failure rate impact of other components on V_{i}, denoted by O_{i} × S_{i}.Where O_{i} is the probability that component V_{i} is affected by the failure of other components, i.e., the corresponding C_{K} value, and is the value of the demerit points of the components that have relevance to it other than component V_{i.} Finally, the following expression can be obtained:
Finally, based on the score S_{ii} of each component, the final level 1 index and the overall score of the bridge are obtained according to the corresponding weight allocation.
4 Case study
4.1 Background information of bridges
This article takes the inspection data of a large span steel cablestayed bridge in Jiangsu Province, China as an example, and evaluates it using the method of considering component correlation. The main bridge has a total length of 1288 m and is a steel tower and box girder structure. The entire bridge steel structure is made of Q345D steel. The cables are arranged symmetrically on the east and west sides, using 7 mm galvanized highstrength low relaxation steel wire. The bridge was officially completed and opened to traffic in 2005.
During the inspection of the background bridge in 2013, some important structural issues were identified (Liu et al. 2017b), as shown in Fig. 3. In the cable system, the main types of diseases include the detachment of antirust paint on the steel casing of the cable, accounting for approximately 16.1% of the total diseases. In addition, there are also issues such as coating deterioration and anticorrosion oil leakage. For the steel box girder, there are multiple problems such as rust, oil leakage, and paint peeling at the internal joints of the coating layer, among which the longest rust length of the coating layer is 3.0 m. In the truss structure of the main girder, common problems include welding hole cracking, cracks, holes, scratches, etc. Among them, 73.6% of the box compartments have diagonal web member cracking problems. As for the cable tower, the main diseases are coating deterioration and bolt corrosion or loosening, which occur in 55.8% and 42.3% of the box compartments, respectively. In terms of the substructure, the support suffered damage to the pad stone concrete. Concrete expansion and cracking and other issues may occur locally on the bridge pier. In addition, in the ancillary facilities, issues such as peeling and scratching of antirust paint on guardrails, blockage of expansion joints, damage to a protective net in the drainage system, and two 5 m long longitudinal cracks on the bridge deck pavement layer have also been detected, with oil stains, water seepage, and other issues present in 0.16% of the area.
In addition, the monitoring of the cable force, main beam alignment, and cable tower alignment of the bridge was as follows: for cable force, the sum of upstream and downstream cable force tests was basically the same, and the unevenness did not exceed 2% of the total cable force on one side. The overall performance of the structural cable force was stable; The deflection of the main beam sank, and by October 2013, the mid span deflection had decreased by 80 mm. For the cable tower alignment, there was currently no abnormal deviation at the top of the tower.
4.2 Comprehensive evaluation process
4.2.1 Solution of correlation values among components
Considering the actual composition structure of the cablestayed bridge system, we can construct a connection diagram between various components based on the Pagerank algorithm, as shown in Fig. 4. The connecting lines in Fig. 4 indicate which components of the tier 1 indicators are relevant, as well as the separate tier 2 indicators under the tier 1 indicators, which are numbered. Through expert consultation and structural stress characteristics, we analyzed the potential fault transmission relationships between various components of the bridge, and obtained a directed graph of fault transmission among various components of the cablestayed bridge system, as shown in Fig. 5. The numbering in Fig. 5 corresponds to each component in Fig. 4, and the color of the squares shows the obvious expression of component correlation. The darkcolored squares in the figure represent that the horizontal coordinate component corresponding to the square has a certain correlation with the vertical coordinate component, on the contrary, the lightcolored squares represent that the two components do not have obvious correlation. For a fault transmission directed graph with n nodes, its adjacency matrix C can be expressed as n × n matrix (Long et al. 2017):
When i ≠ j:
When i = j, C_{ij} = 0.
At the same time, the state equal probability transfer matrix can also be derived as in Eq. (7). Each row of the C′ matrix represents a component, as does each column. If the corresponding element of each component is not zero, it means that there is some correlation between these components.
According to Eq. (3) and d is taken as 0.3 based on experience. The matrix iteration was performed. The C_{K} value was calculated using programming software. A correlation transformation on the adjacency matrix C was performed to obtain the transposed adjacency matrix C_{T} and its equal probability transition matrix [C_{T}] ‘, as shown in Eq.8. According to Eq. (4), C_{I} values of each component was obtained.
Table 1 shows the C_{K} values and C_{I} values of each component. The influencing and influenced degree of the main girder alignment, cable force, and cable tower alignment are relatively large, indicating that these three indicators have a wide range of influence. Therefore, health monitoring of bridges is particularly important, especially the changes in these three monitoring indicators. For the visual inspection of the main girder and bearing, their C_{K} values are large, while the C_{I} values are small. They are most susceptible to other component failures and can easily reflect abnormal structural conditions (Baig et al. 2022; Šomodíková et al. 2016; Zhao et al. 2023; Thakkar et al. 2023). Therefore, during preventive maintenance, attention should be paid to the status of these two components. On the contrary, the C_{K} values of drainage systems and electromechanical facilities are lower, while the C_{I} values are higher, which means they are more likely to become fault source components. When these components fail, it may also affect other components. For example, poor drainage system and power outage of mechanical and electrical facilities can further accelerate the corrosion of steel structures. Therefore, in daily inspections, it is necessary to ensure that these two components remain in good working condition.
4.2.2 Bridge status rating
Based on the disease situation of the background bridge, the relevant components were evaluated according to the ‘Standards for Technical Condition Evaluation of Highway Bridges’ (hereinafter referred to as TCEB). Finally, the independent fault scores of each secondary indicator are obtained. Then, as described in Section 2, the data is substituted into Eq. (5) to obtain the bridge evaluation results based on the Pagerank algorithm, as shown in Table 2.
4.3 Comprehensive evaluation results
The evaluation results were compared using the TCEB and the methods proposed in this article in terms of first level indicators and overall scores, as well as various components of ancillary facilities, as shown in Fig. 6.
The comparison between the primary indicators and the overall evaluation results shows that there are significant differences in the scores of the cablestayed system, main girder, and cable tower, while others are basically the same. This is mainly because the improved indicator system has added health monitoring indicators such as main girder deflection, cable force, and cable tower alignment. The influencing and the influenced degree on the main beam alignment and cable force are both significant, so the scoring values of monitoring indicators will affect the correlation between components, thereby affecting the final evaluation results.
The final overall rating of the bridge in this article is 77.75, which is a Level 3 bridge. The rating in the TCEB is 85.3, which is a Level 2 bridge. The correlation between bridge components has an undeniable impact on the overall state of the bridge.
Compared to the TCEB, the auxiliary facilities have added longitudinal damping devices and electromechanical facilities, and the scores of each component are not significantly different. Among them, the bridge deck pavement, expansion joint devices, drainage systems, and lighting components will have a slight decrease in status scores due to the correlation of other components.
Compared to the TCEB, the method proposed in this article combines the impact of component failures and considers both health monitoring and manual inspection indicators, making the evaluation results more comprehensive and reliable, providing a reliable basis for the subsequent repair and maintenance of bridges.
5 Conclusions
This article takes cablestayed bridges as the research object and proposes a new evaluation method based on component correlation. The following conclusions are drawn:

(1)
A comprehensive evaluation method for cablestayed bridges considering component correlation has been proposed. A directed graph for the fault transmission of each component of the cablestayed bridge is established, and the correlation of each component is analyzed based on the Pagerank algorithm. Finally, a cablestayed bridge condition evaluation method that integrates the fault rate generated by each component separately and the fault rate generated by the fault correlation of other components.

(2)
By solving the correlation between the components, it can be seen that the main girder alignment, cable force and main tower alignment are affected to a greater extent and influence range, in which the C_{K} value is greater than 0.068 and the C_{I} value is greater than 0.063, which indicates that the influence range of these three indexes is larger, and it is particularly important for the monitoring of the bridge’s health condition; Visual inspection of main girder and the bearing C_{K} is larger than 0.069, the C_{I} is smaller than the smaller than 0.055, belonging to the failure appearance components, in preventive maintenance should focus on; drainage system and mechanical and electrical facilities on the contrary, C_{K} is less than 0.047, C_{I} is greater than 0.058, belonging to the failure source components should focus on in the daily inspection.

(3)
Compared with the specifications, the method proposed in this article considers component correlation, making the evaluation results of the main components of cablestayed bridges more reliable; Due to the correlation of components, the evaluation results of ancillary facilities have decreased. This provides reference for bridge maintenance decisionmaking.
Availability of data and materials
Most data and models generated and used during the study appear in the published article. However, some information is proprietary or confidential in nature and may only be provided with restrictions.
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Funding
This research was financially supported by the National Natural Science Foundation of China (no. 51808301), the Scientific Research Fund of Zhejiang Provincial Education Department (no. Y202248860), and the National “111” Centre on Safety and Intelligent Operation of Sea Bridge (D21013).
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Xiaoling Liu: Conceptualization, Methodology, Writing–original draft. Ying Liu: Data analysis. Bing Wang: Data analysis.
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Liu, Y., Wang, B. & Liu, X. Comprehensive assessment of cablestayed bridge based on Pagerank algorithm. ABEN 4, 32 (2023). https://doi.org/10.1186/s43251023001113
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DOI: https://doi.org/10.1186/s43251023001113