Comprehensive durability assessment of in-service concrete bridges based on combination weighting and extenics cloud model

Abstract

In recent years, with the continuous development of urban transportation, the durability of concrete bridge structures has become increasingly prominent. Many in-service concrete bridges exhibit issues such as cracks, excessive deflection, and a significant reduction in load-bearing capacity during their service life. To meet the requirements of sustainable development, it is urgent to assess the durability of bridges accurately. Therefore, this paper establishes a scientific and reliable evaluation index system for the durability of in-service concrete bridges. Based on a review of the literature and relevant standards and specifications, this paper establishes an evaluation index system for the durability of in-service concrete bridges and calculates the weights using a combination weighting method. The durability of the bridges is assessed using the extension cloud model theory, and MATLAB is utilized for efficient and accurate computation and analysis, ultimately determining the durability grades of the bridges. The method was applied to the durability assessment of five in-service concrete bridges, and the results were in high agreement with the measured data, proving its effectiveness and accuracy. The research results provide a scientific basis for the management and maintenance of concrete bridges, which can help extend their service life. Future studies will expand the sample to cover more bridge types to comprehensively assess and enhance the durability of bridges.

Article Highlights

A scientific and reliable durability evaluation index system for in-service concrete Bridges has been established.

Weight optimization: Based on grey correlation weight and entropy weight to find the combined weight.

The perfect integration of extension theory and cloud model.

Introduction

The construction of bridges represents a pivotal aspect of the advancement of national infrastructure and a fundamental element of transportation infrastructure. In recent years, numerous countries have been confronted with the issue of bridge ageing, with the number of old bridges increasing at an alarming rate [1]. It thus follows that an analysis and study of the durability of in-service bridges has become a particularly necessary undertaking. The durability of a bridge can be defined as the ability of its structural components and the natural environment to withstand the effects of gradual deterioration, traffic volume, and other combined factors. This is achieved through the design of the bridge, taking into account the normal use and maintenance conditions, in order to ensure the safety, functionality, and technical performance of the bridge is maintained [2–5]. If these defects can be detected in a timely manner and appropriate repairs and reinforcements can be carried out, the maintenance costs of the bridges can be greatly saved, and at the same time, the huge losses due to the damage of the buildings as well as the bad social impacts can be prevented [6, 7]. A series of domestic and foreign bridge collapse accidents have occurred in rapid succession, including the Liaoning Tianzhuangtai Bridge and the Jilin Fusong Jinshan Bridge. The overall collapse of these structures has been attributed to a number of factors, with the primary cause being the durability of the problem [8].

Mark Alexander [9] focuses on the durability of reinforced concrete structures and reinforcement corrosion. He states that corrosion of reinforcement is the greatest threat to the durability of reinforced concrete. The service life of reinforced concrete structures depends on their ability to resist the main deterioration mechanisms over an acceptable and ‘predictable’ time period. Yu-Chen Ou et al. [10] proposed a seismic assessment method for corroded reinforced concrete bridges based on nonlinear static pushover analysis by seismically evaluating the effects of chloride erosion on 11 reinforced concrete bridges of different structural types in Taiwan Province from both the material level and the structural level, and the results of the study showed that transverse reinforcing bars have a thicker cover layer and that their corrosion begins 2–32 years after the corrosion, which usually occurs before longitudinal reinforcing bars begin to corrode. Martina Šomodíková et al. [11] point out that combining advanced analytical methods with the nonlinear finite element method is an effective tool for assessing existing bridges. The article describes a probabilistic-based method to assess the load-carrying capacity of bridges degraded over time and applies it to a 60-year-old reinforced concrete bridge, demonstrating the importance of studying the degradation patterns of concrete bridges. Alessandro Nettis et al. [12] investigated the vulnerability of simply supported prestressed concrete bridges constructed in Italy and Europe since the 1960s, focusing on the effects of corrosion of reinforcement and traffic loading on the degradation of bridge capacity. The study provides an in-depth discussion of how these factors affect the vulnerability of bridges and provides a valuable reference for transport authorities in assessing the safety of existing bridges. Laura Anania et al. [13] investigated the collapse of post-tensioned prestressed concrete bridges and found that the collapse was caused by the fracture of the steel reinforcement, which was completely corroded by the steel wires inside the reinforcement. Numerical analyses further confirmed the effect of corrosion of reinforcement on concrete bridges. This suggests that future research on concrete bridge durability should focus on bridge degradation induced by reinforcement corrosion. J. R Mackechnie et al. [14] suggested that improving durability can extend the service life of concrete bridges and that the durability performance of bridges can be virtually discerned by understanding the microstructure of concrete and the underlying deterioration mechanisms. According to the above scholars' research, the durability problem of concrete bridges is particularly urgent. Therefore, this paper will provide a comprehensive assessment of the durability of in-service concrete bridges.

Currently, many scholars both domestically and internationally have conducted extensive research on the durability issues of in-service concrete bridges, but these research methods are relatively simple and the research efficiency is low. Huang Ya'e [15] used the extension analytic hierarchy process to calculate the weights of evaluation indices and designed durability assessment software for sea-crossing tied-arch bridges based on extension set theory using Python programming tools. This provides a reference for the durability assessment of in-service concrete bridges. However, its assessment results are greatly influenced by subjective weights. Liu Junli [16] proposed a comprehensive assessment method for concrete bridge durability that combines the analytic hierarchy process with evidence theory. A system for assessing the durability of concrete bridges was established and its feasibility was verified, laying the foundation for subsequent research on concrete bridge durability. Pan Tao [17] based on research into the durability of prestressed concrete bridges, proposed a bridge structure durability assessment method based on a fuzzy neural network. The study analyzed the calculation results of a self-organizing neural network and a BP neural network model written in Matlab, and conducted experimental verification. The results indicated that the fuzzy neural network is well-suited for assessing the durability of bridge structures. Liu Hanbing [18] and colleagues proposed a new method for assessing the condition of the superstructure of reinforced concrete bridges based on particle swarm optimization of fuzzy c-means clustering, the disadvantage of their method is that the study needs to be based on a large number of training samples of existing field test data of old bridges to conduct the study, the advantage is that it combines the advantages of particle algorithms with accelerated convergence, which greatly improves the effectiveness of the clustering. Li Gong [19] and colleagues addressed the prominent issue of concrete durability in cold and arid regions by establishing a concrete durability assessment model based on grey relational theory. They verified that the model's calculation results were consistent with experimental results, effectively solving the problem of insufficient durability of concrete specimens throughout the entire experimental period. Chen et al. [20] adopted the analytic hierarchy process and, based on the principle of fuzzy comprehensive evaluation, used a neural network adaptive fuzzy inference system as the evaluation engine to establish a durability and safety assessment system. Ma Jibing [21] and colleagues established a comprehensive durability assessment model and index system for mid- and low-rise concrete arch bridges based on the analytic hierarchy process. Additionally, they explored the fuzzy closeness evaluation method of this model using fuzzy comprehensive evaluation theory. He Weinan [22] and colleagues emphasized that the durability of bridge suspenders is directly related to the safety of the entire bridge. Based on an investigation into the durability damage of bridge suspender structures, they established a comprehensive evaluation system for the durability of bridge suspenders and defined the evaluation standards. They proposed a durability assessment method for suspenders based on set pair analysis. By calculating the weights of the durability assessment indices using the analytic hierarchy process, they accurately quantified the durability level of bridge suspender structures through set pair analysis operations. Numa J. Bertola et al. [23] proposed a risk-based approach to assessing the condition of bridges using visual inspection data. The methodology combines the state of deterioration of the bridge components with the impact of the damage on the overall structural safety in order to assess the condition of the bridge. However, this approach usually results in an overly conservative assessment of structural damage, which may lead to unnecessary rehabilitation interventions. Liu et al. [7] integrated the topological method, derived from material element theory and correlation functions, into the durability assessment of reinforced concrete structures. They developed a material element model specifically designed to evaluate the durability of these structures. Li Qingfu [24] and colleagues conducted a comprehensive and accurate durability assessment of in-service cable-stayed bridge structures. They established an evaluation index system for the durability of stay cables, using a combined method of IAHP (Improved Analytic Hierarchy Process) and CRITIC (Criteria Importance Through Intercriteria Correlation) to allocate weights to the evaluation indices. Using the confidence criterion, they performed a durability assessment of the stay cables on the Jiahui Bridge as an example. The comparison analysis using SPA (Set Pair Analysis) and MEE (Maximum Entropy Estimation) methods demonstrated that this approach yields more accurate durability assessment results for bridge stay cables. Xu Baosheng [25] and colleagues addressed the issue that most existing evaluation methods rely on single, often biased, weights. They employed a combination of subjective and objective weighting to avoid the limitations of single-method weighting, thus reflecting both the decision-makers subjective intentions and the objective attributes of the data.

Currently, although research on the durability damage mechanism of in-service concrete bridges is relatively mature and progress has been made in detecting and evaluating the durability damage of individual structural components, there are still fewer studies on the durability assessment of the entire concrete bridge. In addition, most of the existing studies suffer from ambiguity and randomness. In this context, it is particularly important to analyze different methods for determining the weights of indicators. The subjective weighting method comprehensively considers various factors, but it relies heavily on the subjective judgment of decision-makers. Consequently, the weight allocation can become unstable or biased due to the decision-maker's personal preferences. Neural networks, while capable of providing good assessments with the help of big data, depend on the learning samples for accuracy. If the sample size is too large or too small, corresponding issues may arise. The fuzzy comprehensive evaluation method can reflect the fuzziness between various evaluation indices but inherently contains a degree of subjectivity. Grey theory is an effective method but has certain limitations. To improve the accuracy of weight quantification, this paper introduces a combination weighting method based on grey relational analysis and the entropy weight method. This approach addresses the mutual influence between multiple indices and the fuzziness and objectivity in determining weights. Given that the bridge durability assessment system is a complex system, this paper employs the extension cloud model to assess the durability of in-service concrete bridges under the condition of combined weighting. This method not only qualitatively and quantitatively analyzes and objectively describes the characteristics of bridge durability but also effectively combines the objective description of extension theory with the dual uncertainty reasoning characteristics of the cloud model, making the durability assessment of in-service bridges more reasonable and reliable. The flow chart of the evaluation method is shown in Fig. 1.

Fig. 1 [Images not available. See PDF.]

Durability assessment flowchart for in-service concrete bridges

Establishment of the durability evaluation index system for in-service concrete bridges

Principles for selecting evaluation indicators

The durability of concrete bridges has always been a significant issue of concern in both the engineering and academic communities. The durability of these structures is the result of the combined effects of multiple influencing factors, and there are often inconsistencies and contradictions among the indicators of these factors. To ensure that the durability of concrete bridges can be accurately assessed, it is necessary to establish a reasonable and comprehensive evaluation index system. By extensively reviewing the literature [26–29] and based on the analysis of the diseases of in-service concrete bridges, as well as relevant documents and standards such as the 'Design Specifications for Durability of Highway Concrete Structures' [30], the 'Technical Condition Evaluation Standards for Highway Bridges' [31], and the 'Highway Bridge and Culvert Maintenance Specifications' [32], the evaluation index system for the durability assessment of in-service concrete bridges has been established.

Determination of the evaluation index system

Based on the principles for selecting evaluation indicators, as well as the main influencing factors of concrete bridge durability and the analysis of bridge diseases, the durability of in-service concrete bridges is set as the target level. The primary indicators include bridge appearance, reinforcement condition, concrete condition, other influencing factors, and environmental factors. Furthermore, there are 13 secondary indicators, including bridge deck pavement, expansion joint condition, and protective layer thickness, among others. The evaluation index system for the durability of in-service concrete bridges is established, as shown in Fig. 2.

Fig. 2 [Images not available. See PDF.]

Durability assessment index system for in-service concrete bridges

Research method

Grey relational analysis method

In real-life and practical engineering applications, systems and problems with incomplete data and information are common. To address these issues, Professor Deng Julong first proposed the Grey System Theory in 1982, introducing the Grey Relational Analysis within it. The core idea of the Grey System Theory is to infer the behavior and structure of a system through partially known information. Grey systems are between white systems (completely determined) and black systems (completely undetermined), dealing with systems where information is incomplete or partially known. Grey Relational Analysis is a method to measure the strength of the relationships among various factors in a system. Calculating the relational degrees between different factors, can determine the relationships and influence levels among variables. This method can not only analyze situations with incomplete data but also provide intuitive and reliable results, thereby offering scientific support for decision-making in real-life and engineering applications [33, 34].

Calculation of grey relational coefficients

The basic idea of grey correlation analysis is to judge the degree of correlation between different sequences by evaluating the geometric shape similarity of the sequence curves. After dimensionless processing of variables, we use grey correlation analysis to determine the relationship between reference sequence (parent sequence) and comparison sequence (subsequence) [35]. Calculate the grey relational coefficient of subsequence relative to the reference sequence on the -th indicator using Matlab programming software.

1

$${\zeta }_{\mathit{ki}}=\frac{\underset{k}{min}\underset{i}{min}|X_{\mathit{ko}}-X_{\mathit{ki}}|+\rho \underset{k}{max}\underset{i}{max}|X_{\mathit{ko}}-X_{\mathit{ki}}|}{|X_{\mathit{ko}}-X_{\mathit{ki}}|+\rho \underset{k}{max}\underset{i}{max}|X_{\mathit{ko}}-X_{\mathit{ki}}|}$$

In the formula, $\underset{k}{min}\underset{i}{min}|X_{\mathit{ko}}-X_{\mathit{ki}}|+\rho \underset{k}{max}\underset{i}{max}|X_{\mathit{ko}}-X_{\mathit{ki}}|$ are the minimum and maximum differences between the two levels, respectively; $\rho$ is the distinguishing coefficient; the larger the distinguishing coefficient, the greater the resolution, and the smaller the distinguishing coefficient, the lower the resolution. Generally, $\rho$ is taken as 0.5.

Calculation of grey relational degree

By averaging the correlation coefficients of each data point, the relational degree of each index can be obtained, that is

2

$$r_k=\frac{1}{n}{\sum }_{i=1}^n{\xi }_{\mathit{ki}}$$where $n$ is the number of data points.

Calculation of weights for grey relational evaluation indicators

By normalizing the relational degrees of each indicator, the weights of the indicators can be obtained. The calculation formula is:

3

$$W_{1k}=r_k\left./\phantom{r_k{\sum }_{j=1}^m{r}_j}\right){\sum }_{j=1}^m{r}_j$$where $m$ is the total number of indicators, and $W_{1k}$ is the weight of the $k$-th indicator.

Entropy weight method

Calculation of evaluation indicator proportions

Due to the presence of logarithms in the calculation formula of the entropy weight method, this paper adopts a non-negative translation method to handle values less than or equal to zero, referencing SPSSAU (Statistical Product and Service Software Automatically) and literature. Based on the standardized data processing results, the proportion of the $i$-th sample value under the $k$-th evaluation indicator is calculated [36], that is

4

$$P_{\mathit{ik}}=\frac{x_i^{\ast }(k)}{{\sum }_{i=1}^{n}x_i^{\ast }(k)}$$where $x_i^{\ast }(k)$ represents the $i$-th sample value under the $k$-th evaluation indicator after standardization.

Calculation of information entropy for each evaluation indicator

5

$$e_k=-\frac{1}{lnn}\left(\sum _{i=1}^n{P}_{\mathit{ik}}ln{P}_{\mathit{ik}}\right)$$

Calculation of entropy weights for each evaluation indicator

6

$$W_{2k}=\frac{1-e_k}{{\sum }_{k=1}^m(1-e_k)}$$

Combination weighting based on grey relational analysis and entropy weight method

Using the Lagrange multiplier method [36] for combined weighting can integrate both the objective information of various indicators and subjective judgments, thereby obtaining a more scientific and reasonable weight distribution. This approach also enhances the model's adaptability and robustness to different types of data and complex decision-making environments.

1. The specific steps to determine the weights of the durability evaluation indicators for in-service concrete bridges based on the Grey Relational-Entropy Weight Method are as follows:

   7

   $$\left\{\begin{array}{c}minF={\sum }_{i=1}^m{\lambda }_i(ln{l}_i-ln{W}_{1k})+{\sum }_{i=1}^m{\lambda }_i(ln{l}_i-ln{W}_{2k})\hfill \\ st.{\sum }_{i=1}^m{\lambda }_{i=1}{\lambda }_i>1,i=1,2,...,m\hfill \end{array}\right)$$

2. The comprehensive weighting using the Lagrange multiplier method is performed as follows:

   8

   $${\lambda }_i=\frac{\sqrt{W_{1k}{W}_{2k}}}{{\sum }_{i=1}^m\sqrt{W_{1k}{W}_{2k}}}$$

Based on the weights obtained from the Grey Relational Analysis and the Entropy Weight Method, the comprehensive weights of the indicators are calculated using formulas (7) and (8).

Extensible cloud model

Extension theory is an emerging discipline founded by scholars such as Cai Wen from Guangdong University of Technology. This theory is primarily based on formal models, exploring new methods for extending things, and is characterized by high logicality and mathematization. Extension theory aims to solve incompatibilities and contradictions in things, providing an innovative approach and regularity for dealing with such issues. It mainly includes three parts: matter-element theory, extension set theory, and extension logic, each contributing to the systematic analysis and resolution of complex problems [37–39]. According to Extension Theory, based on matter-element theory, the matter-element to be evaluated can be represented by an ordered triple $R=(N,C,V)$, where $N$ is the thing, $C$ is the characteristic, and $V$ is the value of characteristic $C$ for $N$ [40]. In traditional matter-element extension, $V$ is a determined value. However, the evaluation of the durability of in-service concrete bridges involves randomness and fuzziness, and its grade boundaries also exhibit randomness and fuzziness. Therefore, the digital characteristics of the cloud model $(Ex,En,He)$ are used to replace the original object characteristic $V$ [41]. In this context, $\mathit{Ex}$ is the expected value, the most representative point within the domain space, essentially the best sample. $\mathit{En}$ is the entropy, which measures the degree of uncertainty of the qualitative concept. It is determined by the fuzziness and randomness of the qualitative concept, reflecting both the dispersion of cloud droplets and the value range recognized by the qualitative concept. $\mathit{He}$ is the hyper-entropy, which measures the uncertainty of entropy, reflecting the thickness of the cloud droplets. Its value can be adjusted based on the fuzzy threshold of the indices.

Extension cloud theory refers to the coupling of the matter-element model and the cloud model, utilizing matter-element theory to analyze the cloud model. It can be represented using a matrix as follows:

9

$$O=\left[\begin{array}{ccc}N& C_1& (E{x}_1,E{n}_1,H{e}_1)\\ \begin{array}{c}\\ \\ \end{array}& \begin{array}{c}{C}_2\\ ⋮\\ \end{array}& \begin{array}{c}(E{x}_2,E{n}_2,H{e}_2)\\ ⋮\\ \end{array}\\ & C_n& (E{x}_n,E{n}_n,H{e}_n)\end{array}\right]$$

In the equation, $O$ is the assessment grade set matrix; $C_i(i=1,2,...,n)$ is the assessment index; $E{x}_{j}E{n}_{j}H{e}_j$ are the cloud parameters for each grade of the assessment index $C_i$.

Determining the grade interval cloud model

All in-service bridge durability assessment indices are divided into 5 grades. Let the grade interval values for each index be $\left[D_{min},D_{max}\right]$, where the expected value $\mathit{Ex}$ is taken as the middle value. Due to the fuzziness and uncertainty of the association degree, $\mathit{En}$ and $\mathit{He}$ are obtained through the relational expression, thus determining the grade interval cloud model.

10

$$\left\{\begin{array}{c}Ex=\frac{D_{max}+D_{min}}{2}\\ En=\frac{D_{max}-D_{min}}{6}\\ He=p\\ \end{array}\right)$$

In the equation, $D_{max}$ and $D_{min}$ are the upper and lower limits of each grade interval, respectively; $p$ is the fuzzy constant, which can be adjusted according to the fuzziness, discreteness, randomness and actual situation of the index. In this paper, 0.01 is adopted.

According to the extension cloud model generation algorithm, the normal cloud model of the indices is simulated using the digital parameters of the cloud model for concrete bridge durability assessment indices.

Determining the correlation degree of the extension cloud model

Each assessment index value $x$ is considered a cloud droplet. A normally distributed random number $E{n}^{\prime }$ with an expected value $\mathit{En}$ and a standard deviation $\mathit{He}$ is generated. The cloud correlation degree between each index value $x$ and the normal cloud model is then calculated.

11

$$k_i=exp\left(-\frac{{(x_i-D)}^2}{{2(E{n}_i^{\prime })}^2}\right)$$

In the equation, $k_i$ is the correlation degree between the index value $D$ and the extension cloud model; $\mathit{Ex}$ is the variance; $E{n}^{\prime }$ is a random number following a normal distribution.

The judgment matrix $G_{\mathit{mn}}$ formed based on the cloud correlation degree $k_i$ is:

12

$$G_{\mathit{mn}}=\left(\begin{array}{cccc}{k}_{11}& k_{12}& \cdots & k_{1n}\\ k_{21}& k_{22}& \cdots & k_{2n}\\ ⋮& ⋮& ⋮& ⋮\\ k_{m1}& k_{m2}& \cdots & k_{\mathit{mn}}\end{array}\right)$$

In the equation, $G_{\mathit{mn}}$ is the correlation degree between the evaluated index and the grade interval extension cloud model; $m$ is the number of evaluation indices; $n$ is the evaluation grade.

Determining the comprehensive evaluation grade

Combining the comprehensive weight values of the concrete bridge durability assessment indices, the comprehensive evaluation vector $B$ is calculated as follows:

13

$$B=\sum _{i=1}^m{V}_{i}G$$

In the equation, $V_i$ is the set of comprehensive weight vectors for each index.

The fuzzy grade characteristic value of the evaluation $r$ is calculated as follows:

14

$$r=b_i{f}_i$$

In the equation, $b_i$ is the maximum value of vector $B$; $f_i$ is the grade corresponding to the maximum component.

Calculating credibility

Due to the fuzziness and randomness of the correlation degree $k$, it is necessary to increase the number of iterations to reduce the impact of random factors. The grade characteristic expectation $E{x}_r$ and the grade characteristic entropy $E{n}_r$ are calculated as follows:

15

$$\left\{\begin{array}{c}E{x}_r=\sum _{i=1}^q{r}_i(x)/q\\ E{n}_r=\sqrt{\frac{1}{q}\sum _{i=1}^q{[r_i(x)-E{x}_r]}^2}\\ \end{array}\right)$$

In the equation, $q$ is the number of iterations, set to 1000; $r_i(x)$ is the characteristic value of index $x$ in the $i$-th iteration.

Considering the credibility $\theta$, the credibility $\theta$ is defined as follows:

16

$$\theta =\frac{E_{\mathit{nr}}}{E_{\mathit{xr}}}$$

The value of $\theta$ reflects the degree of deviation of the evaluation results, and its magnitude is inversely proportional to the credibility. The smaller the value of $\theta$, the higher the credibility of the evaluation results.

Basis for durability assessment of in-service concrete bridges

A reasonable classification of the evaluation grades for the durability of in-service concrete bridges facilitates the assessment of bridge durability. By referring to relevant standards and literature, and considering the degree of damage and defects in the bridges, the durability of the bridges is classified into five grades, as shown in Table 1. The grading criteria for the evaluation indicators of the durability of in-service concrete bridges are determined based on these classifications, as shown in Table 2.

Table 1. Durability evaluation grades of in-service concrete Bridges

Evaluation Grade	Description of Bridge Durability Condition
Grade 1	Basically undamaged, can be used normally, and only requires regular maintenance
Grade 2	Minor damage, no impact on the use of the bridge, minor repairs needed
Grade 3	Moderate damage, still maintain functional use of the bridge, moderate repairs needed
Grade 4	Major damage to key components, significant impact on bridge functionality, major repairs or reinforcement needed
Grade 5	Severe damage to key components, cannot be used normally, endangers bridge safety, reinforcement or demolition and reconstruction needed

Table 2. Classification criteria of durability evaluation indexes of concrete Bridges in service

Evaluation Indicators	Inspection Items	Evaluation Grade				Indicator Type	Indicator Attribute
		1	2	3	4	5
Deck Pavement $C_1$	–	Intact	Local deformation or local cracks, but scattered	Multiple deformations or obvious local cracks, with scattered areas in the cracked regions	Large-area deformation, oil exudation, most parts scattered and exposed	Multiple block cracks, severe scattering in cracked areas, or longitudinal/transverse cracks	Qualitative	Positive
Expansion Joint Condition $C_2$	–	Intact	Local damage, joint pavement layer peeling off, uneven surface, water seepage	Widespread damage, severe edge wear around joints	Severe damage, hard objects stuck	Function failure, unusable	Qualitative	Positive
Protection Layer Thickness $C_3$	Ratio of the measured thickness value of the protective layer to the design value ($D_{\mathit{ne}}/D_{\mathit{nd}}$)	> 0.95	[0.85,0.95]	[0.7,0.85]	[0.55,0.7]	< 0.5	Quantitative	Positive
Rebar Distribution $C_4$	Ratio of the measured spacing value of the rebar to the design value	> 0.95	[0.9,0.95]	[0.85,0.9]	[0.8,0.85]	< 0.8	Quantitative	Positive
Rebar Corrosion $C_5$	Measurement points for rebar corrosion in concrete ($\mathit{mV}$)	≥ − 200	(− 200,− 300]	(− 300,− 400]	(− 400,− 500]	< − 500	Quantitative	Positive
Concrete Strength $C_6$	Estimated strength homogeneity coefficient ($K_{\mathit{bt}}$)	≥ 0.95	[0.9, 0.95)	[0.8, 0.9)	[0.7, 0.8)	< 0.7	Quantitative	Positive
Concrete Carbonation $C_7$	Ratio of the average concrete carbonation depth to the average measured value of the protective layer($K_c$)	< 0.5	[0.5,1)	[1,1.5)	[1.5,2)	≥ 2	Quantitative	Negative
Bearing System $C_8$	–	Slight deterioration	Moderate deterioration, slight aging	Severe deterioration, aging deformation	Bearing severe deterioration, aging cracking	Bearing extremely severe deterioration, aging	Qualitative	Positive
Traffic Volume(Daily Pass)$C_9$	Ratio of the measured traffic value to the design value $Q_m/Q_d$	< 0.1	[1.0,1.3]	[1.3,1.7]	[1.7,2.0]	> 2	Quantitative	Negative
Foundation Settlement $C_{10}$	–	Intact	Subsidence occurring, developing slowly	Slight subsidence occurring, or subsidence tends to stabilize	Excessive settlement, but not meeting design requirements	Severe subsidence, exceeding the specified settlement amount	Qualitative	Positive
Guardrails and Columns Inspection $C_{11}$	–	Intact and clean	Individual components are loose, damaged, or detached	Up to 10% of components are loose, damaged, or detached	More than 10% of components are severely damaged, damaged, or detached	Most components are severely damaged, incomplete, or detached	Qualitative	Positive
Drainage System Condition $C_{12}$	–	Intact	Partial blockage of drainage pipes, water seepage around them	Cracks and leakage in the seam waterproof layer, some drainage pipes are damaged or detached	Localized aging and damage of the waterproof layer, more than half of the leaking pipes are detached or damaged	Waterproof layer failure, cracking, and severe leakage	Qualitative	Positive
Chloride Ion Content $C_{13}$	Free chloride ion content (%)	< 0.15	[0.15,0.4)	[0.4,0.7)	[0.7,1)	> 1	Quantitative	Negative

In Table 2, a positive indicator, i.e., the larger the value of the indicator, the better, and a negative indicator, i.e., the smaller the value of the indicator, the better.

Case study

In this paper, we refer to the literature [42] and analyze the concrete bridge inspection reports, and according to the determination of the evaluation system, we collect the inspection data related to a certain four concrete highway bridges in Wuhan, China. The bridges are summarised in Table 3.

Table 3. General information of the bridge

	Year of completion	Year of testing	Length of bridge (m)	Width of bridge deck (m)	Design loads
Bridge 1	2019	2020	1600	26	Highway Class I
Bridge 2	2000	2023	3586.38	27	Vehicle Load—Super Class 20
Bridge 3	1956	2018	322.37	26	Vehicle Load Class 18
Bridge 4	1995	2015	3971.41	26.5	Vehicle Load—Super Class 20

In Table 3, the design loads, Highway Class I represents the high bearing capacity standard; Vehicle Load—Super Class 20 class specifically refers to allowing more than 100 tons of gross vehicle weight; Vehicle Load Class 18 that the maximum weight of the car can be carried for 18 tons; bridge 2 and bridge 4 are mainly for its main bridge for the durability analysis, the length of the main bridge is 2458 m and 1876.1 m respectively.

A field analysis was conducted on Changfeng Bridge in Wuhan, Hubei Province, to obtain its relevant detection data. This bridge, a large-scale bridge crossing the Han River, is located on the western section of the Third Ring Road in Wuhan and was completed and opened to traffic in 2001. The bridge starts from the Cihui Interchange on the Third Ring Road in the north, crosses the Han River to the south, and lands on the north side of the original bridge toll station. The bridge spans from the Hankou shore abutment No. 0 to the Hanyang shore abutment No. 35, with a total length of 1130 m. The main bridge is a concrete-filled steel tube tied-arch bridge with a total length of 372 m. The overall diagram of the bridge is shown in Fig. 3.

Fig. 3 [Images not available. See PDF.]

Overall picture of Changfeng Bridge

The measured data for the durability assessment indices of the five in-service concrete bridges are shown in Table 4.

Table 4. Measured values of durability evaluation indexes of Bridges to be evaluated

	Bridge 1	Bridge 2	Bridge 3	Bridge 4	Changfeng bridge
$C_1$	92	56	57	52	57.17
$C_2$	91	50	42	50	64.34
$C_3$	0.99	0.82	0.95	0.84	0.72
$C_4$	0.98	0.9	0.94	0.88	0.87
$C_5$	− 38	− 300	− 212	− 326	− 70
$C_6$	1.4	1.2	1.1	0.85	1.24
$C_7$	0	0.42	0.13	0.38	0.05
$C_8$	85	70	82	71	71
$C_9$	1.1	1.25	1.15	1.18	0.1243
$C_{10}$	95	52	95	75.5	82
$C_{11}$	96	87	80	62	92
$C_{12}$	96	50	50	81	80
$C_{13}$	0.08	0.56	0.21	0.45	0.041

In the measured values of the durability evaluation indicators in Table 4, the indicators $C_1$,$C_2$,$C_8$,$C_{10}$,$C_{11}$ and $C_{12}$ involve a quantification process that may be influenced by the subjective judgment of the inspectors. In order to reduce the influence of subjective factors on the results, the final quantified values of these indicators adopt the quantified average values of multiple inspectors.

Determining the weights of assessment indices

1. Using MATLAB programming software and based on Eqs. (1) to (3), the grey relational weights of the indices are calculated, as shown in Table 5.

2. Using the SPSSAU platform and based on Eqs. (4) to (6), the entropy weights of the durability assessment indices for in-service concrete bridges are calculated, as shown in Table 5.

3. The comprehensive weights of grey relational and entropy weights are calculated using the Lagrange multiplier method. Based on Eqs. (7) to (8), the comprehensive weights of each index are obtained. The results are shown in Table 5.

Table 5. Comprehensive weights of indicators

	Grey relational weight $W_{1k}$	Entropy weight $W_{2k}$	Comprehensive weight $W$
$C_1$	0.0747	0.0291	0.0630
$C_2$	0.0739	0.0447	0.0777
$C_3$	0.0840	0.0037	0.0237
$C_4$	0.0840	0.0010	0.0123
$C_5$	0.0366	0.1983	0.1152
$C_6$	0.0839	0.0178	0.0523
$C_7$	0.0839	0.3927	0.2454
$C_8$	0.0777	0.0042	0.0245
$C_9$	0.0836	0.0012	0.0138
$C_{10}$	0.0785	0.0304	0.0660
$C_{11}$	0.0792	0.0140	0.0450
$C_{12}$	0.0762	0.0436	0.0779
$C_{13}$	0.0839	0.2194	0.1834

According to Table 5, the $C_7$ concrete carbonation indicator has the largest proportion of the combined weight, which has a particularly significant impact on the durability of in-service reinforced concrete bridges. This is closely followed by the $C_5$ reinforcement corrosion and $C_{13}$ chloride content indicators, which also have relatively large combined weight values. The results of this weight allocation are in line with practical theoretical knowledge, as in-service concrete bridges are usually exposed to atmospheric environments and are susceptible to carbonation due to direct contact with carbon dioxide in the air. In addition, bridges are also susceptible to chloride ions in such environments, leading to corrosion of the reinforcement and thus affecting the durability of the bridge. Therefore, the results of the calculation of the combined weights can be considered scientific and reliable.

Based on Table 5, draw the index weight analysis chart, as shown in Fig. 4.

Fig. 4 [Images not available. See PDF.]

Weight distribution of indicators

As shown in Fig. 4, there are significant differences between the gray relational weights and the entropy weights for some indices, and certain indices exhibit inconsistent trends. The gray relational analysis involves a degree of subjectivity in its considerations, whereas the entropy weight method determines weights based on the degree of data dispersion, reflecting objectivity. Combining both methods balances subjective and objective factors, resulting in more reasonable final weights. Overall, the comprehensive weights can more accurately reflect the true importance of each evaluation index, avoiding potential biases inherent in a single weighting method and enhancing the rationality and accuracy of the evaluation results.

In-service concrete extensible cloud model

Establish the durability evaluation model of in-service concrete bridge, and grade it according to the durability evaluation index of in-service concrete bridge. According to Eq. (10), calculate the normal cloud model eigenvalues (expectation, entropy, super entropy) of each evaluation index, see Table 6. According to the cloud digital eigenvalues can get the cloud drop diagram, this paper takes the index 1 bridge deck pavement as an example, as shown in Fig. 5.

Table 6. Characteristic values of the cloud model for assessment indices

	Grade I	Grade II	Grade III	Grade IV	Grade V
$C_1$	(95,2.96,0.01)	(79.5,5.62,0.01)	(59.5,5.62,0.01)	(39.5,5.62,0.01)	(15,8.28,0.01)
$C_2$	(90,5.91,0.01)	(69.5,5.62,0.01)	(49.5,5.62,0.01)	(29.5,5.62,0.01)	(9.5,5.62,0.01)
$C_3$	(0.975,0.01,0.01)	(0.9,0.03,0.01)	(0.775,0.04,0.01)	(0.625,0.04,0.01)	(0.25,0.15,0.01)
$C_4$	(0.975,0.01,0.01)	(0.925,0.01,0.01)	(0.875,0.01,0.01)	(0.825,0.01,0.01)	(0.4,0.24,0.01)
$C_5$	(− 100,59.13,0.01)	(− 250,29.57,0.01)	(− 350,29.75,0.01)	(− 450,29.75,0.01)	(− 550,29.57,0.01)
$C_6$	(1.225,0.16,0.01)	(0.925,0.01,0.01)	(0.85,0.03,0.01)	(0.75,0.03,0.01)	(0.35,0.21,0.01)
$C_7$	(0.25,0.15,0.01)	(0.75,0.15,0.01)	(1.25,0.15,0.01)	(1.75,0.15,0.01)	(2.25,0.15,0.01)
$C_8$	(50,29.57,0.01)	(69.5,5.62,0.01)	(49.5,5.62,0.01)	(29.5,5.62,0.01)	(9.5,5.62,0.01)
$C_9$	(0.05,0.03,0.01)	(1.15,0.03,0.01)	(1.5,0.12,0.01)	(1.85,0.09,0.01)	(2.15,0.09,0.01)
$C_{10}$	(94.5,3.25,0.01)	(79.5,5.62,0.01)	(59.5,5.62,0.01)	(39.5,5.62,0.01)	(14.5,8.57,0.01)
$C_{11}$	(97.5,1.48,0.01)	(87,4.14,0.01)	(69.5,5.62,0.01)	(44.5,8.57,0.01)	(14.5,8.57,0.01)
$C_{12}$	(97.5,1.48,0.01)	(82,7.1,0.01)	(59,5.91,0.01)	(24.5,8.57,0.01)	(4.5,2.66,0.01)
$C_{13}$	(0.075,0.04,0.01)	(0.275,0.07,0.01)	(0.55,0.09,0.01)	(0.85,0.09,0.01)	(1.15,0.09,0.01)

Fig. 5 [Images not available. See PDF.]

Cloud Droplet Diagram for Index $C_1$

Table 6 lists the eigenvalues of the normal cloud model for each index of durability evaluation of in-service concrete bridges. Meanwhile, Fig. 5 visualizes the cloud drop diagram of $C_1$ indicator as an example. By analyzing this figure, the grade distribution of $C_1$ indicators and the related correlation characteristics can be clearly understood.

Calculation of index correlation degree

Based on the characteristic values (expected value, entropy, hyper-entropy) obtained from the cloud model, MATLAB is used to perform the calculations according to Eqs. (11) and (12). The cloud correlation degrees for the durability assessment grades of the five in-service concrete bridges are obtained. Taking the correlation degree data of the example bridge as an example, the results are shown in Table 7.

Table 7. Cloud correlation degrees for durability assessment indices of each in-service concrete bridge

	Grade I	Grade II	Grade III	Grade IV	Grade V	Evaluation results
$C_1$	0.0000	0.0909	0.6774	0.8039	0.8592	V
$C_2$	0.6125	0.4833	0.2250	0.4211	0.2791	I
$C_3$	0.0000	0.0000	0.6667	0.4333	0.7000	V
$C_4$	0.3810	0.1875	0.4000	0.1333	0.3500	III
$C_5$	0.2500	0.0000	0.0000	0.3500	0.1570	IV
$C_6$	0.4000	0.3667	0.2875	0.1857	0.1857	I
$C_7$	0.1000	0.9000	0.9500	0.3667	0.2750	III
$C_8$	0.2900	0.4211	0.3927	0.5246	0.6420	V
$C_9$	0.1635	0.8757	0.9044	0.9269	0.9379	V
$C_{10}$	0.3778	0.1053	0.0968	0.4510	0.6056	V
$C_{11}$	0.4242	0.0714	0.2952	0.5366	0.7324	V
$C_{12}$	0.4286	0.4167	0.3548	0.6721	0.7802	V
$C_{13}$	0.2733	0.7267	0.8975	0.7414	0.6759	III

From Table 7, the cloud correlation of the evaluation ratings of each evaluation index of the durability of in-service concrete bridges can be seen, and based on its results, the comprehensive correlation can be further obtained, as shown in Table 8.

Table 8. Durability assessment grades of various in-service concrete bridges

Bridge	Grade I	Grade II	Grade III	Grade IV	Grade V	Credibility	Evaluation level	Actual situation
Bridge 1	0.5214	0.2451	0.2122	0.3615	0.1544	0.0015	I	I
Bridge 2	0.1352	0.2841	0.3457	0.0421	0.1441	0.0019	III	III
Bridge 3	0.1004	0.4873	0.2347	0.1452	0.1154	0.0025	II	II
Bridge 4	0.1244	0.2251	0.3214	0.0612	0.0023	0.0087	III	III
Changfeng Bridge	0.2634	0.4838	0.5629	0.5030	0.4782	0.0045	III	III

Determining the evaluation grade

Based on Eqs. (13) to (16), the comprehensive correlation degree and credibility are obtained, and the evaluation grade is determined. The results are shown in Table 8.

As can be seen from Table 8, the evaluation class of Bridge 1 is Class I, the evaluation class of Bridge 3 is Class II, and the evaluation classes of Bridge 2 and Bridge 4 as well as Changfeng Bridge are all Class III, and the credibility of the calculations is less than 0.01, which indicates that the results are highly credible. The results indicate that relying on the Matlab platform to establish a model to evaluate the durability of each in-service concrete bridge, the evaluation results coincide with the measured data, which verifies the accuracy of the model. According to the classification of durability assessment class of in-service concrete bridges in Table 1, we can determine that the assessment class of Bridge 1 is Class I, which indicates that it is basically free of damage and can be used normally, and it only needs to maintain routine maintenance. The assessment class of Bridge 2, Bridge 4, and Changfeng Bridge is Class III, indicating that there is a moderate degree of damage, and although they can still maintain the normal use of the function, they need to carry out major repairs to the damaged parts. Bridge 3 was assessed as Category II, indicating minor damage, which has no significant impact on the normal use of the bridge and requires only minor localized repairs.

Discussion

With the increasing phenomenon of bridge deterioration, the number of bridges in urgent need of rehabilitation and strengthening has been increasing, which indicates that it has become imperative to carry out durability studies on in-service concrete bridges. Against this background, this study proposes an efficient method for the comprehensive assessment of bridge durability, which has significant theoretical and practical value.

In this study, a combined assignment method based on grey correlation analysis and entropy weight method is proposed on the basis of the construction of an evaluation index system. The results in Table 6 and Fig. 5 clearly show that this method not only overcomes the limitations of the single assignment method but also provides a more scientific and reasonable weight determination method for the field of bridge durability assessment. The introduction of this method not only enriches and improves the theoretical system of bridge durability assessment, but also provides an important reference for future related research and promotes the further improvement of the field of bridge durability assessment.

By carrying out the modeling evaluation on the MATLAB platform, this study significantly improves the evaluation efficiency. The combination of topological theory and cloud modeling is used to construct the model, which not only considers the interrelationships among the evaluation indicators of the durability of in-service concrete bridges but also pays full attention to the relative independence of each indicator. Compared with the traditional evaluation methods, this innovative combined model performs well in dealing with the complex relationships among evaluation indicators and coping with ambiguity and uncertainty, making the evaluation results more scientific and accurate.

As can be seen from the analysis results in Table 8, the durability assessment results of in-service concrete bridges in this study are highly consistent with the measured results, which verifies the effectiveness and practicality of the proposed method. The method can be effectively applied in the durability assessment of actual bridges, providing a scientific decision-making basis for the bridge management department. By accurately evaluating the durability status of bridges, the relevant authorities can rationally arrange maintenance and reinforcement programs, which can effectively extend the service life of bridges and reduce the maintenance costs. This not only optimizes the bridge management strategy but also provides a strong guarantee for the sustainability of the infrastructure.

Conclusions

The durability assessment of in-service concrete bridges is an important component of bridge construction. With the increasing number of deficient bridges each year, the maintenance and renovation of bridges have become urgent. Therefore, proposing a scientific and effective method for evaluating the durability of in-service concrete bridges is of great significance. Based on extensive literature review, this paper adopts the extension model to assess the durability of in-service concrete bridges, with the main conclusions as follows:

1. Based on the analysis of durability defects in in-service concrete bridges, a two-tier evaluation index system for the durability of in-service concrete bridges was established. Thirteen indicators were selected to assess the durability of concrete bridges: deck pavement, expansion joint conditions, protective layer thickness, rebar distribution, rebar corrosion, concrete strength, concrete carbonation, bearing conditions, traffic volume, foundation settlement, guardrail and railing inspection, drainage system conditions, and chloride ion content.

2. In the extension cloud evaluation system, the weight of each index directly affects the effectiveness of bridge durability assessment. Using a single weight to evaluate bridge durability reduces the referential value of the assessment results. Therefore, this study employs the Lagrange multiplier method to combine the weights of grey theory and entropy weight, enhancing the scientific nature of index weighting. This approach also improves the stability and robustness of the extension model, making the evaluation results more reliable and referential.

3. This study employs an extension cloud model with combined weighting to assess the durability of in-service concrete bridges. The evaluation results are compared with actual bridge data, showing high consistency, which ensures the accuracy of the assessment results. The extension cloud model effectively integrates extension theory and cloud model theory, combining the advantages of both. It excels in handling fuzziness and uncertainty and adequately reflects the complexity and dynamism of the evaluated objects.

In this paper, the durability assessment system and assessment methods of in-service concrete bridges have been studied to a certain extent, but there are still some issues that need to be further improved due to the limitations of data collection and the authors' knowledge level.

1. The durability assessment system established in this paper mainly relies on literature collection and standardised theoretical analysis. Future research can combine methods such as finite element simulation to determine the influencing factors of bridge durability in more depth, so as to define and select the assessment indexes more precisely.

2. In the assessment of bridge durability, there is a certain degree of subjectivity in the quantification process of the indicator values, which may affect the accuracy of the assessment results. Subsequent studies should further refine the quantification process of these indicators and explore more objective and standardised quantification methods to improve the accuracy and reliability of the assessment results.

3. Different types of bridges (e.g., steel, prestressed concrete, and reinforced concrete bridges) have significant differences in durability assessment. Due to the limitations of the sample, this study only analysed the durability of concrete bridges in the same region. Future research should establish a more comprehensive assessment system for different types of bridges in order to improve the applicability and effectiveness of the model and to provide a broader reference basis for the durability assessment of various bridge structures.

Acknowledgements

The authors gratefully acknowledge many important contributions from the researchers of all reports cited in our paper.

Author contributions

JC and XL, who designed the study and analyzed the entire evaluation process, are the lead contributing authors of the study. ZW guided the use of MATLAB software, edited the code and analyzed it. XL wrote the manuscript of the paper, which has been examined by all the authors.

Funding

No external funding was used.

Data availability

Data is provided within the manuscript or supplementary information files.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

All authors agreed to publish this manuscript.

Competing interests

The authors declare that they have no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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