# Reliability Assessment of Highway Bridges Based on Combined Empowerment–TOPSIS Method

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Selection of Evaluation Indicators

_{1},X

_{2},X

_{3},X

_{4},X

_{5}} = {grade 1,grade 2,grade 3,grade 4,grade 5} Grade 1 is a beam bridge with intact function; Grade 2 is a beam bridge with minor defects; Grade 3 is a beam bridge with moderate defects; Grade 4 is a beam bridge with major defects in the main components for which normal use cannot be guaranteed; Grade 5 is a beam bridge with serious defects in the main components, with the bridge in a dangerous condition.

## 3. Rating Method

#### 3.1. Hierarchical Analysis

- (1)
- Construction of judgment matrix

- (2)
- Calculation of evaluation index weights

- (3)
- Consistency test

_{max}is the maximum characteristic root of judgment matrix A.

#### 3.2. Entropy

- (1)
- Data standardization

- (2)
- Calculate the weight of the ith evaluation object under the jth indicator

- (3)
- Calculate the entropy of the jth index

- (4)
- Calculate the coefficient of variation of the information entropy of the jth index

- (5)
- Determine the weights of each evaluation index

#### 3.3. Calculation of Portfolio Assignment Weights

#### 3.4. Reasonableness Analysis of Portfolio Empowerment

- (1)
- Summary of weights

- (2)
- Weight ordering

- (3)
- Spearman Consistency Coefficient

#### 3.5. TOPSIS Evaluation Model

- (1)
- Initial decision matrix

- (2)
- Standardized decision matrix

- (3)
- Weighted standardized decision matrix

- (4)
- Determine the positive ideal solution and negative ideal solution

- (5)
- Calculate Euclidean distance

#### 3.6. The Advantages and Disadvantages of the Combined Weighting–TOPSIS Method

- (1)
- This paper adopts AHP and entropy weight to give subjective and objective weights to bridge reliability evaluation indicators, which fully avoids the one-sidedness of single-method weighting so that the obtained comprehensive weight does not only reflect the subjective intention of decision makers but also reflects the objective properties of the data.
- (2)
- TOPSIS is a commonly used comprehensive evaluation method that makes full use of the original data to accurately reflect the relative closeness of each evaluation scheme to the optimal scheme and the worst-case scheme as the basis for evaluating the pros and cons, and this method has no strict restrictions on data distribution and sample size.

- (1)
- When AHP is used for subjective weighting, expert scoring is used to construct the judgment matrix. There are too many qualitative components, which are unconvincing. When there are many evaluation indicators, statistical data is too complicated, weight is difficult to determine, and the consistency test is more complicated.
- (2)
- When using TOPSIS for comprehensive evaluation, the general weighted standard decision-making matrix is more complex, and it is not easy to solve the positive ideal solution and the negative ideal solution, making the calculation is difficult.

#### 3.7. Related Work

## 4. Project Example Analysis

#### 4.1. Project Overview

#### 4.2. Hypothesis and Limitations

- (1)
- The 10 reliability evaluation indicators selected in this paper are independent of each other, there is no significant correlation, and each indicator can be quantified by a certain measurement method.
- (2)
- This paper uses analytic hierarchy to carry out subjective empowerment and adopts expert scoring to construct the judgment matrix. Experts must be familiar with bridge reliability and be able to score objectively and accurately.
- (3)
- In this paper, entropy weight is used for objective weighting, which requires the measured data of each indicator to be known, and for different indicators, different numerical standardization formulas need to be used for dimensionless processing.
- (4)
- When using TOPSIS for reliability evaluation, there must be two or more research objects.

#### 4.3. Bridge Superstructure Reliability Evaluation

- (1)
- Calculation of subjective weights based on hierarchical analysis

- (2)
- Calculation of objective weights based on entropy weight

- (3)
- Portfolio empowerment and rationalization analysis

- (4)
- Combination of weighting and TOPSIS

_{0}is established with the bridge reliability index rating criteria, and the initial decision matrix is obtained according to Table 5 and Equation (19).

- (5)
- Analysis of evaluation results

#### 4.4. Comparative Study

- (1)
- The object element classical domain is denoted as:

- (2)
- Determination of the elements to be evaluated

- (3)
- Calculation of the correlation of evaluation indicators

- (4)
- Determine the reliability evaluation grade of highway bridges

## 5. Conclusions

- (1)
- This article refers to the “Technical Condition Evaluation Standards of Highway Bridges” (JTG/T H21-2011) and “Regulations for the Evaluation of Bearing Capacity of Highway Bridges” (JTG/T J21-2011) to select the corrosion potential level of steel bars, resistivity, chloride ion content, estimated strength homogeneity coefficient of concrete, average value of concrete carbonation depth/average measured protective layer thickness, characteristic value of steel protective layer thickness/design value, main beam crack width, main beam deformation, staggered height and bridge deck crack width, etc. A total of 10 indicators are used to evaluate the reliability of the bridge superstructure.
- (2)
- There are uncertainties in the indicators that affect the reliability of highways and bridges, and it is difficult to quantitatively describe them. This paper uses AHP and entropy weight to carry out subjective weighting and objective weighting, respectively, for each evaluation index, and then uses minimum discriminant information to combine subjective and objective weighting to obtain a combined weighting, which fully avoids single-method weighting. The one-sidedness of the data makes the obtained comprehensive weights not only reflect the subjective intentions of decision makers, but also reflect the objective attributes of the data.
- (3)
- In this paper, combined weighting and TOPSIS are used for highway bridge reliability evaluation, and a complete highway bridge reliability evaluation model is established. In order to determine the reliability level of the upper structure of the three spans of a reinforced concrete bridge, the relative closeness ${G}_{i}$ of each span is calculated using TOPSIS, and the order is: 0.556, 0.562, 0.573. According to the size of the obtained relative closeness ${G}_{i}$, the reliability level evaluation results of the superstructure of the three spans of the bridge are: Grade 2, Grade 2 and Grade 2. It can also be obtained that the reliability of the three spans of the bridge is ranked as follows: ${S}_{3}>{S}_{2}>{S}_{1}$. At the same time, a comparative study was done with the AHP–Extenics method. The evaluation results are consistent with the results obtained by the method used in this paper and with the actual bridge working conditions. The comparison results show that the combined weighting–TOPSIS method is reliable for highway bridges while being more reasonable and effective.
- (4)
- This paper evaluates the reliability of highway bridges. Since there are many factors that affect the reliability of bridges, this paper only selects 10 of the more common and impactful evaluation indicators. There are deficiencies in many aspects, such as the importance. Therefore, more in-depth exploration and research are needed for the reliability assessment of highways and bridges in order to move forward in the direction of sustainable development.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Indicators | Evaluation Grade | ||||
---|---|---|---|---|---|

${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | ${\mathit{X}}_{3}$ | ${\mathit{X}}_{4}$ | ${\mathit{X}}_{5}$ | |

Steel corrosion potential level ${u}_{1}/\mathrm{mV}$ | ≥−200 | −300~−200 | −400~−300 | −500~−400 | <−500 |

Resistivity ${u}_{2}/\left(\mathsf{\Omega}\xb7\mathrm{cm}\right)$ | ≥20,000 | 15,000~20,000 | 10,000~15,000 | 5000~10,000 | <5000 |

Chloride ion content (% of cement content) ${u}_{3}/\%$ | <0.15 | 0.15~0.4 | 0.4~0.7 | 0.7~1.0 | ≥1.0 |

Homogeneity factor of presumed strength of concrete ${u}_{4}$ | ≥0.95 | 0.9~0.95 | 0.8~0.9 | 0.7~0.8 | <0.7 |

Average value of concrete carbonation depth/average value of measured protective layer thickness ${u}_{5}/\mathrm{mm}$ | <0.5 | 0.5~1.0 | 1.0~1.5 | 1.5~2.0 | ≥2.0 |

Reinforcing steel protective layer thickness characteristic value/design value ${u}_{6}/\mathrm{mm}$ | >0.95 | 0.85~0.95 | 0.7~0.85 | 0.55~0.7 | ≤0.55 |

Crack width of main beam ${u}_{7}$/mm | <0.05 | 0.05~0.10 | 0.10~0.15 | 0.15~0.20 | >0.20 |

Main beam deformation (maximum deflection in span/1/1000 of calculated span diameter) ${u}_{8}$ | <0.83 | 0.83~1 | 1~1.67 | 1.67~2 | ≥2 |

Wrong platform height ${u}_{9}$/cm | <1 | 1~2 | 2~5 | 5~8 | ≥8 |

Width of bridge deck cracks ${u}_{10}$/mm | <1 | 1~3 | 3~5 | 5~7 | ≥7 |

Numerical Value | Importance |
---|---|

1 | Equal importance |

3 | Moderate importance of one over another |

5 | Essential or strong importance |

7 | Very strong importance |

9 | Extreme importance |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

R.I. | 0 | 0 | 0.52 | 0.89 | 1.12 | 1.26 | 1.36 | 1.41 | 1.46 | 1.49 |

Evaluation Method | Indicator 1 | Indicator 2 | ⋯ | Indicator n |
---|---|---|---|---|

Hierarchical analysis | ${\beta}_{1}$ | ${\beta}_{2}$ | ⋯ | ${\beta}_{n}$ |

Entropy | ${\gamma}_{1}$ | ${\gamma}_{2}$ | ⋯ | ${\gamma}_{n}$ |

Evaluation Method | Indicator 1 | Indicator 2 | ⋯ | Indicator n |
---|---|---|---|---|

Hierarchical analysis | ${\beta}_{1}^{\prime}$ | ${\beta}_{2}^{\prime}$ | ⋯ | ${\beta}_{n}^{\prime}$ |

Entropy | ${\gamma}_{1}^{\prime}$ | ${\gamma}_{2}^{\prime}$ | ⋯ | ${\gamma}_{n}^{\prime}$ |

Author | Related Work |
---|---|

Y. N. Zhang | Y. N. Zhang established a matter–element model for reliability evaluation of the superstructure of a concrete double-curvature arch bridge, introduced a correlation function and correlation degree in the extension set, and calculated the degree of correlation to the actual reliability. |

H. B. Liu | H. B. Liu evaluated the system reliability of fabricated concrete hollow slab bridges based on an improved AHP method and compared it with the traditional method to verify that this method is more suitable for evaluating the system reliability of such bridges. |

J. Leander | J. Leander used the steel bridge fatigue assessment model in the Eurocode for reliability assessment. The study showed that the assessment model needs to be improved and pointed out the parameters that the model should focus on. |

X. Y. Bian | X. Y. Bian proposed a response-surface bound method based on non-probabilistic reliability theory to solve a non-probabilistic reliability index and evaluate the reliability of bridges in service. |

L. Wang | L. Wang proposed a reliability assessment method for existing reinforced concrete (RC) bridges with small sample datasets using the influence parameters of structural resistance as fuzzy variables. |

R. F. Nie | R. F. Nie proposed a temporary structure state assessment method based on system reliability analysis that can better solve the problem of resistance and load uncertainty and can ensure consistent structural reliability. |

B. S. Xu | B. S. Xu proposed a bridge reliability assessment method based on a combined weighting–TOPSIS method. This method can avoid the one-sidedness of the weighting of a single method, and, when using TOPSIS for evaluation, it can make full use of the information of the original data and accurately evaluate bridge reliability grades. |

Span Number | Evaluation Indicators | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

u_{1}/mV | u_{2}/Ω·cm | u_{3}/% | u_{4} | u_{5}/mm | u_{6}/mm | u_{7}/mm | u_{8} | u_{9}/cm | u_{10}/mm | |

${S}_{1}$ | −245 | 18,273 | 0.16 | 0.92 | 0.40 | 0.80 | 0.08 | 0.90 | 1.6 | 2.0 |

${S}_{2}$ | −281 | 16,483 | 0.12 | 0.85 | 0.56 | 0.88 | 0.05 | 0.88 | 1.4 | 1.6 |

${S}_{3}$ | −253 | 16,769 | 0.17 | 0.84 | 0.65 | 0.82 | 0.03 | 0.84 | 1.1 | 1.1 |

Method | u_{1} | u_{2} | u_{3} | u_{4} | u_{5} | u_{6} | u_{7} | u_{8} | u_{9} | u_{10} |
---|---|---|---|---|---|---|---|---|---|---|

Hierarchical analysis | 10 | 2 | 3 | 1 | 7 | 5 | 6 | 4 | 8 | 9 |

Entropy | 10 | 2 | 3 | 1 | 6 | 4 | 9 | 5 | 7 | 8 |

Durability Grade | ${\mathit{S}}_{\mathit{i}}{}^{+}$ | ${\mathit{S}}_{\mathit{i}}{}^{-}$ | ${\mathit{G}}_{\mathit{i}}$ |
---|---|---|---|

${X}_{1}$ (Grade 1) | 0 | 0.318 | 1 |

${X}_{2}$ (Grade 2) | 0.073 | 0.251 | 0.775 |

${X}_{3}$ (Grade 3) | 0.144 | 0.177 | 0.552 |

${X}_{4}$ (Grade 4) | 0.233 | 0.087 | 0.271 |

${X}_{5}$ (Grade 5) | 0.318 | 0 | 0 |

Indicators | |
---|---|

${X}_{1}$ (Grade 1) | $0.775<{G}_{i}\le 1$ |

${X}_{2}$ (Grade 2) | $0.552<{G}_{i}\le 0.775$ |

${X}_{3}$ (Grade 3) | $0.271<{G}_{i}\le 0.552$ |

${X}_{4}$ (Grade 4) | $0<{G}_{i}\le 0.271$ |

${X}_{5}$ (Grade 5) | 0 |

Span Number | ${\mathit{S}}_{\mathit{i}}{}^{+}$ | ${\mathit{S}}_{\mathit{i}}{}^{-}$ | ${\mathit{G}}_{\mathit{i}}$ | Evaluation Results |
---|---|---|---|---|

${S}_{1}$ | 0.231 | 0.290 | 0.556 | Grade 2 |

${S}_{2}$ | 0.215 | 0.276 | 0.562 | Grade 2 |

${S}_{3}$ | 0.225 | 0.302 | 0.573 | Grade 2 |

Indicators | Classic Domain | Nodal Domain | ||||
---|---|---|---|---|---|---|

${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | ${\mathit{X}}_{3}$ | ${\mathit{X}}_{4}$ | ${\mathit{X}}_{5}$ | ||

${u}_{1}$ | $\left[-200,-100\right)$ | $\left[-300,-200\right)$ | $\left[-400,-300\right)$ | $\left[-500,-400\right)$ | $\left[-600,-500\right)$ | $\left[-600,-100\right)$ |

${u}_{2}$ | $\left[20,000,25,000\right)$ | $\left[15,000,20,000\right)$ | $\left[10,000,15,000\right)$ | $\left[5000,10,000\right)$ | $\left[0,5000\right)$ | $\left[0,25,000\right)$ |

${u}_{3}$ | $\left[0,0.15\right)$ | $\left[0.15,0.4\right)$ | $\left[0.4,0.7\right)$ | $\left[0.7,1.0\right)$ | $\left[1.0,1.3\right)$ | $\left[0,1.3\right)$ |

${u}_{4}$ | $\left[0.95,1\right)$ | $\left[0.9,0.95\right)$ | $\left[0.8,0.9\right)$ | $\left[0.7,0.8\right)$ | $\left[0.6,0.7\right)$ | $\left[0.6,1\right)$ |

${u}_{5}$ | $\left[0,0.5\right)$ | $\left[0.5,1\right)$ | $\left[1,1.5\right)$ | $\left[1.5,2\right)$ | $\left[2,2.5\right)$ | $\left[0,2.5\right)$ |

${u}_{6}$ | $\left[0.95,1.05\right)$ | $\left[0.85,0.95\right)$ | $\left[0.7,0.85\right)$ | $\left[0.55,0.7\right)$ | $\left[0.4,0.55\right)$ | $\left[0.4,1.05\right)$ |

${u}_{7}$ | $\left[0,0.05\right)$ | $\left[0.05,0.10\right)$ | $\left[0.10,0.15\right)$ | $\left[0.15,0.20\right)$ | $\left[0.20,0.25\right)$ | $\left[0,0.25\right)$ |

${u}_{8}$ | $\left[0,0.83\right)$ | $\left[0.83,1\right)$ | $\left[1,1.67\right)$ | $\left[1.67,2\right)$ | $\left[2,2.5\right)$ | $\left[0,2.5\right)$ |

${u}_{9}$ | $\left[0,1\right)$ | $\left[1,2\right)$ | $\left[2,5\right)$ | $\left[5,8\right)$ | $\left[8,10\right)$ | $\left[0,10\right)$ |

${u}_{10}$ | $\left[0,1\right)$ | $\left[1,3\right)$ | $\left[3,5\right)$ | $\left[5,7\right)$ | $\left[7,9\right)$ | $\left[0,9\right)$ |

Evaluation Indicators | $\mathbf{Span}\text{}{\mathit{S}}_{1}$ | $\mathbf{Span}\text{}{\mathit{S}}_{2}$ | $\mathbf{Span}\text{}{\mathit{S}}_{3}$ |
---|---|---|---|

${u}_{1}$ | −245 | −281 | −253 |

${u}_{2}$ | 18,273 | 16,483 | 16,769 |

${u}_{3}$ | 0.16 | 0.12 | 0.17 |

${u}_{4}$ | 0.92 | 0.85 | 0.84 |

${u}_{5}$ | 0.4 | 0.56 | 0.65 |

${u}_{6}$ | 0.8 | 0.88 | 0.82 |

${u}_{7}$ | 0.08 | 0.05 | 0.03 |

${u}_{8}$ | 0.9 | 0.88 | 0.84 |

${u}_{9}$ | 1.6 | 1.4 | 1.1 |

${u}_{10}$ | 2 | 1.6 | 1.1 |

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## Share and Cite

**MDPI and ACS Style**

Xu, B.; Qi, N.; Zhou, J.; Li, Q.
Reliability Assessment of Highway Bridges Based on Combined Empowerment–TOPSIS Method. *Sustainability* **2022**, *14*, 7793.
https://doi.org/10.3390/su14137793

**AMA Style**

Xu B, Qi N, Zhou J, Li Q.
Reliability Assessment of Highway Bridges Based on Combined Empowerment–TOPSIS Method. *Sustainability*. 2022; 14(13):7793.
https://doi.org/10.3390/su14137793

**Chicago/Turabian Style**

Xu, Baosheng, Ningning Qi, Jianpeng Zhou, and Qingfu Li.
2022. "Reliability Assessment of Highway Bridges Based on Combined Empowerment–TOPSIS Method" *Sustainability* 14, no. 13: 7793.
https://doi.org/10.3390/su14137793