# Evaluation of the Durability of Bridge Tension Cables Based on Combination Weighting Method-Unascertained Measure Theory

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Introduction of Research on Durability Damage Mechanism of Pulling Slings

#### 1.2. Introduction of Existing Durability Evaluation Methods

#### 1.3. Introduction to the Proposed Method in this Study

## 2. Establishment of the Durability Evaluation Index System for Pulling Sling Structures

#### 2.1. Causes of Damage to the Durability of Pulling Sling Structures

#### 2.2. Pulling Sling Durability Evaluation Index System

#### 2.3. Determine the Durability Evaluation Level of the Pulling Sling Structure

## 3. Evaluation Methodology and Process of the CWM-UM Method

#### 3.1. Objective Weight Method

#### 3.1.1. The Theory of MEE Method

- (1)
- We determine the classical domain, the section domain, and the matrix of elements [52] to be evaluated.

- (2)
- We calculate the correlation.
- a
- We determine the correlation function for each level of the system to be evaluated.

#### 3.1.2. CRITIC Method

#### 3.1.3. Calculation Steps

- (1)
- We determined the durability evaluation indexes of the system to be evaluated and the grading criteria of each index.
- (2)
- We determined the classical domain, the section domain, and the object element matrix of the system to be evaluated.
- (3)
- We calculated the correlation matrix of the system to be evaluated and normalized the correlation matrix.
- (4)
- We substituted the judgment matrix obtained from Step (3) into Equations (9)–(13) to obtain the weight value of each durability evaluation index.

#### 3.2. IAHP Method

#### 3.3. The Rationality Test

#### 3.4. The Combination Weighting Method

#### 3.5. UM Theory

#### 3.5.1. Single-Index Measure

#### 3.5.2. Multi-Index Measure

#### 3.6. Durability Evaluation

## 4. Example Application

#### 4.1. Determine the Weight Value of the Durability Index of the Pulling Sling Structure

- (1)
- We determine the classical domain, the section domain, and the matrix of elements to be evaluated.

- (2)
- We calculate the correlation.

- (3)
- We then calculate weights using the CRITIC method.

#### 4.2. Calculation Index Weights by the IAHP

#### 4.3. Rationality Test

- (4)
- We use the combination weighting method.

#### 4.4. Determine the Single Indicator Measurement Function and Matrix

#### 4.5. Determine the Multi-Index Comprehensive Measure Evaluation Vector

#### 4.6. Durability Evaluation

## 5. Comparison between Existing and Proposed Method

#### 5.1. SPA Method

- (1)
- We determine the evaluation index and evaluation level.

- (2)
- We determine the number of t-element links for comprehensive evaluation of each index.

- (3)
- We determine the number of $t$-element links for the total index.

#### 5.2. The Matter Element Extension (MEE) Method

## 6. Discussion

## 7. Conclusions

- (1)
- According to the damage mechanism of pulling sling durability, the three-layer durability evaluation index system of the pulling sling structure is established. Seven indexes were selected for the durability evaluation of the pulling sling: PE cracking, PE scratching, corrosion of the rope body, damage to the rope body, damage to the anchorage, damage to the conduit, and damage to the vibration damping device.
- (2)
- There are uncertainties in the factors affecting the durability of pulling sling structures that are difficult to describe quantitatively. The combined weighting method is applied, combining the advantages of IAHP and CRITIC methods to compensate for the bias of a single assignment method. Through the theory of unascertained measure, the multi-indicator comprehensive evaluation vector of pulling sling is obtained, and the results are analyzed according to the confidence criterion.
- (3)
- On the basis of combined weighting method-unascertained-measure theory, a complete durability evaluation model of the pulling sling was established. Taking the Shaoxing Jiahui Bridge pulling sling structure as an example, the durability evaluation was carried out, and the evaluation result was calculated to be level III, which basically matches the actual situation of the bridge. It shows the applicability of the model and provides a new method for the durability evaluation of a pulling cable structure.
- (4)
- Through a comparative study, it was found that the final evaluation results of the three methods are consistent, which proves that the CWM-UM method can make a more accurate evaluation of the durability of the pulling cable structure, and it has the advantages of four theories, the MEE method, the CRITIC method, the IAHP method, and UM theory, with a concise calculation process and more accurate results. It shows the superiority of the CWM-UM method in durability evaluation. In practical application, the method to improve the durability of the cable body can be stated more accurately according to the evaluation results. The proposal and rational use of this method will contribute to the construction of resource-saving highways and the realization of sustainable development.

- (1)
- Since different bridges have different forms of pulling and slinging structures, a more comprehensive durability evaluation system can be established for different forms of pulling and slinging structures to make the evaluation results more reasonable and realistic.
- (2)
- The durability evaluation indexes established in this paper mainly rely on literature collection and theoretical analysis. In subsequent studies, these can be combined with finite element analysis software to identify the factors affecting the durability of the pulling sling structure through real bridge simulation. We can use this standard to establish the evaluation index.
- (3)
- Because of the special structural form and material characteristics of the pulling cable structure compared to the concrete structure, the selection of the environmental index and durability index eigenvalues in this paper is preliminary and rough, and further refinement and research are needed to ensure that the calculation of environmental index eigenvalues and durability index eigenvalues is more accurate and comprehensive.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comprehensive evaluation system for the durability of bridge pulling and lifting cable structures.

Level 1 | Level 2 | |
---|---|---|

Comprehensive evaluation of the durability of bridge tension cable structures | PE | $\mathrm{PE}\text{}\mathrm{scratch}\text{}{B}_{1}$ |

$\mathrm{PE}\text{}\mathrm{cracking}\text{}{B}_{2}$ | ||

Steel wire for cable | $\mathrm{Corrosion}\text{}\mathrm{of}\text{}\mathrm{cable}\text{}\mathrm{body}\text{}{B}_{3}$ | |

$\mathrm{Cable}\text{}\mathrm{body}\text{}\mathrm{damage}\text{}{B}_{4}$ | ||

Anchor device | $\mathrm{Anchorage}\text{}\mathrm{damage}\text{}{B}_{5}$ | |

$\mathrm{Catheter}\text{}\mathrm{damage}\text{}{B}_{6}$ | ||

$\mathrm{Damage}\text{}\mathrm{to}\text{}\mathrm{shock}\text{}\mathrm{absorbers}\text{}{B}_{7}$ |

Durability Level | Pull Sling Structure Durability Status | Classification Criteria |
---|---|---|

I | In good shape, durability failure risk is very small. | (0.8,1) |

II | Slightly damaged, durability failure risk is small. | (0.6,0.8) |

III | Medium damaged, durability failure risk is average. | (0.4,0.6) |

IV | Severely damaged, durability failure risk is high. | (0.2,0.4) |

V | Extremely dangerous, durability failure risk is very high. | (0,0.2) |

Element | Assignment | Meaning |
---|---|---|

${a}_{j{j}^{\prime}}$ | 1 | In a certain level, indicator j is equally important compared to indicator j′ |

2 | At a certain level, indicator j is slightly more important compared to indicator j′ | |

3 | At a certain level, indicator j is important compared to indicator j′ | |

${a}_{{j}^{\prime}j}=1/{a}_{j{j}^{\prime}}$ |

Guideline Layer | PE | Cable Steel Wire | Anchor Device | $\mathit{s}\mathit{w}\left(\mathit{i}\right)$ |
---|---|---|---|---|

0.196 | 0.311 | 0.493 | ||

${B}_{1}$ | 0.667 | 0.131 | ||

${B}_{2}$ | 0.333 | 0.065 | ||

${B}_{3}$ | 0.333 | 0.104 | ||

${B}_{4}$ | 0.667 | 0.207 | ||

${B}_{5}$ | 0.547 | 0.270 | ||

${B}_{6}$ | 0.190 | 0.094 | ||

${B}_{7}$ | 0.263 | 0.130 |

Methods | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | ${\mathit{B}}_{4}$ | ${\mathit{B}}_{5}$ | ${\mathit{B}}_{6}$ | ${\mathit{B}}_{7}$ |
---|---|---|---|---|---|---|---|

IAHP | 3 | 7 | 5 | 2 | 1 | 6 | 4 |

CRITIC | 1 | 6 | 7 | 3 | 4 | 2 | 5 |

Methods | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | ${\mathit{B}}_{4}$ | ${\mathit{B}}_{5}$ | ${\mathit{B}}_{6}$ | ${\mathit{B}}_{7}$ |
---|---|---|---|---|---|---|---|

PE | 0.677 | 0.323 | |||||

Cable steel wire | 0.343 | 0.657 | |||||

Anchor Device | 0.429 | 0.277 | 0.294 | ||||

$cw\left(i\right)$ | 0.169 | 0.080 | 0.099 | 0.190 | 0.198 | 0.128 | 0.136 |

Distribution Form | Graphics | Function Expressions |
---|---|---|

Linear distribution | $\{\begin{array}{c}{z}_{d}\left(x\right)=\{\begin{array}{c}\frac{-x}{{e}_{d+1}-{e}_{d}}+\frac{{e}_{d+1}}{{e}_{d+1}-{e}_{d}},{e}_{d}<x\le {e}_{d+1}\\ 0,x>{e}_{d+1}\end{array}\\ {z}_{d+1}\left(x\right)=\{\begin{array}{c}0,x\le {e}_{d}\\ \frac{x}{{e}_{d+1}-{e}_{d}}-\frac{{e}_{d}}{{e}_{d+1}-{e}_{d}},{e}_{d}<x\le {e}_{d+1}\end{array}\end{array}$ |

Layer | PE | Cable Steel Wire | Anchor Device | Pulling Cable | Actual Level |
---|---|---|---|---|---|

Evaluation Level | II | III | III | III | III |

Pulling Sling Structure Durability Longevity Evaluation System and Index Level Standard Criteria Division | Index (Weight) | Standard Split Point for Each Grade | Measured Value | ||||

I | II | III | IV | V | |||

${B}_{1}$ (0.132) | 90 | 75 | 60 | 50 | 35 | 72 | |

${B}_{2}$ (0.144) | 90 | 75 | 60 | 50 | 35 | 80 | |

${B}_{3}$ (0. 142) | 90 | 75 | 60 | 50 | 35 | 65 | |

${B}_{4}$ (0.151) | 90 | 75 | 60 | 50 | 35 | 55 | |

${B}_{5}$ (0.156) | 90 | 75 | 60 | 50 | 35 | 63 | |

${B}_{6}$ (0.142) | 90 | 75 | 60 | 50 | 35 | 45 | |

${B}_{7}$ (0.133) | 90 | 75 | 60 | 50 | 35 | 85 |

Criterion Layer | Indexes | IAHP | CRITIC | EWM | Combined Weight |
---|---|---|---|---|---|

PE | Cracking | 0.131 | 0.208 | 0.132 | 0.169 |

Scratch | 0.065 | 0.094 | 0.144 | 0.080 | |

Steel wire for cable | Damage | 0.104 | 0.091 | 0.142 | 0.099 |

Corrosion | 0.207 | 0.166 | 0.151 | 0.190 | |

Anchor device | Anchorage damage | 0.270 | 0.139 | 0.156 | 0.198 |

Catheter damage | 0.094 | 0.167 | 0.142 | 0.128 | |

Shock absorbers damage | 0.130 | 0.136 | 0.133 | 0.136 |

Pulling Cable | Comprehensive Unascertained Measurement | Evaluation Results | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | CWM UM | SPA | MEE | Actual Grade | |

Jiahui | 0 | 0.313 | 0.627 | 0.060 | 0 | III | III | III | III |

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**MDPI and ACS Style**

Li, Q.; Zhang, T.; Yu, Y.
Evaluation of the Durability of Bridge Tension Cables Based on Combination Weighting Method-Unascertained Measure Theory. *Sustainability* **2022**, *14*, 7147.
https://doi.org/10.3390/su14127147

**AMA Style**

Li Q, Zhang T, Yu Y.
Evaluation of the Durability of Bridge Tension Cables Based on Combination Weighting Method-Unascertained Measure Theory. *Sustainability*. 2022; 14(12):7147.
https://doi.org/10.3390/su14127147

**Chicago/Turabian Style**

Li, Qingfu, Tianjing Zhang, and Yingqiao Yu.
2022. "Evaluation of the Durability of Bridge Tension Cables Based on Combination Weighting Method-Unascertained Measure Theory" *Sustainability* 14, no. 12: 7147.
https://doi.org/10.3390/su14127147