Advances in Civil Engineering

Volume 2019, Article ID 8267406, 10 pages

https://doi.org/10.1155/2019/8267406

## Reliability Evaluation of Water-Rich Loess Tunnel with Lining Crack Based on Extension Theory

^{1}School of Highway, Chang’an University, Xi’an 710064, China^{2}Key Laboratory of Highway Construction and Maintenance Technology in Loess Region, Shanxi Transportation Research Institute, Taiyuan 030006, China

Correspondence should be addressed to Xinxing Zhou; nc.ude.tuhw@83323490xxz

Received 22 November 2018; Revised 14 February 2019; Accepted 6 March 2019; Published 7 April 2019

Academic Editor: Amir Si Larbi

Copyright © 2019 Xiaohui Xue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Crack is the most prevalent disease of water-rich loess tunnels, which impacts the tunnel safety and reliability. The study aimed to investigate the multi-index of tunnel lining crack based on extension theory and reliability evaluation. A new evaluation model was built, and the matter-element concept was used to describe the reliability of a water-rich loess tunnel with lining crack. Crack width, crack depth, crack length, crack geometry, and water seepage of the Qiaoyuan tunnel were measured by intelligent crack measuring instrument, 10 m tape, laser scanner, and dosing cup. According to the analytic hierarchy process (AHP), the judgment matrix could be used to describe the relative importance of each index. Combining the maximum eigenvalue with checking consistency, the weight coefficients of 5 factors were calculated. A probabilistic statistic method and comparison of specification are performed to confirm the validity of extension theory in reliability evaluation. The results showed that the extension theory provided a reliability evaluation method of a water-rich loess tunnel with lining crack.

#### 1. Introduction

Loess is widely distributed in the central and western China. In recent years, with the rapid development of traffic foundation construction, a large number of highway tunnels had been constructed in the loess region. Owing to the effect of rainfall and other seasonal and transient infiltration, there are many local water-rich loess layers in the loess region. The special engineering properties of loess, such as significant structural characteristic and intense water sensitivity [1, 2], make its physical and mechanical indexes to decline sharply in the water-rich environment, which will cause the lining cracks, leakage, and other problems during construction and operation [3–5]. The lining crack is the most general disease of a water-rich loess tunnel, and it impacts the tunnel safety and reliability [6–8]. Therefore, it is important for the water-rich loess tunnel with lining crack to study the reliability evaluation. Many traditional methods had been used to evaluate the reliability of tunnel lining, including field monitoring [9–11], physical model experiment [12], numerical simulation [13], probability analysis [14], and engineering analogy method. However, these methods did not take the comprehensive influencing factors of tunnel lining into consideration, and they also did not solve the incompatibility problem during the difference factors.

Extension theory has been widely used in the field of engineering technology after 1983 [15], which can solve the contradiction problem of evaluation indexes very well. But, the application of reliability evaluation method of tunnel engineering is relatively hysteretic. Kong et al. [16] established the health diagnosing method for a shield tunnel based on extension theory, and this method was applied to the Shanghai Yangze River Tunnel. Yong et al. [17] presented the comprehensive assessment of tunnel collapse risk based on integration of statistical analysis and extension theory. Hare [18] founded the model of evaluating the tunnel construction environment using the two methods (game theory and extension theory incorporating). Analytic hierarchy process (AHP)—extension synthesis method—was proposed [19], and it was known as an effective tool to evaluate the safety level of an operating tunnel.

In this study, the aim is to investigate the reliability evaluation of water-rich loess tunnel with lining crack based on extension theory. Crack width, crack depth, crack length, crack geometry, and water seepage of the Qiaoyuan tunnel were measured. The reliability evaluation model of a water-rich loess tunnel with lining crack was built, and the matter-element concept had been put forward.

#### 2. Experiments and Methods

##### 2.1. Extension Theory Method

The extension theory is an effective model to solve the contradictory problem of quantitative evaluation indexes. It imports the correlation function in the quantitative calculation process based on the matter-element theory, extension mathematics, and AHP. Therefore, it can transfer each evaluation index into a compatible issue and make the evaluation results and the actual situation consistent.

In the matter-element concept, the symbol of matter is defined as , the character , and the character value . Therefore, the ordered triple denotes the elementary unit to describe the characteristics of things. If the matter has characters, we can express this matter using the following matrix:

For a transition from qualitative description to quantitative analysis, the “distance” parameter was imported in the correlation function. Then, the value of the correlation function can denote different grades of samples with characteristics . Let be one point on the real axis, and let be a given interval, the extension distance of point , and interval is written as follows:

According to the extension theory, the elementary correlation function is established to calculate the rank of things with characteristics , whose formulas are shown as follows:where and are the extension distances between point and intervals and , respectively.

As mentioned above, the correlation parameter of each index of evaluation matter about each rank is shown as follows:

The correlation of each evaluation matter about each rank is defined as follows:where is the weight coefficient of each evaluation character , whose specific calculating method will be elaborated in the follow-on work.

After calculating the correlation value of each evaluation matter about each rank , is set to be the max , that is,

So, the value of can reflect the evaluation grade of matter .

##### 2.2. Standardization of Evaluation Parameter

The dimensions of each in situ monitoring data are not consistent, so it is necessary to weaken the dimensional effect of each index. The threshold is the most common method of standardization [20].

If the evaluation parameter is close to the minimum value, the index will be favorable, and the basic formula is shown as follows:

If the evaluation parameter is close to the maximum value, the basic formula is shown as follows:where is the dimensionless value, is the sample value, is the maximum of the *j*th type sample value, and is the minimum of *j*th type sample value.

##### 2.3. Extension Theory Evaluation Procedure

Based on the extension theory, the evaluation procedure of the water-rich loess tunnel with lining crack is as follows.

First, identify the reliability evaluation indexes and influencing factors of the water-rich loess tunnel with lining cracks based on the comparability principal, integrity principal, and simplified principal. Then, determine the evaluation indexes values and grades by field monitoring, engineering analogy, and literature investigation. Next, calculate the correlation degree of each evaluation matter about each rank. Finally, specify the evaluation rank and provide technology support for the disease treatment of water-rich loess tunnel.

The concrete flowchart of the evaluation procedure is shown in Figure 1.