Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 626527 |

Hujun He, Yumei Yan, Cuixia Qu, Yue Fan, "Study and Application on Stability Classification of Tunnel Surrounding Rock Based on Uncertainty Measure Theory", Mathematical Problems in Engineering, vol. 2014, Article ID 626527, 5 pages, 2014.

Study and Application on Stability Classification of Tunnel Surrounding Rock Based on Uncertainty Measure Theory

Academic Editor: Igor Andrianov
Received20 Jun 2014
Revised02 Oct 2014
Accepted02 Oct 2014
Published19 Oct 2014


Based on uncertainty measure theory, a stability classification and order-arranging model of surrounding rock was established. Considering the practical engineering geologic condition, 5 factors that influence surrounding rock stability were taken into account and uncertainty measure function was obtained based on the in situ data. In this model, uncertainty influence factors were analyzed quantitatively and qualitatively based on the real situation; the weight of index was given based on information entropy theory; surrounding rock stability level was judged based on credible degree recognition criterion; and surrounding rock was ordered based on order-arranging criterion. Furthermore, this model was employed to evaluate 5 sections surrounding rock in Dongshan tunnel of Huainan. The results show that uncertainty measure method is reasonable and can have significance for surrounding rock stability evaluation in the future.

1. Introduction

Entering the new century, with the rapid and sustainable development of national economy and the implementation of the western great development strategy, railway, highway, and hydropower construction gained hitherto unknown development. In particular in recent years, the construction scale and quantity of tunnels were increasing constantly; long tunnels came forth increasingly. According to incomplete statistics, there were more than 1700 highway tunnels built in China [1]; the tunnel played an important role in promoting the development of our national economy and improving the traffic environment and so forth. For the engineering characteristics of tunnel construction are “complicated geological conditions, many influence factors, and high difficulty in construction” [1], especially the fact that tunnel excavation in complex rock conditions may make these characteristics more prominent, the key problems in construction process are to evaluate objectively the surrounding rock and support stability.

As the main basis of engineering design and supporting structure calculation, surrounding rock stability evaluation has attracted widespread attention in engineering field. Many scholars had researched this problem and put forward the evaluation methods of all kinds of engineering which could be classified as the single index method and the comprehensive evaluation method [2]. The single index evaluation is relatively easier to evaluate surrounding rock stability on the basis of single index, but only consider the influence of some factors in the process of evaluation, which is not comprehensive enough; comprehensive evaluation methods include entropy weight coefficient method [3], fuzzy mathematics method [4], artificial neural network [5], the extension method [6, 7], weighted distance discriminant analysis method [8], and support vector machine theory [9]. But all the evaluation methods mentioned above have some limitations in the application process. For example, fuzzy mathematics method cannot solve information overlay caused by the evaluation index dependence; the determination of membership degree and weight should be further explored; the artificial neural network is easy to show result with mean value; the principle and the calculation process are complex; the extension evaluation method can comprehensively consider each factor, but setting interval midpoint as the best value in the correlation calculation may omit the important constraint condition and finally lead to significant differences between evaluation result and actual result [10]; support vector machine method has poor anti-interference ability to determine the boundary and its noise sensitive [11]. However, the evaluation of surrounding rock stability is affected by many factors, and these factors are characterized by diversity, variability, uncertainty, and so forth so that the stability evaluation of surrounding rock is a complex system with uncertainty analysis. In this regard, uncertainty mathematics theory provides a good approach.

The uncertainty information and related mathematical theory were first proposed by Professor Wang [12] of Harbin Institute of Architecture and Engineering in 1990. Uncertainty information is a new concept which is different from random information, fuzzy information, and grey information. On this basis, Liu et al. [13, 14] studied systematically uncertainty mathematics and built uncertainty mathematics theory, and the theory was applied to social science and natural science. In the application of uncertainty mathematics, uncertainty measure evaluation model has been used most widely. Based on the previous studies of comprehensive assessment of tunnel surrounding rock stability classification and the theory and idea of uncertainty measure evaluation model, this study introduced uncertainty mathematics theory to tunnel surrounding rock stability evaluation, which could solve uncertainty problem in classification evaluation of tunnel surrounding rock stability, and improve data reliability and precision; reorder surrounding rock with stability, and provided a new thought for classification evaluation of tunnel surrounding rock stability.

2. Uncertainty Measure Comprehensive Evaluation Model

2.1. Uncertainty Measure

stand for objects, namely, index space ; each object has single evaluation index space; namely, ; can be expressed as -dimensional vector . have evaluation levels, evaluation level space , where indicates the level; level is much stronger than level; namely, ; then it is an ordered partition class in evaluation space .

2.2. Uncertainty Measure of Single Index

is set to indicate measured value belonging to level of the evaluation level; is required to meet

Meeting formula (1) is called uncertainty measure for short measure [1518]. Matrix is single index evaluation matrix having

2.3. Uncertainty Measure Function

In the construction of single index measure evaluation matrix, we should first establish single index measure function. At present, construction methods of single index measure function mainly include linear, exponential, parabola, and sinusoidal. Linear type uncertainty measure function is currently the most widely used and the most simple measure function, so this paper also uses linear type uncertainty measure function. No matter what kind of simulation function is, “nonminus, unitary, additivity” must be satisfied. According to the characteristics of specific indexes, suitable uncertainty measure functions are selected; linear type uncertainty measure function is currently the most widely used and the most simple measure function, so this paper also uses linear type uncertainty measure function.

Matrix is single index evaluation matrix having

2.4. Determining Index Weight

is set to represent importance of relative to other indexes; that is, is the weight of ; the weight of each index is determined by information entropy theory; that is,

2.5. Multiple Indexes Comprehensive Measure

is set to indicate measured value belonging to level of the evaluation level, according to the index weight determined; multi-index comprehensive measure of evaluating object is wherein , then is known as uncertainty measure, and is multi-index comprehensive evaluation measure vector of .

2.6. Recognition Criterion of Credible Degree

If , introduce credible degree recognition criteria, is credible degree (, usually take or 0.7) [13, 14] , then is considered to belong to .

2.7. Order-Arranging

In addition to discrimination which evaluation level belongs to, sometimes has to be ordered according to its important degree. If and command value of is , then , and , where is uncertainty important degree of evaluation factor and is called uncertainty importance vector. Important area of activity is ordered according to the value of .

3. Application Example

In this paper, take tunnel rock mass actual measurement data of Dongshan in Huainan Basin as the research object provided by Liu [1]. Based on correlative quality classification research of engineering rock mass [19, 20], set rock quality designation (RQD), rock uniaxial compressive strength, rock mass integrity coefficient, strength coefficient of structural face, and groundwater seepage as evaluation indexes, 5 indexes represented by , , , , and . The grading standard and assignment value are shown in Table 1. Each evaluation index was classified and valued; evaluation set is , namely, I, II, III, IV, and V, respectively, which expresses stability, relative stability, general stability, instability, and very instability. Actual measurement data of evaluation index in Dongshan tunnel surrounding rock are shown in Table 2.

FactorsStability grade
Relative stability
General stability
Very instability

(1) Rock quality designation (RQD)/%>9075–9050–7525–50<25
(2) Rock uniaxial compressive strength/MPa>12060–12030–6015–30<15
(3) Rock mass integrity coefficient>0.750.45–0.750.3–0.450.2-0.3<0.2
(4) Strength coefficient of structural face>0.80.6–0.80.4–0.60.2–0.4<0.2
(5) Groundwater seepage (L/min·10 m)<55–1010–2525–125>125

SectionStability evaluation index

3.1. Construction of Single Index Measure Function

According to the definition of single index measure function above, combined with grading standards in Table 1 and specific value in Table 2, single index function are constructed in order to obtain the measure value of each evaluation index. Single index measure functions are shown in Figures 1, 2, 3, 4, and 5.

According to single index measure function above and combined with the actual data in Table 2, single index evaluation matrix of 5 sections surrounding rocks can be calculated. Setting as an example for evaluation, single index evaluation matrix shows

3.2. Calculation of Multiple Index Measure Evaluation Matrix

Formula (4) determines the weight of each index. The weight of evaluation index of is . According to single index measure matrix and multiple index calculation formula, multiple index vector is .

3.3. Credible Degree Recognition

Take credible degree ; by multiple index comprehensive evaluation measure vector and credible evaluation criterion formula, , can judge grade as is IV. Similarly, other 4 surrounding rocks sections were evaluated. The evaluation results are shown in Table 3. Table 3 also lists the evaluation results by artificial neural network model [1]. Seen from Table 3, the evaluation results of 5 surrounding rocks sections by uncertainty measure are consistent with artificial neural network, but the principle and calculation process of uncertainty measure are simpler than artificial neural network, and especially information entropy theory in the determination of weight reduces man-made factors; the weight more objective and real, fuzzy comprehensive evaluation [4] is unable to compare, so uncertainty measure theory has very good practicality.

Sections General uncertainty measureEvaluation results of uncertainty measure modelEvaluation results of artificial neural network model


3.4. Order-Arranging of Surrounding Rock Stability

According to order formula, because , take ; then , .

4. Conclusions

(1)Evaluation of surrounding rock stability is controlled and affected by many factors. And these factors have obvious diversity, variability, uncertainty, and so forth; considering the characteristics of surrounding rock stability evaluation, uncertainty measure evaluation method is introduced into stability evaluation of surrounding rock, and uncertainty measure evaluation model of surrounding rock stability is built.(2)According to influence factors of surrounding rock stability and grading standards, uncertainty measure functions of surrounding rock stability evaluation index are built based on uncertainty measure theory, the weight of each index is calculated based on information entropy theory, surrounding rock stability level is evaluated based on credible degree recognition criterion, and finally surrounding rock is ordered based on order-arranging criterion.(3)The stability analysis of 5 surrounding rock sections in Dongshan tunnel of Huainan shows that uncertainty measure evaluation model is more scientific and reasonable. The surrounding rock stability level and level order could be established in this modeling system. It provides a new mentality for surrounding rock stability evaluation and shows important theoretical and practical significance.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This work was financially supported by the major program of the National Natural Science Foundation of China (41030749), the National Natural Science Foundation of China (41072223), and the Special Fund for Basic Scientific Research of Central Colleges (CHD2011JC111).


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Copyright © 2014 Hujun He 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.

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