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

Authors: Hujun He; Yumei Yan; Cuixia Qu; Yue Fan
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

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.


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## Body

## 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

x
1
,
x
2
,
…
,
x
n stand for n objects, namely, index space X={x1,x2,…,xn}; each object has m single evaluation index space; namely, I={I1,I2,…,In}; xi can be expressed as m-dimensional vector xi={xi1,xi2,…,xim}. xi have p evaluation levels, evaluation level space U={C1,C2,…,Cp}, where Ck indicates the k level; k level is much stronger than k+1 level; namely, Ck>Ck+1; then it is an ordered partition class in evaluation space U.


### 2.2. Uncertainty Measure of Single Index

μ
i
j
k
=
μ
(
x
i
j
∈
C
k
) is set to indicate measured value belonging to Ck level of the k evaluation level; μ is required to meet
(1)0≤μ(xij∈Ck)≤1(i=1,2,…,n;j=1,2,…,m;k=1,2…,p)μ(xij∈U)=1(i=1,2,…,n;j=1,2,…,m)μ(xij∈⋃l=1kCl)=∑l=1kμ(xi∈Cl)(k=1,2,…,p).Meeting formula (1) is called uncertainty measure for short measure [15–18]. Matrix (μijk)m×p is single index evaluation matrix having
(2)(μijk)m×p=[μi11μi12⋯μi1pμi21μi22⋯μi2p⋮⋮⋮⋮μim1μim2⋯μimp].


### 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(μijk)m×p is single index evaluation matrix having
(3)μi(x)={-xai+1-ai+ai+1ai+1-aiai<x<ai+10x>ai+1μi+1(x)={0x≤aixai+1-ai-aiai+1-aiai<x<ai+1.


### 2.4. Determining Index Weight

w
i
j
(
0
≤
w
i
j
≤
1
,
∑
j
=
1
m
w
i
j
=
1
) is set to represent importance of Ij relative to other indexes; that is, wij is the weight of Ij(j=1,2,…,m); the weight of each index is determined by information entropy theory; that is,
(4)vij=1+1lgp∑i=1pμijklgμijk,wj=vij∑i=1nvij.


### 2.5. Multiple Indexes Comprehensive Measure

μ
i
k
=
μ
(
x
i
∈
C
k
) is set to indicate measured value belonging to Ck level of the k evaluation level, according to the index weight determined; multi-index comprehensive measure of evaluating object is
(5)μik=∑j=1mwijμijk(i=1,2,…,n;j=1,2,…,m;k=1,2,…,p),
wherein 0≤μk≤1,∑k=1pμik=1, then μik is known as uncertainty measure, and {μi1,μi2,…,μip} is multi-index comprehensive evaluation measure vector of xi.


### 2.6. Recognition Criterion of Credible Degree

IfC1>C2>⋯>Cp, introduce credible degree recognition criteria, λ is credible degree (λ≥0.5, usually take λ=0.6 or 0.7) [13, 14] k0=min⁡(k:∑l=1kpμil≥λ,1≤l≤k), then is considered to belong to Ck0.


### 2.7. Order-Arranging

In addition to discrimination which evaluation levelxi belongs to, sometimes xi has to be ordered according to its important degree. If C1>C2>⋯>Cp and command value of Cl0 is nl0, then nl0>nl+1, and qxi=∑l=1pnlμil, where qxi is uncertainty important degree of evaluation factor xi and q={qx1,qx2,…,qxp} is called uncertainty importance vector. Important area of xi activity is ordered according to the value of qxi.


## 2.1. Uncertainty Measure

x
1
,
x
2
,
…
,
x
n stand for n objects, namely, index space X={x1,x2,…,xn}; each object has m single evaluation index space; namely, I={I1,I2,…,In}; xi can be expressed as m-dimensional vector xi={xi1,xi2,…,xim}. xi have p evaluation levels, evaluation level space U={C1,C2,…,Cp}, where Ck indicates the k level; k level is much stronger than k+1 level; namely, Ck>Ck+1; then it is an ordered partition class in evaluation space U.


## 2.2. Uncertainty Measure of Single Index

μ
i
j
k
=
μ
(
x
i
j
∈
C
k
) is set to indicate measured value belonging to Ck level of the k evaluation level; μ is required to meet
(1)0≤μ(xij∈Ck)≤1(i=1,2,…,n;j=1,2,…,m;k=1,2…,p)μ(xij∈U)=1(i=1,2,…,n;j=1,2,…,m)μ(xij∈⋃l=1kCl)=∑l=1kμ(xi∈Cl)(k=1,2,…,p).Meeting formula (1) is called uncertainty measure for short measure [15–18]. Matrix (μijk)m×p is single index evaluation matrix having
(2)(μijk)m×p=[μi11μi12⋯μi1pμi21μi22⋯μi2p⋮⋮⋮⋮μim1μim2⋯μimp].


## 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(μijk)m×p is single index evaluation matrix having
(3)μi(x)={-xai+1-ai+ai+1ai+1-aiai<x<ai+10x>ai+1μi+1(x)={0x≤aixai+1-ai-aiai+1-aiai<x<ai+1.


## 2.4. Determining Index Weight

w
i
j
(
0
≤
w
i
j
≤
1
,
∑
j
=
1
m
w
i
j
=
1
) is set to represent importance of Ij relative to other indexes; that is, wij is the weight of Ij(j=1,2,…,m); the weight of each index is determined by information entropy theory; that is,
(4)vij=1+1lgp∑i=1pμijklgμijk,wj=vij∑i=1nvij.


## 2.5. Multiple Indexes Comprehensive Measure

μ
i
k
=
μ
(
x
i
∈
C
k
) is set to indicate measured value belonging to Ck level of the k evaluation level, according to the index weight determined; multi-index comprehensive measure of evaluating object is
(5)μik=∑j=1mwijμijk(i=1,2,…,n;j=1,2,…,m;k=1,2,…,p),
wherein 0≤μk≤1,∑k=1pμik=1, then μik is known as uncertainty measure, and {μi1,μi2,…,μip} is multi-index comprehensive evaluation measure vector of xi.


## 2.6. Recognition Criterion of Credible Degree

IfC1>C2>⋯>Cp, introduce credible degree recognition criteria, λ is credible degree (λ≥0.5, usually take λ=0.6 or 0.7) [13, 14] k0=min⁡(k:∑l=1kpμil≥λ,1≤l≤k), then is considered to belong to Ck0.


## 2.7. Order-Arranging

In addition to discrimination which evaluation levelxi belongs to, sometimes xi has to be ordered according to its important degree. If C1>C2>⋯>Cp and command value of Cl0 is nl0, then nl0>nl+1, and qxi=∑l=1pnlμil, where qxi is uncertainty important degree of evaluation factor xi and q={qx1,qx2,…,qxp} is called uncertainty importance vector. Important area of xi activity is ordered according to the value of qxi.


## 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 x1, x2, x3, x4, and x5. The grading standard and assignment value are shown in Table 1. Each evaluation index was classified and valued; evaluation set is {C1,C2,C3,C4,C5}, 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.Table 1
Evaluation factors and grading standard.
Factors
Stability grade
Stability(I)
Relative stability(II)
General stability(III)
Instability(IV)
Very instability(V)
(1) Rock quality designation (RQD)/%
>90
75–90
50–75
25–50
<25
(2) Rock uniaxial compressive strength/MPa
>120
60–120
30–60
15–30
<15
(3) Rock mass integrity coefficient
>0.75
0.45–0.75
0.3–0.45
0.2-0.3
<0.2
(4) Strength coefficient of structural face
>0.8
0.6–0.8
0.4–0.6
0.2–0.4
<0.2
(5) Groundwater seepage (L/min·10 m)
<5
5–10
10–25
25–125
>125Table 2
Survey statistics table of surrounding rock.
Section
Stability evaluation index
x
1
x
2
x
3
x
4
x
5
E
1
40.0
25.0
0.28
0.35
35.0
E
2
70.5
40.5
0.43
0.55
15.5
E
3
40.5
20.0
0.25
0.30
30.0
E
4
72.0
50.0
0.55
0.55
20.0
E
5
20.0
50.0
0.23
0.51
13.0


### 3.1. Construction of Single Index Measure Function

According to the definition of single index measure function above, combined with grading standards in Table1 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.Figure 1
Single index measure function of rock quality designation.Figure 2
Single index measure function of rock uniaxial compressive strength.Figure 3
Single index measure function of rock mass integrity coefficient.Figure 4
Single index measure function of strength coefficient of structural face.Figure 5
Single index measure function of groundwater seepage.According to single index measure function above and combined with the actual data in Table2, single index evaluation matrix of 5 sections surrounding rocks can be calculated. Setting E1 as an example for evaluation, single index evaluation matrix shows
(6)μA1=[000.10.90000.110.890000.240.760000.250.750000.70.30].


### 3.2. Calculation of Multiple Index Measure Evaluation Matrix

Formula (4) determines the weight of each index. The weight of evaluation index of E1 is {w1,w2,w3,w4,w5}={0.22,0.22,0.19,0.19,0.18}. According to single index measure matrix and multiple index calculation formula, multiple index vector is μ1=ϖ1·μA1={0,0,0.27,0.73,0}.


### 3.3. Credible Degree Recognition

Take credible degreeλ=0.6; by multiple index comprehensive evaluation measure vector and credible evaluation criterion formula, k0=1>0.5, can judge E1 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.Table 3
Evaluation results of uncertainty measure model.
Sections
General uncertainty measure
Evaluation results of uncertainty measure model
Evaluation results of artificial neural network model
I
II
III
IV
V
E
1
0
0
0.27
0.73
0
IV
IV
E
2
0
0.21
0.75
0.04
0
III
III
E
3
0
0
0.15
0.8
0.05
IV
IV
E
4
0
0.29
0.7
0.01
0
III
III
E
5
0
0.1
0.49
0.09
0.32
IV
IV


### 3.4. Order-Arranging of Surrounding Rock Stability

According to order formula, becauseI>II>III>IV>V, take nl=12-2q; then q={qx1,qx2,qx3,qx4,qx5}={4.54,6.34,4.2,6.56,4.74}, E3<E1<E5<E2<E4.


## 3.1. Construction of Single Index Measure Function

According to the definition of single index measure function above, combined with grading standards in Table1 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.Figure 1
Single index measure function of rock quality designation.Figure 2
Single index measure function of rock uniaxial compressive strength.Figure 3
Single index measure function of rock mass integrity coefficient.Figure 4
Single index measure function of strength coefficient of structural face.Figure 5
Single index measure function of groundwater seepage.According to single index measure function above and combined with the actual data in Table2, single index evaluation matrix of 5 sections surrounding rocks can be calculated. Setting E1 as an example for evaluation, single index evaluation matrix shows
(6)μA1=[000.10.90000.110.890000.240.760000.250.750000.70.30].


## 3.2. Calculation of Multiple Index Measure Evaluation Matrix

Formula (4) determines the weight of each index. The weight of evaluation index of E1 is {w1,w2,w3,w4,w5}={0.22,0.22,0.19,0.19,0.18}. According to single index measure matrix and multiple index calculation formula, multiple index vector is μ1=ϖ1·μA1={0,0,0.27,0.73,0}.


## 3.3. Credible Degree Recognition

Take credible degreeλ=0.6; by multiple index comprehensive evaluation measure vector and credible evaluation criterion formula, k0=1>0.5, can judge E1 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.Table 3
Evaluation results of uncertainty measure model.
Sections
General uncertainty measure
Evaluation results of uncertainty measure model
Evaluation results of artificial neural network model
I
II
III
IV
V
E
1
0
0
0.27
0.73
0
IV
IV
E
2
0
0.21
0.75
0.04
0
III
III
E
3
0
0
0.15
0.8
0.05
IV
IV
E
4
0
0.29
0.7
0.01
0
III
III
E
5
0
0.1
0.49
0.09
0.32
IV
IV


## 3.4. Order-Arranging of Surrounding Rock Stability

According to order formula, becauseI>II>III>IV>V, take nl=12-2q; then q={qx1,qx2,qx3,qx4,qx5}={4.54,6.34,4.2,6.56,4.74}, E3<E1<E5<E2<E4.


## 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.



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*Source: 626527-2014-10-19.xml*