Study on Seismic Damage Risk Assessment of Mountain Tunnel Based on the Extension Theory

Abstract

In this paper, the extension method is applied to the assessment of seismic damage risk in mountain tunnels. Based on various considerations such as the geological conditions of the tunnel site, the state of the tunnel, the structural situation and the earthquake, the seismic damage risk assessment index for mountainous tunnels is proposed. The range of values of different assessment indexes under each grade is quantified. The seismic damage risk of the tunnel is divided into four classes, and assessed based on the extension theory. The specific assessment process is as follows: Firstly, five tunnels affected by the Wenchuan earthquake were selected to construct the matter element to be assessed, and construct the classical domain and nodal domain; then, the entropy weight method and analytical hierarchy process (AHP) were used to determine the weights of each index; finally, the correlation function values were calculated to obtain the seismic damage risk level of the tunnels. Among them, the seismic damage risk level of the Longchi tunnel and Longxi tunnel is “high risk”, and the seismic damage risk level of the Longdongzi tunnel, Youyi tunnel and Maanshi tunnel is “moderate risk”. The five tunnels are sorted according to the eigenvalues of the seismic damage risk level as follows: Longchi tunnel > Longxi tunnel > Longdongzi tunnel > Youyi tunnel > Maanshi tunnel. Combined with the seismic damage investigation of the Wenchuan earthquake, the actual seismic damage of tunnel is consistent with the assessment results, which verifies the accuracy and effectiveness of this method. The seismic damage risk assessment model for mountain tunnels proposed in this paper has certain reference value for the future seismic damage risk assessment of tunnels.

1. Introduction

The investigation of seismic damage caused by the Tangshan earthquake, Chi-Chi earthquake, Hanshin earthquake, Wenchuan earthquake and other earthquakes shows that the tunnels have suffered varying degrees of damage [1,2,3,4,5]. The most common forms of damage include tunnel deformation and cracking, lining leakage, concrete block falling, and collapse in the tunnel portal section and the cavern section. Once the tunnel collapses, the losses incurred are immeasurable. With the increasing use of underground space, the number of tunnels located in high seismic intensity areas is also increasing. The seismic intensity represents the strength of the impact of an earthquake event on buildings. The higher the seismic intensity, the more severe the earthquake damage will be when it occurs. How to quantify the seismic damage risk of tunnel is a complex work.

Tunnel safety assessment includes many aspects, such as the risk of collapse during the construction period, the risk of water and mud inrush in karst tunnels, etc. Ou et al. selected 11 risk indexes such as excavation span, grade of surrounding rock to evaluate the collapse risk of deep buried tunnel during construction, and successfully applied it to Yuxi tunnel [6]. Kim et al. collected 24 tunnel collapse cases, established a database, and quantified the influencing factors of tunnel collapse using an analytic hierarchy process (AHP) [7]; Chen et al. proposed a mountain tunnel collapse risk assessment model integrating case-based reasoning, rough set theory and um-spa theory, and evaluated the Xiucun tunnel crossing the shattered fault zone [8]. Lin et al. developed a simple and practical software package for assessing the risk of water gushing in tunnels using cloud model and variable weight theory [9]; Luo et al. established a shield tunneling risk assessment model based on the characteristics of shield tunnels in soft soil, and successfully applied it to Fuzhou Metro Line 6 [10]. Zhen and Gong et al. carried out the damage and safety assessment of the tunnel during the operation period, which provided a reference for the reinforcement and maintenance of the tunnel [11,12]. In summary, a significant number of studies have been carried out for collapse risk, water gushing risk and excavation risk in tunnels; however, almost all studies have not considered the seismic damage risk factor when assessing the safety of tunnels. Scholars have also carried out a large number of researches on the dynamic response of tunnels under earthquake action through an investigation method [13,14], theoretical analysis [15,16], numerical simulation [17,18,19] and shaking table tests [20,21], respectively. Their studies have focused on the investigation and classification of seismic damage of tunnels, mechanical behavior of tunnels, and anti-seismic measures, as they are more concerned with the tunnel itself. However, the seismic damage risk of tunnels is also related to geological conditions, tunnel status, tectonic conditions and many other factors. With the increasing number and scale of tunnels in high intensity areas, the safety of tunnels under earthquakes poses a great challenge; a reasonable seismic damage risk assessment of the tunnel is helpful to understand the safety of the tunnel, and can strengthen the tunnel with high seismic damage risk in a targeted manner so as to reduce the loss of life and property caused by the tunnel damage when the earthquake occurs.

In recent years, a large number of assessment methods have been used in engineering safety assessment, including fault tree analysis [22], risk matrix method [23,24], analytic hierarchy process [25], Bayesian network [26,27], artificial neural network method [28], and fuzzy comprehensive assessment method [29], among others; however, these methods have some shortcomings when in use. For example, the fault tree analysis tends to omit potential risk factors. The risk matrix method and analytic hierarchy process need to rank the risk factors, and the selection of weight depends on the experience of experts. The artificial neural network method requires a large amount of data as a learning sample. Bayesian models can make poor predictions due to the assumed prior model. When using the fuzzy comprehensive assessment method, a larger number of indexes in the same index level will lead to smaller weight coefficients and affect the assessment accuracy.

The shortcomings of the above-mentioned assessment methods make them somewhat limited in engineering applications. The risk of seismic damage in tunnels is a multi-objective decision problem that requires the consideration of numerous factors, and the magnitude of the impact of these factors on the tunnel is uncertain. The extension method can transform incompatible conflicts into compatible relationships, thus achieving the best global decision. It can quantitatively describe the seismic damage risk of tunnel and make up for the deficiency of single subjective or objective weighting. It can also consider the fuzziness and uncertainty of risk factors. In addition, to avoid the occurrence of invalid correlation functions, which are caused by the magnitudes of the matter-element out of the range of the nodal domain, the conventional extension method is improved. In the present work, the seismic damage risk factors of mountain tunnels are summarized and a seismic damage risk assessment index system is established based on a large amount of literature research and seismic damage research. Then, the analytic hierarchy process and the improved entropy weight method are used to determine the weight of each risk assessment index, five damaged tunnels in the Wenchuan earthquake are selected to form the matter-elements to be evaluated. The correlation function value is calculated, and the seismic damage risk level of the tunnel is finally obtained. The assessment results are compared with the actual earthquake damage to verify the correctness of the model proposed in this paper.

2. Risk Assessment Model Based on Extension Theory

The assessment model based on extension theory can transform multi-objective decision problems into single-objective decision problems, and quantitatively express the assessment results, as it has strong practicability and has been widely used in the field of engineering safety assessment [30].

The construction process of matter-element extension assessment model generally includes five steps.

• Establishing the classical domain and nodal domain matter-element matrix of assessment indexes;
• Constructing the correlation function;
• Determining the index weights and calculating the comprehensive correlation degree;
• Determining the assessment level.

2.1. Determination of Classic Domains and Nodal Domains

For a particular risk assessment index, the classical domain is a range of values describing the characteristics of the assessment object for different assessment levels, and the nodal domain is the maximum range that the risk assessment index can take. According to the assessment standard, the tunnel seismic damage risk assessment index is divided into several grades; the classical domain is the range of values of the seismic damage risk assessment index for the tunnel at the specified level, which is expressed as shown in Equation (1):

$$R_j=\left(N_j,C_i,U_{ij}\right)=\left(\begin{array}{ccc}{N}_j& c_1& U_{1j}\\ & c_2& U_{2j}\\ & ⋮& ⋮\\ & c_n& U_{nj}\end{array}\right)=\left(\begin{array}{ccc}{N}_j& c_1& \left[a_{1j},b_{1j}\right]\\ & c_2& \left[a_{2j},b_{2j}\right]\\ & ⋮& ⋮\\ & c_n& \left[a_{nj},b_{nj}\right]\end{array}\right)$$

(1)

where $R_j(j=\mathrm{1,2},\dots ,m)$ is the j-th matter-element; $N_j$ is the j-th assessment level of the risk index; $C_i(i=\mathrm{1,2},\dots ,n)$ is the i-th assessment index; $U_{ij}$ is $N_j$ on the range of quantitative values specified by index $C_i$, $U_{ij}=\left[a_{ij},b_{ij}\right]$ is the range taken by the assessment index $C_i$ at the j-th level, that is, the classical domain of $N_j$.

Referring to the way of determining the classical domain, the nodal domain can be obtained as shown in Equation (2):

$$R_p=\left(P,C_i,U_{ip}\right)=\left(\begin{array}{ccc}P& c_1& U_{1p}\\ & c_2& U_{2p}\\ & ⋮& ⋮\\ & c_n& U_{np}\end{array}\right)=\left(\begin{array}{ccc}P& c_1& \left[a_{1p},b_{1p}\right]\\ & c_2& \left[a_{2p},b_{2p}\right]\\ & ⋮& ⋮\\ & c_n& \left[a_{np},b_{np}\right]\end{array}\right)$$

(2)

where $P$ is all assessment levels of tunnel risk indexes; $U_{ip}=\left[a_{ip},b_{ip}\right]$, $U_{ip}$ is all assessment levels which are in proximity the range of values taken by the assessment index $C_i$, that is, the nodal domain of $P$.

The value of the classic domain is generally determined according to the existing specifications in the industry norms, objective laws, industry expert experience judgment, data statistical analysis, etc. The classical domain cannot extend beyond the nodal domain and the value range of the nodal domain is also determined after the classical domain is determined.

2.2. Matter-Element to Be Assessed

For a tunnel to be assessed, the basic information or data analysis results of the tunnel are represented by matter-element R; then, R is called matter-element to be assessed, as shown in Equation (3).

$$R=\left(\begin{array}{ccc}P& c_1& u_1\\ & c_2& u_2\\ & ⋮& ⋮\\ & c_n& u_n\end{array}\right)$$

(3)

where $P$ is all assessment levels for an index of a tunnel to be evaluated; $u_i$ is the value of the tunnel about the assessment index $c_i$, that is, the specific value of the tunnel to be assessed.

2.3. Correlation Functions

The correlation function expresses the degree of affiliation of each index of the object to be evaluated with respect to each assessment level, and the association degree of the index is calculated according to the definition of distance in extension theory; its calculation formula is as shown in Equation (4).

$$K_j(u_i)=\frac{\rho (u_i,U_{ij})}{D(u_i,U_{ij},U_{ip})}$$

(4)

where $\rho \left(u_i,U_{ij}\right)$ represents the distance between $u_i$ and the classical domain $U_{ij}=\left[a_{ij},b_{ij}\right]$; its calculation formula is as shown in Equation (5).

$$\rho \left(u_i,U_{ij}\right)=\left|u_i-\frac{a_{ij}+b_{ij}}{2}\right|-\frac{b_{ij-}{a}_{ij}}{2}$$

(5)

where $D(u_i,U_{ij},U_{ip})$ represents the place value of the point with respect to the interval $U_{ij}$ and the interval $U_{ip}$, $U_{ij}\subset U_{ip}$, place value and distance can quantitatively express the positional relationship between points and intervals; its calculation formula is as shown in Equation (6).

$$D\left(u_i,U_{ij},U_{ip}\right)=\left\{\begin{array}{cc}-\left|U_{ij}\right|=-\left|b_{ij}-a_{ij}\right|& u_i\in U_{ij}\\ \rho \left(u_i,U_{ip}\right)-\rho \left(u_i,U_{ij}\right)& u_i\notin U_{ij}\end{array}\right.$$

(6)

2.4. Determination of the Assessment Level

According to the actual situation, choose the appropriate method to determine the weight coefficient of each index. Commonly used methods include Delphi method, frequency statistical analysis method, entropy method, AHP, etc. The comprehensive correlation degree $K_j\left(p\right)$ under the risk level J is obtained through the correlation degree and the corresponding weight coefficient, as shown in Equation (7).

$$K_j\left(p\right)=\sum _{i=1}^n{\omega }_i{K}_j\left(u_i\right)$$

(7)

where ${\omega }_i$ is the weight coefficient of the i-th assessment index; $K_j\left(u_i\right)$ is the degree of association of the i-th assessment index at level J.

The larger the $K_j\left(p\right)$ value of the matter-element p to be evaluated, the smaller the deviation of the index from the assessment level; if $K_{j0}\left(p\right)=\max \left[K_j\left(p\right)\right(j=\mathrm{1,2},\dots ,m\left)\right]$, then the seismic damage risk of the tunnel to be evaluated is level J₀.

The risk level eigenvalues $j^{*}$ are calculated as follows.

$${\stackrel{-}{K}}_j\left(p\right)=\frac{K_j\left(p\right)-min\left[K_j\left(p\right)\right]}{max\left[K_j\left(p\right)\right]-min\left[K_j\left(p\right)\right]}$$

(8)

$$j^{*}=\frac{{\sum }_{j=1}^{m}j\times {\stackrel{-}{K}}_j\left(p\right)}{{\sum }_{j=1}^m{\stackrel{-}{K}}_j\left(p\right)}$$

(9)

$j^{*}$ corresponds to the extent to which p is biased towards the adjacent level; for example, $j^{*}=2.6$, which indicates that p belongs to the second level and is biased towards the third level.

3. Improvement of Matter-Element Extension Model for Mountain Tunnel

When the conventional matter-element extension method is used for quantitative analysis, once the value of the matter-element $u_i$ is out of the range of the nodal domain, it leads to the invalid of the correlation function to be calculated, and to inaccurate assessment results. On the other hand, in determining the weights of each index, both the subjective and objective weighting methods have certain drawbacks. The subjective weighting method relies too much on subjective judgment, and the weight calculation results will have obvious differences due to different subjective experiences of researchers. The calculation results of the objective weighting method are based on actual data and lack the correction of subjective experience. Therefore, it is necessary to integrate the advantages of the subjective and objective weighting method, combining subjective judgment based on practical problems and theoretical calculation based on objective data to determine the comprehensive weight of indexes, try to make the weight of each index match the actual importance, improve the credibility of the index weight, and improve the accuracy of the assessment results.

3.1. Standardized Processing of Classical Domains and Matter Elements

In response to the problem that the correlation function fails due to the value of the matter-element being outside the range of the nodal domain, this paper normalizes the matter-element matrix, classical domain and nodal domain matrix to be evaluated; the specific improvement method is as follows:

$$R=\left\{\begin{array}{c}1\hspace{1em}\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k} \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x} \mathrm{i}\mathrm{s} \mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{r}, \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k} \mathrm{i}\mathrm{s} \mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}.\\ 0 \mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k} \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x} \mathrm{i}\mathrm{s} \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r}, \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k} \mathrm{i}\mathrm{s} \mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}.\end{array}\right.$$

(10)

Then the matter-element value to be evaluated is normalized as shown in Equation (11).

$$u_i^{\prime }=\left\{\begin{array}{cc}R& u_i\ge b_{ip}\\ \frac{{(b_{ip}-u_i)}^{1-R}{(u_i-a_{ip})}^R}{b_{ip}-a_{ip}}& u_i\in (a_{ip},b_{ip})\\ 1-R& u_i\le a_{ip}\end{array}\right.$$

(11)

where $u_i^{\prime }$ is normalized matter-element value, $u_i$ is the actual value of the matter-element to be assessed; the above processing also solves the problem of non-uniform dimensions.

Normalize the classical domain and nodal domain functions in the same way. The normalized classical domain and nodal domain are as shown in Equations (12) and (13).

$$U_{ij}^{\prime }=\left[\frac{a_{ij}-a_{ip}}{b_{ip}-a_{ip}},\frac{b_{ij}-a_{ip}}{b_{ip}-a_{ip}}\right]$$

(12)

$$U_{ip}^{\prime }=\left[\mathrm{0,1}\right]$$

(13)

3.2. Comprehensive Weight of Improved Extension Method Index

The method of determining the weight of indexes can be divided into subjective weighting method and objective weighting method; subjective weighting method is mainly employs on subjective experience and judgment to assign weights to assessment indexes, such as AHP, Delphi method, etc. The objective weighting method calculates the weight value of each index based on the information reflected by the sample data, such as the entropy weighting method. When using the subjective weighting method to determine the weights, it involves more a reflection of human intention and the decision result may be subjective and arbitrary. The entropy weighting method follows only mathematical theory. Both the subjective and objective weighting methods have certain drawbacks. This paper adopts a comprehensive subjective–objective weighting method which can effectively avoid the excessive influence of subjective factors in the assignment process and fully exploit the information in the sample data.

3.2.1. Analytic hierarchy Process to Determine the Weight of Assessment Index

First, a judgment matrix $\mathrm{Q}={\left(x_{ii}\right)}_{n\times n}(i=\mathrm{1,2},3,\dots ,n)$ is constructed by inviting a number of experienced experts to score. When constructing the judgment matrix, two related factors are compared according to the scale $h_{ab}$; specific standards are as shown in

Table 1. Meaning of scale $h_{ab}$.

Scale hab	Meaning
1	Indexes a and b are equally important
3	Index a is slightly more important than b
5	Index a is obviously more important than b
7	Index a is strongly more important than b
9	Index a is extremely more important than b
2, 4, 6, 8	The degree of importance takes the middle value of two adjacent judgments
1, 1/2, …, 1/9	The ratio of the effects of a to b is opposite to the above

.

Consistency check based on the constructed judgment matrix; check formula is as shown in Equation (14).

$$CR=CI/RI$$

(14)

where $CI=({\lambda }_{max}-n)/(n-1)$, $CI$ is consistency check standard, ${\lambda }_{max}$ is maximal characteristic root of the judgment matrix.

$RI$ is an assessment of the average random consistency index, related to the order n of the judgment matrix (see

Table 2. The value of the random consistency index RI.

n	1	2	3	4	5	6	7	8	9
RI	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45

for details).

When the ratio $CR=CI/RI<0.10$ of random consistency is less than 0.10, it can be considered that the single sorting results of the hierarchy have good consistency; otherwise, the values of the elements of the judgment matrix must be readjusted.

The weight vector is obtained using the geometric average approach, and the elements of each row of the judgment matrix are multiplied separately, as shown in Equation (15).

$${\stackrel{-}{W}}_i=\sqrt[n]{{\prod }_{t=1}^n{x}_t}$$

(15)

The weights of each index $W_s$ are obtained by normalizing ${\stackrel{-}{W}}_i$, as shown in Equation (16).

$$W_i=\frac{{\stackrel{-}{W}}_i}{{\sum }_{t=1}^n{\stackrel{-}{W}}_i}$$

(16)

3.2.2. Determination of the Weight of Assessment Index using Improved Entropy Weight Method

The entropy weighting method is an objective weighting method [31,32], which is a method to determine the weight of an index from the objective data obtained. The entropy value of each index is calculated according to the degree of variation of each index; the index information entropy is inversely proportional to the degree of variation and weight. There are x assessment indexes, y tunnels to be assessed, and the normalized original assessment matrix $\mathrm{X}={\left({\mathrm{r}}_{ij}\right)}_{x\times y}, \left(i=\mathrm{1,2},3\dots ,x\right), \left(j=\mathrm{1,2},3\dots ,y\right)$ is obtained, as shown in Equations (17) and (18). First, the entropy value of each index needs to be found.

$$E_i=-\frac{1}{lnx}\sum _{j=1}^y{P}_{ij}\times ln{P}_{ij}$$

(17)

$$P_{ij}=\frac{r_{ij}}{{\sum }_{j=1}^y{r}_{ij}}$$

(18)

where $P_{ij}$ is the proportion of the j-th tunnel under the i-th assessment index. The entropy weight of each assessment index is calculated according to Equation (19).

$$w_i=\frac{1-E_i}{{\sum }_{i=1}^x(1-E_i)}$$

(19)

When calculating using the above formula, if the entropy value $E_i$ is close to 1, the slight difference of different indexes may cause a large change in the entropy weight. Some improvements to the above equation can reduce the errors arising from similar indexes [33], as shown in Equation (20).

$$w_i=\frac{exp \left({\sum }_{k=1}^x{E}_k+1-E_i\right)-exp E_i}{{\sum }_{l=1}^x[exp \left({\sum }_{k=1}^x{E}_k+1-E_l\right)-exp E_l]}$$

(20)

3.2.3. Determination of Comprehensive Weights

In order to make the difference degree of the index weight calculated by the AHP and the improved entropy weight method consistent with the difference of the corresponding distribution coefficient, in this paper, the distance function is used to calculate the comprehensive weight of the assessment index [34]. Specific steps are as follows:

$$d(w_i,W_i)=\sqrt{\sum _{i=1}^x{(w_i-W_i)}^2}$$

(21)

$$\left\{\begin{array}{c}d(w_i,W_i)=\alpha -\beta \\ \alpha +\beta =1\end{array}\right.$$

(22)

where the comprehensive weight of the assessment index i is the linear combination of the results calculated by the above AHP and the improved entropy weight method.

$$W_i^{*}=\alpha w_i+{\beta W}_i$$

(23)

4. Seismic Damage Risk Assessment Index System of Mountain Tunnel

4.1. The Assessment Index System

The seismic damage in the tunnel is the result of the combined action of many factors, insufficient surrounding rock strength or integrity caused by hazardous environments such as geological structure and unfavorable geology, which can easily destabilize the surrounding rock during earthquakes and cause disasters such as tunnel collapse. The stability of the surrounding rock is greatly affected by its own strength and integrity. Generally speaking, the lower the strength and integrity of the surrounding rock, the more likely it is to become unstable and produce a seismic hazard when an earthquake occurs [35]. Weathering will reduce the mechanical properties of the surrounding rock, and the degree of weathering has a great relationship with the stability of the surrounding rock [36]. In the case of loose stratum or large slope angle at the entrance area of the tunnel, the collapse of the slope and rockfall caused by the earthquake may lead to the cracking and damage of the tunnel portal and the blockage of the tunnel entrance [37].

The condition of tunnel lining itself, including tunnel defect, tunnel burial depth and size, etc., all determine whether the lining can resist the earthquake and maintain its integrity when an earthquake occurs. Tunnel defects, including cavities behind the lining, cracking, water seepage, etc. [38], will reduce the bearing capacity of the structure and thus reduce the seismic performance of the tunnel. It can be seen from the statistics of seismic damage of tunnels around the world that the magnitude of the seismic damage to the tunnel is related to the burial depth of the tunnel [39]. Tunnel earthquake damage survey shows that when the burial depth is greater than 100 m and the tunnel does not cross an active fault, the tunnel will suffer minor or no damage. Most of the tunnels with burial depths of less than 50 m have suffered some degree of damage. The span of the tunnel will also affect its seismic characteristics [16]. The earthquake intensity in the tunnel site area reflects the average level of the ground vibration intensity in the area when the earthquake occurs. The higher the earthquake intensity, the greater the potential for damage in the event of an earthquake, and unfavorable geology can exacerbate the destructive power of earthquakes [40].

In summary, the seismic damage risk assessment factors of mountain tunnels are divided into three categories: geological conditions of the tunnel site, the condition of the tunnel lining, geological structure and earthquakes. The engineering geological conditions include the surrounding rock classification, the weathering of surrounding rock and the slope angle of tunnel entrance. The condition of the tunnel includes the maximum burial depth of tunnel, tunnel defect, tunnel width, structural form and strength of the lining. Unfavorable geology and earthquakes include the length of shattered fault zone, the earthquake intensity of the tunnel site and the unfavorable geology, as shown in Figure 1.

4.2. The Seismic Damage Risk Assessment Level

Based on a large number of tunnel seismic damage investigations [3,4,5,39], combined with relevant specifications for tunnel seismic design and engineering practice, this paper defines the seismic damage risk assessment grades as four categories: low risk (I), moderate risk (II), high risk (III), and extremely high risk (IV), as shown in the

Table 3. Classification of the seismic damage risk level.

NO.	Assessment Index	Seismic Damage Risk Level
		Grade I (Low Risk)	Grade II (Moderate Risk)	Grade III (High Risk)	Grade IV (Extremely High Risk)
$c_1$	Surrounding rock classification (JTG 3370.1-2018) [41]	I, II	III	IV	V
$c_2$	Degree of weathering	No weathering, no impact on structure	Weak weathering, little impact on structure	Strong weathering, great impact on the structure	Full weathering, extreme impact on the structure
$c_3$	Slope angle of tunnel entrance/°	0~20	20~40	40~60	60~90
$c_4$	Maximum burial depth of tunnel/m	300~200	200~100	100~50	50~0
$c_5$	Tunnel defect	Structure in good condition	Structure is damaged in some places, and the degree of damage is small	Structure is damaged in some places, and the degree of damage is moderate	Structure is damaged in many places, the degree of damage is serious
$c_6$	Tunnel width/m	0~8	8~12	12~18	18~25
$c_7$	Unfavorable geological condition	No unfavorable geological condition	There is unfavorable geological condition, but the scale is small, and the impact on the safety of the tunnel is small	The scale of unfavorable geology is large, which has a great impact on the safety of the tunnel	The scale of unfavorable geology is large, which has an extreme impact on the safety of the tunnel
$c_8$	Length of shattered fault zone/m	0~5	5~20	20~50	50~100
$c_9$	Earthquake intensity	≤VI	VII	VIII	≥IX

.

4.3. Normalization of Assessment Index

The assessment indexes are normalized. In Table 3, there are many qualitative descriptions in the assessment indexes, such as $c_1,c_2,c_5,{c_7,c}_9$, which quantify the qualitative indexes; the specific quantitative criteria are shown in

Table 4. Quantitative criteria for qualitative indexes.

Number of Risk Assessment Index	Risk Level
	Grade I	Grade II	Grade III	Grade IV
$c_1$	1~2.5	2.5~3.5	3.5~4.5	4.5~6
$c_2,c_5,c_7,c_9$	0~2.5	2.5~5.0	5.0~7.5	7.5~10

. When quantifying seismic damage risk level according to surrounding rock classification, $c_1$ is the sum of the surrounding rock proportion multiplied by the grade.

In order to make different assessment indexes comparable with each other and facilitate scientific generalization, according to Equations (11)–(13), the indexes were normalized and dimensionless, and the processed ones are shown in

Table 5. Normalization of seismic damage risk assessment index parameters.

NO.	Assessment Index	Seismic Damage Risk Level
		Grade I (Low Risk)	Grade II (Moderate Risk)	Grade III (High Risk)	Grade IV (Extremely High Risk)
$c_1$	Surrounding rock classification (GB/T 50218-2014)	0~0.30	0.30~0.50	0.50~0.70	0.70~1.00
$c_2$	Degree of weathering	0~0.25	0.25~0.50	0.50~0.75	0.75~1.00
$c_3$	Slope angle of tunnel entrance/°	0~0.33	0.33~0.67	0.67~0.83	0.83~1.00
$c_4$	Maximum burial depth of tunnel/m	0~0.25	0.25~0.50	0.50~0.75	0.75~1.00
$c_5$	Tunnel defect	0~0.25	0.25~0.50	0.50~0.75	0.75~1.00
$c_6$	Tunnel width/m	0~0.32	0.32~0.48	0.48~0.72	0.72~1.00
$c_7$	Unfavorable geological condition	0~0.25	0.25~0.50	0.50~0.75	0.75~1.00
$c_8$	Length of shattered fault zone/m	0~0.05	0.05~0.20	0.20~0.50	0.50~1.00
$c_9$	Earthquake intensity	0~0.25	0.25~0.50	0.50~0.75	0.75~1.00

.

5. Application Example of Seismic Damage Risk Assessment Model for Mountain Tunnel Based on Extension Theory

On 21 May 2008, am earthquake with a magnitude of 8.0 occurred in Yingxiu Town, Wenchuan County, Sichuan Province (31.0° N, 103.4° E), which caused severe damage to the surrounding buildings. In this section, the seismic damage risk assessment method proposed in this paper is used to assess five tunnels (Longchi tunnel, Longxi tunnel, Longdongzi tunnel, Youyi tunnel, Maanshi tunnel) affected by the Wenchuan earthquake.

5.1. Matter-Element to Be Assessed, Classical Domain, Nodal Domain

Taking the five mountain tunnels affected by the Wenchuan earthquake as the research objects, the seismic damage risk assessment of the mountain tunnels proposed in this paper is carried out, the specific indexes of the five tunnels to be assessed are shown in the

Table 6. Indexes of the tunnel to be assessed.

Assessment Index	Longchi Tunnel	Longxi Tunnel	Longdongzi Tunnel	Youyi Tunnel	Maanshi Tunnel
Surrounding rock classification	III, IV, V	III, IV, V	III, IV, V	IV, V	III, IV, V
Degree of weathering	Strong weathering, great impact on the structure	Weak weathering, little impact on the structure	Strong weathering, great impact on the structure	Strong weathering, great impact on the structure	Weak weathering, little impact on the structure
Slope angle of tunnel entrance	40~60	35~55	40~70	30~60	25~55
Maximum burial depth of tunnel	470	830	173	215	130
Tunnel defect	Structural is damaged in some places, and the degree of damage is moderate, local leakage water in tunnel	The tunnel is still in the construction stage. No damage	Structural is damaged in some places, and the degree of damage is small	Structural equipment is damaged in some places, and the degree of damage is small	Structural equipment is damaged in some places, and the degree of damage is small
Tunnel width	12	10.4	10.4	8.5	10.5
Unfavorable geological condition	The tunnel traverses a 254 m long coal seam, where moderate-to-large deformation of the surrounding rock may occur	High ground stress, and moderate rockburst may occur	Dangerous rock collapse at the entrance of the tunnel	Large deformation of surrounding rock may occur locally in the shattered fault zone	Small collapse area at the entrance of the tunnel
Length of shattered fault zone	55	65	60	1.5	8
Earthquake intensity	8	8	7	7	7

below.

According to Table 3, Table 4 and Table 5, and Equations (1)–(3), the value range corresponding to the seismic damage risk index of grades I to IV is taken as the classical domain R_j; then, the nodal domain $R_p$ and the matter-element to be assessed, Longchi Tunnel $R_1$, Longxi Tunnel $R_2$, Longdongzi Tunnel $R_3$, Youyi Tunnel $R_4$, and Maanshi Tunnel $R_5$, respectively, are as follows:

$$\begin{gathered} R_p=\left(\begin{array}{ccc}P& c_1& \left[\mathrm{0,1}\right]\\ & c_2& \left[\mathrm{0,1}\right]\\ & c_3& \left[\mathrm{0,1}\right]\\ & c_4& \left[\mathrm{0,1}\right]\\ & c_5& \left[\mathrm{0,1}\right]\\ & c_6& \left[\mathrm{0,1}\right]\\ & c_7& \left[\mathrm{0,1}\right]\\ & c_8& \left[\mathrm{0,1}\right]\\ & c_9& \left[\mathrm{0,1}\right]\end{array}\right)R_1=\left(\begin{array}{ccc}{p}_1& c_1& 0.600\\ & c_2& 0.580\\ & c_3& 0.556\\ & c_4& 1.000\\ & c_5& 0.640\\ & c_6& 0.480\\ & c_7& 0.840\\ & c_8& 0.550\\ & c_9& 0.625\end{array}\right)R_2=\left(\begin{array}{ccc}{p}_2& c_1& 0.440\\ & c_2& 0.350\\ & c_3& 0.500\\ & c_4& 1.000\\ & c_5& 0.000\\ & c_6& 0.416\\ & c_7& 0.620\\ & c_8& 0.650\\ & c_9& 0.625\end{array}\right) \\ {R}_3=\left(\begin{array}{ccc}{p}_3& c_1& 0.560\\ & c_2& 0.600\\ & c_3& 0.611\\ & c_4& 0.432\\ & c_5& 0.320\\ & c_6& 0.416\\ & c_7& 0.380\\ & c_8& 0.600\\ & c_9& 0.375\end{array}\right)R_4=\left(\begin{array}{ccc}{p}_4& c_1& 0.700\\ & c_2& 0.660\\ & c_3& 0.500\\ & c_4& 0.283\\ & c_5& 0.320\\ & c_6& 0.340\\ & c_7& 0.350\\ & c_8& 0.015\\ & c_9& 0.375\end{array}\right)R_5=\left(\begin{array}{ccc}{p}_5& c_1& 0.520\\ & c_2& 0.380\\ & c_3& 0.444\\ & c_4& 0.567\\ & c_5& 0.280\\ & c_6& 0.420\\ & c_7& 0.140\\ & c_8& 0.050\\ & c_9& 0.375\end{array}\right) \end{gathered}$$

5.2. Correlation Function Value and Comprehensive Weight Calculation

According to Equations (4)–(6), the correlation function value of the seismic damage risk level of the tunnel to be assessed is obtained as shown in

Table 7. The value of the correlation function.

	Indicators	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$	${\mathit{c}}_7$	${\mathit{c}}_8$	${\mathit{c}}_9$
Tunnel
Longchi	KI	−0.429	−0.440	−0.337	−1.000	−0.520	−0.250	−0.787	−0.526	−0.500
	KII	−0.200	−0.160	0.335	0.240	−0.280	0.000	−0.680	−0.438	−0.250
	KIII	0.500	0.320	−0.204	−1.000	0.440	0.000	−0.360	−0.100	0.500
	KIV	−0.200	−0.288	−0.382	−1.000	−0.234	−0.333	0.360	0.100	−0.250
Longxi	KI	−0.241	−0.222	−0.254	−1.000	0.000	−0.188	−0.493	−0.632	−0.500
	KII	0.300	0.400	0.500	0.240	−1.000	0.400	−0.240	−0.563	−0.250
	KIII	−0.120	−0.300	−0.254	−1.000	−1.000	−0.133	0.480	−0.300	0.500
	KIV	−0.371	−0.533	−0.398	−1.000	−1.000	−0.422	−0.255	0.300	−0.250
Longdongzi	KI	−0.371	−0.467	−0.419	−0.290	−0.179	−0.188	−0.255	−0.579	−0.250
	KII	−0.120	−0.200	0.174	0.240	0.280	0.400	0.480	−0.500	0.500
	KIII	0.300	0.400	−0.132	−0.310	−0.360	−0.133	−0.240	−0.200	−0.250
	KIV	−0.241	−0.273	−0.360	−0.467	−0.573	−0.422	−0.493	0.200	−0.500
Youyi	KI	−0.571	−0.547	−0.254	−0.104	−0.179	−0.056	−0.222	0.300	−0.250
	KII	−0.400	−0.320	0.500	0.240	0.280	0.125	0.400	−0.700	0.500
	KIII	0.000	0.360	−0.254	−0.402	−0.360	−0.292	−0.300	−0.925	−0.250
	KIV	0.000	−0.209	−0.398	−0.567	−0.573	−0.528	−0.533	−0.970	−0.500
Maanshi	KI	−0.314	−0.255	−0.204	−0.423	−0.097	−0.192	0.440	0.000	−0.250
	KII	−0.040	0.480	0.335	0.240	0.120	0.375	−0.440	0.000	0.500
	KIII	0.100	−0.240	−0.337	−0.305	−0.440	−0.125	−0.720	−0.750	−0.250
	KIV	−0.273	−0.493	−0.465	−0.461	−0.627	−0.417	−0.813	−0.900	−0.500

below.

Using AHP and improved entropy weight method, the subjective weight and objective weight of nine assessment indexes of seismic damage risk level of mountain tunnel are obtained, where the weights of the AHP were calculated from the results of a questionnaire survey carried out by 20 tunneling and geological experts. There is a certain difference in the assessment index weights determined by the improved entropy weight method and the analytic hierarchy process. It is too unilateral to rely solely on pure mathematical theory or pure engineering experience for the seismic damage risk assessment of such complex mountain tunnels. Therefore, combining the subjective weight and the objective weight, the comprehensive weight is calculated according to Equations (20)–(22), which has positive significance for improving the rationality of the assessment results. From the comprehensive weight, it can be seen from

Table 8. Determination of comprehensive weights.

Assessment Indexes	Weights
	AHP	Improved Entropy Weight Method	Comprehensive Weights
Surrounding rock classification	0.026	0.104	0.057
Degree of weathering	0.028	0.105	0.059
Slope angle of tunnel entrance	0.056	0.105	0.075
Maximum burial depth of tunnel	0.084	0.110	0.094
Tunnel defect	0.143	0.123	0.135
Tunnel width	0.060	0.104	0.077
Unfavorable geological condition	0.174	0.113	0.150
Length of shattered fault zone	0.192	0.131	0.168
Earthquake intensity	0.237	0.105	0.185

that the weight of tunnel defect, unfavorable geology, length of shattered fault zone and earthquake intensity is relatively high, and they are more important in assessing seismic damage risk.

5.3. Correlation Function Value and Comprehensive Weight Calculation

According to Equation (7), the correlation degree of the seismic damage risk of each tunnel i with respect to the level j is obtained:

$$K_{ij}\left(p\right)=\left(\begin{array}{cccc}-0.558& -0.232& 0.019& -0.184\\ -0.427& -0.180& -0.169& -0.378\\ -0.328& 0.166& -0.173& -0.344\\ -0.151& 0.101& -0.353& -0.549\\ -0.096& 0.145& -0.412& -0.605\end{array}\right)$$

The larger the $K_j\left(p\right)$ value of the matter-element p to be evaluated, the smaller the deviation of the index from the assessment level; if $K_{j0}\left(p\right)=\max \left[K_j\left(p\right)\right(j=\mathrm{1,2},\dots ,m\left)\right]$, then the seismic damage risk of the tunnel to be evaluated is level J₀. According to the calculation results in

Table 9. Correlation degree of tunnel seismic damage risk.

No.	Tunnel	KI (p)	KII (p)	KIII (p)	KIV (p)
1	Longchi tunnel	−0.558	−0.232	0.019	−0.184
2	Longxi tunnel	−0.427	−0.180	−0.169	−0.378
3	Longdongzi tunnel	−0.328	0.166	−0.173	−0.344
4	Youyi tunnel	−0.151	0.101	−0.353	−0.549
5	Maanshi tunnel	−0.096	0.145	−0.412	−0.605

, the seismic damage risk levels of the five tunnels to be assessed can be obtained. Among them, the seismic damage risk level of Longxi tunnel and Longchi tunnel is Level III (high risk), and the seismic damage risk level of Longdongzi tunnel, Youyi tunnel and Maanshi tunnel is Level II (moderate risk), According to Equations (8) and (9), the eigenvalues of the seismic damage risk level of the tunnels can be obtained: $j_1^{*}=3.04, j_2^{*}=2.64, j_3^{*}=2.22, j_4^{*}=1.84, j_5^{*}=1.78$; this indicates that the seismic damage risk level of Longchi tunnel is “high risk” biased towards “extremely high risk”, the seismic damage risk level of the Longxi tunnel is “high risk” biased towards “moderate risk”, the seismic damage risk level of the Longdongzi tunnel is “moderate risk” biased towards “high risk”, and the seismic damage risk level of the Youyi tunnel and Maanshi tunnel is “moderate risk” biased towards “low risk”. The five tunnels are sorted according to the eigenvalues of the seismic damage risk level as follows: Longchi tunnel > Longxi tunnel > Longdongzi tunnel > Youyi tunnel > Maanshi tunnel.

Longchi tunnel crosses four large-scale shattered fault zones, and large cracks appeared at the retaining wall. Longitudinal cracks, circumferential cracks and diagonal cracks appeared in the lining of the portal entrance, with localized falling blocks and exposed steel reinforcement (Figure 2a). Longitudinal cracks appeared in the vault, arch foot and side walls of the tunnel body section, and diagonal cracks appeared near the tunnel crossing the shattered fault zone (Figure 2b,c). Pavement protrusions and dislocations were observed in some sections (Figure 2d).

Longxi tunnel is located between the Yingxiu Fault (F3) and the Longxi Fault (F2), crossing the F8 Fault, as well as a number of minor secondary shattered faults. The tunnel was damaged for a length of 4500 m. The lining of the portal section was severely cracked (Figure 3a), the reinforcement was exposed (Figure 3b), and obvious dislocations up to 20 cm were produced at the construction joints (Figure 3c). Circular and longitudinal cracks appeared in the tunnel body section (Figure 3d), accompanied by water leakage (Figure 3e). The second lining and waterproof plate collapsed (Figure 3g), and the surrounding rock was destabilized in the severely damaged areas, producing an overall collapse (Figure 3h). The seismic damage in Longchi tunnel and Longxi tunnel is in accordance with the evaluation result of “high risk”.

The slope at the portal of Longdongzi tunnel collapsed; thus, the damage to the entrance of the tunnel was relatively large, and cracks and water leakage appeared in some parts of the lining (Figure 4a–c), which essentially conformed to the assessment results of “moderate risk” biased to “high risk”. The Youyi tunnel (Figure 4d–f) and Maanshi tunnel (Figure 4g–i) were slightly affected by the earthquake, which was mainly manifested in local cracks and water leakage, cracking and spalling of the secondary lining. The seismic damage is in accordance with the assessment result of “moderate risk”.

In general, the seismic damage investigation is essentially consistent with the assessment results. Therefore, the conclusions obtained using the seismic damage risk assessment method proposed in this paper are consistent with the actual situation and have great practical value, which can also be applied to the seismic damage risk assessment of similar projects.

6. Conclusions

In this paper, the extension theory is introduced to evaluate the seismic damage risk of mountain tunnels, taking into account the geological conditions of the tunnel site, the tunnels’ state, structural conditions and seismic damage risk factors. The assessment process includes the establishment of the risk index system, the calculation of the weight of the risk index, the division of the risk level, the quantification of the tunnel to be assessed, and the classification of the assessment results. The following conclusions were obtained:

• A comprehensive approach to weighting using a combination of improved entropy weighting method and AHP method can effectively avoid the excessive influence of subjective factors in the assignment process and fully exploit the information in the sample data. Numerically speaking, tunnel defect, unfavorable geology, length of shattered fault zone and earthquake intensity have higher weights and are more important in assessing seismic damage risk.
• Five tunnels affected by the Wenchuan earthquake were assessed for seismic damage risk. Among them, the Longchi tunnel is locally defective and large deformation of the surrounding rock may occur, leaving it at a significantly higher risk of seismic damage. The seismic damage risk level of Longchi tunnel is “high risk” biased towards “extremely high risk”. The Longxi tunnel crosses the longest shattered fault zone and has high ground stresses; thus, it has a “high risk” biased towards “moderate risk” for seismic damage. The seismic damage risk level of the Longdongzi tunnel is “moderate risk” biased towards “high risk”. The seismic damage risk level of the Youyi tunnel and Maanshi tunnel is “moderate risk” biased towards “low risk”. The five tunnels are sorted according to the eigenvalues of the seismic damage risk level as follows: Longchi tunnel > Longxi tunnel > Longdongzi tunnel > Youyi tunnel > Maanshi tunnel.
• Combined with the seismic damage investigation of the Wenchuan earthquake, the Longchi tunnel and the Longxi tunnel suffered the most serious seismic damage during the earthquake, which conforms to the assessment result of “high risk”. Longdongzi tunnel, Youyi tunnel and Maanshi tunnel suffered less seismic damage, which conforms to the assessment result of “moderate risk”. The assessment results of this paper are essentially consistent with the actual tunnel damage. The accuracy and effectiveness of the method are verified.