Prediction of Tunnel Earthquake Damage Based on a Combination Weighting Analysis Method

Abstract

To reduce or evaluate the damage of tunnels in seismically active areas when earthquakes happen, it is very important to quickly predict the tunnel damage. This study proposes an anti-entropy–fuzzy analytic hierarchy process (FAHP) combination weighting method for tunnel earthquake damage prediction. The tunnel cross section is a symmetrical structure. The method uses tunnel damage data from the tunnels in a region where earthquake disasters have occurred as sample data to calculate the standard earthquake damage index. The weights of evaluation factors are determined by combining the FAHP and anti-entropy weighting. The correction coefficient of each evaluation factor is obtained by considering the degree of each evaluation factor’s influence on the average damage index. Then, the earthquake damage and the corresponding damage degree of each tunnel are obtained by weighting calculation. In this study, 55 tunnels in the Wenchuan earthquake-affected area are taken as analysis cases. In these cases, 45 cases of damage tunnels are used as sample data, and 10 random tunnels are used as training cases. The calculated results are compared with the observed results. The proposed method is confirmed simple and easy to implement, which can greatly reduce the workload of field investigation, calculation and analysis. The results is of great significance to the rapid earthquake emergency assessment and post earthquake recovery of tunnels.

1. Introduction

With the continuous development of economic and national defense construction in various countries, the total number of tunnels constructed worldwide is rapidly rising, and tunnels have become an important part of traffic engineering construction [1]. In the early stages of tunnel construction, seismic design was not considered [2]; it was believed that the constraint of soil layer ensured the safety of a tunnel [3]. However, with the occurrence of a series of earthquake-influenced cases, it has been proven that the impact of earthquakes on tunnels is significant [4]. For example, the Chi-Chi earthquake in Taiwan [5], Wenchuan earthquake in China [6], mid-Niigata Prefecture earthquake in Japan [7], and the latest Norcia earthquake in Italy [8] all resulted in tunnel damage. Figure 1 shows examples of tunnel damage due to the 2008 Wenchuan earthquake. Research for earthquake prediction is required to reduce tunnel damage and economic loss in earthquake-affected areas and provide references for post-earthquake recovery, which has important practical significance for tunnel engineering.

Earthquake damage prediction originated in the 1930s. To establish the earthquake insurance industry, the United States carried out a series of basic studies on earthquake damage prediction [10]. Earthquake damage prediction is a key part of earthquake research [11]. Timely and effective earthquake prediction and evaluation and corresponding prevention and control measures have been considered by scholars [12]. At present, the existing earthquake prediction methods include the historical earthquake damage statistical method, semiempirical and semitheoretical method, fuzzy analogy, expert evaluation, structural calculation, and dynamic analysis. The statistical historical earthquake damage method [13] is used to predict earthquake damage by statistical analysis and empirical judgement. The semiempirical and semitheoretical method uses strict mathematical methods to address the relationship between variables based on observations, statistics, and summaries of earthquake damage. It has a wide range of applications but high requirements for the adaptability of theoretical models. The fuzzy analogy method [14] uses fuzzy mathematics theory for prediction and has the same applicability as the earthquake damage statistical method but provides a more accurate prediction result. Expert evaluation [15] is based on the judgement of experts’ experience on an earthquake. It has high reliability when data samples are not complete [16]. In recent years, a great deal of research work has been performed on structural damage prediction based on artificial neural networks [17,18,19]. Intelligent systems also have received recognition for their ability to solve extremely complicated and multidimensional problems [20].

The influences of different factors are prone to induce error in the earthquake prediction methods, as mentioned above [21]. The calculation method uses ground motion theory to conduct seismic nonlinear analysis on the complex mechanical model of a structure, and soil can elucidate the whole elastic and plastic reaction process of the structure until failure [22,23]. Zheng et al. [24] introduced the strain energy density ratio (SEDR) principle and proposed an earthquake damage prediction method that can accurately predict crack locations and extension directions in lining structures. A method which could predict the exact location of damage by introducing enhanced damage indicator through a flexibility index was also produced [25]. The calculation accuracy of these methods is high, but a considerable amount of theoretical preparation and experimental research is necessary, so these methods are not applicable to many tunnels for quick prediction or assessment. The dynamic analysis method, considering the change in structural resistance with time, is more suitable to be combined with other methods because of the time effect [26]. However, the process of tunnel failure during earthquakes is not clear [27] at present. In terms of the prediction of the tunnel failure process during earthquakes, some research methods include many assumptions or few considered factors, and the mutual relationship between the factors is not deeply discussed, all of which result in certain limitations.

To solve the above problems, this study proposes a prediction method based on the statistical characteristics of tunnel damage in earthquakes. This method uses tunnel damage data as sample data from a region where earthquake disasters have occurred. The standard earthquake damage index of a tunnel is calculated under the tunnel damage classification system, and the anti-entropy–fuzzy analytic hierarchy process (FAHP) combination weighting method is proposed to combine weights of the predicted evaluation factors. The correction coefficient of each evaluation factor is calculated to correct the weights by considering the degree of each evaluation factor’s influence on the average damage index. Finally, the weighted calculation of the average earthquake damage index of the tunnel and its corresponding damage level are calculated. This method can comprehensively consider various factors of tunnels when earthquakes occur and clarify the relationship between these factors and the correction coefficient of a single factor. The study can improve the reliability of tunnel earthquake damage prediction and attract attention for establishing detailed data platforms of tunnel damage due to earthquakes in the future.

2. Prediction Method of Tunnel Earthquake Damage

2.1. The Standard Earthquake Damage Index of Tunnels

The standard earthquake damage index of structures provides a quantitative description of the damage level of buildings under an earthquake and the seismic vulnerability of all kinds of buildings in a city or region [28]. In this study, a tunnel damage quantification standard [29] is used to classify the threshold values of the standard earthquake damage index of different damage grades of tunnels. The method of triangular fuzzy evaluation was used.

Table 1. Tunnel damage classification with description [29].

ID	Damage Level	Damage Description
1	None	No damage detectable by visual inspection
2	Slight	Light damage (w < 3 mm, l < 5 m), seepage of underground water, small rock falls at portals, no effect on traffic
3	Moderate	Small spalling, cracking of linings (3 mm < w < 30 mm, 5 m < l < 10 m), leaking of underground water, rock topple and slide at portals
4	Severe	Large spalling, cracking of linings (w > 30 mm, l > 10 m), exposed reinforcement, displacement of lining (w > 20 cm), heave or differential movement of tunnel pavement and invert (w > 20 cm), inrush of underground water, deep slides and large slumps of rock/soil masses at portals
5	Collapse	Collapse of tunnel

w = width of crack; l = length of crack.

shows the classification description of tunnel damage.

The standard earthquake damage index of a tunnel is a quantitative description of the earthquake damage level of the structure [30]. Generally, a value between 0 and 1 is used to express the earthquake damage degree of a tunnel. A value of 0 means that the tunnel has no damage detectable by visual inspection, and a value of 1 means that the tunnel is completely destroyed. In this section, the comprehensive energy efficiency evaluation is divided into five failure grades (P1, P2, P3, P4, and P5), which are no damage, slight damage, moderate damage, severe damage, and collapse, respectively. A triangular membership function is combined with a semitrapezoidal function to describe the damage. Formula (1) establishes the membership function corresponding to each failure grade.

Table 2. Tunnel earthquake damage grades corresponding to the standard earthquake damage index.

Damage Grade	None	Slight	Moderate	Severe	Collapse
Median standard earthquake damage index	0.015	0.100	0.25	0.5	0.9
Limit values of standard earthquake damage index	[0, 0.05]	(0.05, 0.15]	(0.15, 0.30]	(0.30, 0.65]	(0.65, 1.00]

gives the earthquake damage index corresponding to the five failure levels.

$$\begin{array}{l}{\phi }_{p1}\left(n_i\right)=\{\begin{array}{l}0,\begin{array}{cc}& g_{n_i}\le f_4\end{array}\hfill \\ \left(g_{n_i}-f_4\right)/\left(f_5-f_4\right)\hfill \\ 1,\begin{array}{cc}& g_{n_i}\ge f_5\end{array}\hfill \end{array},\begin{array}{cc}& f_4<g_{n_i}<f_5\end{array}\\ {\phi }_{p2}\left(n_i\right)=\{\begin{array}{l}\left(g_{n_i}-f_3\right)/\left(f_4-f_3\right),\begin{array}{cc}& f_3<g_{n_i}<f_4\end{array}\hfill \\ \left(f_5-g_{n_i}\right)/\left(f_5-f_4\right),\begin{array}{cc}& f_4\le g_{n_i}<f_5\end{array}\hfill \\ 0,\begin{array}{cc}& g_{n_i}\le f_3,g_{n_i}\ge f_5\end{array}\hfill \end{array}\\ {\phi }_{p3}\left(n_i\right)=\{\begin{array}{l}\left(g_{n_i}-f_2\right)/\left(f_3-f_2\right),\begin{array}{cc}& f_2<g_{n_i}<f_3\end{array}\hfill \\ \left(f_4-g_{n_i}\right)/\left(f_4-f_3\right),\begin{array}{cc}& f_3\le g_{n_i}<f_4\end{array}\hfill \\ 0,\begin{array}{cc}& g_{n_i}\le f_2,g_{n_i}\ge f_4\end{array}\hfill \end{array}\\ {\phi }_{p4}\left(n_i\right)=\{\begin{array}{l}\left(g_{n_i}-f_1\right)/\left(f_2-f_1\right),\begin{array}{cc}& f_1<g_{n_i}<f_2\end{array}\hfill \\ \left(f_3-g_{n_i}\right)/\left(f_3-f_2\right),\begin{array}{cc}& f_2\le g_{n_i}<f_3\end{array}\hfill \\ 0,\begin{array}{cc}& g_{n_i}\le f_1,g_{n_i}\ge f_3\end{array}\hfill \end{array}\\ {\phi }_{p5}\left(n_i\right)=\{\begin{array}{l}1,\begin{array}{cc}& g_{n_i}<f_1\end{array}\hfill \\ \left(f_2-g_{n_i}\right)/\left(f_2-f_1\right),\begin{array}{cc}& f_1\le g_{n_i}<f_2\end{array}\hfill \\ 0,\begin{array}{cc}& g_{n_i}\ge f_2\end{array}\hfill \end{array}\end{array}$$

(1)

Thus, the standard earthquake damage index of a tunnel can specifically refer to the average earthquake damage degree of the tunnel in a certain area under a specific intensity. The average earthquake damage index can be used as the standard earthquake damage index to assess tunnel earthquake damage. Under different earthquake intensities, the calculation formula of this standard earthquake damage index is defined as follows:

$$D_{ib}={\displaystyle \sum _{j=1}^5{D}_j\times P\left(D_j/I\right)}$$

(2)

where $D_{ib}$ is the benchmark earthquake damage index of the tunnel under an earthquake intensity of $I$, $D_j$ is the median earthquake damage index with $j$ level damage, and $P\left(D_j/I\right)$ is the probability of grade $j$ failure of the tunnel under an earthquake intensity of $I$.

Figure 2 shows the framework of tunnel earthquake damage prediction proposed in this study. For predicting the earthquake damage of a tunnel under a certain earthquake intensity corresponding to earthquake damage index $D_i$, the following steps can be followed: (1) determine a city or region that exhibits tunnel damage due to earthquakes under the intensity of the standard index $D_{ib}$; (2) obtain the weight of the predicted evaluation factors ${\omega }_m$ by the anti-entropy–FAHP combination weighting method, considering the main influencing factors of the tunnel and the influence degree of different factors on the seismic capacity of the tunnel; (3) analyze the predicted evaluation factors and different impact factors to obtain the correction coefficient of each evaluation factor $k_{jn}$, and then use the comprehensive weighting method to calculate the comprehensive impact factors of the tunnel; and (4) obtain the predicted tunnel earthquake damage index $D_i$ by combining the above-mentioned results with the standardized coefficient of the evaluation factors $X_m$. The calculation formula of the tunnel earthquake damage index $D_i$ is as follows:

$$D_i=D_{ib}\times {\displaystyle \sum _{i=1}^n{\omega }_m{k}_{jn}}{X}_m$$

(3)

2.2. Selection of Predictive Evaluation Factors

The influencing factors of tunnel structures in earthquakes are very complex [31], and there may be correlations among them, so the form and degree of tunnel earthquake damage caused by different factors are also different [32]. This is the main reason why the evaluation results of seismic capacity of a tunnel vary [33]. Other reasons include the complexity of the surrounding soil and the tunnel structure. The earthquake damage site observation and the analysis results of structural mechanics suggest that the designed seismic standard, earthquake intensity, surrounding rock, construction method, epicentral distance, length and width of fault, etc., also influence the tunnel earthquake damage [34]. The presented method in this study can select relevant factors from sample data as evaluation factors according to the actual condition.

3. Combination Weighting Method

3.1. Fuzzy Analytic Hierarchy Process

The fuzzy analytic hierarchy process (FAHP) is a method of fuzzy mathematics with an analytic hierarchy process [35]. In this study, the subjective weight is determined by FAHP. The FAHP establishes the fuzzy complementary judgement matrix. The fuzzy complementary judgement matrix is transformed into the fuzzy consistency judgement matrix through mathematical transformation, and its weight is calculated using the normalization method while the consistency test is carried out.

In the FAHP, when different factors are compared and judged in pairs, the fuzzy judgement matrix can be obtained according to the importance of one factor to another if the matrix has the following properties:

$$\begin{gathered} a_{ii}=0.5,\begin{array}{cc}& i=1,2,\dots ,n\end{array} \\ {a}_{ij}+a_{ji}=1,\begin{array}{cc}& i,j=1,2,\dots ,n\end{array} \end{gathered}$$

(4)

The matrix is called the fuzzy complementary judgement matrix. To quantitatively describe the importance of the comparison between the two different factors, the importance is usually quantified by the scale method, as presented in

Table 3. The meaning of the scale value of the fuzzy judgement matrix.

Intensity of Weight	Definition	Explanation
0.5	Equal importance	Two risk factors contribute equally to the objective
0.6	Weak importance	Experience and judgement slightly favor one risk factor over another
0.7	Essential importance	Experience and judgement strongly favor one risk factor over another
0.8	Demonstrated importance	One risk factor is strongly favored and demonstrated in practice
0.9	Extreme importance	The evidence that favors one risk factor over another is of the highest possible order of affirmation
0.1, 0.2, 0.3, 0.4	Reverse comparison	$\mathrm{If}\mathrm{element}{a}_i$$\mathrm{is}\mathrm{compared}\mathrm{with}\mathrm{element}{a}_j$$\mathrm{to}\mathrm{obtain}\mathrm{the}\mathrm{judgement}{r}_{ij},$$\mathrm{then}\mathrm{the}\mathrm{judgement}\mathrm{obtained}\mathrm{by}\mathrm{comparing}\mathrm{element}{a}_i$$\mathrm{with}\mathrm{element}{a}_j$$\mathrm{is}{r}_{ji}$$=1-r_{ij}$

. Thus, judgement matrix A of the evaluation index is constructed as follows:

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]$$

(5)

This study uses a general formula to solve the weight of the fuzzy complementary judgement matrix, which fully includes the excellent characteristics and judgement information of the fuzzy complementary judgement matrix, requires only a small amount of calculation, and is very convenient in practical applications. Its specific solution formula is as follows:

$${\omega }_i=\frac{{\displaystyle \sum _{j=1}^n{a}_{ij}+\frac{n}{2}-1}}{n\left(n-1\right)},\begin{array}{cc}& i=1,2,\dots ,n\end{array}$$

(6)

Whether the weights calculated by the above formula are reasonable still needs to be tested for consistency. Here, the consistency is tested according to the compatibility of the fuzzy judgement matrix. The method of judging consistency is as follows:

(1)

Assuming that both A and B are fuzzy judgement matrices, then

$$I\left(A,B\right)=\frac{1}{n^2}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n\left|a_{ij}-b_{ij}\right|}}$$

(7)

(2)

The above formula is called the compatibility index of A and B

Let $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)}^T$ be the weight vector of fuzzy judgement matrix A, ${\displaystyle \sum _{i=1}^n{\omega }_i=1,{\omega }_i\ge 0\left(i=1,2,\dots ,n\right)}$, and let ${\omega }_{ij}=\frac{{\omega }_i}{{\omega }_i+{\omega }_j},\left(\forall i,j=1,2,\dots ,n\right)$; then, the n-order matrix ${\omega }^{\ast }={\left({\omega }_{ij}\right)}_{n\times n}$ is the eigenmatrix of judgement matrix A.

When the compatibility index $I\left(A,B\right)$ is less than or equal to the attitude T from the decision maker, the judgement matrix is considered to have satisfactory consistency. The smaller T is, the higher the consistency requirement of decision makers for the fuzzy judgement matrix. Generally, T = 0.1 is recommended.

Generally, because a certain decision requires the participation of more than one expert, the above method can also be extended to group decision-making. When m experts give the fuzzy complementary judgement matrix $A^q={\left(a_{ij}^q\right)}_{n\times n},\left(q=1,2,\dots ,m\right)$. The weight of matrix $A^q\left(q=1,2,\dots ,m\right)$ is calculated, and the consistency test is carried out. It is theoretically proven that when the fuzzy complementary judgement matrix passes the consistency test, its comprehensive judgement matrix also meets the consistency test requirements. Then, the average value of M weight set is taken as the weight distribution vector of the factor set.

3.2. Anti-Entropy Weighting Method

According to fuzzy mathematics theory, a 5-level system can accurately describe the object of evaluation. In this section, tunnel damage is divided into 5 damage levels (P1, P2, P3, P4, and P5), and the evaluation index set is as follows:

$$L=\{None,Slight,Moderate,Severe,Collapse\}=\left\{l_1,l_2,l_3,l_4,l_5\right\}$$

According to the index in the evaluation system, the evaluation matrix of the object to be evaluated can be obtained from the evaluation set:

$$B=\left|\begin{array}{cccc}{b}_{11}^{}& b_{12}^{}& \cdots & b_{1n}^{}\\ b_{21}^{}& b_{22}^{}& \cdots & b_{2n}^{}\\ \cdots & \cdots & & \cdots \\ b_{m1}^{}& b_{m2}^{}& \cdots & b_{mn}^{}\end{array}\right|$$

(8)

Since there are two types of evaluation indices, the larger the better type and the smaller the better type, there are different standardized treatment methods for different types of indicators. For the larger the better type, the standardized treatment indices are as follows:

$$b_{ij}=\frac{b_{ij}^{\prime }-\min b_{ij}^{\prime }}{\max b_{ij}^{\prime }-\min b_{ij}^{\prime }}\begin{array}{cc}& i=1,2,\dots ,m;j=1,2,\dots ,n\end{array}$$

(9)

For the smaller the better type, the standardized treatment indices are as follows:

$$b_{ij}=\frac{\max b_{ij}^{\prime }-b_{ij}^{\prime }}{\max b_{ij}^{\prime }-\min b_{ij}^{\prime }}\begin{array}{cc}& i=1,2,\dots ,m;j=1,2,\dots ,n\end{array}$$

(10)

After standardized treatment, the standardized evaluation matrix is obtained as follows:

$$B^{\prime }=\left|\begin{array}{cccc}{b}_{11}^{\prime }& b_{12}^{\prime }& \cdots & b_{1n}^{\prime }\\ b_{21}^{\prime }& b_{22}^{\prime }& \cdots & b_{2n}^{\prime }\\ \cdots & \cdots & & \cdots \\ b_{m1}^{\prime }& b_{m2}^{\prime }& \cdots & b_{mn}^{\prime }\end{array}\right|$$

(11)

The entropy weighting method with objective weights is adopted in this calculation, but because tunnel earthquake damage data are scarce, the traditional entropy method for determining the index sensitivity will result in a weight that the accuracy is insufficient due to the restriction of sample data [36]. Thus, the anti-entropy weighting method is used for calculation. For each index in the evaluation system, the greater its influence on the degree of tunnel damage, the greater the earthquake damage, and the smaller the corresponding post-earthquake safety entropy, namely, the lower the disorder degree of the system. The disadvantage of each index can be calculated by using the post-earthquake safety entropy of the evaluation index; that is, the objective weight of each index can be obtained according to the existing data of the evaluation index.

According to the definition and calculation formula of entropy, the inverse information entropy of each evaluation factor can be defined as follows:

$$H_n=-{\displaystyle \sum _{i=1}^m{h}_{n_i}\ln \left(1-h_{n_i}\right)}$$

(12)

where $h_{n_i}=\left(g_{n_i}\right)/{\displaystyle \sum _{i=1}^m}\left(g_{n_i}\right)$. Then, the objective weight of each evaluation can be calculated according to the following formula:

$${\omega }_j=\frac{H_n}{{\displaystyle \sum _{j=1}^m{H}_n}}$$

(13)

3.3. Combination of Empowerment

Subjective weighting is easily affected by the difference in decision makers’ subjective judgement, leading to the deviation of weights. However, the basic principles of the objective weighting method are mostly statistical, which may be inconsistent with the indicators. To avoid the limitation of single weighting and improve the reliability of weights, the combination of subjective weight and objective weight is applied. The weight obtained through this way combines the actual measurement samples and considers the requirements of regulations and standards, avoiding the equalization of weight [37]. Combination weighting adopts the multiplication synthesis method. This method multiplies the weights obtained by the subjective and objective weighting method, and then normalizes the weights to obtain the combination weights. The calculation formula is as follows:

$${\omega }_j^{\ast }=\frac{{\displaystyle \prod _{r=1}^R}{\omega }_j^r}{{\displaystyle \sum _{j=1}^k\left(\prod _{r=1}^R{\omega }_j^r\right)}}$$

(14)

To avoid the contradiction of weights calculated by different methods, a consistency test is carried out before combination weighting, and a distance function $d\left({\omega }^{(1)},{\omega }^{(2)}\right)$ is adopted to describe the degree of consistency. When $0\le d\left({\omega }_1,{\omega }_2\right)\le 1$, the weight obtained by the two weighting methods passes the consistency test, and the calculation formula of the distance function is as follows:

$$d\left({\omega }^{(1)},{\omega }^{(2)}\right)={\left[\frac{1}{2}{{\displaystyle \sum _{i=1}^n\left({\omega }_i^{(1)}-{\omega }_i^{(2)}\right)}}^2\right]}^{\frac{1}{2}}$$

(15)

4. Correction of Evaluation Factor

The evaluation factor is predicted to represent the influence degree of a certain influencing factor on the damage of tunnels under a given earthquake intensity. Some factors can improve the seismic resistance of tunnels. For example, a tunnel with a shock absorption layer has stronger seismic resistance than a tunnel without a shock absorption layer. Some factors may reduce the seismic capacity of the structure, such as a low-strength concrete segment that greatly limits the seismic performance of a tunnel. Therefore, the effect of each factor on the seismic capacity of tunnels cannot be generalized.

The earthquake damage matrix of tunnels in a region can comprehensively reflect the overall seismic capacity of the tunnels in the region, and the average earthquake damage index at each intensity can represent the seismic level of such structures at the corresponding seismic intensity. Based on this, it is necessary to consider the influence of each single evaluation factor on the average earthquake damage index. Firstly, the data of two cases in the monomer sample database are calculated; one is to remove the factor and the other is not to remove the factor. Then the contribution of the two cases to the earthquake damage matrix are compared to obtain an impact factor. The ratio of the two cases is the correction factor of the predicted evaluation factor, as shown in Figure 3. The specific calculation steps are as follows:

(1)

Establish an earthquake damage database and earthquake damage prediction database for various building structures in different regions.

(2)

Select different types of tunnels in the database and calculate the average damage index under a certain earthquake intensity $D_i$.

(3)

Under the same conditions as above, consider the contribution of a certain predicted evaluation factor to the whole tunnel; that is, remove the structure containing this factor from the sample data, and then calculate the average earthquake damage index $D_{i\left[j\right]}$.

(4)

Calculate the relative changes in the earthquake damage index under steps (2) and (3); the modification coefficient of the n-th influencing factor and the j-th evaluation factor were considered for the structure $k_{jn}$. The calculation formula is as follows:

$$k_{jn}=\frac{D_i}{D_{i\left[j\right]}}$$

(16)

5. Case Study of Tunnels in Wenchuan Earthquake in China

5.1. Parameter Selecting

(1)

Earthquake parameters

The main earthquake parameters relevant to tunnels are earthquake intensity and epicentral distance. In conjunction, these parameters determine the earthquake intensity in a particular area with a higher earthquake intensity and a shorter epicentral distance. The earthquake will be more intense in an area and exert a stronger influence on tunnels.

(2)

Tunnel parameters

The section design of expressway tunnel is different from that of a common road tunnel. The cross-section area of expressway tunnel is larger than those of common road tunnels. Previous research shows that the peak stress of the tunnel increases with an increase in tunnel diameter, so the type of tunnel structure is also an important factor. There are great differences in stiffness between tunnels under construction and tunnels in operation. The integrity of the operating tunnel is obviously better than that of the tunnel under construction. The tunnel under construction is easy to collapse directly under the earthquake because the tunnel structure has not fully supported the surrounding rock.

(3)

Ground parameters

For mountain tunnels, most sections are located in the intact rock ground with great earthquake resistance ability. However, many tunnel sections have to pass through faults or faulted zones. As can be seen from a tunnel damage survey [13], among the 10 tunnels that suffered severe damages, 8 tunnels passed through faults or faulted zones. Tunnels located in regions of adverse geology are often subjected to loosening or plastic earth pressure before an earthquake takes place. The ground deformations developed in adverse geological regions during an earthquake are generally large and could lead to unsatisfactory seismic performance on tunnels. In these cases, the fault number would increase the risk of tunnel damage under earthquake. Therefore, ground parameters are necessary.

5.2. Prediction Calculation

In this section, the mountain tunnels around the location of the Wenchuan earthquake in 2008 are taken as a case study, and 55 tunnels [13,29] in the earthquake damage area are selected to verify this method. In these tunnels, 45 samples are randomly selected as sample data for analysis.

Table 4. Sample data of tunnel damage in the Wenchuan earthquake.

Earthquake Intensity	Tunnel Number	None	%	Slight	%	Moderate	%	Severe	%	Collapse	%
VI	8	7	87.5	1	12.5	0	0	0	0	0	0
VII	8	2	25	6	75	0	0	0	0	0	0
VIII	11	2	18.2	7	63.6	2	18.2	0	0	0	0
IX	5	1	20	0	0	4	80	0	0	0	0
X	5	0	0	0	0	5	100	0	0	0	0
XI	8	0	0	0	0	0	0	2	25	6	75

and

Table 5. Sample data of tunnel damage in the Wenchuan earthquake tunnel.

Tunnel Name	Adverse Geology	Earthquake Intensity	Operation/Construction	Tunnel Type	Epicentral Distance (km)	Length (m)	Fault Number (m)	Observed Damage
Longxi	Y	XI	C	ET	2.3	3691	1	5
Jiujiaya	Y	XI	C	CRT	9.6	2285	6	5
Youyi	Y	XI	O	CT	2.4	950	3	5
Zipingpu	Y	XI	C	ET	7.3	4081	10	5
Baiyunding	Y	XI	O	CRT	6.5	450	2	5
Shaohuoping	Y	XI	C	ET	1.3	450	1	5
Zaojiaowan	Y	XI	O	CRT	6.6	1926	0	4
Longchi	Y	XI	O	CRT	7.2	1177	4	4
Gengda	Y	IX	C	CRT	17.5	938	/	3
Chediguan	Y	IX	O	CRT	18.2	403	0	3
Taoguan	Y	IX	O	CRT	22.5	625	/	3
Caopo	Y	IX	O	CRT	26	759	0	3
Sanpanzi	Y	VIII	O	CRT	13.2	282	/	3
Guanyazi	Y	VIII	O	ET	43	696	/	3
Maanshi	Y	X	O	CRT	7.3	399	1	3
Maojiawan	N	X	O	CRT	12.3	381	0	3
Panlongshan	Y	X	C	CRT	14	300	/	3
Futang	N	X	O	CRT	18.5	2365	0	3
Dankanliangzi	Y	X	O	CRT	35	/	1	3
Shiziping	Y	VI	C	CRT	>50	601	/	2
Qujiapo	Y	VII	O	ET	33	/	0	2
Mafu	Y	VII	C	CRT	>50	/	/	2
Tianjiaba	N	VII	C	CRT	>50	/	/	2
Caomigang	Y	VII	C	CRT	>50	/	/	2
Hongfu	Y	VII	C	CRT	>50	/	0	2
Wangong	N	VII	C	CRT	>50	/	0	2
Yinxin	N	VIII	O	CRT	12	/	/	2
Huayanzi	Y	VIII	C	CRT	18	570	/	2
Chenjiashan#1	N	VIII	O	CRT	21.1	/	/	2
Fenshuiling	Y	VIII	O	ET	22	1300	/	2
Qiujiapo	Y	VIII	O	CRT	33	/	/	2
Mingyuexia	Y	VIII	O	CRT	38	1285	/	2
Feixianguan	Y	VIII	O	CRT	44.7	384	/	2
Xiquanyan	N	IX	O	CRT	28	150	/	1
Feishaguan	N	VII	O	CRT	30.5	100	/	1
Shitigou	Y	VII	O	ET	43	371	/	1
Xinjiagou	Y	VIII	O	ET	45	713	0	1
Shiwengzi	Y	VIII	O	ET	42	1300	/	1
Xiaoqiudi	Y	VI	O	CRT	>50	/	/	1
Jiujiapeng	N	VI	O	CRT	>50	/	/	1
Zagunao	N	VI	O	CRT	>50	/	/	1
Luobugang	N	VI	C	CRT	>50	/	/	1
Zhegushan	N	VI	O	CRT	>50	/	/	1
Shizikou	N	VI	O	CRT	>50	100	/	1
Xuequ#1	Y	VI	O	CRT	>50	200	/	1

Note: Y = yes, N = no, C = construction, O = operation, CRT = common road tunnel, ET = expressway tunnel.

show the earthquake damage data of the 45 samples. The evaluation factor of the main influencing factors of various tunnels in

Table 6. The standard earthquake damage indices of different grades.

Earthquake Intensity	VI	VII	VIII	IX	X	XI
The Standard Earthquake Damage Indices	0.034	0.081	0.1091	0.185	0.500	0.800

is not selected because the tunnel length and depth data are not complete. The fault data are not complete. Therefore, it is assumed that the fault number is 0 when fault data are blank. Finally, the tunnel damage in the earthquake is predicted by considering the geology, earthquake intensity, tunnel state (construction or operation), tunnel structure type, epicentral distance, and fault number as evaluation factors. Using Formula (2), we calculated 45 tunnels with different levels of baseline earthquake damage indexes, as shown in Table 6.

According to Formulas (9) and (10), the evaluation factors selected in the table are standardized to obtain the following matrix:

$$P^{\prime }=\left(\begin{array}{cccccc}0& 0& 0& 0& 0.0017& 0.0210\\ 0& 0& 0& 0.0263& 0.0071& 0.0093\\ 0& 0& 0.0313& 0.0236& 0.0018& 0.0163\\ 0& 0& 0& 0& 0.0055& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0.0667& 0.0417& 0.03125& 0.0263& 0.0374& 0.0233\\ 0& 0.0417& 0.0313& 0.0263& 0.0374& 0.0233\end{array}\right)$$

According to Formulas (12) and (13), the entropy weight of the influence factor can be calculated as follows:

$$\omega =\left[\begin{array}{cccccc}0.3531& 0.1409& 0.1446& 0.1243& 0.1342& 0.1030\end{array}\right]$$

Three experts who have long been engaged in seismic research of tunnels were consulted to score and judge the influence weights of the selected assessment factors. The subjective ranking of the experts is shown in

Table 7. Fuzzy hierarchical method expert of experts.

Category	Expert 1		Expert 2		Expert 3
	Intensity Importance	Important Ordering	Intensity Importance	Important Ordering	Intensity Importance	Important Ordering
Adverse geology	8	1	7	3	7	3
Earthquake intensity	7	2	9	1	8	1
Operation/construction	5	5	5	5	5	5
Tunnel type	4	6	4	6	4	6
Epicentral distance	7	2	8	2	8	1
Fault number	6	4	6	4	6	4

. According to the scores and order given by the experts, the following fuzzy complementary judgement matrix A is established according to the fuzzy judgement relation:

$$A=\left(\begin{array}{cccccc}0.50& 0.40& 0.80& 0.85& 0.45& 0.70\\ 0.60& 0.50& 0.80& 0.90& 0.85& 0.80\\ 0.20& 0.15& 0.50& 0.60& 0.20& 0.35\\ 0.15& 0.10& 0.40& 0.50& 0.15& 0.30\\ 0.55& 0.45& 0.65& 0.85& 0.50& 0.60\\ 0.30& 0.20& 0.65& 0.70& 0.40& 0.50\end{array}\right)$$

The subjective weights of the impact factors calculated according to Formula (3) are shown as follows:

$$\omega =\left[\begin{array}{cccccc}0.1900& 0.2167& 0.1333& 0.1200& 0.1817& 0.1583\end{array}\right]$$

According to the fuzzy judgement matrix of influencing factors, its characteristic matrix B can be calculated:

$$B=\left(\begin{array}{cccccc}0.500& 0.467& 0.588& 0.613& 0.511& 0.545\\ 0.533& 0.500& 0.619& 0.644& 0.544& 0.578\\ 0.412& 0.381& 0.500& 0.526& 0.432& 0.457\\ 0.387& 0.356& 0.474& 0.500& 0.398& 0.431\\ 0.489& 0.456& 0.577& 0.602& 0.500& 0.534\\ 0.455& 0.422& 0.543& 0.569& 0.466& 0.500\end{array}\right)$$

Formula (7) is used to calculate the compatibility index of the fuzzy judgement matrix of influencing factors and its characteristic matrix for consistency verification, so that $I\left(A,B\right)=0.087$ is obtained. The compatibility index is less than 0.1 after verification because the fuzzy judgement matrix is considered to be satisfactory and consistent, and its weight distribution is reasonable.

To avoid the conflicting weights calculated by different methods, a consistency test is carried out before the combination of weights, and the distance function in Formula (15) is adopted, thus, $d\left({\omega }^{(1)},{\omega }^{(2)}\right)=0.135\in \left(0,1\right)$ passes the consistency test. The multiplicative synthesis method of Formula (14) is adopted to multiply the weights obtained by the subjective and objective weighting method and then normalize the weights to obtain the combined weights, as shown in

Table 8. Subjective, objective, and combined weights.

Category	Weight
	Subjective Weight	Objective Weight	Combination Weight
Adverse geology	0.1900	0.3531	0.3889
Earthquake intensity	0.2167	0.1409	0.1770
Operation/construction	0.1333	0.1446	0.1118
Tunnel type	0.1200	0.1243	0.0865
Epicentral distance	0.1817	0.1342	0.1413
Fault number	0.1583	0.1030	0.0945

.

Formula (16) is used to consider the impact factor correction coefficient because of the limited sample seismic data. The existence of adverse geology and the epicentre distance was considered as a single factor. After removing the factor of adverse geology and epicentre distance from the sample data, only 13 tunnels do not have adverse geology, 13 tunnels have an epicentral distance more than 50 m, and 18 tunnels have an epicentral distance more than 40 m. Thus, this step may induce some error, so the two factors need to be corrected, and the correction coefficient is 1.0. The correction coefficients of each evaluation factor are shown in

Table 9. Correction coefficients of evaluation factors.

	Adverse Geology	Earthquake Intensity	Operation/Construction	Tunnel Type	Epicentral Distance	Fault Number
Correction Coefficient	1.0000	1.0000	0.8002	0.8626	1.0000	0.5044

.

Finally, the earthquake damage index of the tunnel is calculated by using Formula (3), which was verified by the damage degree of another 10 random tunnels in the Wenchuan earthquake.

Table 10. Correction coefficients of evaluation factors.

Tunnel Name	Adverse Geology	Earthquake Intensity	Operation/Construction	Tunnel Type	Epicentral Distance (km)	Fault Number (m)	Damage Index	Predicted Damage	Observed Damage
Xuequ #2	N	VI	O	C	>50	/	0	1	1
Baixionggou	N	VI	O	C	>50	/	0	1	1
Shunhe	N	VII	C	C	>50	/	0.0101	2	2
Banyang	N	VII	C	C	>50	/	0.0101	2	2
Qinlinpo	N	VIII	O	C	40	/	0.0109	1	1
Qipanguan	N	VIII	O	C	>50	/	0.0077	1	1
Chenjiasha #2	Y	VIII	O	C	21.5	/	0.0589	2	2
Futangba	N	X	O	E	13	/	0.1610	3	3
Longdongzi	Y	XI	C	E	8	4	0.6950	5	5
Niujiaya	Y	XI	O	C	4	/	0.5567	4	4

Note: Y = yes, N = no, C = construction, O = operation, CRT = common road tunnel, ET = expressway tunnel.

gives the actual damage degree of selected tunnels after the Wenchuan earthquake, and the predicted damage degree calculated in this study. By comparison, the results of the predicted damage degree are completely consistent with the observed ones.

Due to the lack of tunnel damage data in earthquake-prone areas, the sample dataset in this study is small, and the selection of evaluation factors for the sample data is not sufficient. Due to incomplete data on the tunnel depths and other factors, the tunnel damage index is approximately calculated. However, this method has been shown as an effective tool to deal with uncertainties that arise from a lack of data. The method can reflect a finer model when the number of samples is sufficient. Therefore, the establishment of a tunnel earthquake damage database is very important in the subsequent prediction work of tunnel earthquake damage.

6. Conclusions

In this study, a new tunnel earthquake damage prediction method based on combination weighting is proposed. According to damage classification criteria and previously collected data, the standard earthquake damage indices of tunnels in a specific area are calculated and verified by the selected tunnel damage cases in the Wenchuan earthquake-affected area in China. Such an approach has been proven to be an effective tool to deal with uncertainties that arise from a lack of data. The following conclusions can be obtained:

(1) The tunnel damage prediction model was built using regional earthquake data considering the characteristics of the high complexity in terms of impact factors, and the performance of evaluation factors was determined by adopting the anti-entropy–FAHP combination weighting method. Considering the influence of each single factor, the correction coefficient of each single evaluation factor was obtained, and the tunnel earthquake damage index and the corresponding damage level under a certain intensity was examined. Compared to the traditional prediction approach, the advantage of the proposed anti-entropy-FAHP combination weighting method is that it could deal with imprecise, vague, and fuzzy data including uncertainties, and it is more practical for quick resilience assessment and decision making for the disaster prevention and recovery.

(2) The proposed method is simple and easy to implement, which can greatly reduce the workload of field investigation, calculation and analysis. However, the accuracy of the prediction results is related to the rationality of the selected earthquake damage matrix of the tunnel in the region, and the actual earthquake damage matrix of the region or its adjacent areas is usually adopted. This method can provide better results when the number of samples is sufficient. Therefore, the establishment of a tunnel earthquake damage database is very important in the subsequent prediction work of tunnel earthquake damage.

(3) The prediction of tunnel earthquake damage is a complex process, and the influencing factors and damage mechanisms need to be further analyzed. Therefore, discriminant analysis and comprehensive prediction should be combined with multisource data to make tunnel damage prediction more reliable in earthquakes and ensure valuable results for references.