A Quantitative Evaluation Method Based on Back Analysis and the Double-Strength Reduction Optimization Method for Tunnel Stability

Quantifying tunnel stability using the proposed combination of back analysis and the strength reduction method (SRM) is useful during construction. To feasibly and reliably obtain geotechnical parameters for the surrounding rock (which vary in diﬀerent places), a real-coded genetic algorithm is used in setting the initial parameters of the neural network to improve the prediction accuracy of the parameters via back analysis by reasonably selecting the selection operator, crossover operator, and mutation operator. After obtaining the parameters, the proposed SRM, i.e., the optimization double-strength reduction method (ODSRM), which is based on the optimization method, is used to evaluate stability. By using this method, the cohesion and friction angle have diﬀerent reduction factors that are more reasonable and accurate. The combined method is veriﬁed in an application to the Yu Liao Tunnel, where it is demonstrated that the combined method can use the measured displacements to obtain the safety factor. Compared with the traditional method, the proposed back analysis method can reduce errors in the predicted performance, and unlike the SRM, the ODSRM can avoid overestimating the safety factor with the same reduction factor. Finally, the presented methods can reduce the amount of calculation required and are convenient for evaluating tunnel stability with displacement.

In the traditional SRM, the cohesion c and friction angle φ are reduced by the same reduction factor; for this approach, the cohesion and friction angle are treated identically as damage to the engineered structure is gradually accrued. However, according to some researchers [14][15][16][17], the two indices have different effects and reduction rates in the destabilization process. To address this issue, many scholars have adopted the double-strength reduction method (DSRM), which separately reduces the cohesion and friction angle by different coefficients. For example, the shortest path of strength reduction was proposed by Isakov and Moryachkov [18], Wang has applied this method to shallow buried tunnels [19], and Yuan [20] and Bai [21] each studied the DSRM.
Rock mass characteristics have significant effects on the stability of the tunnel through the rock mass; however, the values of the rock mass parameters are very difficult to determine in practice, and the use of inaccurate values directly affects stability analysis. Although in situ measurement is the most popular method for obtaining these values, it consumes considerable resources and time and can be conducted only in certain engineering projects [22,23]. To obtain these values feasibly and reliably using limited amounts of measurement data, the most commonly used and efficient method is called back analysis to infer "cause" from "result" [24,25]. e back analysis in geotechnical engineering can be divided into displacement back analysis, stress back analysis, strain back analysis, and groundwater seepage field back analysis. Displacements are often measured because of their reliability and convenience; thus, back analysis based on displacements has been studied by many scholars [26][27][28][29][30]. Artificial neural networks (ANNs) are typically used in back analysis and can provide an appropriate approach to representing the nonlinear relationship between rock mass characteristics and displacements [31].
ANNs are made up of many neurons, which reflect the intensity of their connections through weights, and many neurons connect with each other to form complex neural networks. Because of the different types of neurons, the means of connection and the means of learning, ANNs can be divided into many types. e BP-ANN is an algorithm based on the least-squares method, which is realized by minimizing the mean square error between the output value and the expectation of an ANN. Before the error threshold is reached, the network is constantly iterative, and each iteration includes the forward propagation of information and the reverse propagation of error. As shown in Figure 1, datasets u, e, and y are used as samples of input and output, and multiple sets of training data can be collected to train neural networks. Before training, the weight of the network is initialized by a random number. e input data are calculated and processed by the input layer, the hidden layers, and the output layer to obtain the output data. e output data are then compared with the expectation to gain the mean square error, and the inverse propagation stage of the error is conducted until the error is satisfied. During the process, the weights between the neurons of each layer are adjusted. e function of the error decreases along the direction of negative gradient. e iterations continue according to the above steps until the final error is less than the threshold or the number of iterations reaches the upper limit to stop the training. e weight value after training is used as the final weight of the neural network, and the neural network can be used for back analysis and calculation. To search for estimated parameter values in a large space, the genetic algorithm (GA) and the simulated annealing technique are the often used algorithms of greatest efficiency [32,33].
However, conventional GAs are prone to low computational efficiency or local optimal value problems, so it is necessary to improve the efficiency of back analysis and calculation and avoid the emergence of a local optimal value. In the present paper, the real-coded genetic algorithm (RCGA), which is better than GA due to its binary code and is seldom used in the back analysis of geotechnical parameters [34], is selected to determine the initial parameters of an ANN. To avoid a local optimal value and increase computational efficiency, the selection operator, crossover operator, and mutation operator are recombined. e proposed approach can be used to determine the weight value of an ANN and is called the RCGA-ANN. erefore, a new DSRM, coupled with the back analysis, is proposed to evaluate the stability of a tunnel. An optimization method, called the mode search method, is used to iteratively search for reasonable values for the two indices; this approach, called the optimization double-strength reduction method (ODSRM), does not require much calculation or a fixed proportion, which are commonly encountered issues of the shortest-path method. is paper presents the details of a new approach for calculating the indices to evaluate tunnel stability.
A combination of the RCGA-ANN and ODSRM is proposed to feasibly and reliably obtain geotechnical parameters for the surrounding rock and evaluate the stability of the surrounding rock. e descriptions of the methods are provided as follows.

Displacement Back Analysis Based on the RCGA
Displacement back analysis can be a convenient approach for determining rock mass characteristics, which are difficult to determine via field measurement [22,35,36]. Uniform design [37] is used to obtain the training samples for back analysis. In this paper, to improve the performance by solving a shortcoming of an ANN (i.e., that the characteristics of a local minimum could be obtained) [38], the RCGA is used to determine initial network weights; this approach is called the RCGA-ANN.

Method Improvement.
e main purpose of this section is to propose an efficient hybrid GA based on the RCGA.
is method can be used in the process of building the ANN to obtain the reasonable connection weights, has high computing efficiency, and avoids local optimal values. e characteristics of this method can be used to identify the geological parameters.
In a traditional GA, each variable in an individual is generally represented by the binary digits 0 and 1. is encoding method performs well in solving problems with low precision or low dimension, but it takes a long time to calculate and requires a sophisticated computer configuration for problems with high precision or high dimension. Because each variable is represented by real numbers in the RCGA, this method can be used to solve this problem, which has been widely applied and considered. In particular, when addressing the optimization of continuous space, the RCGA is obviously superior to GA binary coding [38].
is section primarily develops a hybrid GA based on real-number coding to optimize the network weight parameters of the neural network back analysis method. Based on the traditional GA, to avoid local optimization and improve computational efficiency, the hybrid algorithm is used to select the population by using the sorting selection method. Two newer crossover operators are fused to form a new crossover strategy. To ensure the diversity of the population, a dynamic stochastic mutation operator is selected in the algorithm. Because the initial weights of neural networks are generated by random numbers, the final weights may lead to a local optimal value. e RCGA is introduced in this paper to optimize the weights of neural networks. e obtained neural network is embedded by the RCGA, which assists in the back analysis. e main ideas are as follows: (1) the weight parameters of the neural network are trained repeatedly using the training dataset to produce the initial population (each individual is one combination of the weight value of the neural network); (2) each individual is taken as the weight value of the neural network to calculate the fitness of the testing dataset; (3) the contemporary population is determined via selection, crossover, and mutation operations, due to the corresponding probability and calculation factors, to obtain a new population; and (4) an iterative calculation is performed until the adaptability of the population can meet the expected standards. Because the next generation of stocks can inherit the excellent characteristics of the parent generation, the overall evolution of the population tends to the optimal solution. e calculation process for this method is shown in Figure 2, and the calculation steps are as follows: (i) Step 1. Initial network weights are determined via the RCGA. At the beginning of the RCGA, the generation with P(g) individuals (i.e., with P(g) choices of initial network weights in the ANN) is randomly generated, where g is the iteration or generation with g � 0. (ii) Step 2. Based on the networks obtained via the training samples, the ANN reads in the training samples, testing samples, and initial network weights (for network training by the training samples) and predicts the testing samples. e fitness of the predicted results is then evaluated according to the given adaptation function.
e individuals undergo reproduction, crossover, and mutation operations according to their fitness, and the individuals of the new generation, P(g + 1), are obtained. e generation is g � g + 1.
(iv) Step 4. Determine whether a predetermined evolutionary number (PEN) is reached (e.g., by reaching the end of a calculation) and return the individual with the highest fitness; otherwise, return to Step 3 until the termination condition is satisfied.
Step 5. Obtain the optimization parameters of the networks.

Selection Operation.
Roulette wheel selection (RWS) and tournament selection (TS) are the two approaches most frequently used in the RCGA. However, RWS is slow because of the heavy computation required in each generation, and TS can miss potential solutions and obtain a local optimum. Ranking selection (RS) is used in this paper because it has the benefits of the above approaches but is more efficient [39]. First, individuals are ranked by their fitness. en, pN low-ranking individuals are discarded, and pN highranking individuals are generated; p is a proportional parameter in interval (0, 0.5]. e RS function is illustrated in Figure 3 on a population of eight individuals with respect to various proportions. e population size is from 10 to 200.

Crossover Operation.
By combining the advantages of various crossover operators, the search ability of the new RCGA can be improved [38]. We selected two real-coded crossovers: the simulated binary crossover (SBX) [40] and directed crossover (DCX) [41]. Crossover probabilities (P c or P s ) are used to select a crossover from 0 to 1.
Based on relevant research [40], the search power and performance of the SBX used in the real-coded algorithm are better than those of the single-point crossover used in the binary-coded algorithm. e steps of the algorithm are as follows: where ξ i and η i are two offspring individuals obtained using the SBX, β i is the spread factor, u is a uniform random number in the range of [0, 1], and x 1 i andx 2 i are two parent individuals produced by the selection function. In this paper, η equals 20 [42].
To improve the accuracy of the solution, the directed crossover operator is introduced because it can conduct a local search among the best individuals of the current generation. X h is an individual selected at random.m individuals that are better than X h are selected. X c is the centroid of the m individuals selected and is calculated as follows: where m is the number of individuals that are better than X h . New offspring X n is randomly generated by equation (3): where α is a random number between 0 and 1.0.

Mutation Operation.
To prevent the population from converging to a local optimum, an excellent new mutation operator-that is, the dynamic random mutation operator (DRM)-is used in the algorithm [39]. e probability of mutation P m is from 0.0001 to 0.1.
where X * i is the offspring after mutation; X U and X L are the upper and lower bounds of the optimization variable, respectively; Φ 0 is a random perturbation vector of n-dimensional space [−ϕ 0 , ϕ 0 ] n ; ϕ 0 is a user-specified number in the range of (0, 1]; and s m is the mutation step size obtained using equation (6)

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where b is the decay rate parameter of s m and k and k max represent the current generation number and maximum generation number, respectively.

Fitness
Computation. e training error of the individual is used as the fitness function to illustrate the difference between experimental values and the corresponding quantities obtained via the finite element module (FEM); it is defined as where n is the number of test values, Y i exp is the ith experimental value of the measurement points, and Y i num is the ith numerical value of the measurement points.
where c is a conservative estimate of f(x). In this paper, c equals 0.

Optimization of the Double-Strength Reduction Method.
Assume that the use of the Mohr-Coulomb failure criterion and the characteristic strength properties of the cohesion c and friction angle φ undergo a subsequent incremental decrease to c ′ and tan φ ′ , respectively. e stability of the surrounding rock of a tunnel is considered to be affected by a combination of c and φ; that is, when either c or φ loses function, the other index acts as a form of resistance. e surrounding rock becomes unstable only when neither of the two indices can resist the sliding force.
Using the Mohr-Coulomb failure criterion in the tunnel project, the shear strength indices, c and tan φ, are separately reduced, as shown as equation (9): where k 1 is the reduction factor of c, k 2 is the reduction factor of tan φ, c ′ is the factored cohesion, and φ ′ is the factored friction angle. Two issues should be solved in the DSRM: (1) separately reducing c and tan φ and (2) calculating F s . eoretically, there are an infinite number of shear strength (c ′ , tan φ ′ ) values; thus, the purpose of this paper is to propose an approach with fewer calculations to obtain the only reasonable shear strength (c ′ , tan φ ′ ).
Based on Pan Jiazheng's extremum principle [43] and the Hooke-Jeeves algorithm [44], a direct search algorithm, the ODSRM, is proposed. e characteristic strength properties, cohesion c and friction angle φ, which are followed by an incremental decrease in c ′ and tan φ ′ , are obtained using the method summarized as follows: (i) Initialization step. (ii) Let u 1 , ·u 2 , . . . , u n be the coordinate direction.
Choose scalar ε > 0 (to terminate the algorithm), initial step length Δ > 0, and starting point Set Y (0) � X (0) and k � j � 1. en, go to the main step. (iii) Main step Step , then the trial is a success; let j , and go to Step b.
Step b: if j < n, then j is replaced by j + 1, and Step a is repeated.
Otherwise, replace k with k + 1, set j � 1, and repeat Step a.
In the above, Steps a and b describe an exploratory search, and Step c is used to determine whether to terminate or continue the algorithm.

Definition of Safety Factor.
According to the research of Bai [45] regarding the definition of the DSRM, the safety factor of SRM (F s ) can be represented by reduction factor k as follows: where the shear strength of the geotechnical materials is denoted as τ, and the shear strength under the limit state after reduction is denoted as τ ′ .

Instability Criterion.
e instability criteria used in the SRM are classified into three main categories [20]: (1) the displacement discontinuity of the characteristic measured location; (2) the interpenetration of the plastic zone; and (3) the divergence of the finite element model iteration. In this paper, the displacement discontinuity is selected as the instability criterion.

Engineering Background.
e Yu Liao Tunnel, one of three long tunnels designed in the Yong-Tai-Wen (Ningbo-

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Taizhou-Wenzhou) Expressway, has a 5870 m left tunnel and a 5865 m right tunnel. is tunnel is located in a hilly region with complex geology. e entry section is shallowly covered by a residual body and weathered bedrock with a loose structure. e level of the surrounding rock in the entry section is level V. For the main body of the tunnel, the geology is optimal because the amount of underground water is low (i.e., the main level of the surrounding rock is level III).
According to the relevant requirements, one inclined shaft is located in each tunnel (i.e., left and right) to divide the main tunnel into two sections for construction. Two inclined shafts are located on the right side of the tunnel. e length of number 1 is 520 m, the shaft intersects the left hole at ZK 365 + 317.8, and the right hole horizontal intersection is at YK 365 + 310. e length of number 2 is 550 m, and the shaft intersects the left hole at ZK 365 + 357.8 and the right hole at YK 365 + 350. e layout of the tunnel is shown in Figure 4.

Field Measurement.
e total station is used to measure the construction process of the tunnel. e total station of the V-grade surrounding rock is 10 m per monitored section, that of the IV-level surrounding rock is approximately 30 m per monitored section, and that of the III-level surrounding rock is approximately 50 m per monitored section. A survey point of each section is arranged at the preset point section after tunnel excavation and at the test pile buried in B, C, D, and E, as shown in Figure 5.

Construction of the Finite Element Model and the Material
Parameters.
e section located at ZK 362 + 400 is selected to build the FEM. is point is in a region with low mountain slopes and is covered with boulders of approximately 2-4 m in thickness. e compressive strength R c ranges from 61.1 to 72.5 MPa, with a mean value of 66.8 MPa.
e surrounding rock is tuff. e search scopes of the five unknown parameters are shown in Table 1.
GTS NX is used to construct the FEM; the size of the tunnel is shown in Figures 6 and 7, which also show the model and the mesh discretization with a hybrid mesher.

Acquisition of Datasets.
e uniform design method, proposed by mathematicians Kaitai Fang and Yuan Wang, is a type of experimental design from the uniformity and is based on the uniform distribution of the test point within the test range [46]. e essence of all test design methods, including uniform design, is to identify representative points within the experimental data range, with the aim of reducing the workload of the test. e method can select the representative test points from the comprehensive test points and distribute them evenly within the test range, which can reflect the main characteristics of the whole. Compared with the orthogonal design, the uniform design method achieves uniform dispersion and reduces the number of tests but does not achieve neat and comparable characteristics. In particular, when the range of test factors is large and many horizontal tests are needed, the uniform design can achieve the effect as long as the test of the level of the factor is equal to the number of factors, which greatly reduces the workload because the number of orthogonal tests is equal to the horizontal number square test.
To reduce the model computation, the uniform design method is selected when determining the training dataset. As shown in Table 2, the value scheme of five geotechnical parameters is determined by using the uniform design method, and the range of values is shown in Table 1. Using the finite element model constructed in Section 4.3 and the geotechnical parameter scheme in Table 2, the deformation parameters including the vault settlement, BC convergence, and DE convergence of each geotechnical parameter scheme are obtained. e obtained dataset is used as the training data of the back analysis, as shown in Table 2. To test the accuracy of the back analysis neural network obtained by using the training dataset, five sets of data are taken as the testing data, and the displacement deformation value of the feature points is obtained by using the numerical simulation method, as shown in Table 3.

Construction of the RCGA-ANN.
e scheme parameters used as the training set and the measuring point deformations are shown in Table 2. e input data include the three displacement parameters, and the output data include the five geotechnical parameters.
To calculate the weight value of the neural network and to prevent the absolute value of a few data points that are too large from overwhelming the small values in the dataset, all training samples are standardized to a range of −1.0 to 1.0. e dimensional data can be converted into dimensionless data, so that the unequal use of variables of different physical meaning and different dimensions can be avoided. us, the link weights range from −1.0 to 1.0. e equation is expressed as equation (11).
wherex ′ is the normalized data, x is the original data, x min is the minimum of the original data, x mean is the mean of the original data, and x max is the maximum of the original data. e training data in Table 2 are normalized using the above method, as shown in Table 4. In addition, the normalized calculation rules established by the data in Table 2 are used to normalize the test data in Table 3, and the results are shown in Table 5.
Based on the normalized training dataset (Table 4), the structure of the optimal neural network structure of 3-20-10-5 is trained; that is, the input parameters are 3, the output parameters are 5, and the intermediate implicit layer is 2 layers that contain 20 and 10 nodes collectively. To improve the accuracy of the calculation and avoid the problem of the local optimal value, the RCGA is calculated to determine the connection weight. 6 Advances in Civil Engineering

Parameters Obtained via Back Analysis.
In this section, the field measurement data of the vault settlement and the horizontal convergence of ZK 362 + 400 are used in the back analysis; A � 6.91 mm, BC � 7.23 mm, and DE � 1.35 mm. rough the above calculation, we obtained the relevant geotechnical parameters of the test point. e proposed strength parameters obtained via back analysis are given as follows: effective friction angle equals 49.5°, effective cohesion equals 104 kPa, unit weight equals 23.6 kN/m 3 , Young's modulus equals 3.8 GPa, and Poisson's ratio equals 0.246. By obtaining the initial parameters using the RCGA, the performance is improved and surpasses 100%, as shown in Figure 8.

Stability Evaluation of the Surrounding Rock.
After obtaining the parameters of the surrounding rock, the ODSRM is used to calculate the safety factor to evaluate the stability of the surrounding rock. As shown in Table 6, Δ � 0.1, and the cohesion and friction angle are reduced one by one. e geotechnical parameters in the FEM (Figure 8) are obtained in Section 4 via the RCGA-ANN. In the first reduction step, denoted as S1, the parameters are set to φ � 49.5°and c � 104 kPa; then, the vault settlement is calculated. Calculation steps numbers 2, 3, and 4 are the cohesion reduction step, friction angle reduction step, and combined reduction step, respectively. According to the comparison of the vault settlement of steps numbers 2, 3, and 4, the sequence number of S2 is number 4. e reduction is performed until the vault settlement has a significant increase. After 25 reduction iterations, the process with Δ � 0.1 is terminated; then, the result of the safety factor of section F s � 2.06 is obtained, as shown in Figure 9.
rough the process with Δ � 0.05 in the ODSRM, Δ � 0.2 in the ODSRM, and Δ � 0.1 in the SRM, as shown in Figures 10-12    According to the comparative analysis of the four conditions, we obtain two results. First, the safety factor gained by using the ODSRM is smaller than that obtained by using the SRM with the same reduction factors; i.e., the SRM overestimates safety. Second, the reduction factor can affect the safety factor because, when a reduction factor is too high, it could result in a suboptimal selection of one of the two reduction parameters.

Conclusions
In conclusion, a new double-strength reduction method, called the optimization double-strength reduction method (ODSRM), is proposed and verified. e proposed method can reduce the cohesion and friction angle by different reduction factors. e ODSRM is applied in the Yu Liao Tunnel, where it is demonstrated that the combined method can use the measured displacements to obtain the safety factor. rough the establishment of a numerical model, the vault and the horizontal convergence under different values of various parameters are calculated. e calculated data are divided into the training and testing dataset, and the proposed method is used to get the parameters of the neural network. And then the safety factor can be calculated. Compared with the previous SRM, the ODSRM can consider the different effects of these parameters and avoid overestimating the safety factor.

Data Availability
e experimental data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.