The Application of the Strength Reduction Shortest Path Method to the Stability Analysis of Shallow Buried Tunnel

To improve the theoretical analysis of surrounding rock stability of shallow buried tunnels. The strength reduction shortest path theory is applied to the stability analysis of shallow buried tunnel surrounding rock, combined with the ultimate equilibrium strength reduction theory. We discussed the influence of the depth and span ratio of tunnel, cohesion, and internal friction angle on the shortest path of the strength reduction, and studied the effects of various factors on shallow buried tunnel safety relationship by using strength reduction factor of safety and the shortest path of the shallow buried tunnel surrounding rock and the grey relational analysis theory. The results show that: in the analysis of shallow buried tunnel in strength reduction, the approximate distribution obeys parabolic between reduction path length and the reduction ratio. When the strength reduction of cohesion is the shortest path the reduction rate is greater than the internal friction angle. The internal friction angle and cohesive force have a great influence on the stability of shallow tunnels under the method of shortest path of strength reduction. Finally, the comprehensive safety factor of shallow buried tunnels calculated by the finite element strength reduction shortest path method is greater than that calculated by the limit equilibrium method.


Introduction
The stability analysis of a shallow buried tunnel is an important theoretical basis to guide practical engineering. According to relevant statistics, some analysis methods proposed at present mainly include the limit equilibrium method, slip line method, limit analysis method and finite element method. The limit analysis and finite element analysis are the research hotspots at present (Wenxi and Liao 2010;Yiliang et al. 2016;Qing-qing 2009).
These two methods are mainly based on the strength reduction theory (Zienkiewicz et al. 1975;Wang et al. 2021;Sun et al. 2020), and the ultimate equilibrium state of the tunnel is achieved by reducing the shear strength parameters of rock-soil mass (Tschuchnigg et al. 2015). The traditional strength reduction method generally uses the same reduction coefficient for c and tanu of the shear parameters of rock and soil mass (Yonglin et al. 2011;Xinliang 2013). However, Fen Tang and Yingren Zheng et al.(Fen et al. 2007;Fen and Ying-ren 2008) believe that when the rock mass is damaged by sliding, the cohesion and internal friction angle of the soil mass play different roles, order, and degrees. On the other hand, the attenuation degree and speed of cohesive force c and internal friction angle u are also different, and different reduction coefficients can accurately reflect the respective safety reserves of c and u, and the concept of double reduction method is proposed accordingly. Wei et al. (2013), Bai bing, Yuan Wei, Li Xiaochun (2014), Zhao et al. (2015), etc. also proposed their own strength double reduction methods in the slope stability analysis. Bai Bing et al. (2015) pointed out that there should be an infinite number of reduction paths for double strength reduction, but ISAKOV (Isakov and Moryachkov 2014) believed that among all the reduction paths, there should be a path L min to make the slope from the initial state to the limit state fastest, that is, the shortest path of strength reduction. Xiaochun and Wei et al. (2016) also believed that in the slope stability analysis, cohesion and internal friction angle should be reduced along the path with the shortest distance from the ctanu critical curve in the process of attenuation failure.
At the same time, considering the complexity of soil parameters, the precision of the strength reduction method is affected by many factors. For this reason, Hammah et al. (xxxx) compared and analyzed the strength reduction method with the limit equilibrium method, and proved that Poisson's ratio and Young's modulus had little influence on slope stability. Jing Zhoubao (2015) used the ABAQUS strength reduction method to study and showed that the cohesion and internal friction angle have significant effects on slope stability, the dilatancy angle has relatively little effect, and the elastic modulus and Poisson's ratio have almost no effect. Liu et al. (2015) analyzed the influence of dilatancy angle and Poisson's ratio on the safety coefficient of the slope by strength reduction method and found that the influence of dilatancy angle and Poisson's ratio on the safety coefficient of the slope was small. However, at present, the strength reduction shortest path method is mainly used in the slope (Abra and Ftima 2020;Sun et al. 2016), and there are little researches on the tunnel.
Based on this, the strength reduction shortest path method is applied to the finite element strength reduction and ultimate equilibrium strength reduction analysis of shallow buried tunnels. The relationship between the length of the strength reduction path and the reduction ratio and the reasonable strength reduction ratio between the cohesion and the angle of internal friction under different parameters are discussed. The influence laws of buried depth and hole span ratio H/D, cohesion force c, and internal friction angle u on the shortest path of strength reduction was studied. Finally, the relationship between the reduction path, the safety coefficient of surrounding rock, and the grey correlation analysis theory are used to study the influence of various factors on the safety of the shallow buried tunnel. The research results enrich the analysis theory of the surrounding rock stability of shallow-buried tunnels and provide some guidance for the construction of the shallow-buried tunnel.

Strength Reduction Shortest Path Theory
The strength reduction shortest path theory (Ma et al. 2017;Peng et al. 2020) assumes that the cohesion and internal friction angle of rock and soil mass is reduced by different proportions. The reduction coefficient of cohesion isF c , and the reduction coefficient of internal friction angle is F u , the formula is as follows: where, c 0 and u 0 are the initial cohesion and internal friction angle, c SRT and u SRT are the reduced cohesion and internal friction angle. As shown in Fig. 1, assuming that the abscissa is 1=F c and the ordinate is 1=F u , the initial state M 0 of Fig. 1 Schematic diagram of the shortest path of strength reduction the tunnel and the line connecting the limit state line are reduced paths. There are countless strength reduction paths, and each reduction path is L 1 Á Á Á L n . There is a reduction path L k , the intersection point M k of which is the limit state line. Under this reduction path, the cohesion and the angle of internal friction adopt the same reduction coefficient. In this case, the reduction path is the traditional strength reduction path. Among all the reduced paths, there should be a path L min that causes the tunnel to reach the limit state at the earliest from the initial state. This path is the shortest reduced strength path.
The limit states under different reduction paths can be combined to obtain the limit state line shown in Fig. 1, which can be expressed as a function of At the same time, considering the uncertainty of the reduction path, we cannot make a quantitative analysis of it. At this time, the reduction ratio k ¼ F c F u is introduced, so that different strength reduction paths have a specific reduction ratio k corresponding to it. At this time, we can artificially assume different reduction ratios k to make the rock and soil mass be reduced according to different reduction paths, to quantitatively analyze the stability of the surrounding rock of the shallow buried tunnel.
From Fig. 1, it is easy to get the reduced path length from the initial state to the limit state of the tunnel under different reduced paths, the expression can be expressed as: The area outside the limit state line near the origin is the instability zone, while the area inside the limit state line near the initial state of the tunnel is the safety zone. At this time, according to the path length distribution, a functional expression of the strength reduction path length and the tunnel comprehensive safety factor can be established, as in Eq. (4). When the strength reduction path takes the smallest value L min , the minimum comprehensive safety factor can be obtained (Yonglin et al. 2011).

Finite Element Strength Reduction Shortest Path Analysis
According to the existing mechanism of strength reduction, its principle is to gradually reduce the c and u of rock and soil mass Nie et al. 2019). Regarding the non-proportional reduction of the cohesive force c and the internal friction angle u of the rock and soil, we can specify the magnitude of the ABAQUS midfield variable, and then determine the limit state of the tunnel surrounding rock according to the instability criterion. Finally, the safety coefficient of the surrounding rock of the tunnel is obtained by using the relation between the length of the strength reduction path and the comprehensive safety coefficient of the tunnel. As for the instability criterion of rock and soil mass, the commonly used methods at present mainly include the non-convergence criterion, plastic zone penetration criterion, and displacement mutation criterion (Wei et al. 2020;Aijun and Haihong 2013). In this paper, the sudden change in the settlement of the tunnel vault is selected as the criterion for the failure of the tunnel, and the safety factor at this time is selected for calculation and analysis.

Finite Element Example Analysis
For a shallow tunnel, the ratio of the initial buried depth H to the span D is 1, and both are 12 m, the tunnel height is 9.72 m, the surrounding rock mass density is 18 kN/m 3 , the elastic modulus is 1.3 Gpa, and the Poisson's ratio is 0.35. In the initial state, the cohesive force c 0 ¼ 120kPa, and the internal friction angle is u 0 ¼ 27 . As shown in Fig. 2, the overall model length 9 depth = 120 m 9 80 m, and the tunnel height 9 width = 9.72 m 9 12 m. The Mohr-Coulomb constitutive model is used in the calculation. The nodes at the bottom of the model limit the displacement in the X and Y directions, and the nodes on both sides limit the displacement in the X direction.
To make the distribution of strength reduction path more uniform as far as possible, this paper assumes that the value range of reduction ratio is 0.5 * 2.0, to realize the reduction of rock-soil mass parameters according to different reduction paths. Combining the results of finite element limit calculation and instability criterion, the strength reduction path length (expressed as the square of the length) under different reduction ratios can be obtained as shown in Fig. 3. With the increase of the reduction ratio, the length of the reduction path first decreases and then increases, and the minimum value occurs when k ¼ 1:5. At this time, the corresponding reduction path L min is the shortest path mentioned above, and the comprehensive safety factor of the tunnel is 1.5. By fitting the relevant data, the distribution between the reduction ratio and the path length is approximated by a parabola. Figure 4 shows the distribution of the plastic zone obtained by reducing the strength of the rock mass extracted in the finite element calculation according to the reduction ratio k ¼ 1:5. By connecting the points with the largest plastic deformation of the surrounding rock into a line, the potential failure surface of the surrounding rock can be approximated.

Effects of Different Parameters on Tunnel Strength Reduction Shortest Path
Previous studies have shown that elastic modulus, dilation angle, and Poisson's ratio have little influence on rock mass stability (Yuanfu et al. 2014;Ma et al. 2020a,b). This paper mainly studies the influence law of buried depth and hole span ratio H/D, cohesion c, and internal friction angle u on the shortest path of finite element strength reduction. Buried depth and hole span ratio (H/D) select five working conditions of 0.5, 0.75, 1.0, 1.25, and 1.5. Cohesion c 0 selects five working conditions of 80, 100, 120, 140, and 160 kPa. Five operating conditions of 17°, 19.5°, 22°, 24.5°, and 27°are selected for the initial internal friction angle u 0 . The specific calculation conditions are shown in Table 1, and the values It can be seen from Fig. 5 that as H/D increases, the reduced shortest path L 2 min gradually decreases, and the corresponding reduction ratio k gradually decreases.
When H/D is larger, the reduction ratio k is closer to 1, that's to say, as the H/D increases, the shortest path of strength reduction is closer to the traditional strength reduction path.
It can be seen from Fig. 6 that as the initial cohesion c 0 increases, the reduced shortest path L 2 min gradually increases, and the corresponding reduction ratio k becomes larger.
It can be seen from Fig. 7 that when the initial internal friction angle increases, the shortest path length L 2 min for strength reduction gradually increases, while the reduction ratio k at this time remains unchanged.
This means that the change of the internal friction angle will not affect the reduced path of the shortest path of strength reduction.
By counting the reduction rate under the shortest path of strength reduction, the reduction rate under the shortest path of strength reduction is all greater than 1.
This shows that when the rock and soil reach the limit state according to the shortest reduction path, the reduction of cohesion is greater than the reduction of the internal friction angle (Figs. 8, 9, 10 and 11).

Limit Equilibrium Analysis Model
According to Terzaghi's theory and the potential fracture surface in Fig. 4, the ultimate equilibrium failure mode of shallow buried tunnels is obtained. As shown in the figure, the potential fracture surfaces are ABC and A 1 B 1 C 1 , and the following assumptions are made: (1) The surrounding rock near the shallow buried tunnel slides along potential fracture planes AB and A 1 B 1 , and the sliding plane forms an angle a (a = 45°-u/2) to the vertical, and u is the initial internal friction angle; (2) The fracture surface BC and B 1 C 1 are vertical to the ground surface, BDB 1 is parallel to the ground surface, and D is located at the vault.
Considering the symmetry, only the limit equilibrium analysis of the sliding body ABD can be carried out, and its safety factor K can be expressed as: In the formula: F R and F T respectively represent the anti-skid force and the sliding force.
According to the static balance conditions in the horizontal direction: Apply the Moore Coulomb strength criterion to the potential failure surface AB: Substituting Eq. (7) into Eq. (6) can get: Considering that BD is not a sliding surface, therefore: Suppose p ¼ cH, H is the buried depth of the tunnel, p is the surrounding rock pressure of a shallow tunnel, and a = 45°-u/2. when K = 1.0, the shallow tunnel is in a state of limit equilibrium. The working conditions in Table 1 are still used for calculation, and the influence of the three major factors of buried depth and hole-span ratio, cohesion, and internal friction angle on the shortest path of ultimate equilibrium strength reduction is obtained. Through comparison, it can be found that the calculation results of the limit equilibrium method are basically consistent with the finite element numerical calculation results: (1) As H/D increases, the shortest path L min of strength reduction gradually decreases, and the corresponding reduction ratio k gradually decreases.
(2) With the increase of initial cohesion force c 0 , the reduction shortest path L min gradually increases, and the corresponding reduction ratio k also gradually increases. (3) With the increase of the initial internal friction angle, the reduced shortest path L min gradually increases, and the corresponding reduction ratio k remains unchanged.

Comparative Analysis of Comprehensive Safety Factors Under Different Methods
It can be seen from Eq. (4) that the length of the shortest path by strength reduction can indirectly represent the safety factor. To further reflect the difference and relation between the shortest path by strength reduction by finite element method and the shortest path by ultimate equilibrium strength reduction, the comprehensive safety factor of the two different methods under different parameters is obtained as shown in Fig. 12, 13, 14. It can be seen from Figs. 12, 13 and 14 that the greater the internal friction angle and cohesion of the tunnel in the initial state, the greater the overall safety factor, and the two are approximately in a linear relationship. When the buried depth and span ratio are small, the safety factors obtained under the finite element method and the limit equilibrium method are relatively close. However, the finite element method of safety coefficient calculated under different parameters is larger than the limit equilibrium method, which indicates that shallow-buried tunnels are more likely to be damaged under the limit equilibrium method. This is because in the calculation of the limit equilibrium method, the value of the surrounding rock pressure P is larger, and the actual surrounding rock pressure P is smaller than ch, but in general, the two laws are consistent (Tables 2, 3, 4).

Sensitivity Gray Correlation Analysis of İnfluencing Factors
The sensitivity analysis of factors affecting the safety of shallow tunnels belongs to the category of gray correlation analysis (Zhijia et al. 2013). The main principle of grey correlation analysis is to take the minimum comprehensive safety factor as the reference sequence, take each influencing factor as the subsequence, and quantify their sensitivity. The main influencing factors are determined from the buried depth and span ratio, cohesion, and internal friction angle. In actual engineering, these factors should be paid attention to. Combined with the calculation of the comprehensive safety factor of the finite element method in the previous section, the specific working conditions are shown in Tables 5, 6 and 7. According to different influencing factors and working conditions, the initial matrix of gray correlation calculation is obtained as follows: (1) Perform averaging on the above X and Y matrix sequence, let: where: i = 0,1,2,…,m (2) Evaluation of sequence of difference In the difference sequence matrix D, the elements need to be compared to extract the maximum and minimum values: (3) Find the gray correlation coefficient matrix Before the analysis, the distance between the comparison point and the reference point should be sorted out, and the overall analysis should be used to calculate the difference and correlation between the factors. Finally, the correlation coefficient between the comparison factor and the reference factor is obtained through formula (15), which reflects its correlation.
In the formula: f is the resolution coefficient, which is generally 0.5.
(4) Find the degree of grey correlation As in formula (16), the degree of relevance is obtained by calculating the average value of the relevant coefficient, and the relevant comparison of the impact size is made accordingly.
Using the above formula to calculate, the gray correlation degrees of H/D, cohesion c, and internal friction angle u of the shallow tunnel are 0.5347, 0.7805, and 0.7661, respectively. Therefore, in the process of calculating the stability of shallow tunnels by the strength reduction shortest path method, the internal friction angle u and cohesive force c have a greater impact on the safety of shallow tunnels.

Conclusion
(1) In shallow tunnels, the reduced path length (expressed by the square of the length) is obtained by the finite element strength reduction method and the ultimate equilibrium strength reduction method and the reduction ratio approximately obeys the parabolic distribution law.
(2) The strength reduction shortest path L min and the corresponding reduction ratio k gradually decrease with the increase of H/D, and gradually increase with the increase of the initial cohesive force c 0 ; with the increase of the initial internal friction angle, the reduction in the shortest The path L min gradually increases, and the corresponding reduction ratio k remains unchanged. (3) The comprehensive safety factor of shallow tunnel calculated by the finite element strength reduction shortest path method is greater than the comprehensive safety factor calculated by the ultimate equilibrium strength reduction shortest path method. The strength reduction shortest path method can be a new method to analyze the stability of shallow tunnels. (4) Among the factors affecting the stability of shallow tunnels calculated by the strength reduction shortest path method, the internal friction angle u, and cohesion c have a greater impact on the safety of shallow tunnels.

Declarations
Conflict of interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, ''The application of the strength reduction shortest path method to the stability analysis of shallow buried tunnel''.