Distribution Features of the Minimum Rock Cover Thickness of the Surrounding Rock Self-Stability of the Metro Tunnel in the Soil-Rock Dualistic Stratum

Based on the maximum value of the self-stability of surrounding rock, the key issues affecting the distribution of the minimum mock cover thickness of the surrounding rock self-stability (MRCT-SRS) of the metro tunnel in the soil-rock dualistic stratum are discussed.+eMRCT-SRS was studied for different soft stratum thicknessesHs and excavation spansD based on cusp catastrophe theory. Moreover, the 3D spatial distribution was determined, and the fitted curves were constructed according to theMRCT-SRS, Hs, and D. +is facilitated the assessment of the stability of the surrounding rock in the excavation of a metro tunnel. Combined with the construction practices used for the Qingdao metro project, the validity of the obtained MRCT-SRS has been verified by the inverse analysis of the monitored data from the metro tunnel. Furthermore, the application of the MRCT-SRS in determining the reasonable buried depth of a metro line has been explored.+e research results have provided theoretical support and scientific basis for the preparation, revision, and improvement of the relevant codes, standards, and guidelines for metro tunnel planning and design.


Introduction
e fundamental goal of an excellent metro tunnel design scheme is to make full use of the self-stability of the surrounding rock. Recently, many subsurface metro tunnels have been carried out in the soil-rock dualistic stratum in cities, such as Qingdao [1], Dalian [2], Guangzhou [3], Chongqing [4], and Guiyang [5]. When a metro tunnel is excavated in the urban area where the buried depth of the bedrock is shallow and the soft soil stratum thickness is thin, the tunnel should be placed in the rock stratum and maintained such that the minimum rock cover thickness meets the requirements of the self-stability of the rock surrounding. In this case, the construction of a metro tunnel can greatly reduce the tunnel support measures, save construction costs, reduce the tunnel construction safety risk and construction difficulty, and reduce the impact on the surrounding environment.
Many scholars have studied the relationship between the stability of the surrounding rock and the rock cover thickness. Sun et al. [1] used an example of a metro station in Qingdao to determine the safety factor of a tunnel in surrounding rock at different buried depths by using the strength reduction method. ey also considered the safety factor equal to 1 as the basis for judging the surrounding rock self-stability of the metro tunnel and determined the relationship between the minimum rock cover thickness and the soil stratum thickness. Yang and Xiao [6] established the cusp catastrophe model of the safety thickness of a karst tunnel floor and derived the expression for the critical safety thickness. Xu et al. [7] designed a study based on the Qingdao Jiaozhou Bay Subsea Tunnel using two mutually verified discrimination methods: the minimum vertical displacement of the tunnel vault and the maximum safety factor of the tunnel surrounding rock. ey then studied the minimum rock cover thickness of the subsea tunnel and considered that the minimum rock cover thickness of a subsea tunnel is the reasonable and effective thickness of the tunnel. Wang et al. [8] pointed out that the reasonable buried depth of the metro station in the soil-rock dualistic stratum should be determined by using the double index system of the overburden thickness-span ratio and the rock cover thickness-span ratio and noted that the rock cover thickness-span ratio is more sensitive to the safety of tunnel construction than the overburden thickness-span ratio. Qiu et al. [9] studied the relationship between the minimum rock cover thickness and the surrounding rock stability of the subsea tunnel by the numerical method. Zhang et al. [10] discussed and expounded upon the issues of the surrounding rock self-stability of the metro tunnel in the soil-rock dualistic stratum from the three aspects of the stress, the displacement, and the safety factor. Wang et al. [11] studied the conversion timing of tunnel excavation method in upper soft and lower hard strata based on displacement direction angle theory under different working conditions. e catastrophe theory was put forward by om, a French scholar, in the 1970s [12] and was then expanded into a perfect system by Trotman and Zeeman [13], Zeeman et al. [14], and Poston and Stewart [15]. Catastrophe theory is a mathematical method used to study the phenomenon of discontinuity in nature. e cusp catastrophe model is one of seven primary catastrophe models in catastrophe theory, which has been widely used in the study of tunnel surrounding rock stability, and it has achieved ideal results [16][17][18][19][20][21][22]. Xia et al. [16] used cusp catastrophe theory and the discontinuous deformation analysis method to study the stability of the tunnel surrounding rock and obtain the safety factor. Based on the measured deformation data of the surrounding rock, Ren et al. [17] established the catastrophe model of the tunnel surrounding rock cusp by using catastrophe theory and deduced the instability criterion of the surrounding rock. Wang et al. [18] used the cusp catastrophe model to study the stability of gas storage pillars in layered salt caverns. Zhang and Han [19] used catastrophe theory to study the collapse mechanism and possible collapse block shape of a shallow, unlined tunnel. Zhou [20] et al. established a model based on the cusp catastrophe theory and analyzed the stability of the support system in the goaf of room pillar gypsum. Zhang et al. [21] and Huang and Zhang [22] used catastrophe theory to study the instability mechanism of a shallow tunnel. e problem of the minimum mock cover thickness of the surrounding rock self-stability (MRCT-SRS) of the tunnel has been studied by many scholars, who have used numerical calculations to analyze the stress, the deformation, and the safety factor. However, based on cusp catastrophe theory, rarely no relevant report has been found on the MRCT-SRS of the metro tunnel in the soil-rock dualistic stratum. is paper based on the calculation and analysis of the MRCT-SRS under different soft stratum thickness Hs and excavation span D was studied based on cusp catastrophe theory and numerical calculation, the 3D spatial distribution was drawn, and the fitted curves was constructed among MRCT-SRS, Hs, and D, which facilitated the assessment on the stability of the surrounding rock in excavation of metro tunnel. Combined with the construction practice of Qingdao metro project, the validity of the obtained MRCT-SRS has been verified by the inverse analysis on the data monitored for metro tunnel, and the application of MRCT-SRS in determining the reasonable buried depth of metro line has been expounded. e research results can provide theoretical support for the design of reasonable depth of the vertical section of metro tunnel in soil-rock dualistic stratum and further offer a scientific basis for the preparation, revision, and improvement of relevant codes, standards, and guidelines.

Cusp Catastrophe eory.
e cusp catastrophe model uses the simplest catastrophe mode with lag, divergence, and other behaviors, which is unique to the high-order catastrophe model. e critical interface is easy to construct and has a strong geometric rationale. Of the seven primary catastrophe models used in catastrophe theory, the cusp catastrophe model is the one with the most extensive application [23][24][25][26][27][28][29].
e potential function V of the cusp catastrophe model is a 3D space (x, u, v) composed of the control variables u and v and the state variable x as follows: An equation for the cusp catastrophe model for the balanced curved surface has been obtained after first deriving the potential function (1) with respect to x as follows: (2) e cusp catastrophe model uses a continuous balanced surface with a 3D phase space, the interior of which is folded into upper, middle, and lower pages, as shown in Figure 1. Different areas have different numbers of balance points. On the middle page, the balance point falls on the maximum value of the potential function, so it is unstable. However, on the upper and lower pages, the potential function takes its minimum value, so the balance point is stable. e boundary between the upper and lower pages, on the one hand, and the  Advances in Civil Engineering middle page, on the other hand, is composed of points with vertical tangents, which is a set of mutation points as follows: Combining formulas (2) and (3) results in a bifurcation set, which is the projection of the fold of the balance-curved surface in the u-v plane as follows: When applying catastrophe theory, scholars have used formula (4) as a direct criterion to determine the cusp mutation. When Δ > 0, the system is in a stable state; when Δ � 0, the system is in a critical equilibrium state; when Δ < 0, the system is in an unstable state [6]. Figure 2, for the analysis model of the MRCT-SRS of the metro tunnel in the soil-rock dualistic stratum. In the figure, H represents the buried tunnel depth; D, the excavation span; h, the excavation height; and Hs and Hr, respectively, the soft soil stratum thickness and the rock cover thickness.

Analysis Model. See
A statistical analysis was performed of the geotechnical test results of 3227 geological boreholes in nearly 2000 sets of soil-rock strata along the Qingdao metro tunnel. Accordingly, the values of the physical and mechanical parameters of the soft soil stratum and the rock stratum in the soil-rock dualistic stratum structure are shown in Table 1.

Target.
e vault is the point at which the deflection is the largest around the cavern of the shallow metro tunnel. erefore, the vertical displacement of the vault is the key indicator when evaluating the stability of the surrounding rock of the tunnel [30,31]. If the rock cover thickness is large over the vault of the tunnel in the soil-rock dualistic stratum and it is gradually reduced, the vertical displacement of the vault at first is slowly reduced. en, the displacement gradually increases after reaching its minimum as the thickness continues to decrease. When the thickness decreases to a certain value, the displacement sharply rises [7]. Currently, the vertical displacement of the tunnel vault is selected as the target for evaluating the self-stability of the surrounding rock of the tunnel. Furthermore, the corresponding rock cover thickness at the mutation point is used as the MRCT-SRS of the tunnel.

Process.
For the MRCT-SRS of the metro tunnel in the soil-rock dualistic stratum based on featured point displacement in cusp catastrophe theory, the calculation is performed as follows: (1) First, a numerical calculation model is constructed of the metro tunnel in the soil-rock dualistic stratum. In the numerical calculation, the stratum is considered according to the plane strain of ideal elastic-plastic materials, which should conform to the Mohr-Coulomb strength criterion and the large deformation hypothesis. ANSYS numerical calculation software and the inscribed circle DP4 yielded criterion based on the Mohr-Coulomb conditions, which were adopted for the one-time excavation of the whole section without considering the influence of groundwater and other factors. e upper boundary of the calculation model is set at the ground surface, and the lower boundary is set from the vault bottom of the excavation section down to 3.5 times the excavation height. Moreover, the distance between the left and right boundaries is more than 7 times the excavation span. e initial geostress incorporates not only the gravity stress of the rock and soil but also a uniformly distributed load of 20 kPa imposed on the ground surface. A large deformation mode is used during the calculation. e failure of the surrounding rock by large deformation can be reproduced by updating the coordinates. e vertical and horizontal ranges of the unit grid around   (3) e calculation should begin starting with the rock cover thickness with respect to the smallest vault vertical displacement of the tunnel surrounding rock as the thickness gradually increases. e end interpolation method is used to fit the data of vertical displacement S and overburden rock thickness Hr, producing a quartic function as follows: S � a 0 + a 1 Hr + a 2 Hr 2 + a 3 Hr 3 + a 4 Hr 4 , where a 0 , a 1 , a 2 , a 3 , and a 4 are undetermined coefficients. In this research, Hr with respect to Δ � 8u 3 + 27v 2 � 0 is selected as the MRCT-SRS. Its calculation is shown in Figure 3.     Table 2. e data of S and Hr are used, so Hr is set to 11 m to perform the polynomial fitting. When Hr is 1 m, the fitting is carried out on 11 sets of data with Hr vales from 1 m to 11 m, producing the result as follows: S � 47.75224 + 28.81764Hr + 6.90942Hr 2 + 0.69678Hr 3 + 0.02495Hr 4 .

(6)
Substitute the values of the four parameters of a 1 ∼a 4 from formula (6) into formulas (1)

MRCT-SRSs of the Metro Station Tunnel and the Running
Tunnel.
e example of the engineering of the Qingdao metro tunnel is used in this research to investigate the MRCT-SRS of the station tunnel and the running tunnel with two main types of sections in a soil-rock dualistic stratum (see Figure 5). e results of the MRCT-SRS calculations for the metro station tunnel and the running tunnel under different soft soil stratum thicknesses Hs are shown in Table 3 (7). e adjusted R-square value is 0.9991, and the MRCT-SRS is approximately linear along with Hs. e fitted equation of the running tunnel MRCT-SRS and Hs is shown in formula (8), and the adjusted R-square value is 0.99258. With formulas (7) and (8), the MRCT-SRSs for the station tunnel and the running tunnel can be derived with respect to different Hs.
e MRCT-SRS of the metro tunnel increases approximately linearly with the increase in Hs. is result occurs because, when excavating the metro tunnel in the soil-rock dualistic stratum under the requirements of the MRCT-SRS, the overlying soft soil stratum bears a higher load, while the rock cover stratum bears more of the structure. e rock mechanics performance around the tunnel is similar to the force mode of a composite arch or a composite beam. erefore, when the Hs is greater, the corresponding load and the MRCT-SRS become larger.     Advances in Civil Engineering

Discussion of How MRCT-SRS Works for the Metro
Tunnel. e stability characteristics of the tunnel surrounding rock are closely related to the shape and size of the tunnel section. By focusing on the metro station tunnel, the general rule of how the MRCT-SRS varies with the excavation span D is discussed in the present study while maintaining the original section shape and high-span ratio but using only the scaling tunnel section size. e tunnel section size is shown in Figure 7. e MRCT-SRS calculation results of the metro tunnel are shown in Table 4 Figure 9 and Table 5). e metro running tunnel mainly adopts the horseshoe-shaped subterranean excavation form with a single span of 5.8 m∼6.2 m for a one-way single lane.

Stratigraphic Features.
e stratum distribution along Qingdao Metro from top to bottom includes the Quaternary stratum as well as the strongly weathered, the moderately weathered, and the microweathered (or nonweathered) granite strata. To simplify the analysis and highlight the main points, the Quaternary stratum and the strong weathering stratum were considered soft soil stratum, and the strata in the middle weathering strata and below are considered rock stratum, thus constructing a distribution model for the soil-rock dualistic stratum. e structural characteristics of the dualistic stratum along the two metro lines in Qingdao are shown in Figure 10.
According to the statistical analysis of 3227 geological borehole samples along the two metro lines, the average thickness of the soft soil stratum is 10.2 m; the maximum is 38.0 m; the median is 10.0 m; and the lower and upper quartiles are, respectively, 5.0 m and 14.8 m (as shown in Figure 11 and Table 6). Further analysis shows that the number of borehole samples with soft soil stratum thicknesses of 9 m or less along the metro lines accounts for 45.6% of the samples; those with less than 12 m, 61.4%; those with less than 15 m, 77.5%; and those with less than 18 m, 89.7% (as shown in Table 7).

Data Resource.
e monitoring data come from six station tunnels of the two metro lines, which have approximately the same section size, excavation depth and technique, support measures, etc. e number of monitoring points, the soft soil stratum thickness Hs, the rock cover thickness Hr, and the MRCT-SRS are shown in Table 8. e statistical data are from a third party. e monitoring points for the ground surface settlement are selected within the main area of the tunnel, among which the abnormal points should be excluded in the statistics. Table 9 shows that, with regard to the overburden thicknesses exceeding the MRCT-SRS requirements for the stations of Zhiquan Road, Haichuan Road, and Junfeng Road, the average surface settlement is between 2.29 mm and 6.37 mm, with a maximum value between 6.22 mm and 12.59 mm, which are small in general. Regarding those for the stations of Zhongshan Park, Jiangxi Road, and Zhanshan Road, the average surface settlement is between 37.5 mm and 54.75 mm, with a maximum value between 71.54 mm and 80.96 mm, which are large in general.

Inversion Analysis.
Surface settlement can comprehensively reflect the effect of metro construction quality control and the degree of construction influence on the surrounding rock as well as the soil and neighboring environment. For a metro tunnel with shallow buried depth under the approximately same conditions of section size, excavation depth and technique, support measures, etc., the rock cover thickness on the vault meets the MRCT-SRS requirements or not, the surface settlement caused by the tunnel excavation is different by an order of magnitude. is situation fully demonstrates that the self-stability of the surrounding rock has a good correlation with the rock cover thickness (as shown in Figure 12). e rationality of the conclusions obtained in this study has been fully verified.

Engineering Application.
e fundamental goal of an excellent metro line design is to make full use of the selfstability of the surrounding rock and to minimize supplementary engineering measures. Costs should be considered  Advances in Civil Engineering only when engineering security is guaranteed. Furthermore, geological risks and construction difficulty should be reduced. e degree to which the surrounding rock can be selfstabilized depends mainly on the distribution conditions of the strata along the metro lines, the understanding of these conditions, and the ability to execute the required tasks. erefore, the Geologically Easily Stable Area (GSEA) in this study refers to the area that meets the MRCT-SRS requirements for the excavation of the tunnel along the metro in the soil-rock dualistic stratum, otherwise referred to as the Geologically Difficultly Stable Area (GSDA). e metro station and the running tunnel are considered, respectively, as a single-arch large-span excavation structure with the excavation span of 20.8 m and a horseshoe-shaped structure with the excavation span of 6.2 m. e buried depth is set at equal intervals of 3 m falling within the scope of 6 m∼30 m. Table 10 and Figure 13 show the calculation results of the proportions of the GSEA lengths of the station (considering only 16 subterranean excavation stations) and the running tunnels, respectively, as well as the lengths of the station and the running tunnel of Qingdao's two metro lines under different buried depths.
As the buried depth increases, the proportion of the GSEA in the station tunnel rises first rapidly but then slowly. When the buried depth of the station tunnel is at 6 m, the proportion is 49.2%. When the depth increases to 12 m, the proportion increases rapidly to 76.4%. However, when the depth reaches 18 m, the proportion increase to 90.7%. e change of GSEA with buried depth in running tunnel is similar to that in station tunnel. When the buried     depth of the running tunnel is 6 m, the proportion is only 26.3%. However, when the depth increases to 15 m, the proportion rapidly increases to 71.6%. As the depth continues to rise to 18 m, the proportion increases to 87.5%. When the depth reaches 21 m, the proportion slowly increases to 93.9%. e metro line is mainly composed of the station and the running tunnel. e buried depth of the line depends on the depth of the station tunnel. e depth of the running tunnel is mainly determined by factors such as the buried depth of the station tunnel and the longitudinal slope. e height difference between the station and the running tunnels is set as 12 m. erefore, when the buried depth of the Qingdao metro station falls within 12m∼15 m, the GSEA of the station tunnel accounts for 76.4%∼84.1%, and the GSEA of the running tunnel accounts for 96.6%∼ 98.5%.

Conclusion
(1) e MRCT-SRS of the metro tunnel approximately linearly increases or decreases with respect to the soft stratum thickness Hs in the soil-rock dualistic stratum. When the Hs is larger, so is the corresponding MRCT-SRS. (2) e 3D spatial distribution was determined, and the fitted curves were constructed according to the MRCT-SRS, Hs, and D. is facilitated the assessment of the stability of the surrounding rock in the excavation of a metro tunnel.

(3) e variation rule of the Geologically Easily Stable
Area with respect to buried depth has been revealed by examining the distribution features of the strata along the Qingdao metro lines. Moreover, the application of the MRCT-SRS in engineering has been expanded to determine the reasonable buried depth of the metro line.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.