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Article

Pressure Arch Effect of Deeply Buried Symmetrically Distributed Triple Tunnels

by
Ran Li
1,2,3,
Dingli Zhang
2,
Yuan Song
2,4,*,
Ao Li
2 and
Jiwei Luo
2
1
School of Civil Engineering, Hefei University of Technology, Hefei 230009, China
2
Key Laboratory for Urban Underground Engineering of the Ministry of Education, Beijing Jiaotong University, Beijing 100044, China
3
China Tiesiju Civil Engineering Group Co., Ltd., Hefei 230023, China
4
School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan 232001, China
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(3), 673; https://doi.org/10.3390/sym15030673
Submission received: 30 December 2022 / Revised: 22 February 2023 / Accepted: 6 March 2023 / Published: 7 March 2023
(This article belongs to the Section Engineering and Materials)
Browse Figures
Figure 1
<p>Excavation sequence and geometric dimensions of closely spaced triple tunnels in Badaling [<a href="#B27-symmetry-15-00673" class="html-bibr">27</a>]. (①–⑨ represents the tunnel excavation sequence).</p> "> Figure 2
<p>Crown settlement of triple tunnels in the class-V ground (mm).</p> "> Figure 3
<p>Surrounding rock pressure distribution of triple tunnels in the class-V ground (kPa). (①–⑨ represents the tunnel excavation sequence).</p> "> Figure 4
<p>Selection of measuring points (lines) of surrounding rock deformation.</p> "> Figure 5
<p>Elastic-plastic stress distribution of circular cavity in the hydrostatic stress field.</p> "> Figure 6
<p>Pressure arch boundary distribution after tunnel excavation.</p> "> Figure 7
<p>Characterization parameters of surrounding rock stability of deeply buried, closely spaced triple tunnels.</p> "> Figure 8
<p>Three-dimensional numerical model of closely spaced triple tunnels.</p> "> Figure 9
<p>Progressive development of ground settlement of triple tunnels. (<b>a</b>) Step 1; (<b>b</b>) Step 2; (<b>c</b>) Step 3; (<b>d</b>) Step 4; (<b>e</b>) Step 5; (<b>f</b>) Step 6; (<b>g</b>) Step 7; (<b>h</b>) Step 8; (<b>i</b>) Step 9.</p> "> Figure 10
<p>Progressive development of plastic zone of triple tunnels. (<b>a</b>) Step 1; (<b>b</b>) Step 2; (<b>c</b>) Step 3; (<b>d</b>) Step 4; (<b>e</b>) Step 5; (<b>f</b>) Step 6; (<b>g</b>) Step 7; (<b>h</b>) Step 8; (<b>i</b>) Step 9.</p> "> Figure 10 Cont.
<p>Progressive development of plastic zone of triple tunnels. (<b>a</b>) Step 1; (<b>b</b>) Step 2; (<b>c</b>) Step 3; (<b>d</b>) Step 4; (<b>e</b>) Step 5; (<b>f</b>) Step 6; (<b>g</b>) Step 7; (<b>h</b>) Step 8; (<b>i</b>) Step 9.</p> "> Figure 11
<p>Progressive development of max principal stress of left tunnel. (<b>a</b>) Location of the left tunnel excavation face; (<b>b</b>) Variation curve of max principal stress above the left tunnel crown.</p> "> Figure 12
<p>Progressive development of max principal stress of right tunnel. (<b>a</b>) Location of the right tunnel excavation face; (<b>b</b>) Variation curve of max principal stress above the right tunnel crown.</p> "> Figure 13
<p>Progressive development of max principal stress of middle tunnel. (<b>a</b>) Location of the middle tunnel excavation face; (<b>b</b>) Variation curve of max principal stress above the middle tunnel crown.</p> "> Figure 14
<p>Progressive development of max principal stress of triple tunnels. (<b>a</b>) Step 1; (<b>b</b>) Step 2; (<b>c</b>) Step 3; (<b>d</b>) Step 4; (<b>e</b>) Step 5; (<b>f</b>) Step 6; (<b>g</b>) Step 7; (<b>h</b>) Step 8; (<b>i</b>) Step 9.</p> "> Figure 15
<p>Pressure arch boundary height in V1 grade surrounding rock.</p> "> Figure 16
<p>Middle line deformation of the left and right pillars in the V1 grade surrounding rock. (<b>a</b>) Horizontal displacement of left and right rock pillars; (<b>b</b>) Vertical displacement of the left and right rock pillars.</p> "> Figure 17
<p>Deformation curves of triple tunnels with different depths <span class="html-italic">H<sub>d</sub></span>. (<b>a</b>) Crown settlement of tunnels at different buried depths <span class="html-italic">H<sub>d</sub></span>; (<b>b</b>) Horizontal convergence of tunnels with different depths <span class="html-italic">H<sub>d</sub></span>.</p> "> Figure 18
<p>Left rock pillar deformation with different depths <span class="html-italic">H<sub>d</sub></span>. (<b>a</b>) Horizontal displacement of the left rock pillar with different buried depth <span class="html-italic">H<sub>d</sub></span>; (<b>b</b>) Vertical displacement of the left rock pillar with different buried depth <span class="html-italic">H<sub>d</sub></span>.</p> "> Figure 19
<p>The distribution of plastic zone in different depths <span class="html-italic">H<sub>d</sub></span>. (<b>a</b>) <span class="html-italic">H<sub>d</sub></span> = 40 m; (<b>b</b>) <span class="html-italic">H<sub>d</sub></span> = 50 m; (<b>c</b>) <span class="html-italic">H<sub>d</sub></span> = 60 m; (<b>d</b>) <span class="html-italic">H<sub>d</sub></span> = 70 m; (<b>e</b>) <span class="html-italic">H<sub>d</sub></span> = 80 m; (<b>f</b>) <span class="html-italic">H<sub>d</sub></span> = 90 m; (<b>g</b>) <span class="html-italic">H<sub>d</sub></span> = 100 m; (<b>h</b>) <span class="html-italic">H<sub>d</sub></span> = 120 m; (<b>i</b>) <span class="html-italic">H<sub>d</sub></span> = 140 m.</p> "> Figure 19 Cont.
<p>The distribution of plastic zone in different depths <span class="html-italic">H<sub>d</sub></span>. (<b>a</b>) <span class="html-italic">H<sub>d</sub></span> = 40 m; (<b>b</b>) <span class="html-italic">H<sub>d</sub></span> = 50 m; (<b>c</b>) <span class="html-italic">H<sub>d</sub></span> = 60 m; (<b>d</b>) <span class="html-italic">H<sub>d</sub></span> = 70 m; (<b>e</b>) <span class="html-italic">H<sub>d</sub></span> = 80 m; (<b>f</b>) <span class="html-italic">H<sub>d</sub></span> = 90 m; (<b>g</b>) <span class="html-italic">H<sub>d</sub></span> = 100 m; (<b>h</b>) <span class="html-italic">H<sub>d</sub></span> = 120 m; (<b>i</b>) <span class="html-italic">H<sub>d</sub></span> = 140 m.</p> "> Figure 20
<p>The max principal stress programs of different buried depths <span class="html-italic">H<sub>d</sub></span>. (<b>a</b>) <span class="html-italic">H<sub>d</sub></span> = 40 m; (<b>b</b>) <span class="html-italic">H<sub>d</sub></span> = 50 m; (<b>c</b>) <span class="html-italic">H<sub>d</sub></span> = 60 m; (<b>d</b>) <span class="html-italic">H<sub>d</sub></span> = 70 m; (<b>e</b>) <span class="html-italic">H<sub>d</sub></span> = 80 m; (<b>f</b>) <span class="html-italic">H<sub>d</sub></span> = 90 m; (<b>g</b>) <span class="html-italic">H<sub>d</sub></span> = 100 m; (<b>h</b>) <span class="html-italic">H<sub>d</sub></span> = 120 m; (<b>i</b>) <span class="html-italic">H<sub>d</sub></span> = 140 m.</p> "> Figure 21
<p>Pressure arch range of triple tunnels with different buried depths. (<b>a</b>) Outer boundary height of pressure arch; (<b>b</b>) Inner boundary height of pressure arch; (<b>c</b>) Pressure arch thickness.</p> ">
Versions Notes

Abstract

:
Compared with single or twin tunnels, the pressure arch effect of deeply buried, symmetrically distributed triple tunnels are more complex and less studied. In this paper, the arching responses are in-situ measured in the deeply buried, symmetrically distributed triple tunnels of Badaling Great Wall station. Numerical research is subsequently conducted to investigate the formation and development of the pressure arch of triple tunnels. Then, the influencing law of buried depth on pressure arch behavior is systematically studied. Based on monitoring data, the rock pressure distribution is asymmetric about the axis of the triple tunnels, and the arching response of the middle tunnels is more significant than that of the left and right tunnels. According to numerical analysis, a combined large pressure arch may be easily formed across the triple tunnels. The pre-arching and double-arching effects are also numerically observed during triple tunnel excavations. The inner boundary of the pressure arch of the middle tunnel is 14.0 m, nearly two times higher than those of the left and right tunnels. This simulation result indicates that the mechanical state of the middle tunnel is the least desirable. Moreover, the critical arching depth of closely spaced tunnels is 1.75 times that of a single tunnel. Compared with a single tunnel, the support of triple tunnels should be additionally strengthened.

1. Introduction

The arch structure is commonly recognized as a stable structure in engineering. It has been successfully applied to overground buildings, such as church domes and stone bridges. The significant arching effect also exists in underground engineering and has been widely observed and described for nearly a century. The pressure arch effect has been widely recognized as a critical factor for the roof stability of deeply buried tunnels. In 1882, Engesser [1] observed the “arching action” above the sandy tunnel roof and revealed the deformation mechanism of the tunnel lining influenced by ground pressure. In 1939, IME mentioned that rock deformation and stress redistribution might lead to the generation of a stable pressure arch and a disturbed loosening zone. In 1946, Terzaghi [2] described the pressure arch development in tunnel excavation, and then he clarified that the overburden load is supported by the ground arch. In 1946, Chappell [3] stated the pressure arch is the stress concentration area after excavation, mainly for transferring loose rock weight and resisting tunnel collapse.
The pressure arching effect has been further studied and discussed in recent decades. In 2002, Huang et al. [4] proposed that the inner boundary of the pressure arch is the tunnel roof and the outer boundary could be determined by stress deflection points. In 2011, Chen et al. [5] established a three-dimensional excavation model. They found the inflection point height of the vertical stress-buried depth curve might be a reliable mechanical parameter to show the pressure arch development. In 2016, Zhang et al. [6] modified the Terzaghi arching model and proposed an arching coefficient to investigate soil pressure acting on the jacked pipes. This theoretical method was then verified by in-situ measured data. In 2018, Kong et al. [7] performed 2D numerical investigations to study the pressure arch’s initial formation and subsequent development. The internal boundary height was found to be an effective index to describe rock stability, and it is significantly influenced by the initial stress state. The arching mechanism has also been further investigated by theoretical analysis [8,9,10], numerical simulation [11,12], and experimental tests [13].
Based on previous studies, it could be concluded that the pressure arch refers to the arched compression area above the tunnel where the surrounding rock’s initial stress is disturbed by tunnel excavation. The pressure arch is a mechanical rock structure with an arch function to protect the tunnel from collapsing. The rock masses in the pressure arch wedge each other due to the differentiated displacement. The pressure arch effect represents the self-stabilizing ability of the surrounding rock. With the existence of the pressure arch, the upper load is converted into compressive stress and transferred to the arch feet on both sides, and the tunnel lining only bears the loosened rock weight under the pressure arch [14].
Compared with the arching mechanism of a single tunnel, the pressure arch effect of closely spaced tunnels is more complex and less studied [15]. Khairil [16] explored the relationship between the pressure arch and clearance in closely spaced tunnels, and he concluded that a combined large pressure arch is formed when clearance becomes sufficiently small. Conversely, with the space becoming larger, two independent small pressure arches tended to be separately formed and developed. Yang et al. [17] conducted theoretical derivations and field monitoring, and they found the significant load bias characteristics of shallow buried triple tunnels. Based on Protodyakonov’s equilibrium arching theory, Li et al. [18] proposed an analytical method to calculate the rock pressure on closely spaced twin tunnels. The deeply buried twin tunnels were divided into three mechanical models according to the different distributions of the pressure arch. They observed that the failure pattern and loosening pressure of twin tunnels are profoundly influenced by pillar width and strength. Seo [19] studied the rock pillar behavior through the experimental method and found significant arching behavior in the steel-rib-reinforced pillar.
The excavations of adjacent tunnels inevitably lead to complex ground deformation [20,21] and stress redistribution [22,23,24], especially for closely spaced, symmetrically distributed triple tunnels. The slip of the rock pillar may cause the formation of a combined asymmetric pressure arch across the triple tunnels, and the overall collapse of the integrated pressure arch may seriously threaten the tunnel excavation safety. There are very few studies about the mechanical behavior of closely spaced triple tunnels [25,26]. Therefore, it is urgent to study the mechanical behavior and evolution law of deeply buried, symmetrically distributed triple tunnels.
This paper aims to gain in-depth insight into the arching mechanism of deeply buried, symmetrically distributed triple tunnels. The field monitoring of excavation responses in triple tunnels is first conducted to show the significant arching behavior. A series of 3D numerical analyses are performed to reveal the arching mechanism of triple tunnels. With the whole process of excavation simulation of triple tunnels, the formation and development of the pressure arches are illustrated by analyzing the stress redistribution and rock deformation around underground excavations. Furthermore, the influence of buried depth on the arching behavior of triple tunnels is investigated in detail. Based on pressure arch stability, some practical suggestions are proposed for the design and construction of the triple tunnels. This pressure arch study might serve as a preliminary reference for similar triple tunnel constructions.

2. Engineering Background and Field Monitoring

2.1. Overview of Deeply Buried Tripe Tunnels of Badaling Great Wall Station

The Badaling Great Wall Station is an underground high-speed railway station located at the scenic Badaling Great Wall in Beijing. Closely spaced triple tunnels are symmetrically distributed in the platform layer of this station (Figure 1). The maximal buried depth of the triple tunnels is 102 m, and the total length is 398 m (from DK67 + 851 to DK68 + 249). The clearance between neighboring tunnels is narrowly 6 m in the main section. The excavation sequence is left tunnel—right tunnel—middle tunnel, and the length of the steps is generally 8–14 m. The excavation area of both side tunnels is 161.9 m2, the middle tunnel is 144.2 m2, and the clearance is narrowly 6 m. Each tunnel is divided into three parts: the top heading, the bench, and the invert. Thus, the steps are marked 1–9.
With the New Austrian Tunneling Method (NATM), the triple tunnels are excavated by the drilling and blasting method. These tunnels are constructed in mountainous terrain with considerable land undulations. The surrounding rock mass is generally weak and broken, with a saturated uniaxial compressive strength of 40 MPa and a density of 1700 kg·m−3. The cohesion and friction angles of the surrounding rock are 100 kPa and 22°, respectively. According to the Chinese basic quality (BQ) rock mass classification system, the surrounding rock grade is mainly class V, with a relatively high excavation risk. The corresponding conversion relationship between the Chinese BQ classification system and the international Q classification system is explained in Table 1 [28]. The groundwater is not developed; it is mainly composed of bedrock fissure water, and most of it was drained before excavation. Based on the previous studies [29,30], the mechanical parameters of the surrounding rock have a decisive influence on the mechanical behavior of the tunnel. It should be noted that these geotechnical parameters were directly obtained according to the geotechnical investigation report provided by the tunnel design institute. The authors have not conducted additional geotechnical tests.
The Code for Design of Railway Tunnels (TB10003-2016) is used to design the triple tunnels. Composite lining, including primary and secondary lining, is used for the support structure in this project. The primary lining consists of the I-section steel frame, the 8 mm diameter reinforced mesh, the 300 mm thick shotcrete, and the 4.5 m long prestressed anchor. The shotcrete (primary lining) and the reinforced concrete (secondary lining) mainly bear pressure, and those compressive strengths are required at 25 MPa and 35 MPa, respectively. There is a 12 mm thick EVA waterproofing membrane between the primary and secondary linings to prevent water leakage. The primary lining is constructed immediately after the tunnel excavation, while the secondary lining is commonly built with a 50 m lag behind the primary lining.

2.2. Field Monitoring of Triple Tunnels

Since the class-V ground is the soft and broken surrounding rock with an extremely high excavation risk, it is selected as the main monitoring section to perform field tests. Pressure arch is commonly recognized as the mechanical behavior of load transfer when rock encounters uneven deformation. The arching characteristics of triple tunnels can be obtained by monitoring the crown settlement and the rock pressure. The detailed monitoring layout can be found in the previous study by the authors [31].
The box-line diagram of the crown settlement of the triple tunnels in the class-V ground is shown in Figure 2. Since the deformation monitoring is carried out after the tunnel excavation, the advanced settlement could not be measured. The average crown settlement of the middle tunnel is 63.4 mm, significantly larger than the 42.2 mm of the left tunnel and the 34.7 mm of the right tunnel. The crown settlement is positively related to the pressure arch height. Thus, the arching characteristic of the middle tunnel is the most significant. It can also be inferred that in addition to the independent pressure arches of each tunnel, there is an enormous combined pressure arch above the triple tunnels.
The final distributions of the surrounding rock pressure in triple tunnels in the class-V ground are obtained by pressure cells (Figure 3). The positive value represents compressive stress, while the negative value represents tensile stress. It should be noted that there are essential differences between original rock stress and surrounding rock pressure. The original rock stress is a natural stress existing in the stratum without engineering disturbance. The original rock stress is closely related to the dead weight and tectonic stress. However, the surrounding rock pressure is the rock load on the tunnel support after tunnel excavation. The surrounding rock pressure mainly depends on the geological conditions, tunnel support, and support time [32].
The tunnel support mainly bears compressive stress, with only tensile stress observed at the left feet of the middle tunnel. The rock pressure distribution of each tunnel is generally arched in shape and asymmetrically distributed on the axis. The rock load at the inner monitoring points of the left and right tunnels was generally larger than that at the outer monitoring points. Yang et al. [17] also analytically found the asymmetrical rock load distribution pattern in shallow-buried, symmetrically distributed triple tunnels. The middle tunnel bears the largest ground load compared to the left and right tunnels. This distribution shows that an ultimate combined load-carrying pressure arch tends to be formed and developed across the triple tunnels.

3. Mechanical Parameters of Pressure Arch of Triple Tunnels

The ground deformation, plastic zone, and pressure arch boundary are key mechanical parameters to characterize the pressure arch stability of deeply buried triple tunnels. The chosen reason and significance of each parameter will be discussed in detail and explained as follows:

3.1. Deformation Parameters of Surrounding Rock

With the excavation of the tunnel, the original equilibrium is destroyed. In order to achieve a new steady-state equilibrium, the surrounding rock of the tunnel is squeezed inward with substantial ground deformation [33]. The deformation parameters of the surrounding rock can be subdivided into crown settlement, horizontal convergence, and pillar displacement (Figure 4).
The crown settlement represents the vertical deformation characteristics of the surrounding ground. In practical engineering, the measured data of crown settlement can provide feedback and guide tunnel design and construction. The crown settlement is the most important index to reflect the tunnel roof stability [34], and the larger the crown settlement is, the worse the surrounding rock safety state is. With each tunnel’s crown settlement development curve, the longitudinal influence of adjacent tunnels can be clearly analyzed. When the crown settlement exceeds the allowed value, the tunnel is in an unstable state, and more strong support should be adopted immediately.
Horizontal convergence represents the shrinkage of surrounding rock towards the tunnel axis. As the horizontal convergence increases, the arch foot of the triple tunnels continuously slides inward, which may lead to the overall collapse of the triple tunnels. Especially in deeply buried tunnels with a significant lateral pressure coefficient, the horizontal stress is the max principal stress, so the lateral shrinkage of the tunnel is more severe. According to Chinese Technical Regulations for Monitoring and Measurement of Railway Tunnels, horizontal convergence and crown settlement must be measured in tunnel monitoring. Therefore, more attention should be paid to horizontal convergence in deeply buried tunnels with a high initial stress field.
Pillar displacement can also reflect the deformation stability of the triple tunnels. The essential characteristic of the closely spaced tunnels is the narrow pillar width, and rock pillars are severely disturbed during neighboring tunnel excavation. The core of the triple tunnel designation is ensuring the safety and stability of the rock pillars. Moreover, the tunnel pressure arch safety is mainly related to the support force provided by both abutments. The rock pillars are the stable abutments of three independent pressure arches in triple tunnels. Either rock pillar acted as the substantial abutment twice in neighboring two individual pressure arches. Therefore, the rock pillar displacement must be monitored during triple tunnel construction. In this paper, the horizontal and vertical displacements of the pillar axis are selected to measure the rock pillar deformation.

3.2. Plastic Zone Parameters

(1)
Plasticity of the rock pillar
The rock pillar can protect the pressure arch from collapse for the closely spaced triple tunnels. Therefore, the safety state of the rock pillar is directly related to the triple tunnels’ stability. In this paper, the plasticity ratio of the rock pillar is taken as the core parameter to characterize the safety state.
(2)
Range of plastic zone
Suppose the surrounding rock is not in a plastic state after stress redistribution. In that case, the surrounding rock is only slightly affected by adjacent tunnel excavation and still has further potential bearing capacity. If the plastic zone develops excessively in the deep surrounding rock, it indicates that the tunnel excavation has a great disturbance and failure effect on the surrounding rock. Therefore, the extended range of the plastic zone (mainly above the tunnel crown) can also be used as an additional parameter to evaluate the rock safety state.

3.3. Boundary Parameters of Pressure Arch

The stress distribution in the pressure arch significantly differs from that in the original rock and the loose area. This difference implies that the pressure arch can be determined by the stress redistribution characteristics. Compared with the original rock’s (zone 4 in Figure 5) stress, the max principal stress of the surrounding rock in the pressure arch (zones 2–3) is proudly larger. In contrast, the max principal stress in the loosening zone (zone 1) is smaller. Accordingly, the inner boundary of the pressure arch is selected as the intersection line where the max principal stress increases from below the original rock stress to the initial rock stress (the inner boundary height is Hint, Figure 6). The outer boundary of the pressure arch is determined to be the intersection line where the max principal stress decreases to the original rock stress (the outer boundary height is Hout). In addition, the core boundary of the pressure arch is the line connecting the max principal stress deflection points (the core boundary height is Hcor). For a single tunnel, the core boundary is ideally an elastoplastic boundary. However, the cross-section of the closely spaced triple tunnels is a non-axisymmetric horseshoe shape, and the numerical mesh cannot be divided too densely. Due to these factors, the stress redistribution of triple tunnels is uncertain, and it is very challenging to define the pressure arch boundary accurately. Therefore, this paper only gives the pressure arch boundary directly above the triple tunnel crowns to quantitatively reflect the pressure arch development.
The characterization parameters for pressure arch stability of triple tunnels are listed in Figure 7. Among the first-level characterization parameters, the surrounding rock deformation and plastic zone have precise physical meanings and strict judging criteria. They should be given more attention when compared with the pressure arch boundaries. Among the second-level characterization parameters, the crown settlement and the plasticity rate of the rock pillar are the most critical, and they need to be analyzed in detail.

4. Progressive Evolution of Pressure Arch of Triple Tunnels

Arching behavior is the performance of the self-supporting capacity of the surrounding rock. The pressure arch evolution of the deeply buried tunnel is gradual and progressive, and there is a particular omen before the failure of the surrounding rock. Tunnel arching behavior can manifest in three forms: settlement arch [35], plastic arch [27], and stress arch [7]. Settlement arch refers to the apparent arch-shape deformation boundary between the pressure arch, the loose area, and the undisturbed original rock in the ground settlement contour. Plastic arch refers to the arch-shaped plastic zone above the tunnel. Stress arch refers to the compressive, stress-increased zone where the stress path deflects. Overall, these arching forms result from the heterogeneous displacement and stress path deflection of the surrounding rock in reaching the new steady state.

4.1. Numerical Model

A three-dimensional numerical simulation is performed to study the mechanical behavior of closely spaced triple tunnels. The overall size of the model (Figure 8) is 250 m × 200 m × 100 m (horizontal width × vertical height × longitudinal length). The top surface is free, the vertical displacement of the bottom surface is limited, and the horizontal displacement of the four vertical surfaces is restricted. Based on the monitoring data [36], the primary lining bears 75% of the surrounding rock pressure, whereas the secondary lining is the safety reservation. Moreover, the secondary lining is commonly constructed more than 60 m behind the excavation face, and the surrounding rock has achieved a new steady state during the secondary lining construction. The following basic assumptions are made:
(1)
The surrounding rock is isotropic and ideally elastoplastic under the Mohr–Coulomb strength criterion.
(2)
The stress of the original rock is uniformly distributed, and the direction of the principal stress is vertical or horizontal.
(3)
The surrounding rock is simulated with the solid element, and the tunnel lining is modeled with the shell element.
The detailed material parameters of the surrounding rock and tunnel lining are listed in Table 2. The differences in section size and ground grade are not considered in the numerical model. The standard high-speed double-track tunnel section (the same as the middle tunnel in Badaling’s closely spaced triple tunnels) is taken for all three tunnels, with a buried depth of 100 m, a lateral pressure coefficient of 2.0, and a clearance of 5.0 m. The surrounding rock grade is class V. The tunnel adopts a three-stage excavation method; the length of the upper and lower stages is 8 m, and the invert length is 10 m. The excavation sequence is left tunnel—right tunnel—middle tunnel. The lag between the right tunnel and the left tunnel is 26 m, and the lag between the middle tunnel and the left tunnel is 52 m.

4.2. Progressive Development Process of Settlement Arch

Based on the nine excavation steps of the closely spaced triple tunnels, the development process of vertical subsidence of the strata at the monitoring section (y = 50 m) is accordingly given. The ground settlement contour is arched (Figure 9), and there is an apparent arched boundary between the disturbing surrounding rock and the undisturbed original rock. As a form of pressure arch effect, the settlement arch’s internal rock is loose and broken with greater deformation than the outer rock. Therefore, the mechanical behavior of the pressure arch can be described macroscopically by analyzing the ground settlement. When the left tunnel is excavated (step 3), a small asymmetric settlement arch is formed above the tunnel, and the max settlement occurs at the tunnel crown with a value of 33.3 mm. After the right tunnel is constructed (step 6), the independent settlement arches above the side tunnels tend to be connected, and the max settlement also increases to 34.7 mm. After the middle tunnel is excavated (step 9), the settlement arch above the middle tunnel is much larger than the side tunnels’, and the settlement arches above the side tunnels are offset to the middle. Finally, a large combined-settlement arch is formed over the triple tunnels. The crown settlement of the middle tunnel is the largest, with a value of 47.2 mm. There are small independent-settlement arches and large combined-settlement arches above each tunnel. It is implied that each tunnel has a double-arched effect.

4.3. Progressive Development Process of Plastic Arch

When the surrounding rock’s shear stress reaches shear strength, it will enter the plastic state without any further shear stress growth. For the single tunnel in hydrostatic pressure and ideally elastoplastic condition, the radial stress gradually increases from zero at the tunnel contour to the outer surrounding rock, returning to the initial stress in the deep surrounding ground. With the increase in the radial stress, the tangential stress of the surrounding rock in the plastic zone gradually increases and reaches its max value at the elastoplastic interface. Then the surrounding rock enters the elastic state, and the tangential stress continually decreases to the initial stress. The arching behavior of triple tunnels can be investigated by studying the plastic zone development (Figure 10).
In this numerical simulation, the rock in the plastic state is a total shear failure. Without considering the residual strength, the plastic zone can reflect the rock’s relative safety. After the left tunnel is excavated (step 3), the plastic zone is mainly distributed around the left tunnel, with a horizontal expansion depth of 4.9 m and an upward expansion depth of 7.2 m. With the excavation of the right tunnel (step 6), the failure zone increases sharply, and the rock pillar is broken in a large area of the plastic zone. The plastic zone above the middle tunnel extends to 23.0 m. After the middle tunnel is excavated (step 9), a much larger plastic arch is observed above the middle tunnel, and the plastic zone depth increases to 27.6 m. The rock pillars may slip and fail, leading to the overall collapse of the upper pressure arch. In practical triple tunnel construction, improved pillar-reinforcing cables and radial grouting should be timely adopted to protect the vulnerable rock pillars.

4.4. Progressive Development Process of Stress Arch

4.4.1. Longitudinal Development of Stress Arch

The mechanical influence of the time-space effect of tunnel construction can be studied by analyzing the variation of the max principal stress during triple tunnel excavation. Ll, Lm, and Lr are defined as the distances between the monitoring section (y = 50 m) and the excavation faces of the left, right, and middle tunnels. When the tunnel excavation face does not reach the monitoring section, Ll, Lm, and Lr are negative. The corresponding values are positive when the monitoring section is behind the tunnel excavation face. The max principal stress path above each tunnel crown varies with the excavation steps, and the following development process can be obtained. It should be noted that the monitoring section and buried depth are totally the same in this section. The monitoring section is located in the middle of the numerical model (y = 50 m). The buried depth is 100 m.
(1)
For the first-excavated left tunnel (Figure 11):
① Figure 11a is merely the graphical illustration of the excavation stage of Figure 11b. When Ll is −16 m, the max principal stress above the left tunnel crown decreases linearly from the tunnel contour to the deep rock, roughly the same as the original rock stress. This trend indicates that when the tunnel excavation face is 1B (B represents the excavation span of each tunnel) behind the monitoring section, the excavation disturbance does not transfer to the monitoring section. The pressure arch at the monitoring section has not yet begun to form.
② When Ll is −6 m, the max principal stress above the left tunnel crown is greater than the original rock, increases first, gradually decreases outward, and finally returns to the initial stress state. The inner boundary of the pressure arch is located in the tunnel contour, and the outer boundary is far away from the tunnel wall. There is no loosening zone at this stage.
③ When Ll gradually changes from −6 m to +16 m, the max principal stress in a specific range is lower than the initial rock stress. The rock mass within this range is the loosening zone, and the upper boundary of the loosening zone is the inner boundary of the pressure arch. With the continuous excavation of the left tunnel, the core boundary height of the pressure arch increased from 0 to 9.9 m. During this stage, the max principal stress rises first and then decreases from the tunnel contour to the ground surface, and it is implied that the pressure arch is slowly developing towards the deep surrounding rock.
④ When Ll continually changes from +26 m to +42 m, the core boundary of the pressure arch of the left tunnel only slightly varies from 9.9 m to 10.4 m. Due to the thick rock pillar between the right and left tunnels, the right tunnel’s excavation has little effect on the left tunnel.
⑤ When Ll slowly changes from +52 m to +68 m, the core boundary of the pressure arch of the left tunnel increases significantly, and the core boundary height grows from 10.4 m to 16.3 m. This trend indicates that the middle tunnel excavation greatly influences the existing left tunnel, and the loose zone of the left tunnel will further extend upward.
(2)
For the second-excavated right tunnel (Figure 12):
① When Lr changes from −26 m to −10 m, the max principal stress above the right tunnel crown remains linear, implying the arching effect is weak. This distribution is because the distance between the left and right tunnels is 24.38 m (nearly 2B). When the relative distance increases further, the left tunnel excavation will have little influence on the right tunnel.
② When Lr increases from −6 m to +16 m, the variation of the max principal stress above the right tunnel crown is similar to that of the left tunnel. The core boundary of the pressure arch develops outward from the tunnel contour at 9.9 m.
③ When Lr varies from +26 m to +42 m, the core boundary of the pressure arch of the right tunnel significantly increases to 14.0 m, indicating that the right tunnel is severely disturbed due to the middle tunnel excavation. In addition, there are two extreme values of the max principal stress above the right tunnel crown. This effect is called the double arching effect, which can be understood as the combined function of the independent small pressure arch of the right tunnel and the combined large arch across the triple tunnels.
(3)
For the last-excavated middle tunnel (Figure 13):
① When Lm increases from −52 m to −36 m, the max principal stress above the middle tunnel crown changes significantly, and the inner boundary height of the pressure arch of the middle tunnel reaches 5.0 m. This trend implies that the left tunnel excavation severely disturbs the middle tunnel, and the middle tunnel’s advanced arching effect (incomplete arching effect) is observed 36 m ahead of the palm face.
② When Lm changes from −26 m to −10 m, the max principal stress above the middle tunnel crown varies more significantly, and the core boundary height of the pressure arch of the middle tunnel reaches 10.4 m. This development shows that the right tunnel excavation also severely influences the middle tunnel. The stress state of the surrounding rock in front of the middle tunnel is the worst. The pressure arch of the middle tunnel has been formed in front of the excavation face.
③ When Lm varies from −6 m to +16 m, the pressure arch of the middle tunnel continues to expand into the deep surrounding rock, and the core boundary height of the pressure arch finally reaches 25.3 m. The double arching effect of the middle tunnel is the most significant among triple tunnels, and the stress path change of the middle is also the most complex.

4.4.2. Lateral Development of Stress Arch

The development of the max principal stress of the monitoring section (Figure 14) is presented to study the arching behavior of triple tunnels. When the left tunnel is excavated (step 3), only a tiny pressure arch (where rock stress is increased) is observed above the left tunnel, and the loosening zone (stress reduction zone) is also limited around the tunnel contour. With the excavation of the right tunnel (step 6), there are two relatively independent pressure arches above the left and right tunnels, and the loosening zone area of the left tunnel generally remains unchanged. After the middle tunnel is excavated (step 9), a large combined-pressure arch is formed across the triple tunnels, and the loosening zone also sharply expands upward.
The mechanical behavior of the pressure arch of triple tunnels is qualitatively studied by roughly analyzing the stress nephogram. Accurate pressure arch boundaries (Figure 15) should be proposed to evaluate pressure arch stability quantitatively. The inner, outer, and core boundaries of the pressure arch in the middle tunnel are 14.0 m, 50.1 m, and 25.3 m, which are 107.1%, 15.6%, and 55.5% higher than those in the left tunnel, and 121.8%, 9.9%, and 80.5% higher than those in the right tunnel.

4.5. Different Safety States of the Triple Tunnels

According to the progressive evolution of the pressure arch, there are significant arching stability differences among the left, middle, and right tunnels. The different crown settlements, plastic zone heights, and inner boundaries of the pressure arches of the triple tunnels are listed in Table 3. The mechanical responses of the middle tunnel are the most prominent with the most significant parameter values, and the right tunnel is the least influenced in the triple tunnel excavations. This comparison shows that the safety state of the middle tunnel is the worst, and the left tunnel is the second-worst.
The different safety states of the triple tunnels can be attributed to the excavation sequence. The surrounding rock of the last excavated middle tunnel is severely damaged by the construction of the leading side tunnels, and considerable pre-settlement and pre-disturbance occur ahead of the excavation face of the middle tunnel. Therefore, the middle tunnel is the most vulnerable tunnel among triple tunnels. The stress state of the first excavated left tunnel is worse than the second excavated right tunnel because the left tunnel is affected by the double excavation disturbance caused by the right and middle tunnels. The right tunnel is only once affected by the middle tunnel excavation.
The above inference of the triple tunnel safety state can also be verified by the different ground deformations between the left and right rock pillars (Figure 16). The left rock pillar’s max horizontal and vertical displacements are 15.9 mm and 33.6 mm, respectively, significantly larger than the 11.9 mm and 29.6 mm of the right rock pillar. It implies that the left rock pillar is more seriously flexed inward than the right rock pillar and that the loosening zone of the left rock pillar is broader. In practical engineering, the support structure of the middle tunnel should be additionally strengthened, and the tunnel deformation should be strictly monitored in the middle tunnel.

5. Effect of Buried Depth on Pressure Arch

The buried depth strongly influences the arching behavior, and the pressure arch effect cannot be observed in the shallow buried tunnel. Different construction cases with a buried depth ranging from 20 m to 140 m are numerically simulated to reveal the influence of buried depth. The main excavation parameters are the same as the numerical model in Chapter 4. The final mechanical state after triple tunnel excavations is analyzed.

5.1. Deformation of Surrounding Rock with Different Buried Depth

The crown settlement of triple tunnels increases approximately linearly with buried depth (Figure 17). During the process of increasing buried depth from 20 m to 140 m, the average crown settlement of triple tunnels develops rapidly from 2.4 mm to 70.3 mm. The buried depth rises 7.0 times, while the average crown settlement increases 30.4 times. With the increase in buried depth, the initial stress of surrounding rock accordingly rises, and the stress concentration effect is remarkably prominent. The variation curve of horizontal displacement of triple tunnels also shows similar influencing behavior. With the buried depth increasing from 20 m to 140 m, the average horizontal convergence rises rapidly from the initial 6.8 mm to the final 105.0 mm, increasing to 15.4 times the initial value.
The rock pillar deformation is also significantly influenced by the change in buried depth (Figure 18). When the buried depth changes from 40 m to 140 m, the max horizontal displacement varies from 3.8 mm to 32.8 mm, and the max vertical displacement increases from 6.8 mm to 49.6 mm. The greater the buried depth, the greater the rock pillar deformation. It indicates sufficient expansion space should be reserved in deep triple tunnels, or the building clearance will be threatened after the tunnel lining is constructed.

5.2. Plastic Zone Distribution with Different Buried Depth

With the increased buried depth, the plastic zone extends outwards along the tunnel shoulder and tends to be integrated and connected (Figure 19). The plastic zone of the middle tunnel is significantly larger than the side tunnels. It can be explained that the large, buried depth will increase the stress concentration degree of the rock state. The middle tunnel is disturbed most seriously, with the largest failure zone. When the buried depth is greater than 70 m, the combined plastic arch is formed, and increasing the buried depth will not change the shape of the plastic zone. This phenomenon indicates that the triple tunnels have entered a deeply buried state and that a complete and adequate combined pressure arch structure has been formed to bear the rock load.

5.3. Pressure Arch Boundary with Different Buried Depth

The buried depth has an essential influence on the formation and development of the pressure arch of the triple tunnels. When the buried depth is less than 50 m, the arching behavior is not apparent (Figure 20). The internal and external boundary curves are close to the tunnel contour. These pressure arch boundary curves steadily extend upward with the increased buried depth. This trend is because when the buried depth increases, the load borne by the supporting structure increases, and the pressure arch also grows towards the deep surrounding rock.
When the buried depth is not more than 30 m, the max principal stress above the tunnel crown is always less than the initial rock stress, and there is no combined pressure arch. When the buried depth is 40 m, the inner boundary of the integrated pressure arch begins to appear (Figure 21), but there is no apparent outer boundary. It indicates that the combined pressure arch is not stable. When the buried depth reaches 70 m, clear internal and external boundaries are gradually formed. The inner boundaries of the left, middle, and right tunnels are 4.5 m, 7.2 m, and 3.4 m, while the outer boundaries are 41.1 m, 47.9 m, and 43.3 m. When the buried depth increases to 140 m, the internal and external boundaries of the combined pressure arch also slightly increase. Though the depth doubles, the boundary height change does not exceed 17%. This trend implies that the pressure arch shape does not change significantly after reaching a certain buried depth.
Based on the pressure arch effect, the combined pressure arch of closely spaced triple tunnels needs a certain depth. The critical arching depth is proposed to refer to the buried depth in a crucial state where the internal and external boundaries of the pressure arch start existing. For the closely spaced triple tunnels in class-V ground, the critical arching depth for pressure arch formation in deep and shallow buried is determined to be 70 m. In addition, the arching depth of a single tunnel is 40 m, and the critical arching depth of closely spaced tunnels is 1.75 times that of a single tunnel. The stress state of closely spaced triple tunnels is significantly worse than that of a single tunnel, so the design and construction method of the single tunnel cannot be directly applied to closely spaced tunnels.

6. Conclusions

This paper analyzes the arching mechanical responses of deeply buried, symmetrically distributed triple tunnels by field monitoring and numerical modeling. The main conclusions are summarized below.
  • The noticeable asymmetric pressure arch effect of triple tunnels is in-situ observed above triple tunnels. The arching behavior of the middle tunnels is more significant than the side tunnels.
  • Due to the complicated mechanical disturbances, pre-arching and double-arching effects are also observed in triple tunnel excavations.
  • The pre-deformation and pre-failure are considerable in the middle tunnel, so the safety state of the middle tunnel is the worst. Improved pillar-reinforcing cables and radial grouting should be timely adopted in the middle tunnel.
  • The left and right rock pillars are recognized as the two stable abutments of three pressure arches in triple tunnels. Pillar-reinforcing cables and radial grouting should be used to prevent the overall collapse of the combined large pressure arch.
  • Enough buried depth is crucial for forming a complete pressure arch in triple tunnels. The critical arching depth of symmetrically distributed triple tunnels is proposed based on whether the internal and external boundaries of the pressure arch exist.

Author Contributions

R.L., D.Z. and Y.S.: conceptualization, methodology, investigation, formal analysis, and writing; A.L. and J.L.: data curation, visualization, review, and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Anhui Province Postdoctoral Research Activities Funding Project under Grant 2022B642, and Key Project of the High-Speed Rail Joint Fund of the National Natural Science Foundation of China under Grant U1934210.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Symmetry 15 00673 g001 550
Figure 1. Excavation sequence and geometric dimensions of closely spaced triple tunnels in Badaling [27]. (①–⑨ represents the tunnel excavation sequence).
Figure 1. Excavation sequence and geometric dimensions of closely spaced triple tunnels in Badaling [27]. (①–⑨ represents the tunnel excavation sequence).
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Figure 2. Crown settlement of triple tunnels in the class-V ground (mm).
Figure 2. Crown settlement of triple tunnels in the class-V ground (mm).
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Figure 3. Surrounding rock pressure distribution of triple tunnels in the class-V ground (kPa). (①–⑨ represents the tunnel excavation sequence).
Figure 3. Surrounding rock pressure distribution of triple tunnels in the class-V ground (kPa). (①–⑨ represents the tunnel excavation sequence).
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Figure 4. Selection of measuring points (lines) of surrounding rock deformation.
Figure 4. Selection of measuring points (lines) of surrounding rock deformation.
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Figure 5. Elastic-plastic stress distribution of circular cavity in the hydrostatic stress field.
Figure 5. Elastic-plastic stress distribution of circular cavity in the hydrostatic stress field.
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Figure 6. Pressure arch boundary distribution after tunnel excavation.
Figure 6. Pressure arch boundary distribution after tunnel excavation.
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Figure 7. Characterization parameters of surrounding rock stability of deeply buried, closely spaced triple tunnels.
Figure 7. Characterization parameters of surrounding rock stability of deeply buried, closely spaced triple tunnels.
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Figure 8. Three-dimensional numerical model of closely spaced triple tunnels.
Figure 8. Three-dimensional numerical model of closely spaced triple tunnels.
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Figure 9. Progressive development of ground settlement of triple tunnels. (a) Step 1; (b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (f) Step 6; (g) Step 7; (h) Step 8; (i) Step 9.
Figure 9. Progressive development of ground settlement of triple tunnels. (a) Step 1; (b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (f) Step 6; (g) Step 7; (h) Step 8; (i) Step 9.
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Symmetry 15 00673 g010a 550Symmetry 15 00673 g010b 550
Figure 10. Progressive development of plastic zone of triple tunnels. (a) Step 1; (b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (f) Step 6; (g) Step 7; (h) Step 8; (i) Step 9.
Figure 10. Progressive development of plastic zone of triple tunnels. (a) Step 1; (b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (f) Step 6; (g) Step 7; (h) Step 8; (i) Step 9.
Symmetry 15 00673 g010aSymmetry 15 00673 g010b
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Figure 11. Progressive development of max principal stress of left tunnel. (a) Location of the left tunnel excavation face; (b) Variation curve of max principal stress above the left tunnel crown.
Figure 11. Progressive development of max principal stress of left tunnel. (a) Location of the left tunnel excavation face; (b) Variation curve of max principal stress above the left tunnel crown.
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Figure 12. Progressive development of max principal stress of right tunnel. (a) Location of the right tunnel excavation face; (b) Variation curve of max principal stress above the right tunnel crown.
Figure 12. Progressive development of max principal stress of right tunnel. (a) Location of the right tunnel excavation face; (b) Variation curve of max principal stress above the right tunnel crown.
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Figure 13. Progressive development of max principal stress of middle tunnel. (a) Location of the middle tunnel excavation face; (b) Variation curve of max principal stress above the middle tunnel crown.
Figure 13. Progressive development of max principal stress of middle tunnel. (a) Location of the middle tunnel excavation face; (b) Variation curve of max principal stress above the middle tunnel crown.
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Figure 14. Progressive development of max principal stress of triple tunnels. (a) Step 1; (b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (f) Step 6; (g) Step 7; (h) Step 8; (i) Step 9.
Figure 14. Progressive development of max principal stress of triple tunnels. (a) Step 1; (b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (f) Step 6; (g) Step 7; (h) Step 8; (i) Step 9.
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Figure 15. Pressure arch boundary height in V1 grade surrounding rock.
Figure 15. Pressure arch boundary height in V1 grade surrounding rock.
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Figure 16. Middle line deformation of the left and right pillars in the V1 grade surrounding rock. (a) Horizontal displacement of left and right rock pillars; (b) Vertical displacement of the left and right rock pillars.
Figure 16. Middle line deformation of the left and right pillars in the V1 grade surrounding rock. (a) Horizontal displacement of left and right rock pillars; (b) Vertical displacement of the left and right rock pillars.
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Figure 17. Deformation curves of triple tunnels with different depths Hd. (a) Crown settlement of tunnels at different buried depths Hd; (b) Horizontal convergence of tunnels with different depths Hd.
Figure 17. Deformation curves of triple tunnels with different depths Hd. (a) Crown settlement of tunnels at different buried depths Hd; (b) Horizontal convergence of tunnels with different depths Hd.
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Figure 18. Left rock pillar deformation with different depths Hd. (a) Horizontal displacement of the left rock pillar with different buried depth Hd; (b) Vertical displacement of the left rock pillar with different buried depth Hd.
Figure 18. Left rock pillar deformation with different depths Hd. (a) Horizontal displacement of the left rock pillar with different buried depth Hd; (b) Vertical displacement of the left rock pillar with different buried depth Hd.
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Figure 19. The distribution of plastic zone in different depths Hd. (a) Hd = 40 m; (b) Hd = 50 m; (c) Hd = 60 m; (d) Hd = 70 m; (e) Hd = 80 m; (f) Hd = 90 m; (g) Hd = 100 m; (h) Hd = 120 m; (i) Hd = 140 m.
Figure 19. The distribution of plastic zone in different depths Hd. (a) Hd = 40 m; (b) Hd = 50 m; (c) Hd = 60 m; (d) Hd = 70 m; (e) Hd = 80 m; (f) Hd = 90 m; (g) Hd = 100 m; (h) Hd = 120 m; (i) Hd = 140 m.
Symmetry 15 00673 g019aSymmetry 15 00673 g019b
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Figure 20. The max principal stress programs of different buried depths Hd. (a) Hd = 40 m; (b) Hd = 50 m; (c) Hd = 60 m; (d) Hd = 70 m; (e) Hd = 80 m; (f) Hd = 90 m; (g) Hd = 100 m; (h) Hd = 120 m; (i) Hd = 140 m.
Figure 20. The max principal stress programs of different buried depths Hd. (a) Hd = 40 m; (b) Hd = 50 m; (c) Hd = 60 m; (d) Hd = 70 m; (e) Hd = 80 m; (f) Hd = 90 m; (g) Hd = 100 m; (h) Hd = 120 m; (i) Hd = 140 m.
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Figure 21. Pressure arch range of triple tunnels with different buried depths. (a) Outer boundary height of pressure arch; (b) Inner boundary height of pressure arch; (c) Pressure arch thickness.
Figure 21. Pressure arch range of triple tunnels with different buried depths. (a) Outer boundary height of pressure arch; (b) Inner boundary height of pressure arch; (c) Pressure arch thickness.
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Table
Table 1. Relationship between the BQ System and the Q System.
Table 1. Relationship between the BQ System and the Q System.
ValueClass I
(Very Good)		Class II
(Good)		Class III
(Fair)		Class IV
(Poor)		Class V
(Very Poor)
BQ>550451–550351–450251–350<250
Q>4010–404–101–4<1
Table
Table 2. Material parameter for numerical model of triple tunnels.
Table 2. Material parameter for numerical model of triple tunnels.
MaterialDensity
ρ/kg·m−3		Elastic Modulus
E/GPa		Poisson Ratio
ν		Cohesion
c/kPa		Friction Angle
φ/°		Thickness
H/m
Class V ground17001.30.4110022200
Tunnel lining240026.40.2--0.4
Table
Table 3. Comparison of Three-tunnels Stability Characterization Parameters.
Table 3. Comparison of Three-tunnels Stability Characterization Parameters.
TunnelCrown Settlement/mmHeight of Plastic Zone/mInner Boundary of Pressure Arch/m
Left tunnel39.310.96.8
Right tunnel35.09.96.3
Middle tunnel47.227.6 m14.0
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Li, R.; Zhang, D.; Song, Y.; Li, A.; Luo, J. Pressure Arch Effect of Deeply Buried Symmetrically Distributed Triple Tunnels. Symmetry 2023, 15, 673. https://doi.org/10.3390/sym15030673

AMA Style

Li R, Zhang D, Song Y, Li A, Luo J. Pressure Arch Effect of Deeply Buried Symmetrically Distributed Triple Tunnels. Symmetry. 2023; 15(3):673. https://doi.org/10.3390/sym15030673

Chicago/Turabian Style

Li, Ran, Dingli Zhang, Yuan Song, Ao Li, and Jiwei Luo. 2023. "Pressure Arch Effect of Deeply Buried Symmetrically Distributed Triple Tunnels" Symmetry 15, no. 3: 673. https://doi.org/10.3390/sym15030673

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