Research on the Rockbursts Prevention Theory and Method for Elliptical Tunnels with Optimal Aspect Ratio Based on Prestress

Boyi Zhang¹, Cheng Zhao¹,²,³,, Yuan Qian⁴, Zeyuan Sun¹, Jinquan Xing¹, Yinfeng Luo¹, Qinyuan Luo¹*

¹ Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
² Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China
³ School of Engineering, Tibet University, Lhasa, Tibet 850000, China
⁴ Key Laboratory of Geotechnical Mechanics and Engineering of Ministry of Water Resources, Changjiang River Scientific Research Institute, Wuhan, Hubei 430010, China

Abstract

Rockbursts, as a severe geological hazard, often pose a significant threat to tunnel stability and construction safety. Traditional circular tunnels under non-hydrostatic pressure conditions are prone to forming localized stress concentration zones in the surrounding rock, which can easily trigger rockbursts. In contrast, elliptical tunnels with specific axis ratios have the potential to avoid this issue, making them a focus of rockburst prevention research. This paper analyzes the stress distribution and localized tangential stress concentration characteristics in conventional circular and elliptical tunnels. Based on this analysis, a theoretical method for rockburst prevention using prestressed elliptical tunnels with optimal aspect ratio is proposed, and a theoretical solution for the optimal aspect ratio of prestressed elliptical tunnels is derived. Using a tunnel in a high-altitude region as a case study, numerical results confirm that designing the prestressed tunnel shape according to the optimal aspect ratio can completely eliminate localized tangential stress concentration, achieve a uniformly distributed stress state, and significantly reduce the overall tangential stress level of the tunnel. Theoretically, this approach also offers the advantage of achieving excavation stress compensation. This provides a theoretical reference for stress control and rockburst prevention in high-stress tunnel engineering.

Keywords: high-stress tunnels; rockbursts prevention; tunnel cross-section shape; optimization design

1. Introduction

Rockbursts, as a severe geological hazard, typically pose significant threats to tunnel stability and construction safety, seriously affecting production, damaging equipment, and potentially causing casualties \cite{1}. The occurrence of rockbursts is closely related to stress redistribution after excavation, surrounding rock strength, and rock brittleness. In hard and brittle rock masses, tunnel shape and surrounding stress distribution have critical influences on rockburst initiation. Under non-hydrostatic pressure conditions, the tangential stress σθ distribution around conventional circular tunnels is non-uniform. Particularly under strong tectonic conditions, obvious relative stress concentration zones form around the tunnel perimeter. These stress concentration zones typically reach plasticity and damage first, becoming potential rockburst sources. Elliptical tunnels have become a research focus for rockburst prevention due to their favorable stress distribution characteristics \cite{2}.

Numerous scholars have investigated the failure mechanisms of elliptical tunnel surrounding rock. Guo Jiaqi et al. \cite{3} analyzed the plastic zone distribution around elliptical tunnel openings using complex variable function methods and applied the results to stability evaluation of karst tunnels and rock pillars in concealed dry cavities. Li Guichen et al. \cite{4} pointed out that for high-stress roadways, appropriate circular or elliptical tunnel cross-section shapes should be selected based on the lateral pressure coefficient and principal stress direction, with elliptical shapes generally better suited to high-stress environments. Ma Depeng et al. \cite{5} comprehensively analyzed the energy release characteristics, stress distribution, and plastic zone development patterns of surrounding rock, studying the influence of horizontal tectonic stress on tunnel stress states. Their results demonstrated that elliptical tunnels exhibit superior post-excavation mechanical behavior with the lowest rockburst risk. Fan Jiyue et al. \cite{6}, considering the interaction between lining and surrounding rock, indicated that when the aspect ratio of elliptical tunnels ranges between certain values, the lining experiences more reasonable stress distribution and higher structural safety. Yu Xuefu \cite{7,8} noted that elliptical tunnels have the capacity to relieve surrounding rock pressure. By modifying the traditional circular tunnel shape to an elliptical configuration, stress concentration around the tunnel can be effectively mitigated, stress distribution becomes more uniform, and rockburst risk is reduced. Additionally, rockburst-prone tunnels are typically located in hard and brittle rock characterized by failure at small deformations. He Manchao \cite{9} argued that tangential stress around tunnels primarily results from stress redistribution caused by excavation unloading. If stress compensation can be performed promptly after tunnel excavation, the magnitude of stress adjustment can be reduced, thereby effectively decreasing tangential stress development.

Therefore, combining two measures—optimized tunnel shape design and prestressed support—can provide internal pressure through prestressed support and calculate the elliptical tunnel aspect ratio that achieves uniform tangential stress distribution based on internal pressure and lateral pressure coefficient. This approach enables determination of the optimal tunnel cross-section shape for prestressed support according to geological conditions, further improving tunnel stability and reducing rockburst risk. This study theoretically proposes this method and discusses its feasibility, particularly for special tunnels in strongly tectonic regions such as shafts, ventilation wells, water diversion tunnels, and even some traffic tunnels with less stringent clearance requirements. To this end, the tangential stress distribution and local stress concentration characteristics of circular tunnels, circular prestressed support tunnels, and elliptical tunnels with optimal aspect ratio based on "axis transformation theory" are examined. Based on this analysis, the optimal shape for prestressed elliptical tunnels is derived, and the advantages of this method and its effectiveness in eliminating local high stress concentration are investigated.

2. Stress Distribution in Conventional Circular and Elliptical Tunnels

The tunnels addressed in this study are linear engineering structures whose length far exceeds their width and height, satisfying the plane strain assumption. Since tunnel siting must favor safety and stability, construction typically occurs in intact, non-fractured rock masses with good integrity, which exhibit strong hard-brittle characteristics. Therefore, the surrounding rock can be assumed as a homogeneous, isotropic elastic body and analyzed using elastic theory. To ensure tunnel stability, the tunnel axis direction is assumed to align with the maximum horizontal principal stress direction, with the tunnel cross-section subjected to minimum horizontal principal stress and vertical stress. Furthermore, prestressed rock bolts are assumed to be uniformly and densely distributed along the tunnel cross-section, with each bolt applying equal prestress. In theoretical calculations, the prestress is simplified as a uniform internal pressure acting on the tunnel perimeter.

2.1 Local Stress Distribution in Non-Prestressed Circular Tunnels

After tunnel excavation, rock mass disturbance gradually decreases with distance from the excavation face, approaching the original in-situ stress state. During construction, the tunnel wall is the most vulnerable area requiring special attention. For the most widely used circular tunnels without active prestressed support, assume the planned circular tunnel radius is R₀, the rock mass elastic modulus is E, Poisson's ratio is ν, vertical stress is p₀, and the lateral pressure coefficient is λ (assuming λ > 1). Under these conditions, the secondary stress distribution in surrounding rock has been extensively studied. Under elastic conditions, the expressions for tangential stress σθ (where θ is defined as the angle between a point on the tunnel perimeter and the tunnel crown, positive clockwise) and radial stress σr are given by \cite{10}:

$$
\sigma_\theta = p_0[(1+\lambda) + 2(1-\lambda)\cos 2\theta]
$$

$$
\sigma_r = 0
$$

Analysis shows that σθ reaches its maximum at the crown and minimum at the sidewalls, with a difference of:

$$
\Delta\sigma_\theta = 4p_0(\lambda-1)
$$

This demonstrates that under non-prestressed active support conditions, tangential stress distribution around circular tunnel walls is non-uniform. The greater the deviation of λ from 1, the larger this difference becomes, with more significant stress concentration at the crown and invert. The σθ at the crown increases and may satisfy the conditions for high stress concentration, causing the crown and invert to become the first areas of compressive stress damage and potential rockburst sources.

2.2 Stress Distribution in Prestressed Circular Tunnels

For prestressed circular tunnels, assume the prestress is uniformly distributed and acts perpendicular to the tunnel wall with magnitude pₛ. The boundary conditions are:

$$
\sigma_r = p_s
$$

$$
\tau_{r\theta} = 0
$$

From equation (4), after adding prestressed support, the σθ values are all reduced to some extent, which is effective for mitigating high stress concentration at the crown and invert. σθ still reaches its maximum p₀(3λ-1) and minimum p₀(3-λ) at the crown and sidewalls respectively, with the difference Δσθ remaining 4p₀(λ-1). The relative difference in stress remains unchanged. Moreover, this approach further increases the risk of tensile stress at the sidewalls. Since rock has very weak tensile strength, the appearance of tensile stress is extremely detrimental to tunnel stability and safety. For λ > 3, tensile stress will appear at the sidewalls even without prestress; for 1 < λ ≤ 3, the compressive stress at the sidewalls is minimal and decreases with increasing λ. Therefore, while circular active prestressed support can reduce highly concentrated maximum tangential stress to some degree, it has no effect on the difference between maximum and minimum tangential stress and increases the risk of tensile stress at minimum stress locations.

2.3 Stress Distribution in Non-Prestressed Elliptical Tunnels with Optimal Aspect Ratio

Optimizing tunnel shape to adjust the stress field is an economical and efficient approach, particularly applicable to non-traffic openings without cross-section clearance requirements. This study can draw upon "axis transformation theory," focusing more on reducing maximum compressive stress—that is, the weakening effect on stress concentration—while ensuring no tensile stress appears in the tunnel. Assume the tunnel is elliptical with horizontal semi-axis a and vertical semi-axis b, aspect ratio m = b/a, and lateral pressure coefficient λ > 1.

The stress distribution around elliptical tunnels without prestressed support \cite{7} is:

$$
\sigma_\theta = \frac{p_0}{m^2\cos^2\theta + \sin^2\theta} \left[ \frac{m^2(1+\lambda) + (\lambda-1)(m^2-1)\sin^2\theta}{m^2\cos^2\theta + \sin^2\theta} \right]
$$

When designed with the aspect ratio m = 1/λ, the elliptical tunnel perimeter not only has no tensile stress but also exhibits uniformly distributed compressive stress, which is most favorable for roadway stability. This aspect ratio is defined as the optimal aspect ratio \cite{8}.

The uniformly distributed compressive stress is:

$$
\sigma_\theta = p_0(1+\lambda)
$$

The difference between the maximum tangential stress in circular tunnels σθ = p₀(3λ-1) and this value is 2(λ-1)p₀ > 0, indicating that the uniformly distributed tangential stress under optimal aspect ratio reduces the maximum stress concentration in circular tunnels by 2(λ-1)p₀. The difference between equation (10) and the minimum tangential stress in circular tunnels σθ = p₀(3-λ) is 2(2λ-1)p₀ > 0, showing this value increases compared to the circular tunnel minimum, reducing the risk of local tensile stress. Thus, the obtained uniformly distributed stress eliminates stress concentration, but the stress reduction magnitude is limited. According to equation (10), when λ and p₀ are large, σθ remains substantial. Furthermore, without active prestressed support in engineering practice, excavation stress cannot be compensated, making it difficult to control deformation at the excavation face. Rockbursts can easily occur without obvious surrounding rock deformation, particularly in tunnels with high rockburst risk.

The above series of formulas are obtained with λ > 1 as an example, and the related theories and principles apply equally to cases where λ < 1.

3. Prestressed Elliptical Tunnel with Optimal Aspect Ratio for Rockburst Prevention

3.1 Calculation Method for Optimal Aspect Ratio of Prestressed Elliptical Tunnels

Analysis of the three methods above reveals that none can simultaneously satisfy multiple requirements: eliminating local stress concentration at the tunnel wall, comprehensively reducing stress levels, achieving stress compensation, and resolving the contradiction between passive support and rockbursts. Therefore, their effectiveness is limited. The design concept for prestressed elliptical tunnels with optimal aspect ratio can combine the advantages of non-prestressed elliptical tunnels with optimal aspect ratio and circular prestressed support, potentially meeting all the above requirements. This approach can overcome the static conditions for rockbursts and, through prestress compensation forming internal pressure support, anchor rock masses that may experience elastic rebound, controlling the dynamic conditions for rockbursts.

In the author's previous patent \cite{11}, theoretical analysis was used to derive the optimal aspect ratio m for elliptical tunnels under combined external and internal pressure:

$$
m = \frac{p_0 + p_s}{\lambda p_0 - p_s}
$$

The elliptical shape calculated through this aspect ratio enables uniform tangential stress around the tunnel perimeter, all in compression. This formula provides the optimal aspect ratio for prestressed elliptical tunnels. Designing the tunnel according to this aspect ratio yields a shape closest to uniform stress distribution, maximizing the elimination of stress concentration.

3.2 Numerical Validation of Effectiveness

The effectiveness of the proposed prestressed elliptical tunnel with optimal aspect ratio and its advantages compared to the other three methods can be directly demonstrated through engineering case studies. Numerical computation provides an effective verification approach, allowing different tunnel design methods to be input into numerical software to calculate their respective plastic zones for comparison in terms of range and magnitude. This validates the effectiveness of each design method in mitigating local stress concentration in surrounding rock. Plastic zones form due to local stress concentration, within which surrounding rock will experience damage and failure. In the rockburst development process, plastic zones that appear first may become subsequent rockburst sources. Therefore, observing the presence of plastic zones in numerical results for this hard rock engineering project helps directly determine whether local stress concentration conditions that could lead to rockbursts exist. If no plastic zone exists in a particular case, it indicates that the tunnel wall is unlikely to form subsequent rockburst sources.

A finite element model considering the elastoplastic characteristics of rock strata was established for this case study. Four scenarios were simulated: conventional circular tunnel, non-prestressed elliptical tunnel with optimal aspect ratio, conventional circular prestressed tunnel, and prestressed elliptical tunnel with optimal aspect ratio determined by the proposed method. The differences in plastic zones among these three tunnel shapes were compared to evaluate the engineering effectiveness of the proposed design method. Using a tunnel from a high-altitude railway project \cite{12,13} as an example for trial calculations, the tunnel site is located in the Yarlung Zangbo River bend area. At a burial depth of 1221 m, the measured in-situ stresses are: S_H = 33.58 MPa, S_h = 31.71 MPa, and S_v = 25.70 MPa. Assuming the most favorable condition where the tunnel axis aligns with the S_H direction, the vertical stress on the tunnel cross-section is p₀ = 25.70 MPa and the horizontal stress is λp₀ = 31.71 MPa, giving λ = 1.234. Assuming a circular tunnel radius R₀ of 6 m (the aspect ratio m from equation (11) is independent of R₀), the cross-sectional area S is 113.04 m². Under the condition of constant cross-sectional area, the specific dimensions of the semi-axes for each scenario are shown in Table 1 [TABLE:1]. The circular internal pressure support value uses pₛ = 10 MPa as a representative case.

Table 1 Tunnel semi-axis dimensions for four simulation scenarios

The rock type at the tunnel location is diorite, with physical and mechanical parameters shown in Table 2 [TABLE:2]. A Mohr-Coulomb elastoplastic constitutive model was adopted to consider elastoplastic characteristics. The lateral pressure coefficient and in-situ stresses were consistent with measured data. The computational sequence consisted of: initial in-situ stress equilibrium, tunnel excavation, and application of prestressed internal pressure support. Based on this model, the cumulative plastic deformation zones for the three tunnel design methods were calculated.

Table 2 Basic characteristics and physical-mechanical parameters of rock strata \cite{12}

Figure 1 [FIGURE:1] shows the cumulative plastic deformation of tunnel surrounding rock for the four scenarios. In Figure 1(a), the conventional non-prestressed circular tunnel exhibits plastic deformation around the entire perimeter, indicating surrounding rock damage, most pronounced near the crown and invert, which are most likely to become high stress concentration zones for rockbursts and subsequent rockburst sources. In Figure 1(b), the non-prestressed elliptical tunnel with optimal aspect ratio also shows a plastic zone around the perimeter, indicating that without prestressed support, although stress distribution is very uniform, high stress levels still create plastic zones around the entire tunnel. Compared to Figure 1(a), the plastic zone in Figure 1(b) is more uniform with smaller values, with concentration at the crown, invert, and sidewalls, related to the uniform stress distribution. Therefore, while the non-prestressed elliptical tunnel with optimal aspect ratio eliminates stress concentration, the overall stress level remains high, easily forming uniformly distributed plastic zones that could still become rockburst sources. In Figure 1(c), after applying 10 MPa prestress to the conventional circular tunnel, the plastic zone is significantly reduced while the stress level around the tunnel also decreases. Prestress application effectively mitigates local stress concentration, reduces the extent of plastic deformation, makes stress distribution more uniform, and lowers rockburst risk. In Figure 1(d), the prestressed elliptical tunnel with optimal aspect ratio, due to uniform and low stress levels around the tunnel, shows no plastic deformation throughout the process, with surrounding rock maintaining good original condition. This demonstrates that the design method effectively avoids premature damage in local tunnel areas. Under these secondary stress conditions, no location will become a rockburst source due to strain concentration and prior damage. Thus, the proposed theoretical method for rockburst prevention based on prestressed elliptical tunnels with optimal aspect ratio is validated as effective in solving the problem of high local stress concentration in tunnel surrounding rock, providing an optimized design approach for rockburst prevention.

3.3 Discussion on Rockburst Prevention Approach Using Prestressed Elliptical Support

The significance of the optimized design method based on prestressed elliptical tunnels with optimal aspect ratio lies in its ability to ensure low and uniformly distributed stress levels throughout the tunnel, eliminating local high stress concentration conditions for rockbursts so that no area around the tunnel becomes a preferential rockburst source. In engineering practice, especially in projects with strong tectonic stress, secondary stress becomes more concentrated in local areas under tectonic influence, making this method more necessary. Additionally, the introduction of prestressed internal pressure support enables active force application to surrounding rock immediately after excavation during the initial support stage, providing stress compensation and resolving the dilemma where passive support requires large rock deformation to develop anchoring force. This avoids the contradiction between hard-brittle surrounding rock experiencing rockburst at small deformation and passive support requiring large displacement to generate support force. Therefore, prestressed elliptical support can eliminate local stress concentration at the tunnel wall, comprehensively reduce stress levels, achieve stress compensation, and resolve the contradiction between passive support and rockbursts, offering a promising theoretical method for rockburst prevention.

The elliptical tunnel calculated using this method can ensure elimination of stress concentration while maintaining constant cross-sectional area. Primary application scenarios include ventilation tunnels/airways, mine shafts, water diversion tunnels, and traffic tunnels with less stringent clearance requirements. Moreover, mature smooth blasting technology can precisely control tunnel shape without significantly damaging surrounding rock \cite{14}, solving the construction challenges of elliptical tunnels. Due to its construction feasibility, elliptical tunnels have been applied in the underground engineering section structure design study of the Three Gorges Project \cite{7,10} and in the U1 section of Canada's URL tunnel \cite{15}.

4. Conclusions

From the perspective of active rockburst control, this study proposes a theoretical method for rockburst prevention based on prestressed elliptical tunnels with optimal aspect ratio and validates it through numerical simulation. The following conclusions are drawn:

(1) Under high in-situ stress conditions where the lateral pressure coefficient is not equal to 1, non-prestressed circular tunnels, prestressed circular tunnels, and non-prestressed elliptical tunnels with optimal aspect ratio all exhibit local tangential stress concentration at the tunnel wall and high overall tangential stress levels, creating favorable conditions for rockburst occurrence.

(2) The proposed theoretical method for rockburst prevention based on prestressed elliptical tunnels with optimal aspect ratio can precisely calculate tunnel shape according to in-situ stress and prestress levels. Its effect enables uniform tangential stress distribution around the tunnel and effectively reduces the overall tangential stress level, overcoming the rockburst condition of high local stress concentration and facilitating stress compensation.

(3) Numerical simulation of the engineering case study validates the effectiveness of the proposed method. Compared to circular tunnels and non-prestressed elliptical tunnels, only the prestressed elliptical tunnel with optimal aspect ratio showed no surrounding rock plastic zone caused by stress concentration during excavation, with the surrounding rock remaining in an elastic state.

The proposed method demonstrates significant theoretical effectiveness, but its assumptions remain relatively idealized. Further research is required before engineering application.

References

[1] He Manchao, Jia Xuena, Miao Jinli, et al. Experimental study on rockburst mechanism and its control measures[C]//Innovation and Practice in Rock Mechanics and Engineering: Proceedings of the 11th National Conference on Rock Mechanics and Engineering. Wuhan, Hubei, China, 2010.

[2] Wang Yang, He Manchao, Liu Dongqiao, et al. Experimental study on rockburst in surrounding rock of deep elliptical chambers[J]. Chinese Journal of Rock Mechanics and Engineering, 2021, 40(11).

[3] Guo Jiaqi, Qiao Chunsheng. Plastic zone of elliptical openings and its application in karst tunnel engineering[J]. Journal of the China Railway Society, 2013, 35(3): 108-114.

[4] Li Guichen, Zhang Nong, Wang Cheng, et al. Numerical simulation study on optimization of roadway cross-section shape in high ground stress[J]. Journal of China University of Mining & Technology, 2010, 39(5): 652-658.

[5] Ma Depeng, Yang Yongjie, Cao Jisheng, et al. Optimization of deep shaft roadway cross-section shape based on energy release[J]. Journal of Central South University (Science and Technology), 2015, 46(9): 3354-3360.

[6] Fan Jiyue, Su Zongxian. Study on reasonable excavation cross-section of underground caverns[J]. Railway Engineering, 2007(3): 51-53.

[7] Yu Xuefu. Blasting Excavation Teaching and Research Group, Beijing Institute of Iron and Steel Industry. Axis Transformation Theory—A Theory on Ground Pressure Problems[M]. Beijing: Metallurgical Industry Press.

[8] Yu Xuefu, Qiao Duan. Axis transformation theory and three laws of surrounding rock stable axis ratio[J]. Nonferrous Metals, 1981(3): 8-15.

[9] He Manchao, Ren Shulin, Tao Zhigang. Disaster prevention and control methods for deep-buried tunnels[J]. Journal of Engineering Geology, 2022, 30(6): 1777-1797.

[10] Cai Meifeng, He Manchao. Rock Mechanics and Engineering (2nd Edition)[M]. Beijing: Science Press, 2013.

[11] Zhao Cheng, Sun Zeyuan, Qian Yuan, et al. A layout design method for hard rock underground compressed air energy storage cavern group: CN202310850441.X[P].

[12] Wang Dong. Study on typical engineering geological problems of tunnels in high-altitude and large-height-difference areas of Sichuan-Tibet Railway[D]. Chengdu University of Technology.

[13] Wang Dong, Li Tianbin, Jiang Liangwen, et al. Analysis of in-situ stress characteristics and rockburst in an ultra-deep buried tunnel of Sichuan-Tibet Railway[J]. Journal of Railway Engineering Society, 2017, 34(4): 46-50.

[14] Lu Wenbo, Geng Xiang, Chen Ming, et al. Comparative study on excavation sequence and contour blasting method for deep underground powerhouse[J]. Chinese Journal of Rock Mechanics and Engineering, 2011, 30(8).

[15] Read R S. 20 years of excavation response studies at AECL's Underground Research Laboratory[J]. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(8): 1251-1275.