Preamble

Study on Calculation Method of Tunnel Earth Pressure Considering Expanding Arch Effect

Zhang Erfeng¹,²
(1. China Coal Technology Engineering Group Chongqing Research Institute, Chongqing 400039;
2. State Key Laboratory of Coal Mine Disaster Prevention and Control, Chongqing 400039)

Abstract

In the process of tunnel excavation, considering the form of tunnel potential slip surface and combined with the development law of stratum arch effect, a triangular slip surface model comprehensively considering the expanded arch effect is proposed, and the tunnel earth pressure under limit equilibrium conditions is derived and calculated using this model. Through an example, the influence of model parameters such as cohesion reduction coefficient, sliding surface friction angle, and slip surface camber angle on the earth pressure calculated by the model is compared and analyzed, and various limit equilibrium theory tunnel earth pressure analyses are carried out to verify the applicability of the calculation model. The results show that the smaller the cohesion reduction coefficient, the greater the earth pressure calculated by the model; the smaller the friction angle of the sliding surface, the greater the earth pressure calculated by the model; the greater the camber of the slip surface, the smaller the calculated earth pressure. With increasing depth, the earth pressure first increases and then decreases. In this model method, the selection of cohesion reduction coefficient plays a decisive role in the calculation of tunnel earth pressure, and the earth pressure calculated by selecting cohesion reduction coefficient λ=0.2 is similar to that calculated by other theories, showing stronger applicability. At the same time, combined with the analysis of the limitations and applicable critical depth of various loose earth pressures, and according to the use conditions and influencing factors of various earth pressures, the appropriate calculation method is selected, which provides a basis for accurate calculation of tunnel earth pressures.

Keywords: Soil arching effect; Expanding arch; Triangular slip surface; Limit equilibrium; Tunnel earth pressure

Author Biography: Zhang Erfeng (1992-), male, from Weinan, Shaanxi, engineer, master's degree, mainly engaged in research related to underground geotechnical engineering and geological disasters.

Funding Project: Key Independent Project of China Coal Technology Engineering Group Chongqing Research Institute (2023zdyf15)

Introduction

Tunnel engineering plays a crucial role in urban infrastructure construction; however, the complexity of underground environments often makes earth pressure calculation complicated and difficult. In tunnel engineering, non-uniform deformation occurs above the tunnel, leading to stress path transfer in the soil and the development of soil arching effects, which influence the surrounding rock pressure on the tunnel. Different stress paths generate geotechnical pressures that affect the selection of tunnel support systems and safe construction. Therefore, research on tunnel surrounding rock earth pressure calculation considering soil arching effects is of great significance for guiding tunnel design and construction.

In terms of model tests and numerical calculations, Chevalier [1-3] utilized trapdoor tests and discrete element software simulations to verify the existence of a tower-shaped slip surface in soil failure through analysis of arching effects and load transfer mechanisms. Takemura [4] employed centrifuge tests to study the failure mode of shallow circular tunnels as a triangular arch, considering the variation law of soil shear strength. Lei [5] conducted model experiments to investigate the failure mechanism and lining stress characteristics of shallow tunnels under unsymmetrical loading, analyzing the variation law and distribution of surrounding rock pressure, and obtained an inverted cone-shaped collapse body above the tunnel.

Ma Lin [6] et al. considered the structural characteristics of surrounding rock and used FLAC3D to simulate the excavation process of loess tunnels, studying the influence of surrounding rock structure on tunnel earth pressure during excavation. Wang Zhiwei [7] utilized UDEC discrete element software and the limit analysis upper bound method to study calculation methods for loose rock and soil pressure under different boundary conditions. Xu Weizhong et al. [8] used finite element software ABAQUS to study the formation mechanism and influence of soft soil arching effects under different shield burial depths. Bai Yanhui [9] conducted numerical calculations to study the differences in soft soil arching effects caused by stratum loss during shield excavation in Shanghai soft soil strata, obtaining the relationship between stratum loss and soft soil arching effects.

In theoretical calculations, the earth pressure above the tunnel changes with the soil arching effect. Engesser [10] proposed a calculation model for tunnel surrounding rock pressure based on an upward concave arch, which is only applicable to conditions with good surrounding rock. Lou Peijie [11] considered the influence of downward concave arch curves on tunnel earth pressure and established a calculation method for loose earth pressure of shallow tunnels under such arch curves. Zhu Menglong et al. [12] studied the loose earth pressure of shallow tunnels with tower-shaped slip surfaces based on incomplete soil arching effects. Wang Jiaquan et al. [13] determined the variation law of triangular slip surfaces through trapdoor tests and, using the triangular slip surface as a mechanical model, derived an earth pressure calculation model through differential soil strip stress state analysis. Xu Changjie et al. [14] investigated the loose earth pressure of major principal stress trajectories with circular, catenary, and parabolic arch shapes.

This study aims to propose a triangular slip surface model that comprehensively considers the expanding arch effect of soil arching for the calculation of tunnel earth pressure. Considering that the progressive development of soil arch slip surfaces significantly affects the loose earth pressure above the tunnel, the limit equilibrium method is used to establish a loose earth pressure calculation model considering triangular slip surfaces, analyzing the variation law and applicability of earth pressure under triangular slip surfaces. Through theoretical calculations and actual engineering case analyses, the accuracy and practicality of this method are verified, providing an important reference for tunnel design and construction.

2.1 Development Law of Stratum Arching

Terzaghi [15-16] verified the existence of soil arching effects through trapdoor tests, demonstrating that the arching effect develops with stratum changes. Iglesias et al. [17-18] analyzed the development of arching effects in soil layers through centrifugal model tests, proposing that the stratum arching effect can be roughly divided into four stages with increasing stratum deformation: elastic stage, maximum arch, expanding arch, and limit arch, as shown in Figure 1 [FIGURE:1].

Figure 1 Development law of stratum arching effect

In the elastic stage, affected by slight stratum disturbance, the soil arching effect has not yet formed. As deformation further increases, the soil at both edges of the trapdoor begins to yield, and a stratum arch develops upward along the trapdoor edges, entering the maximum arch stage. Under the action of the stratum arch, the stress of the upper soil transfers along the arch. In the maximum arch stage, shear surfaces develop at the trapdoor edges forming semi-circular or triangular stratum arches at an angle α to the vertical plane. With further deformation, the system enters the expanding arch stage, where the angle between the shear surface and vertical plane further decreases, developing into triangular slip surfaces that penetrate to the ground surface, and further evolving into trapezoidal shapes. When the shear surface becomes vertical, the limit arch stage is reached, and the slip surface becomes rectangular.

2.2 Tunnel Slip Surface Forms

Figure 2 [FIGURE:2] Calculation model for loose earth pressure of shallow tunnels

For the tunnel construction process, the form of slip surfaces varies at different stages under the influence of stratum deformation, while tunnel burial depth also significantly affects the slip surface form. The slip surface forms resulting from stratum deformation can be referenced from Iglesias' [17-18] stratum arch development, with the progressive stratum arch model shown in Figure 2. For shallow tunnels with relatively small displacement, the soil arching effect cannot be fully developed, and the soil mass is not affected by arching effects. For deep tunnels where soil arching effects occur, the earth pressure above the tunnel changes with the development of slip surface shapes.

3.1 Bierbaumer Theory

The Bierbaumer theory [19] proposes that when excavating shallow tunnels in loose rock mass, the subsidence of soil at the tunnel crown drives the subsidence of soil on both sides, ultimately forming a sliding surface. For calculation convenience, it is assumed that the sliding surface extends upward from the tunnel spring line at a rupture angle θ, with the slip surface assumed to be planar, as shown in Figure 3 [FIGURE:3].

Figure 3 Bierbaumer earth pressure calculation analysis diagram

Assuming that the vertical pressure on the tunnel crown is mainly borne by the weight of the overlying soil, a vertical sliding surface forms in the soil above the tunnel. During the sliding process of the vertical soil mass, a total vertical pressure Q is created above the tunnel, and the vertical pressure q acting on the tunnel structure crown is given by the calculation formula in Equation (2).

Where: a is half of the collapse width at the tunnel crown, γ is the unit weight of soil, c is the soil cohesion, φ is the internal friction angle; Ka is the Rankine earth pressure coefficient; H is the burial depth at the tunnel crown.

3.2 Expanded Arch Earth Pressure Theory

Figure 4 [FIGURE:4] Expanded arch earth pressure analysis diagram

According to the development law of stratum arching, the expanded arch is the most important stage in the deformation above the tunnel between the elastic stage and the limit deformation stage, and the triangular arch expansion theory was first proposed by Bierbaumer [20]. Based on the triangular arch expansion theory, this paper designs a calculation model considering expanded arch earth pressure. It is assumed that a triangular slip surface forms along the expanded arch crown above the tunnel. The expanded arch earth pressure analysis and collapse body force analysis are shown in Figure 4, with slip lines JG and JH above the tunnel. Let the collapse body JGH above the tunnel have weight W, and the symmetric wedge bodies AHJ and BGJ on both sides of the tunnel have weight G, with calculation formulas given in Equation (2).

Figure 5 [FIGURE:5] Force analysis diagram of collapse body

Taking the collapse body JGH above the tunnel as the research object, the forces on collapse body JGH are shown in Figure 5. According to Coulomb earth pressure theory, using wedge bodies AHJ and BGJ as objects, the force E₁ acting on the collapse body from both wedge bodies is calculated as shown in Equation (3). Decomposing E₁ along shear surfaces JH and JG yields normal stress N₁ and tangential stress T₁, directed obliquely downward, as shown in Equations (4) and (5). Applying the Mohr-Coulomb strength criterion on the rupture surface gives the shear force T, directed obliquely upward, as shown in Equation (6).

Where: G is the weight of wedge body AHJ, δ is the internal friction angle of surfaces JH and JG, taken as δ=φ within the soil mass, θ is the angle between the slip surface AD at the tunnel foot and the horizontal plane, L is the length of sliding inclined surface AH, α₁ is the angle between the expanded arch slip line and the horizontal plane, λ is the cohesion reduction coefficient introduced to fully consider the influence of cohesion on the slip surface. As the earth pressure loosens, the friction angle on the slip surface cannot be fully mobilized, resulting in a slip surface friction angle smaller than the soil internal friction angle, i.e., δ≤φ.

Substituting Equation (4) into (6) gives the shear strength T₁ on rupture surfaces JH and JG as shown in Equation (7).

Where: λ is the cohesion reduction coefficient, δ is the friction angle on slip surfaces JH and JG, with the friction angle on the slip surface being smaller than the soil internal friction angle.

According to the static equilibrium condition of collapse body JGH, the total earth pressure Q above the tunnel is obtained, as shown in Equation (8).

Where: the weight of collapse body is W. Substituting (3) and (7) into Equation (8) yields the total earth pressure Q above the tunnel, and the vertical pressure q acting on the tunnel structure crown, as shown in Equation (9).

3.3 Limit Arch Earth Pressure Theory

Under the action of the limit arch, vertical slip surfaces form above the tunnel. The sliding soil mass above is constrained by the soil on both sides, and the earth pressure above the tunnel equals the weight of the overlying soil mass minus the side friction resistance on both sliding surfaces.

Figure 6 [FIGURE:6] Limit arch earth pressure analysis diagram

Considering the limit arch earth pressure theory, micro-element analysis is conducted on the vertical sliding surface soil at the tunnel crown, yielding the Terzaghi theoretical calculation formula.

3.4 Other Earth Pressure Theories

According to the Code for Design of Highway Tunnels, the surrounding rock pressure of shallow tunnels is divided into two cases:

(1) When the burial depth H is less than or equal to the equivalent load depth hq, the vertical pressure on the tunnel crown is treated as uniform pressure q = γH acting on the tunnel crown. This theory ignores the friction resistance on the slip surface and is applicable to very shallow tunnel earth pressure calculations.

(2) When the burial depth is greater than hq but less than Hp, the calculation theory of Xie Jia-shu [21] is used.

4 Case Study

In the calculation method for earth pressure of triangular slip surfaces considering the expanded arch effect, the vertical pressure at the tunnel structure crown is mainly influenced by the burial depth H, the cohesion reduction coefficient λ of the triangular slip surface, and the friction angle δ of the triangular slip surface. Taking an urban metro tunnel as an example, the tunnel excavation width is 2a=6.2m, the lining structure height is h=6.5m. Assuming a slip surface at the tunnel foot with angle θ between the slip surface and horizontal plane, the soil above the tunnel is a single layer with unit weight γ=19 kN/m³, cohesion c=15 kPa, internal friction angle φ=30°. The cohesion reduction coefficient on the slip surface is λ, and the friction angle is δ.

4.1 Influence of Slip Surface Cohesion Reduction Coefficient λ

There are certain differences between the triangular slip surface and the actual tunnel slip surface. The cohesion reduction coefficient λ is introduced for correction, assuming that the cohesion is not fully mobilized during the deformation process of the triangular slip surface. To further investigate the influence of the triangular slip surface cohesion reduction coefficient λ on the earth pressure above the tunnel, the friction angle on the slip surface is assumed to be fully mobilized, i.e., δ=φ. The variation law of earth pressure with burial depth under different slip surface cohesion reduction coefficients λ is studied, as shown in Figure 7 [FIGURE:7].

Figure 7 Influence of cohesion reduction coefficient on earth pressure

As shown in Figure 7, the magnitude of earth pressure from the triangular slip surface model first increases and then decreases with increasing depth. When the tunnel burial depth is very shallow, the influence of cohesion mobilization on earth pressure is minimal. As the burial depth increases, the smaller the cohesion mobilization coefficient, the greater the earth pressure. Full cohesion mobilization reduces the earth pressure above the tunnel. Therefore, in practical engineering, considering methods to fully mobilize the cohesion on the slip surface to reduce tunnel earth pressure can ensure safe tunnel construction.

4.2 Influence of Slip Surface Friction Angle

The friction angle δ of the triangular potential slip surface directly affects the magnitude of anti-sliding force on the sliding surface. Without considering the influence of cohesion, different values are taken for the slip surface friction angle δ to analyze its influence on the earth pressure above the tunnel, obtaining the variation curve of earth pressure, as shown in Figure 8 [FIGURE:8].

Figure 8 Variation curve of earth pressure with slip surface friction angle

As can be seen from Figure 8, with increasing tunnel burial depth, the extension of the triangular slip surface and the increase in lateral earth pressure of the lateral soil wedge cause the frictional resistance on the slip surface to increase. When the generated frictional resistance exceeds the weight of the overlying sliding soil wedge, the earth pressure reaches its peak. With further increase in burial depth, the earth pressure gradually decreases. Overall, the earth pressure above the tunnel first increases and then decreases with increasing tunnel burial depth. In the shallow burial section, the influence of slip surface friction angle δ on earth pressure is relatively small, with earth pressure mainly provided by the weight of the overlying triangular wedge. As δ increases, the frictional resistance generated by the normal stress on the slip surface increases, and the earth pressure gradually decreases.

4.3 Influence of Slip Surface Camber Angle θ

The camber angle θ of the slip surface above the tunnel directly affects the range of sliding soil mass above the tunnel, which inevitably influences the vertical pressure above the tunnel. Without considering the influence of cohesion reduction and friction angle on the slip surface, the variation law of earth pressure with depth is calculated for different slip surface camber angles, as shown in Figure 9 [FIGURE:9].

Figure 9 Variation curve of earth pressure with slip surface camber angle

As shown in Figure 9, within a burial depth of 20m, the influence of slip surface camber angle θ on earth pressure is not significant. With further increase in depth, the larger the slip surface angle, the slower the increase in earth pressure, which eventually reaches its maximum value. As the camber angle increases, a second rupture surface may develop inside the soil wedge on both sides above the tunnel, making this model no longer applicable for calculating tunnel earth pressure. Therefore, using an expanded camber angle to calculate earth pressure above the tunnel has certain limitations and uncertainties.

4.4 Analysis of Various Earth Pressure Influences

Through the analysis of the influence of slip surface cohesion reduction coefficient, slip surface friction angle, and slip surface camber angle on earth pressure, it is evident that slip surface cohesion has the most significant impact on earth pressure. To further compare and analyze various earth pressures, several existing limit equilibrium methods for tunnel rock and soil pressure calculation are compared and analyzed. Taking cohesion reduction coefficients λ=0, 0.1, 0.2, 0.3 and slip surface friction angle δ, a comparative analysis is conducted on the variation law of various limit equilibrium theoretical earth pressures with depth, as shown in Figure 11 [FIGURE:11].

Figure 10 [FIGURE:10] Analysis of limit equilibrium theoretical earth pressure influences

As shown in Figure 10, the earth pressures calculated by the Bierbaumer and Xie Jia-shu formulas first increase and then decrease with increasing depth. When exceeding the critical depth, the calculated surrounding rock pressure results will not match reality. In contrast, the earth pressure calculated by the Terzaghi formula increases slowly with depth and eventually stabilizes. When λ=0 (i.e., without considering cohesion influence), as the value increases, the growth rate of earth pressure calculated by this paper's method gradually slows down. When λ=0.2, the calculated earth pressure is similar to that calculated by other theories.

Conclusions

(1) Based on the development law of tunnel stratum arching and considering the influence of soil arching effects, this study uses limit equilibrium analysis methods to propose a triangular slip surface model for stratum development law and derives the calculation of tunnel earth pressure considering the expanded arch effect.

(2) Based on the tunnel earth pressure calculation method established in this paper, the influence of slip surface cohesion reduction coefficient λ, slip surface friction angle δ, and slip surface camber angle θ on calculated earth pressure is analyzed. The smaller the cohesion reduction coefficient, the greater the calculated tunnel earth pressure; the smaller the slip surface friction angle, the greater the calculated earth pressure; the larger the slip surface camber angle, the smaller the calculated earth pressure. With increasing depth, earth pressure first increases and then decreases.

(3) Limit equilibrium analysis for earth pressure calculation shows that with increasing tunnel burial depth, the calculated earth pressure first increases and then decreases. Through comparative analysis of several limit equilibrium earth pressure calculation theories, taking the slip surface friction angle equal to the rock-soil friction angle and the slip surface camber angle as 45°+φ/2, the selection of cohesion reduction coefficient plays a decisive role in earth pressure calculation. Selecting the cohesion correction coefficient λ=0.2 yields earth pressure calculations similar to other theoretical methods, demonstrating stronger applicability.

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