Study on the Rationality of Tunnel Plane Strain Model Based on Stress Release Principle

Abstract

Based on the finite element difference software FLAC3D, a three-dimensional refined simulation model for tunnel excavation and a plane strain model based on the stress release principle were established. By comparing the deformation characteristics of surrounding rock and the stress distribution of support structures, it is concluded that, apart from slight differences in the stress distribution of the steel arch, the results of the plane strain model and the three-dimensional refined model show high consistency in the deformation characteristics of surrounding rock and the stress distribution of anchor bolts. The calculated results for the maximum load on a single anchor bolt differ by 13.25%, and the differences in surrounding rock deformation and the maximum load on the steel arch are both less than 8%. The plane strain model based on the stress release method can well match the three-dimensional refined model, indicating its feasibility for practical engineering analysis.

1 Introduction

Tunnel excavation construction is a complex three-dimensional and temporal problem [1]. During the construction process, the front of the construction face is the unexcavated stratum, and the back is the excavated tunnel. Excavation disrupts the equilibrium state of the original stress field in the rock mass, leading to unloading rebound and stress redistribution in the surrounding rock. The excavation continuously generates dynamic disturbances in the surrounding rock. Part of the stress and deformation in the rock mass near the excavation face is gradually released as the excavation face advances, causing changes in the stress state of the surrounding rock induced by excavation. This ultimately affects the deformation and failure modes of the tunnel surrounding rock [2]. Therefore, the evolving stress process in the surrounding rock throughout the entire tunnel excavation process has always been a key concern in today’s tunnel engineering field [3].

With the gradual development of numerical simulation technology, three-dimensional numerical simulation can now accurately simulate the process of tunnel excavation. However, when a large number of numerical calculations are required, true three-dimensional simulation calculations are very complex and time-consuming. Therefore, some scholars have proposed a method using the principle of stress release to simulate the process of tunnel excavation [4]. The stress release rate, i.e., after tunnel excavation, the stress in the rock mass is released due to unloading, resulting in a corresponding reduction in nodal forces on the tunnel face [5]. The ratio of the reduction in nodal forces on the tunnel face compared to the pre-excavation support force is called the stress release rate. This simulation method, by selecting a reasonable stress release rate and applying the corresponding support force, simulates the spatial effects of the excavation process, achieving the goal of using a two-dimensional plane strain model to simulate the three-dimensional tunnel excavation process [6]. This method is commonly used in the simulation of tunnel excavation and support processes, the release of ground pressure, the calculation of support structure forces, and discussions on the timing of tunnel support [7, 8].

While the calculation speed of a two-dimensional plane strain model is faster and more conducive to controlling variables when solving the deformation of surrounding rock or the force characteristics of support structures based on the stress release method [9], whether the two-dimensional model can match the calculation results of the three-dimensional model well and reproduce the spatiotemporal effects of the three-dimensional problem of tunnel excavation is still a topic without a unified viewpoint among scholars [10]. Based on the aforementioned issues, in order to investigate whether the two-dimensional model based on the stress release theory is suitable for addressing the three-dimensional spatiotemporal problem of tunnel excavation, this study utilizes the finite element simulation software FLAC3D. The stress release rates corresponding to excavation and unloading in the three-dimensional model are input into the two-dimensional plane strain model for analysis. A comparative analysis of the calculation results of the two-dimensional and three-dimensional models is conducted from the perspectives of surrounding rock deformation and the forces on support structures. This aims to validate the rationality and practicality of the tunnel excavation simulation method based on the stress release principle in actual engineering applications.

2 Three-Dimensional Refined Model and Plane Model Based on Stress Release Method

2.1 Three-Dimensional Refined Simulation Model

In the context of a tunnel project in Xinjiang, a specific tunnel section was selected for investigation. The selected tunnel section has a burial depth of 240 m, with the predominant rock type being Carboniferous volcanic breccia, classified as Class IIIb surrounding rock, and exhibiting a horseshoe-shaped cross-section. The geological survey data provided the physical–mechanical parameters for the rock mass and material properties for the support structures, as presented in Tables 1 and 2, respectively.

Table 1 Physical mechanical parameters of rock mass

Table 2 Material parameters of support structures

Leveraging the finite element finite difference software FLAC3D, a three-dimensional refined simulation model was established (Fig. 1). The boundaries of the model were set at a range of 30 m (approximately 4 times the tunnel diameter) around all sides, with a longitudinal length of 90 m and an overall size of 60 × 60 × 90 m. Normal constraints were applied to the boundaries and bottom, and vertical uniformly distributed loads were added to the top to simulate the gravity load of the overlying rock mass. To analyze the deformation characteristics of the surrounding rock during tunnel excavation, a monitoring section is established at Y = 39 m along the excavation axis. Due to the relatively large deformation values at the top, sidewalls, and bottom positions of the horseshoe-shaped section during excavation, monitoring points are set at these positions. The average deformation completion rate of each monitoring point is taken as the final deformation completion rate of the tunnel.

Fig. 1

Three-dimensional refined simulation model

2.2 Stress Release Method and Plane Strain Model

The computation of stress release rates encompasses methodologies such as the reverse stress approach, unbalanced force stress method, and incremental stress technique. Among these, the reverse stress method stands out for its adept control of variables, particularly in simulating the spatial effects on the excavation face during tunneling. The procedural sequence involves: 1. Imposing constrained boundary conditions on the computational model to ascertain the initial stress equilibrium under gravitational influence; 2. Post-tunnel excavation, calculating a single step and determining the unbalanced forces (Fi) at various nodes along the excavation face; 3. Achieving a targeted alteration in the surrounding rock stress release rate (λ) by applying reverse virtual support forces (Fn = (1 − λ)Fi) to individual nodes; 4. Eliminating virtual support forces, introducing initial support structures, and iteratively computing until reaching a state of equilibrium.

Based on the reverse stress method, the established plane strain model is illustrated in Fig. 2. The model extends along the tunnel axis for a length of 2.7 m, with overall dimensions of 60 × 60 × 2.7 m. The midsection along the tunnel axis is selected as the monitoring cross-section. A single excavation of the tunnel is performed, and after excavation, the initial surrounding rock stress is incrementally released into 20 equal parts (i.e., gradually adding corresponding virtual support forces Pi). Starting with the initial release of 5% of the surrounding rock stress, deformation data for the rock mass are obtained at this stress release rate. Subsequently, the surrounding rock stress release rate is increased in 5% increments, following a gradient, until the surrounding rock stress is fully released.

Fig. 2

Plane strain model based on the stress release method

3 Comparison of Computational Results

The key to simulating the three-dimensional spatiotemporal effects using the stress release method lies in the correspondence between the stress release rate and the control distance of the retaining surface. In this study, the completion rate of surrounding rock deformation is used as a bridge, aligning the three-dimensional refined model’s curve of surrounding rock deformation completion rate against the distance to the retaining surface with the curve of surrounding rock deformation completion rate against the stress release rate in the plane strain model. Taking the control distance of the retaining surface as 1 m, as shown in Fig. 3, the corresponding stress release rate is 78.35%. The following compares the characteristics of surrounding rock deformation and the force characteristics of support structures between the three-dimensional refined model and the plane strain model.

Fig. 3

The correspondence process between the control distance of the retaining surface and the stress release rate

3.1 Deformation Characteristics of Surrounding Rock

The deformation characteristics of tunnel surrounding rock in the two models are illustrated in Figs. 4 and 5, respectively.

Fig. 4

Deformation in the Z-direction of surrounding rock

Fig. 5

Deformation in the X-direction of surrounding rock

It can be observed that the calculated results of surrounding rock deformation in both models exhibit similar patterns of variation. Specifically, in the Z-direction (vertical direction) of the tunnel, there is a slight settlement at the top greater than the uplift at the bottom. In the X-direction, the convergence deformation on both sides is essentially consistent. While the numerical values of the surrounding rock deformation show slight deviations, the maximum settlement of the arch crown in the plane strain model is 12.163 mm, compared to 11.32 mm in the three-dimensional model, with a difference of 7.45%. The maximum uplift at the bottom in the plane strain model is 10.958 mm, which is 7.92% larger than the 10.154 mm in the three-dimensional model. In the X-direction, the maximum convergence deformations in the two models are 13.192 and 12.917 mm, with a difference of only 2.13%. From the perspective of surrounding rock deformation, although the calculated results of the plane strain model are slightly larger, the error does not exceed 8%, which meets the allowable error for engineering purposes. When considering deformation results from two directions, the difference in deformation in the Z-direction is relatively significant. This difference may be attributed to the fact that the Z-direction represents the maximum principal stress direction in the initial stress field. The disparity in the maximum principal stresses between the two models is greater in the vertical stress direction than in the horizontal stress direction. This implies that in practical engineering, special attention should be given to the differences in settlement at vault of the tunnel.

3.2 Mechanical Characteristics of Segment Structure

The stress conditions of the tunnel support structures in the two models are illustrated in Figs. 6 and 7, respectively.

Fig. 6

Forces on rock bolts

Fig. 7

Forces on steel arches

It can be observed that in the distribution of forces on rock bolts, the calculation results of the plane strain model and the three-dimensional refined model are essentially the same. Both models show that the maximum tensile stress is experienced in the middle of the top rock bolt on the front side of the tunnel, while the forces on the rock bolts on the rear side are more evenly distributed. In terms of specific values, the maximum tensile stress on a single rock bolt in the plane strain model is 34.348 kN, while the result from the three-dimensional model is 30.361 kN, resulting in a difference of 13.25%. In the distribution of forces on the steel arches, there is a slight difference in the calculation results between the two models. The steel arches are simulated using two layers of calculation elements. In the plane strain model, the forces on the two layers of elements are basically consistent. In the distribution of forces on the steel arches, there is a slight difference in the calculated results between the two models. The steel arches are simulated using two layers of calculation elements. In the plane strain model, the forces on the two layers of elements are basically consistent. In the three-dimensional model, however, the inner elements experience slightly higher compressive stresses than the outer elements. In terms of specific values, the maximum compressive stress on a single steel arch in the plane strain model is 21.819 MPa, while the result from the three-dimensional model is 20.97 MPa, resulting in a difference of only 4.05%. The cause of this difference may be that the calculation elements in the Y-direction are identical in the plane strain model, leading to convergence in the results for the forces on the steel arches. In contrast, the elements in the Y-direction in the three-dimensional model have different sizes, resulting in more pronounced differences between the calculation elements.

In summary, considering the comprehensive comparison results, the calculation results of the plane strain model and the three-dimensional refined model exhibit a high degree of consistency in the deformation characteristics of surrounding rock and the distribution of forces on rock bolts. While the forces on steel arches show slight differences due to the subdivision of calculation elements, the overall agreement is notable. In terms of specific values, the maximum force on a single rock bolt differs by 13.25% between the two models. However, both values are significantly below the design strength for tensile capacity of the rock bolt, minimizing the safety risks associated with model selection. The differences in surrounding rock deformation and the maximum force on the steel arches are both below 8%.

In conclusion, the use of the plane strain model based on the stress release method demonstrates good compatibility with the three-dimensional refined model. In situations where extensive computations are required, the plane strain model can be considered feasible for practical engineering analysis.

4 Conclusion

This paper is set against the engineering context of a tunnel project in Xinjiang. Leveraging the finite element numerical software FLAC3D, we constructed a three-dimensional refined simulation model for tunnel excavation and a plane strain model based on the stress release method. Through an examination of surrounding rock deformation and the forces acting on support structures, we validated the rationale of the plane strain model in simulating the three-dimensional spatiotemporal challenges of tunnel excavation. In conclusion, the following findings were obtained:

• (1) The curve of surrounding rock deformation completion rate against the distance to the retaining surface obtained from monitoring points in the three-dimensional refined model aligns with the curve of surrounding rock deformation completion rate against the stress release rate obtained from monitoring points in the plane strain model. This alignment is achieved through the common indicator of surrounding rock deformation completion rate, enabling the plane strain model to simulate the spatiotemporal challenges of three-dimensional excavation.

• (2) The calculation results of surrounding rock deformation in the plane strain model and the three-dimensional refined model exhibit a fundamentally similar pattern of variation. While there are slight numerical deviations, with a difference of 7.45% in the maximum settlement at the arch crown, 7.92% in the maximum uplift at the bottom, and only 2.13% in the maximum convergence deformation in the X-direction.

• (3) Regarding the maximum tensile stress on a single rock bolt, there is a difference of 13.25% between the calculation results of the plane strain model and the three-dimensional refined model. However, the numerical values are well below the design strength for the tensile capacity of the rock bolt, minimizing the safety risks associated with model selection. In the distribution of forces on the steel arches, although there is a slight difference in the calculation results, the specific numerical values differ by only 4.05%.

In conclusion, considering both the deformation of surrounding rock and the forces on support structures, although there are slight differences in the numerical values between the plane strain model and the three-dimensional model, they fall within the acceptable range of engineering error. Moreover, the trends in the calculation results are consistent between the two models. Therefore, in practical engineering analysis where extensive computations are required and three-dimensional or refined modeling is impractical, using the plane strain model based on the stress release method as a substitute for the three-dimensional refined model to analyze the three-dimensional spatiotemporal problem of tunnel excavation is feasible. It’s important to note that during the tunneling process, the stress on the surrounding rock in the unexcavated portion ahead of the working face also undergoes changes, but a two-dimensional model may struggle to simulate the state of the rock mass before excavation. This is an aspect that needs to be carefully considered in future research efforts.