# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 1 Introduction
proposed an experimental program to analyze the change in and distribution of tunnel pressures [35]. Based on a 3D large-scale geomechanical model test of excavation and support in a large cross-section tunnel, the progressive failure of the surrounding rock mass, as well as the evolution of deformation and stress in the rock mass, was studied [36; 37]. Compared with theoretical analysis and model tests, numerical simulation offers several advantages, including shorter timeframes, lower costs, and higher calculation accuracy. As a result, it has become widely utilized in the calculation of small clear-distance tunnels [20; 38; 39]. [PERSON] et al. employed a 2D elastoplastic numerical calculation method to investigate the stress and deformation characteristics of the MRP in parallel twin tunnels with surrounding rocks graded III, IV, and V. Their findings indicated that the stress and deformation of the MRP were adversely affected by the presence of a parallel tunnel with a smaller interval [39]. [PERSON] and [PERSON] proposed the concept of rock-column zone division and calculated the yield proximity of surrounding rocks on the basis of numerical simulation results. This enabled them to assess the stability of the MRP via yield proximity criteria [40]. A novel nonlinear reinforcement analysis method was proposed for analyzing the stability of a large-span tail water branch tunnel with complex three-dimensional geometry [41]. [PERSON] et al. developed the yield approach index (YAI) function, which is based on the Mohr-Coulomb shear yield criteria and tensile yield criteria, to analyze the mechanical characteristics of rock that is shared between two styles of neighboring tunnels [42]. These studies were conducted via numerical analysis and centrifugal tests. Considering the instability mode of the MRP, [PERSON] et al. conducted a numerical simulation to analyze the process of loading until destabilization occurred [43]. When examining the stability of the surrounding rock, the strength reduction method (SRM) is commonly employed [44; 45; 46]. The SRM, as a limit analysis method, offers the advantage of not requiring prior knowledge of the failure surface of the project. Consequently, it is effective in assessing overall failure in geotechnical engineering [47]. By utilizing the SRM, it is possible to determine the stability safety factor of the overall failure of a rock tunnel as a basis for tunnel design. The finite element SRM has gained wide popularity for analyzing the stability of rock surrounding tunnels [48; 49; 50]. Both the finite element limit analysis method and the SRM are employed to derive the stability factor formula of the tunnel face, which includes the penetration force, and apply it to actual tunnel engineering [51; 52]. [PERSON] et al. investigated the stability of tunnel-surrounding rock via the DDA method, the SRM, and catastrophe theory [53]. The stability coefficient of shallow tunnels was studied under the upper and lower bound theories of limit analysis by [PERSON] and [PERSON], employing the strength reduction theory [54]. [PERSON] et al. used finite element SRM, physical model tests, and numerical simulations to study the influence of the principal stress direction on the stability of sandstone tunnels [55]. [PERSON] et al. proposed a quantitative calculation method for the stability coefficient of shallow tunnel faces and developed a support optimization system, clarifying the effectiveness of various reinforcement measures on face stability [56]. Recently, [PERSON] et al. [57] systematically investigated the evolution of surrounding rock stability in multi-line parallel tunnels using the upper bound method with rigid translational elements, revealing that the middle rock pillar plays a dominant role in failure under small spacing conditions. [PERSON] et al.