# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 1 Introduction  
Rapid urbanization has led to a significant increase in the urban population, causing a growing demand for urban transportation that cannot be met by the limited ground area available. This has resulted in a worsening problem of ground traffic bottlenecks. As a solution, underground urban transportation has emerged as an excellent alternative because it is not restricted by ground conditions. Today, it has become one of the main approaches for developing and utilizing underground spaces in cities [1, 2, 3, 4, 5, 6]. The critical nodes that connect the underground transportation network are underground interchange tunnels. These tunnels can be classified into two types: planar and three-dimensional intersections. The planar intersection type, known as bifurcated tunnels, typically consistsof a separated portion, a small clearance section, a continuous arch section, and a large span section [7]. A specific type of tunnel layout is the small clear-distance tunnel, where the double tunnel clear distance is less than the recommended value [8]. Small clear-distance tunnels are widely used because they offer advantages such as easier line extension, reduced land occupation, lower costs, simpler construction techniques, and fewer terrain constraints [9; 10; 11; 12; 13; 14]. However, the middle rock pillar (MRP) between small-clearance tunnels is relatively thin, which means that it is subjected to multiple disturbances during construction, leading to more complex stresses on the surrounding rock [15; 16; 17; 18]. The MRP plays a crucial role in supporting small clear-distance tunnels, and excavating tunnels inevitably damages and affects their mechanical behavior. Therefore, the key technology in constructing small clear-distance tunnels lies in strictly controlling the damage caused during tunnel construction. Ensuring that the MRP has sufficient strength and stability is vital [19; 20]. Consequently, it is necessary to determine the bearing state of the MRP to ensure the safety of small-clearance tunnel construction [21; 22]. During the construction process of small-clearance tunnels, the excavation of the advance tunnel alters the initial stress state of the surrounding rock. This, in turn, affects the excavation of the backward tunnel and can potentially lead to the deformation of the MRP. Failure to implement appropriate protective measures promptly can further escalate the occurrence of engineering accidents [23; 24]. Numerous scholars have employed various methods, such as the complex function method [25], the bipolar coordinate method [26; 27], the Schwarz alternation method [28; 29], and other approaches [30], to derive the stress distribution in the surrounding rock of small-clearance tunnels. In addition, [PERSON] et al. [31], [PERSON] et al. [27], [PERSON] [13], and [PERSON] et al. [32] established different types of models to investigate the stress characteristics of MRPs and proposed judgment criteria for the stability of continuous arches and small-clearance tunnels. Moreover, the instability and failure characteristics of mid-adjacent rock under various construction conditions have been analyzed [13]. The stress distribution rule of the MRP in small interval tunnels can be approximated by calculating the interface normal stress and shear stress analysis via the concept of the reserved support hypothesis, along with considering the average stress and an additional bending moment of the MRP, ultimately leading to the final stress distribution of the rock wall [33]. In recent years, numerous experimental analyses, in situ monitoring studies, and numerical simulations have been conducted to evaluate pressures and deformations around small clear-distance tunnels. [PERSON] et al. investigated the stability of small-clearance tunnels and the soil arching effect through centrifugal model tests and numerical simulations and revealed that the failure mode of the interbeds in small-clearance tunnels is similar to that of slopes, mostly involving shear failure [34]. [PERSON] et al.