## 1 Introduction  
With the continuous growth of the population and the ongoing development of cities, urban areas are facing increasingly severe challenges of traffic congestion and limited land resources. To address these issues, urban planners and engineers are actively exploring new transportation infrastructure and urban development models. Against this backdrop, the construction of underground tunnels, an efficient, environmentally friendly, and minimally disruptive mode of urban transportation, is rapidly advancing and finding widespread application. The proper evaluation of the supporting force of the tunnel face and the extent of disturbed soil is crucial for ensuring the safety and stability of underground constructions, especially during tunnel excavation. Extensive efforts have been directed towards addressing the stability of geotechnical structures, with a specific focus on tunnel face stability. This has been approached through various methods, including experimental, analytical, and numerical techniques. [PERSON] and [PERSON] (1994) [1] conducted experimental tests to investigate the failure of a shallow tunnel, providing valuable insights into stability challenges. Similarly, [PERSON] et al. (2011) [2] performed centrifugal model tests to assess the face stability of shallow tunnels. [PERSON] (2013) [3] adopted numerical analysis to study the pile response to open-face tunneling in stiff clay. Among these methodologies, the analytically kinematic approach based on limit analysis stands out due to its distinct advantages. Compared to experimental and numerical methods, limit analysis offers a rigorous and efficient means of evaluating tunnel face stability. This approach provides a systematic and theoretical framework for assessing failure mechanisms and optimizing support strategies, making it a valuable tool for ensuring the reliability and safety of underground constructions. The core process in limit analysis involves creating a valid failure mechanism to compute work, providing an upper or lower bound solution at the moment of failure for geotechnical structures. [PERSON] and [PERSON] (1990) [4] introduced a kinematically based failure mechanism using two conical rigid blocks to represent active and passive failures during tunnel excavation. Building upon this, [PERSON] et al. (2009) [5] introduced a failure mechanism with five conical blocks in homogeneous soil using limit analysis. Expanding the scope, [PERSON] et al. (2011) [6] introduced a spatial discretization technique, dividing the failure block into multiple elements based on a specified rule. This technique helps realize a 3D failure surface generated point by point, improving the precision of limit analysis for evaluating geotechnical structures' stability. In recent years, significant advances have been made in the context of limit analysis. A number of key studies have laid the structures for understanding stability mechanisms in such complex environments. For example, [PERSON] et al. (2004) [7] explored the stability of slopes using a limit analysis method, providing valuable insights into the dynamic and static forces affecting slope stability. In a follow-up study, with a limit analysis approach, a nonlinear failure criterion was considered in different engineering contexts ([PERSON] and [PERSON] (2004) [8]; [PERSON] and [PERSON] (2005) [9]). Most studies on tunnel stability have focused on homogeneous soils. In the area of tunnel stability, [PERSON] and [PERSON] (2011) [10] investigated the collapse mechanism of shallow tunnels under the nonlinear Hoek-Brown criterion, enhancing the design considerations for tunnel stability under complex loading conditions. However, in actual tunnel excavation, the tunnel face typically does not traverse homogeneous soil layers due to the complex and long geological processes ([PERSON] and [PERSON] (2011) [11]). Many scholars have thus analyzed tunnel face stability in spatially variable soils.