# Limit Equilibrium Models for Tunnel Face Stability in Composite Soft-Hard Strata
## 1 Introduction
[18; 19] investigated the active and passive failures of circular tunnel excavation faces using spatial discretization techniques and presented two rotational failure mechanisms for active and passive failures. In the limit equilibrium method, the limit support pressure is calculated by examining the static and moment balances of individual wedges within the sliding soil [23; 24; 25; 26; 27; 28; 29; 30; 31]. [PERSON] [23] initially developed a limit equilibrium model, based on silo theory, to analyze the failure of the tunnel face. This model assumes that the failure zone consists of two parts, namely the wedge in front of the tunnel face and the prism above the wedge. Subsequently, several scholars have made a series of improvements and optimizations to [PERSON]'s model, taking into account the specific characteristics of practical scenarios. [PERSON] and [PERSON] [24] considered the arching effect of the overlying soil, while [PERSON] and [PERSON] [25] studied the influence of seepage. [PERSON] [26] further enhanced the wedge failure model by incorporating the impact of horizontal stresses on the stability of the excavation surface using the strip method. [PERSON] [27] analyzed the influence of arching effects on the vertical pressures at the top of the wedge. [PERSON] et al. [28] proposed a failure model that takes into account the relationship between the height of the silo and the depth of the tunnel. The above study provided substantial contributions to the understanding of the failure mechanisms of tunnel faces under different working conditions. However, most of these studies were based on the assumption of a homogeneous soil layer in front of the tunnel face. Few studies based on theoretical analysis have been conducted to investigate tunnel face stability in composite strata. [PERSON] et al. [32] extended a rotational face collapse mechanism to calculate the limit support pressure of a tunnel face in layered or stratified ground. The findings suggested that the upper weak layers of the tunnel are susceptible to partial failure. Based on the upper bound theorem of kinematic limit analysis and the nonlinear Hoek-Brown yield criteria, [PERSON] et al. [33] investigated the influence of rock layer inclination and weak interlayers on tunnel face stability. [PERSON] et al. [34] used discretization technology to improve the understanding of the collapse mechanism of a rotating rigid body based on limit analysis to evaluate tunnel face stability in inclined layered soil. Summarizing the above research findings, it is worth noting that there have been limited investigations conducted on tunnel face stability in horizontal composite layers. Moreover, there is currently a lack of a comprehensive limit equilibrium model capable of effectively predicting the limit support pressure under various ratios of soft and hard layers. Addressing this gap with further research could provide valuable insights for the field of tunneling engineering. Tunnel face stability in soft upper-hard lower composite strata was investigated in this study. First, the impact of different ratios of soft and hard layers, denoted as the soft-hard ratio (SA), on the limit support pressure is analyzed using numerical simulation. Second, a limit equilibrium model was established to predict the limit support pressure of the tunnel face by analyzing the patterns of the variation in the displacement and shear strain contours. Finally, the model was compared and analyzed alongside numerical simulations and relevant theoretical models.