# Limit Equilibrium Models for Tunnel Face Stability in Composite Soft-Hard Strata
## 5 Analysis of Theoretical Model  
### Overview  
In this study, the limit equilibrium method has been adopted to predict the limit support pressure ratio of composite strata with various SAs. The Horn's classical wedge model is predominantly composed of a wedge and an overlying prism, as depicted in Figure 8a. However, in both the traditional and modified wedge models [23; 24; 25], the assumed planar failure surface is questioned due to experimental tests [5] and numerical simulations [21] indicating that the actual failure surface appears to be a logarithmic spiral curve in profile. [PERSON] et al. were the first to predict the limit support pressure of the tunnel face using a two-dimensional (\(2D\)) logarithmic spiral model, as shown in Figure 8b. Due to the fact that the problem of tunnel excavation is three-dimensional, many subsequent studies have expanded [PERSON]'s model to a three-dimensional log-spiral model [10; 39]. Furthermore, the failure mechanism of tunnel face in soft upper-hard lower layers has been investigated by researchers through a combination of theoretical studies and experimental tests. The relevant theoretical models have been proposed, such as the log-spiral model [40] and partial-wedge model [41], to predict the limit support pressure ratio of composite strata with various SAs, as shown in Figure 9. In this study, numerical simulations were conducted to investigate the failure mechanism at the limit state under different ratios of soft and hard layers. The relevant theoretical models were  
Figure 8: Schematic diagram of classic model. (**a**) [PERSON]’s wedge model. (**b**) [PERSON]’s log-spiral model. Figure 7: Plastic shear strain contours under the limit conditions. optimized based on the phenomena shown in Figures 6 and 7. An optimized model is shown in Figure 10.  
Figures 6 and 7 show that when the tunnel face fails, slip predominantly occurs within the soft soil layer. The slip surface is depicted by incorporating a log-spiral model, whereby the initiation of slip originates from the tunnel crown and terminates at the interface between the soft soil and hard rock. This optimized model is referred to as a truncated log-spiral limit equilibrium model (TLSM). In Figure 10, a collapse mechanism is shown, similar to classical wedge models, which involves the presence of a truncated log-spiral (TLS) slip surface in front of the tunnel face and an overlying prism. Furthermore, \(D\) represents the diameter of the tunnel, and \(C\) signifies the cover depth of the tunnel. Here are some basic assumptions for the TLSM:  
1. The failure only occurs in areas of the upper soft soil layer [40], and the geometric shape of the slip surface is a logarithmic spiral. 2. The upper soft soil layer is considered to be homogeneous and isotropic, adhering strictly to the Mohr-Coulomb failure criterion on each failure surface. 3. The soil pressure generated by the overlying prism acts vertically on the top of TLS. To facilitate the implementation of this straightforward model, a square with an equivalent area was used to approximate the circular cross-section of the tunnel face [39]. This simplification is acceptable as the pressure obtained from both the experimental tests and numerical simulations reasonably corresponds to the predicted limit support pressure [5; 10]. From the force analysis of the longitudinal section of the TLS model, as shown in Figure 11, we established a coordinate system with point \(O\) as the origin. The Y and Z axes are parallel to the horizontal and vertical directions, respectively. The slip surface of  
Figure 10: Limit equilibrium model of the truncated log-spiral. Figure 9: The optimization model for the tunnel face. (**a**) The log-spiral model. (**b**) The partial-wedge model. the log-spiral model initiates at point \(a\), terminates at point \(d\), and is intercepted at point \(b\). In Figure 6, the variables \(R_{a}\), \(R_{b}\), and \(R_{d}\) denote the lengths of lines \(Oa\), \(Ob\), and \(Od\), respectively.