# Limit Equilibrium Models for Tunnel Face Stability in Composite Soft-Hard Strata
## 5 Analysis of Theoretical Model
### Comparison and Analysis  
The limit support pressure ratios obtained from the proposed model (TLSM), numerical simulation (NS), log-spiral model (LSM) [40], and partial-wedge model (PWM) [41] are shown in Figure 13. A general similarity can be observed in the trends of the changes in the four curves, with a gradual decrease in the limit support pressure ratio as SA reduces. However, substantial deviations can be noted between the predictions of the partial-wedge model and those of the numerical simulations. Notably, when SA ranges from 0.25 to 1.0, the limit support pressure ratio is consistently underestimated by the partial-wedge model, resulting in a serious risk of face instability. Similarly, as SA decreases within this range, the log-spiral model exhibits an increasing disparity from the numerical results, thereby indicating its relatively unreliable predictions. In contrast, the proposed TLSM demonstrates a close alignment of its predictions with the numerical simulations. Furthermore, the self-stability of the tunnel face occurs at SA \(\leq\) 0.125.  
As shown in Table 2, seven cases involving SA = 0, 0.125, 0.25, 0.5, 0.75, 0.875, and 1, with corresponding theoretical and numerical simulations were analyzed. When SA = 0 and 0.125, the results predicted by the three models are consistent with the numerical simulation. When SA ranges between 0.25 and 1.0, the results predicted by TLSM are closer to the numerical simulation compared to LSM and PWM. In addition, the difference between the predicted results of TLSM and the numerical simulation is controlled at around 10%. Through a comprehensive comparison of the different models, it was found that the superiority of the proposed model is mainly demonstrated in the range of SA = 0.25 to 1.0. The fitting of various theoretical models is depicted in Figure 14. In Figure 14a, the X axis corresponds to the limit support stress ratio predicted by TLSM for different SAs, while the Y axis represents the numerical simulation results. The detailed values can be found in Table 2. The coordinate meanings of Figure 14b,c are analogous to Figure 14a. The prediction outcomes of the three models were subjected to a proportional function  
2. In the event of tunnel face instability, displacement predominantly occurred in the soft soil layer, whereas the deformation of the hard rock layer was relatively minimal. Additionally, an increase in the SA led to increased overall strata deformation and subsequently increased surface ground disturbance.
3. The comparative analysis of different theoretical models for predicting the limit support pressure ratio reveals that the TLSM exhibits predictions that closely match those obtained from numerical simulation, with a difference of approximately 10%.
4. The prediction results of the three models were subjected to a proportional function (y = x) fitting. The fitting accuracy (\(R^{2}\)) for the three models (TLSM, LSM, and PWM) were 0.991, 0.934, and 0.446 respectively. This indicates that the proposed model (TLSM) outperforms both LSM and PWM in accurately predicting different SAs.  
Conceptualization, X.Z. and D.S.; methodology, Q.S.; data curation, Z.Y. and Y.Z.; data analysis, Q.L. and Q.S.; writing--original draft preparation, Q.S.; writing--review and editing, D.S.; supervision, X.Z.; project administration, Z.Y.; funding acquisition, D.S. and X.Z. All authors have read and agreed to the published version of the manuscript.  
This research work was funded by the Shenzhen Science and Technology Program (grant no.: 20220808143139001), the Key Research and Development Project of Guangdong Province (grant no.: 2019B11105001), and the Shenzhen University 2035 Program for Excellent Research (2022B007).  
Not applicable.  
All data included in this study are available upon request by contact with the corresponding author.  
No potential conflicts of interest were reported by the authors.