# Prediction of Tunnel Face Stability Using a Naive Bayes Classifier
## 3 Application Example  
This section presents an application example to illustrate the framework of the proposed method. In addition to the four features mentioned above, there are several other features that can affect tunnel face stability, such as the unit weight of soil, the surcharge load located on the ground surface, and the underground water. However, it is difficult to take all the influences into account simultaneously. Consequently, several simplifications have been made. The surcharge load was not included because existing literature found that this type of load has neglect effects on tunnel face stability when the ratio of the cover depth to the diameter of the tunnel \(C/D\) is larger than 0.6. The underground water was not considered due to the fact that in most cases the tunnel face is unstable in the presence of underground water. As a result, the method aims to predict the stability of a tunnel face for cases where the underground water table is lower than the bottom of the tunnel, or for cases where dewatering measures have been taken to lower the underground water table before the excavation stage. The unit weight of the soil was considered to be a relatively large constant, 19 kN/m\({}^{3}\). Under this circumstance, the prediction results are conservative for cases where the unit weights of the soil are less than 19 kN/m\({}^{3}\). With these simplifications, a number of tunnels with cover depth \(C\ =\ \{5\,\text{m},\ 10\,\text{m},\ 15\,\text{m},\ 20\,\text{m},\ 25\,\text{m},\ 30\,\text{m}\}\), and diameter \(D\ =\ \{5\,\text{m},\ 7.5\,\text{m},\ 10\,\text{m},\ 12.5\,\text{m},\ 15\,\text{m}\}\) were considered to be driven in a soil molded as an isotropic Mohr-Coulomb material with cohesive strength \(c\ =\ \{5\,\text{kPa},\ 10\,\text{kPa},\ 15\,\text{kPa},\ 20\,\text{kPa},\ 25\, \text{kPa}\}\), and friction angle \(\varphi\ =\ \{5^{\circ},\ 10^{\circ},15^{\circ},20^{\circ},25^{\circ}\}\). General cases of tunnels are within the ranges bounded by the specified values, beyond which are cases which usually require special analyses. For example, if the diameter of a tunnel exceeds 15 m, it can be taken as a large cross-section tunnel. Among these factors, it is well known that cohesion and friction angle are correlated. Herein, we assumed that these two parameters are independent of each other to be in accordance with the assumption of the naive Bayes classifier. In fact, the naive Bayes Classifier has been used to solve many classification problems that involve correlated factors. Despite the assumption, naive Bayes Classifier is still competitive when compared to other algorithms because of its convenience, lower computational cost, and comparable performance [29; 32; 42]. Figure 2 gives the diagram of the numerical models, including the geometry parameters and the boundary conditions. Normal fixities were applied on the left and right boundaries to constrain the horizontal displacements while full fixities were applied on the bottom boundary to constrain both the vertical and horizontal displacements. In terms of full factorial design, the number of the features and the corresponding levels presented in Figure 2 involves \(6\times 5\ =\ 30\) combinations regarding the cover depths and the diameters. This means that 30 numerical models are needed. In conjunction with the number of the values associated with the cohesive strength (5) and the friction angle (5), the total number of strength reduction analyses can be identified as follows:  
\[n\ =\ 30\times 5\times 5\ =\ 750. \tag{12}\]  
It should be emphasized that tunnel face stability is more appropriate to be analyzed in 3D simulations, which may be performed using OptumG3. However, the reality is that OptumG3 has not yet provided a built-in strength reduction analysis function. As a result, the strength reduction process needs to be conducted manually rather than automatically. This requires huge computational cost, particularly when there are 750 3D numerical models to be constructed and simulated.