## 6 Conclusions  
In this study, a novel method for the installation time of the initial support is proposed. The point safety factor is introduced to represent the stability of each rock zone on the excavation boundary. To describe the overall surrounding rock and avoid selecting the special monitoring points, the global  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**The Fitting Results of Equation (8)**} & \multicolumn{3}{c}{**The Installation Location of Initial Support**} \\ \cline{2-7}  & \(\mathbf{\lambda_{0}}\) & \(\mathbf{X}\) & \([\overline{\mathbf{F}}]\) & \(\mathbf{r_{s}}\) & \(\mathbf{\lambda_{s}}\) & \(\mathbf{x_{s}}\) \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{0.309} & \multirow{2}{*}{3.416} & 1.0 & 90.9\% & 0.698 & 4.09 \\  & & & 1.2 & 76.0\% & 0.544 & 2.06 \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{0.381} & \multirow{2}{*}{3.944} & 1.0 & 89.6\% & 0.634 & 3.49 \\  & & & 1.2 & 75.6\% & 0.463 & 1.22 \\ \hline \multirow{2}{*}{3} & \multirow{2}{*}{0.248} & \multirow{2}{*}{6.427} & 1.0 & 68.1\% & 0.310 & 0.73 \\  & & & 1.2 & 26.9\% & 0.074 & \(-\)2.64 \\ \hline \multirow{2}{*}{4} & \multirow{2}{*}{0.309} & \multirow{2}{*}{2.082} & 1.0 & \(--\) & \(--\) & \(--\) \\  & & & 1.2 & \(--\) & \(--\) & \(--\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of the support installation.  
Figure 16: Comparison of results between different cases: (**a**), case 0, case 1 and case 2 with different lateral pressure coefficient, (**b**) case 4, case 0 and case 3 with different lithology.  
safety factor is defined as the geometric mean of all point safety factors to estimate the stability of surrounding rock at a certain tunnel section. With the releasing of excavation load on the excavation boundary, the evolution of the global safety factor can be obtained by two-dimensional analysis. Based on the in-situ measurement, the complete longitudinal displacement profile can be fitted by the existing LDP equation. Combing the evolution of global safety factor with the LDP, the comprehensive graphic relationship between the global safety factor and the distance from the monitoring tunnel section to tunnel face can be established. When the global safety factor decreases to an admissible value, the surrounding rock is considered in a critical state and the corresponding distance in the above relationship is the recommended location to install the initial support. Meanwhile, several cases with diverse lateral pressure coefficients and geological conditions have been presented to validate the proposed method with good effect.  
Although the proposed method has some restrictions, it also provides quick feedback for the initial support design at the typical section in practical engineering.  
Conceptualization, K.S.; methodology, Y.-J.Z. and K.S.; software, Y.-J.Z.; validation, Y.-J.Z.; writing--original draft preparation, Y.-J.Z.; writing--review and editing, Y.-J.Z. and H.-Z.Z.; supervision, Z.-D.Q.; funding acquisition, K.S. and H.-G.W. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Key Research and Development Program of China [grant number 2016 YFC0401803]; and the National Natural Science Foundation of China [grant number 51879207].  
The authors declare that they have no conflict of interest.