# Calculation Method of Loose Pressure in Surrounding Rock Mass
## 4 Boundary Function of the Surrounding Rock's Loose Zone  
The boundary curve of the loose zone was fitted by a non-linear function based on the scope of the loose zone in the surrounding rock mass. Owing to the symmetry of the loose zone in the surrounding rock, half of the boundary curve was taken as the research object. The Cartesian rectangular coordinate system was adopted, taking the arch vertex of the loose zone as the coordinate origin. The positive direction on the X-axis was vertically downward, and the positive direction on the Y-axis was horizontal and to the left. Ten characteristic points were selected from the coordinate information on the boundary curve \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Influencing Factors** & \(\mathbf{H}\) & \(\mathbf{B}\) & \(\mathbf{\lambda}\) & \(\mathbf{K}\) & **GSI** \\ \hline Weight value & 0.12 & 0.37 & 0.15 & 0.14 & 0.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Weight value of influencing factors. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Model Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \multirow{3}{*}{Evaluation index} & \(L(\text{m})\) & 30.46 & 30.56 & 48.34 & N/A & 50.90 & 41.34 & 25.46 & 35.70 \\  & \(H_{1}(\text{m})\) & 17.25 & 9.77 & 11.19 & N/A & 7.04 & 11.30 & 3.11 & 12.67 \\  & \(S\left(\text{m}^{2}\right)\) & 239 & 207 & 336 & N/A & 470 & 203 & 104 & 228 \\ \hline \multirow{3}{*}{**Model Number**} & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** \\ \cline{2-9}  & \(L(\text{m})\) & 41.02 & 50.50 & N/A & 61.58 & 45.30 & 58.50 & 39.78 & 65.46 \\ \cline{1-1}  & \(H_{1}(\text{m})\) & 23.74 & 28.17 & N/A & 24.38 & 25.33 & 9.81 & 17.23 & 11.20 \\ \cline{1-1}  & \(S\left(\text{m}^{2}\right)\) & 517 & 597 & N/A & 887 & 552 & 467 & 392 & 772 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of evaluation indexes of the loose area. and were fitted by the built-in function in MATLAB according to the changing trend of data points to determine the boundary function of the loose zone. After data sorting and analysis, the quadratic function had the highest degree of fit for the dataset. Therefore, this paper took it as the boundary function with \(a\), \(b\), and \(c\) being undetermined coefficients. Taking Model 7 as an example, the fitting effect and relevant information are represented in Figure 11 and Table 7.  
The following can be concluded from the correlation between the evaluation indexes of the loose zone and the boundary function:  
\[\left\{ \begin{array}{l}x=0\Rightarrow y=0\\ x=-\frac{b}{2a}\Rightarrow y=\frac{1}{2}L\\ x=H_{1}+h\Rightarrow y=\frac{1}{2}B\end{array} \right. \tag{7}\]  
Taking Correlation 5 into the boundary function \(y=ax^{2}+bx+c\), the undetermined coefficients \(a\), \(b\), and \(c\) can be approximated:  
\[\left\{ \begin{array}{l}a=\frac{-L\times\left[1+\left(1-\frac{B}{L}\right)^{ \frac{1}{2}}\right]}{\left(H_{1}+h\right)^{2}}\\ b=\frac{L\times\left[1+\left(1-\frac{B}{L}\right)^{\frac{1}{2}}\right]}{H_{1}+h} \\ c=0\end{array} \right. \tag{8}\]  
Using the data in Tables 3-5, the multiple regression equations of \(\mathbf{L}\) and \(\mathbf{H}_{1}\) can be concluded via the process described below:  
(1) Based on the principle of the least squares method, the univariate regression equation between the influencing factors of the loose zone with \(\mathbf{L}\) and \(\mathbf{H}_{1}\) are established, as shown in Table 8. According to the manual, the correlation coefficient threshold value is 0.83. As illustrated in Table 8, the correlation coefficients of the five influencing factors all exceed 0.83, indicating that the univariate regression equations of \(\mathbf{L}\) and \(\mathbf{H}_{1}\) are correct.