## 3 Results and Analysis
### Characteristics of Stress Distribution around a Single Roadway
#### 3.2.1 Solution of the Mapping Function for a Plane Containing a Single Tri-Center  
Arch Roadway  
Utilizing the method of conformal mapping, the plane with an opening (\(Z\)-plane) can be mapped onto the plane with a unit circle (\(\xi\)-plane), achieved through the following mapping function:  
\[Z=\omega(\zeta)=\sum\limits_{k=1}^{\infty}C_{k}\zeta^{2-k} \tag{1}\]  
Under general circumstances, \(C_{k}\) is a complex number. Therefore, taken with the first \(m\) terms, \(C_{k}\) can be expressed as follows:  
\[C_{k}=A_{k}+iB_{k}k=1,\ 2,\ 3,\ \ldots,\ m\]  
where both \(A_{k}\) and \(B_{k}\) are the real-valued constants. For \(l\) points located on the unit circle on the \(\zeta\)-plane with coordinates of (1, \(\theta_{j}\)), their corresponding mapped points on the \(Z\)-plane are denoted as (\(r_{j}\), \(\alpha_{j}\)). Leveraging the orthogonality of trigonometric functions, the expressions for \(A_{k}\) and \(B_{k}\) can be derived as follows:  
\[\left\{ \begin{array}{l}A_{k}=\frac{1}{l}\sum\limits_{j=1}^{l}\left\{r_{j} \cos\alpha_{j}\cos\left[(k-2)\theta_{j}\right]-r_{j}\sin\alpha_{j}\sin\left[( k-2)\theta_{j}\right]\right\}\\ B_{k}=\frac{1}{l}\sum\limits_{j=1}^{l}\left\{r_{j}\cos\alpha_{j}\sin\left[(k- 2)\theta_{j}\right]+r_{j}\sin\alpha_{j}\cos\left[(k-2)\theta_{j}\right]\right\} \end{array} \right. \tag{2}\]  
According to the research by [PERSON] et al. [8], triangle interpolation can be utilized to solve the problem for the parameters \(A_{k}\) and \(B_{k}\) of a given mapping function for the plane with a single opening, using a specific solving process not detailed in this study. Notably, since the roadway shape in this study is axisymmetric, it was assumed that the roadway itself is symmetric about the x-axis. In such a case, \(C_{k}\) is the real number, meaning \(B_{k}=0\) to improve the solving efficiency. Based on the triangle interpolation method, the planar mapping function for a section with a tri-center arch roadway was solved. Here, the function \(f_{dure}\) is incorporated to assess the accuracy of the mapping function, whose expression is provided below:  
\[f_{dure}=\frac{1}{n}\sum\limits_{j=1}^{n}\frac{\left|r_{\sigma}-r_{\sigma}^{(0 )}\right|}{r_{\sigma}^{(0)}} \tag{3}\]  
where (\(r_{\sigma}\), \(\alpha_{\sigma}\)) are the coordinates of the opening boundary mapped points corresponding to \(n\) sample points on the unit circle, which are obtained based on the mapping function; (\(r_{\sigma}^{(0)}\), \(\alpha_{\sigma}\)) are the coordinates of their actual corresponding points;\(f_{dure}\) represents the average absolute relative error (average ARE) of the \(n\) points; the closer it is to 0, the higher is the accuracy of the mapping function. Figure 5 presents the variation curve of \(f_{abre}\) as the number of terms \(C_{k}\), denoted by \(m\), increases. It is observed that as \(m\) increases from 4 to \(9\), \(f_{abre}\) rapidly decreases, indicating that the accuracy of the mapping function significantly improves. When \(m\) is greater than \(9\), \(f_{abre}\) essentially stabilizes within a range close to 0, signifying that the mapping function has achieved a high level of accuracy at this point. Figure 6 further illustrates the comparison between the mapped shapes and the original shapes under different \(m\) values. It is noticeable that at \(m=4\), the discrepancy between the mapped shape and the original shape is very pronounced. When \(m\) reaches 8, the difference between the two becomes minimal. For cases where \(m>8\) in the figure, there is virtually no difference between the mapped shape and the original shape, showcasing a high degree of congruence. In such cases, the mapped and original shapes are nearly overlapping entirely as seen in Figure 6d-f. Therefore, combining Figures 2 and 3, in this study, for the calculation model of the planar section containing a tri-center arch roadway as shown in Figure 1, the number of terms \(m\) for the mapping function \(C_{k}\) was chosen to be 9.