## 3 Elastic Solutions to Stress Surrounding Rock of Rectangular Roadways with Super-Large Sections
### Solution of Mapping Function Coefficients  
According to the theory on complex conformal transformation, the mapping determined by the solution function is conformal mapping. When it comes to engineering, the Schwarz-Christoffel equation is generally adopted to transform a complex shape into a simple one, as shown in Equation (8):  
\[Z=\omega(\zeta)=R\int_{0}^{\zeta}\prod_{k=1}^{n}\left(z-x_{k}\right)\frac{ \alpha_{k}}{\pi}^{-1}dz+A \tag{8}\]  
where \(x_{k}\) is the mapping position from the rectangle on plane \(z\) to the unit circumference on plane \(\xi\); \(\alpha_{k}\) is the mapping angle from the rectangle on the plane \(z\) to the unit circumference on plane \(\xi\); \(R\) is the constant reflecting the dimensional features of rectangular roadways; and \(n\) is the number of mapping function terms, whose values are related to both the complexity of the excavation tunnel shape and the requirement for mapping function accuracy. \[Z=\omega(\zeta)=R(\frac{1}{\zeta}+c_{1}\zeta+c_{3}\xi^{3}+c_{5}\xi^{5}+c_{7} \xi^{7}+\cdots) \tag{9}\]  
where \(R\), \(c_{1}\), \(c_{3}\), \(c_{5}\) and \(c_{7}\) are all real constants. \[ \begin{array}{l}c_{1}=\cos 2k\pi;\\ c_{3}=-\frac{1}{6}\sin^{2}2k\pi;\\ c_{5}=-\frac{1}{10}\sin^{2}2k\pi\cos 2k\pi;\\ c_{7}=\frac{1}{896}(10\cos 8k\pi-8\cos 4k\pi-2);\\ \cdots\cdots\end{array} \]  
When the aspect ratio was equal to 1, the error between the mapping value and actual value at the corner point was defined as the reference error. When an error is larger than the reference error, the mapping accuracy can be improved by increasing the number of integral expansion terms, thus narrowing the boundary error of rectangular roadways with super-large sections. Through analysis of the mapping results from different mapping terms in Table 1, we determined the maximum error of conformal transformations of different terms. When studying Figure 4, which presents the effect of mapping with different accuracies, it is not difficult to see that an increase in the number of terms taken by mapping function \(Z\) can improve the mapping effect. According to Figure 4a, when the aspect ratio _b/h_ increased from 1 to 6, the mapping effect weakened gradually; when the number of mapping function terms was taken as 3, there were large errors between the roof, the base plate and these two sides in the mapping section and the straight boundary in the theoretical section, and the error was even larger at the roadway corner; it was more like an arc shape. This error can be narrowed by increasing the number of terms. According to Figure 4b,c, when the number of terms is increased by one each time, the mapping value will be more like the outline of rectangular roadways, and the subsequent laws will be more like those on-site. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{_b/h_} & \multicolumn{3}{c}{**Number of Conformal Transformation Terms**} \\ \cline{2-4}  & **3** & **4** & **5** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: The maximum error of conformal transformations of different terms. Figure 4: Effect of mapping with different accuracy levels. Since a three-term mapping function can meet the requirement for accuracy when \(b/h\) is smaller than 2, a three-term mapping function option can solve the problem when it comes to the research on conventional roadways. However, when it comes to the research on roadways with super-large sections, the accuracy of a three-item mapping function cannot meet the requirement. In this case, the four-term mapping function option is needed. When \(b/h\) is less than 10, the error of the four-term mapping function is less than 1.01% and, therefore, it can meet the requirement for accuracy. However, when \(b/h\) is not smaller than 10, the error is more than 1.01%. In this case, a five-term mapping function option is necessary. Mapping accuracy selection is relevant to both the objective and purpose of research.