# Modeling of Tunneling Total Loads Based on Symbolic Regression Algorithm
## 3 Constructing \(\pi\) Quantities Based on Dimensional Analysis
### Selection of the Basic Quantities  
Basic quantities should be the core parameters reflecting the physical relationship. The selection of basic quantities directly determines the functional form of each \(\pi\) quantity. However, their selection method has some degree of freedom. Therefore, the basic quantities need to be determined by combining the characteristics of the specific problem. The basic quantity selection process in this paper is shown in Figure 4.  
As the number of basic quantities should be equal to the number of basic dimensions of the research problem, and as there are three basic dimensions in a mechanical system (mass dimension M, length dimension L, and time dimension T), all the influencing parameters are divided into three groups according to their basic dimensions, as listed in Table 2. One parameter is then selected from each group as the basic quantity. Finally, based on the selected basic quantities, a dimensionless operation is performed on the remaining parameters to construct the \(\pi\) quantities. For the selection of the basic quantities in each dimensional group, this paper mainly considers three aspects: (i) whether it is the core parameter that has a key influence on the target quantity; (ii) whether it is a constant parameter that has a key influence on the target quantity, because it is difficult to identify the  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Physical Parameters** & **Dimension** \\ \hline Machine Parameters & Cutterhead diameter \(D\) (m) & L \\ \hline Operating Parameters & Driving speed \(v\) (m/s) & L T\({}^{-1}\) \\  & Cutterhead rotation speed \(\omega\) (r/min) & T\({}^{-1}\) \\ \hline  & Rock density \(\rho\) (g/cm\({}^{3}\)) & M L\({}^{-3}\) \\  & Shear modulus \(G\) (MPa) & M L\({}^{-1}\) T\({}^{-2}\) \\ Geologic Parameters & Poisson’s ratio \(\mu\) & — \\  & Cohesive force \(c\) (kPa) & M L\({}^{-1}\) T\({}^{-2}\) \\  & Compressive strength \(\sigma_{\mathrm{c}}\) (MPa) & M L\({}^{-1}\) T\({}^{-2}\) \\  & Tensile strength \(\sigma_{\mathrm{t}}\) (MPa) & M L\({}^{-1}\) T\({}^{-2}\) \\ \hline Target Parameters & Total thrust \(F\) (kN) & M L T\({}^{-2}\) \\  & Total torque \(T\) (kN \(\cdot\) m) & M L\({}^{2}\) T\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters and their dimensions.  
Figure 4: The basic quantity selection process.  
influence of constant parameters when exploring the physical laws among the parameters through data analysis, and this could easily result in analysis blind spots; and (iii) whether the dimensionality of a parameter is \"concise,\" because the construction of a \(\pi\) quantity essentially uses the basic quantity as a unit system to measure each parameter [12], which is a switch of measurement methods. It is found that choosing basic quantities with relatively simple dimensions is beneficial for obtaining more characteristic information. Considering the above principles, for the basic dimension M, the shear modulus \(G\) is selected as the basic quantity, as it is the core influencing parameter of the machine tunneling. For the basic dimension L, the cutter head diameter \(D\) is selected as the basic quantity. \(D\) is a constant parameter that has a direct influence on the total driving load and its dimension is very concise. For the basic dimension T, the cutter head speed \(\omega\) is selected as the basic quantity, as it has a relatively simple dimension.