## 6 Conclusions  
1. The equivalent force on the back of the concrete retaining wall was calculated based on the Rankine earth pressure theory. The concrete retaining wall was regarded as a four-sided fixed support plate, and the ultimate load was solved by establishing the virtual work equation. A novel method for determining the safe thickness of the concrete retaining wall was proposed, and quantitative criteria for the safe thickness of concrete retaining wall were given.
2. The theoretical formula derived in this paper was compared with the thickness of the concrete retaining wall obtained by the traditional empirical formula and the Kalmykov formula. The traditional empirical formula ignores the formation stress and groundwater pressure behind the concrete retaining wall. The Kalmykov formula has poor applicability when the final grouting pressure is small or large, and the influence of groundwater pressure is not considered. By considering the impacts of different working conditions on the concrete retaining wall, the rationale of the safe thickness of the concrete retaining wall obtained in this paper was further clarified.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Thickness /m** & **Condition 1 (\(P_{j}\) = 0.6 MPa)** & **Condition 2 (\(P_{j}\) = 1 MPa)** & **Condition 1 (\(P_{j}\) = 1.4 MPa)** \\ \hline Empirical formula solution & 0.52 & 1.04 & 0.67 & 1.34 & 0.80 & 1.60 \\ Theoretical formula solution & 0.93 & & 1.15 & & 1.33 & \\ Experimental design & Left tunnel & 0.50 & Left tunnel & 0.70 & Left tunnel & 0.80 \\  & Right tunnel & 0.90 & Right tunnel & 1.20 & Right tunnel & 1.30 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Calculation of thickness of concrete retaining wall and test selection.  
3. There is a connection between the method in this paper and the traditional empirical formula. The quantitative transformation relationship between the traditional empirical formula and the theoretical formula was established. In a shallow tunnel, the value of \(K_{0}\) is relatively reasonable when between one and two. When the final grouting pressure is smaller, the safety factor should be larger. When the final grouting pressure is larger, the safety factor should be smaller. 4. The safe thicknesses calculated by the traditional empirical formula with a \(K_{0}\) of one and by the theoretical formula in this paper were analyzed through field tests to verify the rationale of the theoretical formula. The test results showed that the thickness of the concrete retaining wall designed according to the theoretical formula met the construction requirements. Under the traditional empirical formula (\(K_{0}=1\)) design, the concrete retaining wall failed to meet the construction requirements. We suggest that the theoretical formula established in this paper can be adopted in grouting engineering for shallow tunnels in weakly water-rich strata. If the traditional empirical formula is used, the value of \(K_{0}\) needs to be solved theoretically.  
Conceptualization, Y.L. and M.C.; methodology, Y.L. and W.L.; writing--original draft preparation, Y.L.; writing--review and editing, Y.L. and M.C.; supervision, M.C. and B.C.; project administration, W.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program, grant number 2018 YFB1600100; the National Natural Science Foundation of China Joint Project, grant number U1906229; the Key R&D Program of Shandong Province, grant number 2019 JZZY010427; the National Natural Science Foundation of China Youth Project, grant numbers 5210090249, 51908329 and 52009075; and the Natural Science Foundation of Shandong Province of China, grant numbers ZR2020 QE290 and ZR2020 QE262.  
Not applicable.  
The authors declare no conflict of interest.