## 6 Conclusions  
In this paper, for shallow rectangular pipe jacking excavation, the cross-section model, which considers an excavation boundary's displacement condition and the surface's stress condition, is established clearly. The analytical solution of the complex function of the ground displacements and stresses is obtained using the approximate mapping function and the approximate potential function of the series form. The finite element simulation and engineering-measured data are used for verification and comparison. Then, the influence of different cases of mapping function control points on the accuracy of the analytical solution is calculated. Finally, for the pipe jacking excavation boundary, the influence of the layer loss rate, span, depth of buried, and layer gravity on the settlement and major principal stress is analyzed and studied. The conclusions are as follows:  
1. The displacements and stresses of the analytical solution are consistent with the corresponding finite element solution, and the differences between the analytical solution and the engineering-measured data are acceptable in surface subsidence patterns, maximum values, and ranges.
2. For the top pipe excavation, the midpoints and corners of the top boundary and the midpoints and corners of the bottom boundary can be used as the four control points of the mapping function. The accuracy of the calculation results can meet engineering requirements.  
Figure 14: The effect of different stratum unit weights \(\gamma\): (a) displacements \(v\) of the surface and (b) major principal stress \(\sigma_{1}\) around the excavation boundary.  
3. The surface subsidence increases with the increase in the stratum loss rate, the span, and the stratum unit weight. Moreover, it decreases with the increase in the buried depth, but it will eventually tend to a constant value.
4. The major principal stress at the top boundary of the pipe jacking excavation and near the corner points changed obviously with the change in the stratum loss rate, span, buried depth, and the stratum unit weight. Nevertheless, the change in the major principal stress at the other locations was not obvious. The stress concentrations at the four corners and the center of the top boundary of the excavation were more obvious.  
In order to simplify the calculation process, several assumptions were made in the model. For example, the grouting process of pipe jacking was not considered. In actual pipe jacking construction, the grouting process affects the final displacements and stress of the stratum. And then the convergence model of tunnel excavation boundary is still assumed. The real convergence situation helps to improve the displacement boundary conditions and improve the accuracy of analytical solutions. There is currently no research on the convergence deformation of non-circular tunnel excavation. These problems need to be studied further.  
Y.X.: conceptualization; Y.W.: formal analysis; software; and writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Fundamental Research Funds for China Central Universities, grant number 2021 YJS105.  
Not applicable.  
The data are contained within this article.  
The authors declare no conflicts of interest.