## 6 Conclusions  
In this study, the stress and deformation behavior of circular shafts was analyzed by explicitly incorporating the radial distribution of circumferential stress induced by the spatial arching effect. An analytical expression for the inclination angle of the sliding surface under active limit conditions was derived based on the global force equilibrium of the sliding soil. A novel method for calculating the active earth pressure on circular shafts was proposed, which accounts for both the spatial arching effect and interlayer shear interactions. The validity of the proposed approach was confirmed through comparisons with theoretical solutions, model test results, and field measurements. Furthermore, an elastic mechanics-based model was developed to evaluate the stress and deformation response of pipe segments during mechanical shaft excavation. The main conclusions are summarized as follows:  
Figure 15: Analysis results of internal force and deformation along depth direction. (**a**) Stage I; (**b**) Stage II; (**c**) Stage III. (1) Under active limit conditions, the inclination angle of the sliding surface behind the circular shaft wall is generally larger than that of the Coulomb sliding surface. As the radius-to-height ratio increases, the spatial arching effect--particularly the circumferential arching effect--gradually diminishes, leading to a decrease in the sliding surface angle that asymptotically approaches the theoretical prediction. Additionally, higher soil cohesion increases the shear stress required to satisfy limit equilibrium, thereby resulting in a greater sliding surface angle. This effect is more pronounced in soils with lower internal friction angles. (2) The active earth pressure of circular shafts is significantly influenced by the radius-to-height ratio and interlayer shear stress: a smaller radius-to-height ratio enhances the circumferential arching effect, improving soil stability and reducing active earth pressure, while a larger diameter-to-height ratio amplifies the impact of interlayer shear stress on the pressure distribution. By accounting for the radial variation of the circumferential stress coefficient and the spatial arching effect, the proposed calculation method shows good agreement with model test results and effectively captures the depth-dependent variation of active earth pressure under the translation mode. (3) The stress and deformation behavior of the shaft structure is fundamentally governed by the combined action of internal and external water and soil pressures. Owing to the spatial arching effect of the surrounding soil mass, the distribution of stresses and deformations in Stages II and III demonstrates a distinct nonlinear variation with depth. The analytical predictions show excellent agreement with the measured data, thereby confirming the validity and reliability of the proposed earth pressure calculation framework and the structural stress-deformation analysis approach developed in this study. (4) The stresses experienced by the shaft structure are strongly governed by the pressure differential between its inner and outer sides. An excessive pressure differential can induce elevated structural stresses, while overly high internal pressures may subject the structure to significant tensile forces. Therefore, in practical engineering applications, it is crucial to carefully control both the pressure differential and the internal pressure to ensure the structural integrity and long-term safety of the shaft system. In summary, the proposed calculation framework provides a theoretical basis for the optimization and safety evaluation of circular shaft wall structures under translational mode conditions. Moreover, the underlying concepts offer potential reference value for investigating displacement behavior under rotational failure modes, with prospects for broader application and further development. Writing--original draft preparation, D.L. and Y.L. ; writing--review and editing, Z.X. and L.Y. ; conceptualization and supervision, Y.L.