## 6 Conclusions  
This paper proposes a simplified calculation method to estimate the effect of capsule grouting expansion on the deformation and internal force of neighboring piles, employing a two-stage method. The main conclusions drawn from this study are as follows:  
1. The expression for the lateral displacement field of soil induced by capsule grouting is obtained utilizing the image source method, wherein the cylindrical cavity expansion with finite height is transferred to a series of spherical cavity expansions. Meanwhile, the pile is considered as an Euler-Bernoulli beam on a Winkler foundation to establish a theoretical analytical framework for controlling the lateral displacement of the pile resulting from capsule grouting expansion. The reliability of the proposed method is confirmed through comparisons, demonstrating a good fit between the calculated lateral displacement of the pile and soil and their measured values.
2. The lateral displacement of the pile can be controlled by enlarging the capsule radius, with the maximum lateral displacement of the pile being proportional to the cross-sectional area of the capsule grouting. Increasing the distance between the capsule and the pile diminishes the deformation of the pile, thereby compromising the effectiveness of capsule grouting in mitigating pile deformation. Additionally, expanding the pile diameter significantly curtails the lateral displacement of the pile, albeit leading to a gradual rise in the bending moment.
3. Increasing the capsule length results in heightened lateral displacement and a broader deformation range of the pile. Concurrently, the bending moment initially increases and then decreases with the augmentation of the capsule length. In scenarios where the depth of the capsule head is 0 m, substantial lateral displacement occurs at the pile head under free end constraints, while a significant bending moment manifests at the pile head under fixed constraints. However, when the capsule depth exceeds 8 times that of the pile diameter, the deformation and bending moment of the pile remain unaffected by the constraint at the pile head. The effect of parameter variations on the pile is consistent, regardless of the end constraints imposed on the pile.
4. The theoretical analysis method framework of capsule grouting expansion is effective. Nevertheless, the following limitations require attention in further research. (a) The soil is assumed to be incompressible; (b) the nonlinear behaviors of the soil-pile interaction are not taken into consideration.  
Conceptualization, J.Y. and J.G.; Data curation, J.G.; Formal analysis, J.Y. and J.G.; Funding acquisition, J.Z.; Investigation, J.Y. and J.Z.; Methodology, J.Y., J.G., J.Z., H.H. and X.G.; Project administration, J.Z. and X.G.; Resources, J.Z., H.H. and X.G.; Software, J.G.; Supervision, J.Y.; Validation, J.G.; Visualization, J.G.; Writing--original draft, J.G.; Writing--review and editing, [PERSON], [PERSON], [PERSON], H.H. and X.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (52108350).  
Not applicable.  
The data are available upon reasonable request.  
The authors declare no conflicts of interest.