## 5 Conclusions  
Through the establishment of a composite model incorporating linear and nonlinear shear displacement at the anchorage interface in distinct stages, analytical solutions for the axial force, shear stress, displacement, and maximum bearing capacity of the anchor at each stage have been derived. These solutions have been validated against field-measured data. Concurrently, an analysis of the load transfer mechanism throughout the entire process of the anchor under varying pull-out loads has been conducted. The primary conclusions drawn from this analysis are as follows:  
1. The step-wise linear-nonlinear bond-slip composite model is presented in the form of a single-peak curve. As the pull-out load increases, the anchor surface sequentially transitions through the elastic stage, the plastic stage, and the debonding stage. The model aeptly captures the elastic characteristics of the elastic stage, the softening and nonlinear attributes of the plastic stage, and the residual characteristics of the debonding stage.
2. In the pulling process, when the pull-out load is small, the anchorage interface remains in an elastic state, and the maximum shear stress is situated at the pulling end of the anchor. Consequently, the axial force experiences a rapid decline along the anchoring depth. As the pull-out load surpasses the elastic limit load, the plastic state initiates from the pulling end of the anchor, and the maximum shear stress is located at the boundary point between the elastic and plastic states. Upon exceeding the plastic limit load, the debonding state begins from the pulling end of the anchor, leading to the continued development of axial force into deeper sections. Furthermore, it is noteworthy that most measured values of axial force are less than the calculated values, indicating that the prediction method in this paper tends to be on the side of safety.
3. The load-displacement curves and axial loading distribution of the anchor were determined using the load transfer method based on the step-wise linear-nonlinear bond-slip composite model. The theoretical values were extensively validated against fieldwork published in the literature. The calculation method was applied to an actual project, revealing the shear responses of the anchor during the pull-out test. The output safety factor met the requirements of relevant specifications, affirming the reliability and practicality of the method.  
This paper was accomplished based on collaborative work by the authors. Conceptualization, Z.C.; Funding acquisition, Y.W.; Data curation, K.Z. and D.W.; Investigation, D.W.; Project administration, Y.W. and D.W.; Formula derivation, Z.C.; Validation, Z.C. and D.W.; Writing--original draft, Z.C.; Writing--review and editing, Y.W. and K.Z. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the Guangdong Province science and technology innovation plan project (2021 ZHJKJ-13). We are grateful for the support of the science and technology innovation plan project.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available because they came from a project report.  
[PERSON] was employed by Guangzhou Chemical Grouting Engineering Co., Ltd., CAS. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.