###### Abstract  
In order to explore the problems of load transfer and anchorage mechanisms of tensile anchors under pull-out load for geotechnical anchoring systems, a step-wise mathematical model is established which considers the linear-nonlinear shear stress and shear displacement of the anchorage segment, using an elasto-plastic constitutive model. The displacement, axial force, and shear stress of the anchorage interface in different stages (elastic, plastic, and debonding) are analyzed and solutions are derived. And the theoretical solutions for the ultimate pull-out load of the anchor at each stage are also presented. Two in situ pull-out tests are used to verify and apply these findings in engineering. The results show that the stepwise composite model could reflect the bonding, softening and residual characteristics of the anchoring interface. In the process of the pull-out load increasing, the pulling end of the anchor initially enters the plastic stage and the debonding stage, respectively, and the failure of the anchor occurs at the pulling end, and as the axial force transfers down deeper, the damage gradually spreads deeper. The axial force distribution of the anchorage section is a monotonically decreasing curve, and the peak point of the shear stress gradually moves deeper. The calculation results of the axial force distribution curve and load-displacement curve of the anchor are in good agreement with the measured values, which verifies the rationality and reliability of the theoretical prediction method. This method can provide a theoretical reference for the load transfer analysis and design of tension anchors for geotechnical anchoring systems. a +
Footnote †: journal: Applied sciences  
1  
To ascertain the genuine stress state of the anchor, numerous scholars have conducted extensive research on the load transfer mechanism within the anchorage section. [PERSON] discovered an exponential distribution of shear stress along the anchor based on measured data [8]. [PERSON] et al. compared the impact of different grouting materials and anchoring lengths on bolt performance, providing an empirical equation for pull-out resistance and a straightforward bond-slip constitutive model [9]. [PERSON] et al. introduced an analytical model for predicting the axial force of grouting bolts, elucidating the interaction mechanism between bolts and soft rock masses. They identified a position of maximum anchoring length, beyond which the axial force ceases to increase [10]. [PERSON] et al. and [PERSON] et al. established a hyperbolic model for load transfer in the anchorage section, deriving the functional expression [11; 12]. These studies collectively reveal that shear stress at the anchorage interface is non-uniform and does not follow a linear distribution. However, peak shear stresses from these models are typically highest at the pulling end, gradually decreasing along the depth of the anchorage. These methods overlook the gradual onset of plastic deformation or debonding as pull-out stress increases. Subsequently, [PERSON] established a tri-linear bond-slip model of shear stress and shear displacement based on the stress softening of the anchorage body. He found that the peak shear stress does not necessarily occur at the pulling end [13]. [PERSON] et al., [PERSON] et al., and [PERSON] et al. employed exponential function models, double exponential function models, and exponential-hyperbolic composite models, respectively [14; 15; 16]. They aimed to depict the relationship between shear stress and shear displacement, simulating the entire load transfer process at the anchorage interface. Results indicated that the peak shear stress initially appeared at the pulling end when the pull-out load was small. With increasing tensioning load, yielding phenomena manifested at the pulling end, causing the shear stress peak to shift towards the depth of the anchorage section. [PERSON] et al. categorized the anchor pull-out process into three stages: elastic, plastic, and residual.