## 1 Introduction
### Analysis of Uplift Bearing Capacity  
According to the \"Technical code for geotechnical anchor and shotcrete support engineering\" [32], there are the following provisions for the calculation of the pull-out bearing capacity of an anchor under the strength limitation of the surrounding soil in the anchor structure design:  
\[N_{d}\leq\frac{f_{mg}}{K}\cdot\pi\cdot D\cdot L_{a}\cdot\psi \tag{1}\]  
where \(N_{d}\) denotes the pull-out limit of the bolt (N); \(f_{mg}\) is the standard value of the ultimate bond strength (MPa or kPa) between the grouting body and the soil in the anchorage section; \(K\) represents the pull-out safety factor between the grouting body and the soil of the bolt section; \(D\) represents the borehole diameter (mm) of the anchoring section of  
Figure 4: The influence range of expansion body expansion on loose circle ((**a**) The sleeve is not expanded, (**b**) The sleeve began to expand, (**c**) Expansion of soil compaction zone, (**d**) The soil compaction zone is completely formed). the bolt; \(L_{a}\) represents the length of the anchoring section of the bolt (m); \(\psi\) denotes the influence coefficient of anchorage length on ultimate bond strength. In addition, if a pressure type or pressure dispersion type anchor rod is used, the bearing area of the grouting body in the anchoring section of the anchor rod should be checked in the design. The specific checking formula is as follows:  
\[N_{d}\leq 1.35\cdot A_{p}.\frac{A_{m}^{0.5}}{A_{p}}.\eta\cdot f_{c} \tag{2}\]  
where \(A_{p}\) is the net contact area of a cross-section between the anchor bearing body and the grouting body in the anchorage section; \(A_{m}\) represents the cross-sectional area of the grouting body in the anchorage section; \(\eta\) represents the amplification coefficient of axial compressive strength of the confined grouting body; \(f_{c}\) represents the axial compressive strength limit of the grouting body in the anchorage section. Since there is no exact and reasonable calculation method for the ultimate bearing capacity of the enlarged head bolt, the ultimate bearing capacity is generally determined by a basic test. Therefore, this paper extensively refers to the domestic and international scholars' use of different methods to discuss and analyze the calculation formula of the maximum pull-out force of the enlarged head bolt [33; 34; 35; 36; 37; 38], combined with the provisions of the \"Technical Code for Rock and Soil Anchor and Shotcrete Support Engineering\" on the ultimate pull-out bearing capacity of the bolt. Finally, the calculation method of the pull-out load \(T\) of the umbrella-shaped enlarged-head hollow grouting bolt with an expansion pipe is determined as follows:  
\[T=T_{1}+T_{2}+T_{3} \tag{3}\]  
where \(T_{1}\) is the pull-out resistance provided by the friction resistance of the side wall of the ordinary anchorage section; \(T_{2}\) is the sum of the maximum pull-out resistance provided by the friction resistance of the side wall of the umbrella-shaped enlarged head and the pull-out resistance provided by the positive pressure of the soil on the end face of the umbrella-shaped enlarged head; \(T_{3}\) is the pull-out resistance provided by the friction resistance of the side wall of the expansion body section. The calculation method of three-stage pull-out resistance will be summarized in the following sections. First of all, the pull-out resistance provided by the friction resistance of the side wall surface of the ordinary anchorage section is the simplest, which can be expressed as follows:  
\[T_{1}=\pi D_{1}L_{1}\tau_{f} \tag{4}\]  
where \(D_{1}\) is the drilling diameter of the ordinary anchorage section; \(L_{1}\) is the length of ordinary anchorage section; \(\tau_{f}\) is the frictional resistance strength between the wall of the anchorage body and the soil in the ordinary anchorage section. The pull-out load of the umbrella-shaped enlarged head section is more complicated.