## 1 Introduction
### Analysis of Influence Range of Expansion Body Section  
After the excavation of the roadway, the original in situ stress is released, which causes the surrounding rock to enter the elastic-plastic stress redistribution state and form a loose circle. The boundary of the loose circle can be expressed by the radius \(R_{d}\) of the loose circle. In practical work, the plate nut at the tail of the bolt is tightened to give a certain preload \(P_{0}\) to the bolt, the expansion body is driven by the cone angle \(\theta_{c}\), and then the fastening pressure is applied to the surrounding rock and soil in the loose circle. The working principle is shown in Figure 7.  
Figure 7: Expansion tail mechanical model. The prestress \(P_{0}\) at the end of the bolt acts on the conical expansion body, which can be decomposed into the oblique component force \(F_{b}\) parallel to the conical surface and the normal component force \(F_{n}\) perpendicular to the conical surface and represented as follows:  
\[ \begin{cases}F_{b}=Sec\frac{\theta}{2}P_{0}\\ F_{n}=P_{0}tan\theta_{c}\end{cases}  \tag{32}\]  
The contact area \(A_{s}\) between the expansive body and the soil is as follows:  
\[A_{s}=\pi\frac{(D_{c}+D_{0})}{2}R_{d} \tag{33}\]  
Thus, the normal stress (directly acting on the loose circle) can be calculated as follows:  
\[\sigma_{n}=\frac{F_{n}}{A_{s}}=\frac{P_{0}tan\theta_{c}}{\pi\frac{(D_{c}+D_{0} )}{2}R_{d}} \tag{34}\]  
In the loose circle, the stress distribution of surrounding rock or soil is determined by the Mohr-Coulomb criterion and elastic-plastic mechanics. There is an elastic-plastic zone (far away from the bolt) and a plastic zone (near the bolt expansion body) in the loose circle. According to the Mohr-Coulomb yield criterion, which is as follows:  
\[\tau=c+\sigma_{n}tan\varphi \tag{35}\]  
It is assumed that the active failure condition is reached at the plastic zone boundary \(R_{p}\), which is presented as follows:  
\[\sigma_{r}\big{(}R_{p}\big{)}=K_{p}\sigma_{n} \tag{36}\]  
\[K_{p}=\frac{1+sin\varphi}{1-sin\varphi} \tag{37}\]  
The normal pressure \(\sigma_{n}\) is as follows:  
\[R_{p}=R_{d}\bigg{(}\frac{\sigma_{n}}{c+\sigma_{n}tan\varphi}\bigg{)}^{\frac{ 1}{k_{p}-1}} \tag{38}\]  
The reinforcement effect of the bolt expansion body on the loose circle can be expressed by the stress enhancement ratio \(\eta_{p}\), which is as follows:  
\[\eta_{p}=\frac{\sigma_{n}+\sigma_{0}}{\sigma_{0}} \tag{39}\]  
where \(K_{p}\) is the limit active earth pressure coefficient; \(\varphi\) is the internal friction angle of soil; \(\sigma_{r}\) is the radial stress of soil; \(\sigma_{0}\) is the initial confining pressure before the formation of loose circle. When \(\eta_{p}>1\), it indicates that the stability of the loose circle has been improved. In engineering design, \(\eta_{p}\) is generally required to reach at least 1.5 or above to ensure that the surrounding rock is no longer unstable. On a theoretical basis, the stress process analysis and pull-out bearing capacity research of an umbrella-shaped enlarged-head hollow grouting bolt with an expansion pipe is performed and the structure of an umbrella-shaped enlarged-head hollow grouting bolt with an expansion pipe is optimized. The overall design structure is shown in Figure 8.  
The whole umbrella-shaped expansion-head hollow anchor with an expansion pipe is composed of an anchoring section, an expansion section and a free section. The anchoring section includes the umbrella-shaped expansion-head anchoring section and the conventional anchor anchoring section. The conventional bolt anchorage section is the same as the conventional left-handed, non-longitudinal, rib thread steel. On the basis of high-strength thread steel, it is combined with a left-handed thread with excellent self-locking performance. After the anchorage body (grouting body) is solidified and formed, it can produce a good relaxation and self-locking function.