## 1 Introduction
### Analysis of Uplift Bearing Capacity
In order to facilitate the calculation, the umbrella-shaped expansion head is simplified as a double cone (cone cap) for research, as shown in Figure 6. According to the equilibrium conditions of the infinitesimal body, the following relations can be obtained:  
\[P_{u}+dP_{u}-\left(P_{u}+dS_{1}\tau_{fd}+dS_{2}p_{d}\right)=0 \tag{13}\]  
where \(dS_{1}\) is the area of the outer surface of the micro-element subjected to friction; \(dS_{2}\) is the inner surface area of the micro-element under positive pressure. In the simplified cone state, the diameter \(D(z)\) of the umbrella-shaped expansion head changes with the height z, and the relationship is set as follows:  
\[D(z)=D_{2}+kz \tag{14}\]  
\[k=2 tan\theta \tag{15}\]  
where \(D_{2}\) is the diameter at the beginning of the bottom of the umbrella-shaped enlarged head (cone); \(k=2 tan\theta\) is the taper change rate. Figure 6: Umbrella enlarged head section calculation diagram. From Equation (11) to Equation (14) it can be obtained that under the action of drawing force \(T_{2}\), the displacement \(\omega_{s}\) of the soil at the anchorage section port of the umbrella-shaped expansion head is as follows:  
\[\omega_{s}=\int_{0}^{z^{\prime}}\frac{\left(3+v-2v^{2}\right)(D_{2}+k_{1}z)\tau _{fd}}{2 Ez}dz+\int_{0}^{z^{\prime}_{1}}\frac{\left(3+v-2v^{2}\right)(D_{2}+k_{2 }z)p_{d}}{2 Ez}dz \tag{16}\]  
After simplification, the following can be found:  
\[\omega_{s}=\frac{\left(3+v-2v^{2}\right)}{2E}\Big{[}\tau_{fd}\big{(}D_{2}lnz^{ \prime}+2 tan\theta_{1}z^{\prime}\big{)}+p_{d}\big{(}D_{2}lnz^{\prime}_{1}+2 tan \theta_{2}z^{\prime}_{1}\big{)}\Big{]} \tag{17}\]  
where \(v\) is the Poisson's ratio of soil; \(E\) is the elastic modulus of soil; \(\tau_{fd}\) is the frictional resistance of the outer wall of the umbrella-shaped expansion head; \(p_{d}\) is the positive pressure of the umbrella-shaped enlarged head on the inner end soil under the drawing force. It can be seen from Equation (13) that within the length \(dz\) of the micro-element of the anchorage section, the axial force of the bolt changes as follows:  
\[dP_{z}=-\Big{(}dS_{1}\tau_{fd}+dS_{2}p_{d}\Big{)} \tag{18}\]  
The outer surface area of the cone element and the inner surface area under positive pressure can be derived from the inner and outer surface area of the cone, which is as follows:  
\[\left\{ \begin{aligned} S_{1}=\pi(D_{2}+kz)dz\\ S_{2}=\frac{\pi}{4}(D_{2}+kz)^{2}\end{aligned} \right. \tag{19}\]  
After being substituted into Equation (18), the following simplification can be obtained:  
\[P_{z}(z)=T_{2}-\pi\tau_{fd}\bigg{(}D_{2}z+\frac{k}{2}z^{2}\bigg{)}-\frac{\pi} {4}p_{d}\bigg{(}D_{2}^{2}z+kD_{2}z^{2}+\frac{k^{2}}{3}z^{3}\bigg{)} \tag{20}\]  
According to the elongation formula of the anchorage section, the elongation \(\omega^{\prime}\) can be solved as follows:  
\[\omega^{\prime}=\frac{T_{2}}{E_{eq}A_{u}}L_{2}-\frac{\pi\tau_{fd}}{E_{eq}A_{u }}\bigg{(}\frac{D_{2}}{2}L_{2}^{2}+\frac{k_{1}}{6}L_{2}^{3}\bigg{)}-\frac{\pi p _{d}}{4E_{eq}A_{u}}\bigg{(}\frac{D_{2}^{2}}{2}L_{2}^{2}+\frac{k_{2}D_{2}}{3}L _{2}^{3}+\frac{k_{2}^{2}}{12}L_{2}^{4}\bigg{)} \tag{21}\]  
\[A_{u}(z)=\frac{\pi}{4}\Big{[}2(k_{1}-k_{2})D_{2}z+\Big{(}k_{1}^{2}-k_{2}^{2} \Big{)}z^{2}\Big{]} \tag{22}\]  
where \(A_{u}(z)\) is the effective cross-sectional area of the umbrella-shaped enlarged head; \(E_{eq}\) is the equivalent elastic modulus. Because the bolt, the anchorage body (grouting body), and the surrounding soil form a composite material system, their elastic modulus is different. Therefore, according to different stratum conditions, it can be evolved into the following three calculation methods [41]:  
\[\left\{ \begin{aligned} \frac{1}{E_{eq}}=\frac{1}{E_{u}}+\frac{1}{E_{g}}+ \frac{1}{E_{s}}(Applicable\ to\ weak\ strata)\\ E_{eq}=\frac{E_{u}A_{u}+E_{g}A_{g}+E_{s}A_{s}}{A_{u}+A_{g}+A_{s}}(Suitable\ for\ hard\ strata)\\ E_{eq}=\alpha_{u}E_{u}+\alpha_{g}E_{g}+\alpha_{s}E_{g}(Applicable\ to\ general\ situation)\end{aligned} \right.