## 1 Introduction
### Analysis of Uplift Bearing Capacity
\tag{23}\]  
where \(E_{u}\) is the elastic modulus of bolt material; \(E_{g}\) is the elastic modulus of anchor material; \(E_{s}\) is the elastic modulus of soil, that is, \(v\) above; the weight coefficients of \(\alpha_{u}\) (0.6-0.8), \(\alpha_{g}\) (0.1-0.3), and \(\alpha_{s}\) (0.05-0.2) are determined by experiment or numerical simulation. Thisstudy uses the third method (applicable to general situations) for the theoretical analysis and numerical simulation verification. According to the previous hypothesis, it can be seen that the displacement \(\omega_{s}\) of the soil in the umbrella-shaped enlarged head section is equal to the elongation \(\omega^{\prime}\) of the anchorage section of the umbrella-shaped enlarged head, that is, the displacement coordination equation \(\omega_{s}=\omega^{\prime}\). The maximum drawing force \(T_{2,max}\) of the umbrella-shaped enlarged head section can be obtained by solving the following:  
\[T_{2,max}=P_{u,max}=\frac{E_{eq}A_{u}}{L_{2}}\left\{ \begin{array}{c}\frac{ \left(3+v-2v^{2}\right)}{2E}\left[\tau_{fd}(D_{2}lnz^{\prime}+2 tan\theta_{1}z ^{\prime})+p_{d}\big{(}D_{2}lnz_{1}^{\prime}+2 tan\theta_{2}z_{1}^{\prime}\big{)} \right]\\ +\frac{\pi\tau_{fd}}{E_{eq}A_{u}}\left(\frac{D_{2}}{2}L_{2}^{2}+\frac{k_{1}}{6 }L_{2}^{3}\right)+\frac{\pi p_{d}}{4E_{eq}A_{u}}\left(\frac{D_{2}^{2}}{2}L_{2} ^{2}+\frac{k_{2}D_{2}}{3}L_{2}^{3}+\frac{k_{2}^{2}}{12}L_{2}^{4}\right)\\ \hskip 28.452756 pt+\pi\tau_{fd}(D_{2}+k_{1}z)L_{2}+\pi p_{d}(D_{2}+k_{2}z)L_{2} \end{array} \right\}\right. \tag{24}\]  
Similarly, it is assumed that the maximum drawing force \(T_{3,max}\) (generated by the frictional resistance of the side wall surface) of the expansion body section can be solved. Since the tail expansion body section can be approximately considered as an inverted cone, the side wall area can be expressed as follows:  
\[S_{3}=\pi(D_{3}+k_{3}z)L_{3} \tag{25}\]  
\[D_{3}(z)=D_{3}-k_{3}z \tag{26}\]  
where \(D_{3}\) is the bottom diameter of the tail expansion body; \(k_{3}\) is the taper change rate of the inverted cone (expansion body). It should be noted that the coordinate system used when calculating the tail expansion body is different from the previous one, and the origin \(O\) is the center point of the head of the tail expansion body. By solving the displacement coordination equation, the drawing force \(T_{3}\) of the expansion body section can be obtained as follows:  
\[T_{3}=\frac{E_{eq}A_{e}}{L_{3}}\left\{\frac{\left(3+v-2v^{2}\right)}{2E}[\tau _{s}(D_{3}lnz^{\prime\prime}+2 tan\theta_{3}z^{\prime\prime})]+\frac{\pi\tau_ {s}}{E_{eq}A_{e}}\left(\frac{D_{3}}{2}L_{3}^{2}+\frac{k_{3}}{6}L_{3}^{3} \right)\right\} \tag{27}\]  
Combined with the above equation, the maximum drawing force of the expansion body section can be solved as follows:  
\[T_{3,max}=\frac{E_{eq}A_{e}}{L_{3}}\left\{\frac{\left(3+v-2v^{2}\right)}{2E} [\tau_{s}(D_{3}lnz^{\prime\prime}+2 tan\theta_{3}z^{\prime\prime})]+\frac{\pi \tau_{s}}{E_{eq}A_{e}}\left(\frac{D_{3}}{2}L_{3}^{2}+\frac{k_{3}}{6}L_{3}^{3} \right)\right\} \tag{28}\]  
From Equations (3), (4), (24), and (28), the maximum pull-out force of the hollow growing anchor with the umbrella-shaped expansion head with the expansion pipe can be obtained as follows:  
\[T_{max}=T_{1,max}+T_{2,max}+T_{3,max}=\pi D_{1}L_{1}\tau_{f}+ \tag{29}\] \[\frac{E_{eq}A_{u}}{L_{2}}\left\{ \begin{array}{c}\frac{\left(3+v -2v^{2}\right)}{2E}\left[\tau_{fd}(D_{2}lnz^{\prime}+2 tan\theta_{1}z^{\prime} )+p_{d}\big{(}D_{2}lnz_{1}^{\prime}+2 tan\theta_{2}z_{1}^{\prime}\big{)}\right] \\ +\frac{\pi\tau_{fd}}{E_{eq}A_{u}}\left(\frac{D_{2}}{2}L_{2}^{2}+\frac{k_{1}}{6 }L_{2}^{3}\right)+\frac{\pi p_{d}}{4E_{eq}A_{u}}\left(\frac{D_{2}^{2}}{2}L_{2} ^{2}+\frac{k_{1}D_{2}}{3}L_{2}^{3}+\frac{k_{2}^{2}}{12}L_{2}^{4}\right)\\ \hskip 28.452756 pt+\pi\tau_{fd}(D_{2}+k_{1}z)L_{2}+\pi p_{d}(D_{2}+k_{2}z)L_{2 }\end{array} \right\}\] \[+\frac{E_{eq}A_{e}}{L_{3}}\left\{