# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable
## 3 Upper-Bound Solution of Ultimate Pullout Capacity of Expanded Anchor Cable
### Ultimate Lateral Resistance Calculation  
Through the power calculation of the anchorage segment region and several algebraic operations, the ultimate lateral resistance could be obtained (Equation (8)). In the anchorage segment region, the source of internal energy dissipation was the expansion deformation of the soil and the relative motion between the anchorage segment and its surrounding soil, and the rate of work was attributed to the gravity of the anchorage segment. The calculations are discussed in detail in this section. Figure 9: Cross-sections of the failure mechanisms for different coordinates of point O: (**a**) intersection with the ground surface, (**b**) all intersection with the potential slip surface. (1) Rate of internal energy dissipation of the anchorage segment region  
The rate of internal energy dissipation depended on the soil expansion deformation, which could be computed as follows:  
\[\dot{W}_{\varepsilon}=T\dot{u}\tan\varphi \tag{15}\]  
where \(T\) is the resultant force of the normal pressure acting on the side surface of the anchorage segment; \(\dot{u}\) is the axial velocity along the cable; and \(\varphi\) is the internal friction angle of the contact surface between the anchorage segment and its surrounding soil, which could be taken as approximately equal to that of the surrounding soil. As depicted in Figure 10, the resultant normal force \(T\) could be decomposed into two parts: the resultant normal pressure generated by the vertical component of the soil pressure, denoted as \(Q\) herein, and the force generated by the horizontal component of the soil pressure, denoted as \(P\) herein. Thus, the vertical component of the soil pressure \(q_{v}\) could be written as:  
\[q_{v}=\gamma[h-R_{m}\cos\theta\cos\alpha+L_{m}\sin\alpha] \tag{16}\]  
Then, the differential of \(Q\), which denoted the resultant normal pressure corresponding to an infinitesimal area d\(A\), could be computed through the following equation:  
\[ \begin{array}{ll}dQ&=q_{v}R\mathrm{d}\theta\cos\theta\cos\theta\mathrm{d}L \cos\alpha\cos\alpha\\ &\quad=q_{v}R\cos^{2}\theta\cos^{2}\alpha\mathrm{d}\theta\mathrm{d}L\end{array}  \tag{17}\]  
Figure 10: Integral diagram of vertical and horizontal soil pressure: (**a**) model diagram; (**b**) vertical component of soil pressure; (**c**) horizontal component of soil pressure. Finally, \(Q\) could be obtained through the integral of d\(Q\) over the side surface of the anchorage segment:  
\[ \begin{array}{ll}Q&=\iint_{\Sigma}\mathrm{d}Q\\ &=4\int_{0}^{\frac{\pi}{2}}\int_{0}^{Ln}q_{p}R_{m}\cos^{2}\theta\cos^{2}\alpha \mathrm{d}L\mathrm{d}\theta\\ &=\frac{LnR_{m}T\cos^{2}\alpha(6\pi\hbar-16R_{m}\cos a+3\pi L_{m}\sin(\alpha) )}{6}\end{array}  \tag{18}\]  
Similarly, the horizontal component of the soil pressure \(q_{h}\), d\(P\), and \(P\) could be computed as follows:  
\[q_{h}=K_{0}\gamma[h-R_{m}\cos\theta\cos\alpha+L_{m}\sin\alpha] \tag{19}\]  
\[\mathrm{d}P=q_{h}R_{m}\mathrm{d}\theta\sin\theta\sin\theta\mathrm{d}L=q_{p}R_{ m}\sin^{2}\theta\mathrm{d}\theta\mathrm{d}L \tag{20}\]  
\[ \begin{array}{ll}P&=\int_{0}^{2\pi}\int_{0}^{Ln}q_{h}R_{m}\sin^{2}\theta \mathrm{d}L\mathrm{d}\theta\\ &=\int_{0}^{2\pi}\int_{0}^{L_{m}}K_{0}\gamma[h-R_{m}\cos\theta\cos\alpha+L \sin(\alpha)]R_{m}\sin^{2}\theta\mathrm{d}L\mathrm{d}\theta\\ &=\frac{K_{0}L_{m}R_{m}\gamma\tau(2h+L_{m}\sin\alpha)}{2}\end{array}  \tag{21}\]  
where \(K_{0}\) is the lateral pressure coefficient, which could be taken as \(K_{0}=1-\sin\varphi\). By summing \(Q\) and \(P\), the total normal force \(T\) could be obtained. According to Equations (18) and (21), \(T\) could be expressed as:  
\[ \begin{array}{ll}T&=Q+P\\ &=\frac{L_{m}R_{m}\gamma\cos^{2}\alpha}{6}[6\pi h-16R_{m}\cos\alpha+3\pi L_{m} \sin(\alpha)]\\ &+\frac{K_{0}L_{m}R_{m}\gamma\tau(2h+L_{m}\sin\alpha)}{2}\end{array}  \tag{22}\]  
Hence, the internal energy dissipation caused by the soil expansion deformation could be computed as:  
\[