# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable
## 2 Construction of Failure Model
### Failure Model of the Soil Region at the Front Surface of the Anchorage Segment  
(1) Failure mechanism  
Figure 1: Schematic diagram of expanded anchor cable. Under the end resistance of an expanded anchor cable, the strained condition of the soil located in the front-end surface of the anchorage segment (Figure 2) is similar to the condition of the soil in front of a shield tunnel face when the shield pressure is uniformly distributed on the tunnel face. Therefore, referring to the failure mechanism of the soil in a tunnel face [20; 21], a three-dimensional logarithmic spiral failure model was adopted for the soil at the front surface of the anchorage segment. The soil region at the front surface of the anchorage segment could be renamed the logarithmic spiral region. To be specific, the logarithmic spiral region represents the soil zone at the front end of the anchorage segment, which is surrounded by three-dimensional logarithmic spirals. Figure 3 shows the cross-sections of the failure model of the logarithmic spiral region in the vertical plane (YAZ) passing through the cable axis. As shown in Figure 3, the expanded anchor cable with a buried depth of \(h\) in a homogeneous stratum is destroyed under the action of pullout capacity \(P\), the cross-section is bounded by two log spirals emerging from A and B with a common center O, the intersection of the side surface of the anchorage segment and the YAZ plane is AA' and BB', and D1D2 is the intersection of the potential slip surface and the YAZ plane. (2) Velocity field  
The velocity field was considered in this study as follows: the logarithmic spiral region was damaged according to the above failure model under the action of pullout capacity \(P\), leading to velocity discontinuity on the rupture surface or the so-called velocity discontinuity surface. Specifically, the logarithmic spiral region presented a three-dimensional rotation failure mechanism generated by rotation around the horizontal axis of point O with a uniform angular velocity \(\omega\). Its velocity discontinuity surface intersected the symmetric  
Figure 3: Failure model diagram. Figure 2: Stress diagram of anchorage segment front-end surface. plane at two logarithmic spirals, AN and BM, with a common rotation center O, as shown in Figure 3. In the YAZ plane, the possible region of the rotation center point O was the region formed by the intersection of the semi-infinite region on the right side of the straight line BA and above the straight line AA\({}^{\prime}\), as shown in Figure 4.  
The expressions of the logarithmic spirals AN and BM could be written as follows:  
\[r_{AN}=r_{A}\cdot\exp[(\beta_{A}-\beta)\cdot\tan\varphi] \tag{1}\]  
\[r_{BN}=r_{B}\cdot\exp[(\beta-\beta_{B})\cdot\tan\varphi] \tag{2}\]  
where  
\[r_{A}=\sqrt{Z_{0}^{2}+Y_{0}^{2}} \tag{3}\]  
\[r_{B}=\sqrt{\left(D_{m}\sin\alpha-Z_{0}\right)^{2}+\left(D_{m}\cos\alpha+Y_{0 }\right)^{2}} \tag{4}\]  
\[\beta_{A}=\arctan(-\frac{Z_{0}}{Y_{0}}) \tag{5}\]  
Figure 3:0  
In the above expressions, \(r_{A}\) and \(r_{B}\) are the lengths of OA and OB, respectively; \(\beta_{A}\) and \(\beta_{B}\) are the angles between the lines OA or OB and the vertical direction, respectively; \(\varphi\) is the internal friction angle of the soil stratum; and \(Y_{0}\) and \(Z_{0}\) are the \(Y\)-axis and \(Z\)-axis coordinates of the rotation center O in the cartesian coordinate system YAZ, respectively. (3) Geometrical construction of the 3D velocity discontinuity surface in logarithmic spiral region  
The spatial discretization technique proposed in [20,21] was adopted to generate the 3D velocity discontinuity surface in the logarithmic spiral region. The basic principles were as follows: the failure model was discretized by several radial planes that all intersected at one point, and the intersection lines of the radial planes and velocity discontinuity surfaces were discretized into a series of points.