# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable
## 1 Introduction  
With the rapid growth of urbanization, the development and utilization of underground space resources has become an important means for the construction of sustainable urban developments [1]. In order to make better use of underground space and allow the construction of high-rise structures, foundation-pit engineering has developed rapidly, with support technology playing an important role in ensuring construction safety and reducing environmental impacts. As one of the main supporting methods in foundation-pit engineering, pile-anchor support structures are widely used because of their high safety, good stability, economy, flexible arrangement, and adaptability [2]. In recent years, the appearance of many deep and large foundation pits has introduced higher requirements for the ultimate pullout capacity of anchor cables. However, conventional anchor cables are not able to achieve a sufficient pullout capacity in cases where the surrounding environment has certain constraints that necessitate a longer anchor or where soft soil is widely distributed. This seriously limits the wider application of pile-anchor supports [3; 4; 5]. Compared with ordinary anchor cables, expanded anchor cables can provide greater lateralresistance and extra end resistance due to the expanded anchorage segment, which can substantially increase the ultimate pullout capacity. Therefore, these cables have been widely used in practical engineering [6]. The bearing mechanism of expanded anchor cables has been studied and investigated by several authors, including in studies based on various numerical approaches. [PERSON] et al. [7] used the finite element method (FEM) to study the supporting effect of expanded anchor cables and ordinary anchor cables in a foundation pit and concluded that expanded anchor cables have a high bearing capacity and a remarkable effect on controlling the structure and surrounding rock deformation. [PERSON] and [PERSON] [8] used the discrete element method (DEM) to simulate plate anchors and studied the influencing factors of the ultimate bearing capacity. [PERSON] et al. [9; 10] used the finite difference method (FDM) to analyze the bearing mechanism of underreamed anchors. At the same time, studies based on theoretical analyses have also been carried out. [PERSON] et al. [11] deduced a formula to calculate the pullout capacity of expanded anchor rods by referring to the formula for the ultimate bearing capacity of piles. [PERSON] et al. [12] established a mechanical model of an expanded anchor cable in the process of soil deformation based on the elastic theory and presented a formula for calculating its ultimate pullout capacity. Based on the limit equilibrium principle, [PERSON] et al. [13] presented a formula for the theoretical calculation of the ultimate pullout resistance of horizontally arranged expanded anchor cables, with the influence of the strength of the surrounding rock being taken into consideration. Furthermore, many experimental tests, either in laboratories or in situ, have also been performed to study this problem and similar problems. [PERSON] et al. [14] studied the axial stress distribution regulation of an anchor bolt through a laboratory model experiment. Using grating sensors and photogrammetry methods, [PERSON] et al. [15; 16] conducted a series of pullout experiments on expanded anchors embedded in sand and obtained the corresponding ultimate bearing capacity and the proportion of end resistance in the ultimate bearing capacity. Meanwhile, [PERSON] [17], [PERSON] et al. [18], and [PERSON] et al. [19] carried out in situ pullout experiments on an expanded anchor cable, analyzing its bearing mechanism and pullout bearing capacity. On the one hand, the theoretical research on the ultimate pullout capacity of expanded anchor cables ignores the influence of the failure mechanism of the soil at the front end of the anchorage segment on the end resistance of the expanded anchor cable.