# A Modified Bearing Capacity Model for Inclined Shallow Anchor Cable with Experimental Verification
## 1 Introduction  
Anchor cables play an important role in various geotechnical engineering applications, including foundation stabilization, slope protection, and the support of retaining walls. The ability of anchors to transfer forces to deeper, more stable layers of soil or rock is crucial for maintaining the integrity of structures subjected to uplift or lateral forces [1]. Despite their widespread application, certain aspects of anchor behavior, particularly the effect of inclination on shallow anchors, remain underexplored [2, 3]. Anchor inclination is an important factor influencing load distribution, stress concentration, and failure mechanisms [4]. Previous studies demonstrated that anchor inclination significantly affects both pull-out resistance and the disturbance of the surrounding soil. For instance, in sandy soils and soft clay, the bearing capacity tends to decrease with increasing inclination [5, 6, 7]. Research on rock anchors showed that steeper angles can result in stress concentration and localized failure, particularly in harder materials [8, 9]. Although researchers have some understanding of the mechanisms behind anchor inclination, the accurate calculation of its bearing capacity remains an urgent issue to be addressed [10]. The theoretical analysis of anchor cables is usually developed with the limit equilibrium principles [11, 12, 13, 14]. [PERSON] et al. [15] presented three-dimensional lower-boundsolutions for the stability of plate anchors in sand based on the limit equilibrium method. [PERSON] et al. [16] employed the Hoek-Brown failure criterion along with the limit equilibrium approach to assess the bearing capacities of rock anchors in different geological conditions. [PERSON] et al. [17] further refined limit equilibrium solutions for the uplift capacity of anchors in multi-layered soil, offering critical insights into how the stratification of soils impacts anchor behavior. [PERSON] and [PERSON] [18] utilized upper-bound limit analysis to calculate the vertical uplift capacity of horizontal strip anchors in sand. Their findings suggest significant improvements in predicting failure mechanisms in shallow anchor systems. [PERSON] and [PERSON] [19] evaluated the pull-out capacity of anchors in layered sand using finite-element and limit equilibrium methods, providing a comparative analysis of their applicability to complex soil structures. [PERSON] and [PERSON] [20] used the upper-bound theorem of limit analysis to derive the ultimate pull-out capacity of expanded anchor cables, focusing on the influence of key parameters like anchorage length, diameter, and depth on the total resistance. It also incorporates particle swarm optimization for more precise analysis. [PERSON] et al. [21] present an analytical framework using limit equilibrium to assess earth pressures and bearing capacities in non-homogeneous soils behind retaining walls. It provides an improved method for predicting foundation stability under varying soil conditions. [PERSON] et al. [22] applied a three-dimensional upper-bound limit analysis to evaluate the uplift bearing capacity of anchor plates in non-homogeneous materials, using the Hoek-Brown failure criterion. It highlights how material variability influences bearing capacity predictions. [PERSON] et al. [23] applied limit equilibrium analysis to calculate the uplift capacity considering the influence of edge-distance ratio and slope angle of anchor plates in the sand. [PERSON] [20] proposed a new analytical solution using limit equilibrium methods for calculating the ultimate pull-out capacity of anchor cables in complex underground environments. [PERSON] and [PERSON] [24] examined the pull-out capacity of vertical plate anchors in unsaturated clay, considering variable surface flow conditions, using limit equilibrium methods to model complex soil interactions. These analyses often assume vertical loads, with little attention given to the effects of different inclinations on the bearing capacity of shallow anchors.