Theoretical solutions for the anchoring force of 3-D anchored slope and optimization strategy for anchoring force distribution

Based on the three-dimensional limit equilibrium method with modifying normal stress over the slip surface, this study presents a theoretical solution for the anchoring force of three-dimensional slopes and proposes a new optimization strategy for anchor force distribution. The method employed the Boussinesq solution from the field of elasticity. It led to the derivation of an equilibrium equation set, which incorporated the anchoring force increment. As a result, the approach yielded a more accurate theoretical value for a three-dimensional anchoring force. In practical engineering applications, the prestress of anchor cables is often uniformly set, which may lead to discrepancies in the anchoring force requirements at different locations on the slope, potentially resulting in resource waste. The research introduces a linear programming approach to optimize resource allocation, aiming to minimize the total anchoring force under fixed anchor cable quantities, installation angles, and locations. Nevertheless, the results from the Simplex Method could lead to an overly concentrated distribution of anchoring force. Therefore, the study further employs the normal distribution theory to reallocate anchoring forces, thus achieving a more reasonable distribution. The proposed method has been validated through three example studies and a practical engineering case study, resulting in significant reductions in the total anchoring force following optimization. Moreover, such an approach allows for optimized handling of the anchoring force for each anchor cable. The work provides a new perspective and tool for practical engineering applications.