Quasi-rigorous and non-rigorous 3D limit equilibrium methods for generalized-shaped slopes

Adopting the limit equilibrium theory to obtain the solutions of 3D limit equilibrium is an effective way to solve the stability analysis of 3D slopes. After analyzing the stress of columns under general conditions, three parameters are chosen: inter-force parameter λ₁ of column’s front and back side faces, scaling parameter λbetween column's left and right side faces and their front and back ones, angle ρbetween directions of shear force on bottom of columns and the sliding plane, and a quasi-rigorous 3D limit equilibrium formula that can meet three-force equilibrium equation and three-moment equilibrium equation is established. When the relative 2D inter-force assumptions are used, 3D Spencer method, 3D M-P method and 3D Sarma method are obtained. By making some of the above-mentioned three parameters equal to zero, the proposed quasi-rigorous method can be transformed into three kinds of non-rigorous methods that only meet part of the mechanical equilibrium conditions. Compared with the classical examples and by comparing the difference of the calculated results between the quasi-rigorous method and the non-rigorous method in two asymmetric cases of width and curve surface of left and right sliding surface's, some conclusions can be drawn as follows: (1) the results calculated by 3D Spencer method, 3D M-P method and 3D Sarma method are quite close to those by other methods, indicating the feasibility of the proposed method; (2) for the case of asymmetric sliding surfaces, except that those by the non rigorous method that gets the limit equilibrium equation by ignoring vertical shear force of front and back side faces of columns are smaller, the calculated results by the quasi-rigorous method and two kinds of non-rigorous methods are the same, showing that the non-rigorous methods are also applicable to cases of 3D asymmetric sliding surface; and (3) the established three-dimensional limit equilibrium solutions are all applicable to the quasi-rigorous method and non-rigorous method, and the formulas are simple, and the convergent