Solving the general 3-D safety factor by combining Sarma's idea with the assumption of normal stress distribution over the slip surface.

This study proposes a method for determining 3-D limit equilibrium solutions. The method, inspired by Sarma, introduces the horizontal seismic coefficient as a slope failure parameter and implements a modification of the normal stress over the slip surface. Four equilibrium equations are used to solve the problem without compromising the accuracy of the calculations: three force equilibrium equations in the x, y, and z directions and a moment equilibrium equation in the vertical (z) direction. The reliable factor of safety can be determined by calculating the minimum value of the horizontal seismic coefficient. Furthermore, we analyzed several typical examples of symmetric and asymmetric slopes, finding good consistency with the existing literature. This consistency indicates the reliability of the factor of safety we obtained. The proposed method is favored due to its straightforward principle, convenient operation, fast convergence, and ease of programming.