Unified Variational Solution for Seismic and Static Active Earth Pressures on Rigid and Flexible Walls in the Limit State

Flexible retaining walls, such as sheet pile walls and secant pile walls with small diameters, can take on significant lateral deformation to possibly reach the active limit state if subjected to surcharge or seismic action, which influences the active earth pressure on the walls not theoretically solved currently. This work presents a unified solution for the seismic and static active earth pressure on both flexible and rigid walls under the limit state of the retained soil, derived through the variational calculus principles and pseudostatic tactics. The proposed solutions are compared with some classical results, and the relative error between them is within 4% under the movement modes of rigid wall translation and rotation around its base. The active earth pressure on the flexible wall exhibits a similar profile as that of the rigid wall in terms of the soil internal friction angle and wall-soil friction angle. The earth pressure is also more sensitive to the horizontal seismic acceleration coefficient than the vertical one. As the flexible rigidity of pile wall increases, the active earth pressure decreases nonlinearly to some extent. Compared with the round-pile wall, the square-pile wall exhibits lower active earth pressure with a slightly higher application point. Besides, the proposed method can identify the static and dynamic distribution of the seismic earth pressure by decomposing the total profile into the sum of linear and parabolic patterns. The proposed method is of computational simplicity to serve as a useful analytical tool for engineers to evaluate the active earth pressure in various cases.