Quantifying the Role of Vertical Seismic Forces in Pseudo-Static Slope Stability: A Parametric Study with Practical Analytical Indicators

Vertical seismic forces may exceed horizontal components under certain conditions; however, they are often neglected in engineering practice. The primary justification in pseudo-static slope stability analysis is that the simultaneous increase/decrease in resisting and driving forces due to vertical loading results in a negligible net change in the factor of safety (FS); peak horizontal and vertical accelerations rarely occur simultaneously; and standard horizontal coefficients are generally sufficiently conservative. The present study systematically quantifies the combined influence of horizontal and vertical seismic forces on slope stability. Three representative slope geometries with inclinations of 1:1.5, 1:2, and 1:3 were analyzed using the pseudo-static limit equilibrium method (PS-LEM); the framework was additionally validated using the pseudo-static finite element method (PS-FEM) against a documented landslide case history. Horizontal seismic coefficients (Kh) of 0.025, 0.05, 0.10, and 0.20 were evaluated in combination with PVA/PHA ratios of 0, 0.25, 0.50, and 1.0 under both upward and downward vertical loading. Three analytical parameters are introduced: (i) Factor of Safety Reduction Index (FSRI), (ii) Vertical Force Sensitivity Ratio (VSR), and (iii) Asymmetry Index (AI). The results show FS reductions up to 10.6% under critical downward loading and a consistently asymmetric response. A preliminary validation against the 2008 Sichuan Daguangbao landslide supports the predictive capability of the framework, although broader validation across additional case histories is required to establish general applicability. The findings support the explicit inclusion of vertical seismic forces in slope stability assessments, particularly in near-field regions with PVA/PHA ratios exceeding 0.5.