Differential analysis of slope stability

Despite significant advances in computational methods, limit equilibrium and slice methods remain widely used for analysing slope stability. Among the slice methods, the models of Morgenstern–Price and Janbu accurately consider the equilibrium of forces and moments but face numerical challenges, as often reported by researchers. This analysis thoroughly examines the system of ordinary differential equations (ODEs) that governs these slice methods. It reveals the presence of local indeterminacy and shows that the eigenvalues of the Jacobian matrix of the ODE system can become positive under certain conditions for the Morgenstern–Price model, but are almost always positive for the Janbu model. These findings indicate that the differential equations exhibit numerical instability, which is particularly significant near points of indeterminacy, and that the Morgenstern–Price model exhibits greater numerical stability than the Janbu model. A new model is proposed to address the numerical challenges associated with positive eigenvalues. This model integrates the thrust line concept with a failure criterion for internal forces and moments. These findings enhance the understanding of slope stability analyses and bridge the gap with advanced computations.