This paper presents a nonlinear stability analysis of piles under bilateral contact constraints imposed by a geological medium (soil or rock). To solve this contact problem, the paper proposes a general numerical methodology, based on the finite element method (FEM). In this context, a geometrically nonlinear beam-column element is used to model the pile while the geological medium can be idealized as discrete (spring) or continuum (Winkler and Pasternak) foundation elements. Foundation elements are supposed to react under tension and compression, so during the deformation process the structural elements are subjected to bilateral contact constraints. The errors along the equilibrium paths are minimized and the convoluted nonlinear equilibrium paths are made traceable through the use of an updated Lagrangian formulation and a Newton-Raphson scheme working with the generalized displacement technique. The study offers stability analyses of three problems involving piles under bilateral contact constraints. The analyses show that in the evaluation of critical loads a great influence is wielded by the instability modes. Also, the structural system stiffness can be highly influenced by the representative model of the soil.