A rigid foundation of arbitrary shape resting on a stratified half space (soil), with specific geometrical and elastic properties for its horizontal layers, is examined under the action of a vertical load placed anywhere on the foundation area. On the basis of a purely analytical treatment of the deformability of the soil surface, i.e. without using a finite element discretization of the layered soil mass, a general method is developed in order to determine the contact soil pressures and the resulting settlements through an easily implemented numerical procedure. The possible presence of “inactive” foundation parts not being in contact with the soil surface is taken into account. Parametric studies are performed showing the influence of the variability of the soil layers, as well as of the vertical load location, on the contact pressures developed. In this context, for the cases of a rectangular and a circular foundation the area borders for the location of a vertical load are determined, either for the development of compressive soil pressures all over the footing (core), or for allowing the existence of some inactive part without contact with the soil (zero contact pressure), not exceeding the half of the total foundation surface (limit core).