Alumina ceramics as well as zirconia ceramics have been used to manufacture hip replacement components since the 1990’s. These materials have been used for the ceramic balls and cups in the replacements for the following reasons: in the case of alumina for its excellent wear behavior; and in the case of zirconia, for its good wear behavior but with greater fracture toughness than alumina. Nevertheless, since ceramics are brittle, the design of the components must consider ways to avoid detrimental stress concentrations. In this study the stress concentration on the rim of the ceramic cup is inspected by means of a two-dimensional axis symmetric finite element model. This model simulates contact between the components to obtain the stresses that are transferred from the ball to the cup. Different geometries of the cup’s edge are simulated to evaluate the incidence of the geometry over the stress pattern. Although this model concentrates on the contact between the ball and the cup, the loading is made from the stem to the ball, which means that the load is transferred from the stem to the ball and then to the cup. This approach may offer a more realistic output. The data from this study may be useful for hip replacement designers as well as for surgeons because it shows a direct relationship between the stress patterns and the geometrical parameters of the design.