Summary: A new method is proposed for the solution of frictionless unilateral contact problems, when the elastic bodies in contact have regular boundaries. This method relies on the fact that, on the contact area edge, the distance between the two bodies and the contact force are simultaneously zero. The unilateral problem is reduced to a nonlinear equation. We study two problems: the cylinder compressed between two rigid planes, and the half-cylinder crushed on a support by a uniform pressure on its plane face. The robustness and accuracy of the method are underlined.