A simplified semi-analytical method, which considers advantages of analytical and numerical approaches, is developed to compute the response of a rigid strip and circular machine foundations-subjected to a harmonic excitation-resting on a layer of soil deposit with a noncompliant rock or rock-like material at the base. The method is based on variational principles and minimization of energy using Hamilton's principle. Nondimensional equations are developed for both type of footing resting on a soil layer. Dynamic response characteristics are plotted by using nondimensional parameters for both types of footing. (C) 2001 Elsevier Science Ltd. All rights reserved.