In the selection of materials for machine and structural parts subjected to vibrations it is common practice to consider the fatigue strength of materials. In some cases the damping constant of the material may be considered in a qualitative manner. A theoretical analysis is given in this paper to show that the load resistance of resonant members is a function not only of the fatigue strength but also of the damping constant and modulus of elasticity of the material. Test data on various materials are presented to show that the load resistance of a vibrating member may be appreciably changed by a correct selection of the material when this new design criterion is used. Fatigue strength - damping design criteria are presented in this paper for vibrating beams and plates subjected to both concentrated and distributed resonant forces. Results are given for both external and internal damping. The theory developed may be extended for other members where the possible condition of resonant vibrations may occur. Turbine blades, airplane propeller blades, and shafts of various kinds are applications where the design criteria presented herein may be found useful.