- The present work presents further development of the linking relationships between vibration frequencies predicted by different theories, and they are extended from a flat plate to a spherical shallow shell. In analogy with the membrane vibration problem, exact correspondences are found for vibration frequencies of a functionally graded spherical shallow shell using the classical theory and the first-order and third-order shear deformation theories. Only the predominantly stretching and thickness-shear vibration of dilatational type and predominantly flexural vibration are considered in this work. They are decoupled from the predominantly stretching and thickness-shear vibration of rotational type. These results apply to a simply supported functionally graded spherical shallow shell of polygonal planform with arbitrarily varying material properties in the thickness direction. A Winkler-Pasternak elastic foundation and rotary inertias are incorporated. It is proved that the mathematical analogy warrants positive free vibration frequencies for the shallow shell. Mori-Tanaka's scheme is used to estimate the material properties in the numerical results. 2002 Elsevier Science B.V. All rights reserved.