Curvature effects on shallow shell vibrations

Abstract

The primary objective of the present work is to assess the effect of curvature upon the vibration frequencies of shallow shells. For this purpose, the shell is chosen to have a rectangular boundary supported by shear diaphragms and yields exact, closed form solutions for convenient comparison in the linear, small deflection regime and, at the same time, represents a situation which can readily occur in practical application. The shell is taken to have two independent radii of curvature, R x and R y whose planes are parallel to the edges. The linear eigenvalue problem is solved, both when tangential inertia is included and when it is neglected. Extensive tabular and graphical results are presented which show the effect of curvature ratio R x / R y in the interval −1·5 ⩽ R x / R y ⩽ 1·5 and the “average curvature” upon the frequencies. The analysis is extended into the non-linear, large deflection régime by assuming mode shapes and satisfying the non-linear field equations of motion and compatibility approximately by means of the Galerkin procedure. The effect of large deflections upon frequencies and phase plane plots as the curvature ratio changes is shown.