Analytic Solution and Benchmark Results for the Free Vibrations of Thin Shallow Shells with Rectangular Planform

In this work, a new analytic solution for vibrations of shallow shells is presented. The equations of motion consist of three coupled partial differential equations. Solutions to such complex coupled equations were only available for the Navier and Levy cases of boundary conditions, which is a small part of the scope of the problem. A superposition of two solutions enables to satisfy both the equations of motion and any combination of boundary conditions. For isotropic square shallow shell, there are 9316 different combinations of support conditions. For isotropic rectangular shell or square orthotropic shell, the number is 18496. These numbers apply for a single type of curvature and aspect ratio. For all these a general solution is derived. The functions for the solution are obtained by using carefully chosen series that solve the coupled partial differential equations of motion for in-plane and out-of-plane deformations for all possible combinations of edge conditions. The number of terms in the series is taken such that convergence is assured to the number of digits as shown. Examples of the new solutions are given and compared with available solutions in the open literature.