Numerical analysis for the nonlinear free vibration of shallow shells

A variational approach for the nonlinear free vibration of shallow shells having a quadrilateral boundary is presented in this paper. Natural coordinates xi and eta are used to map the prescribed geometry in the x-y plane. Displacement fields corresponding to mu, upsilon, w, beta1, and beta2 are expressed in terms of the product of two algebraic functions, the form of which is so chosen that the displacement boundary condition can be imposed by manipulating the coefficients. In arriving at the stiffness matrix, no simplification is applied to the nonlinear strains and the variation of the complete energy equation is considered. For the plate problems numerical results are obtained and compared with approximate analytical results by other researchers. Numerical results for the shallow shells are also presented and their characteristics are found to be significantly different from the results for the plates.