Exact vibration results for stepped circular plates with free edges

A pontoon-type, very large floating structure (VLFS) is often modeled as a huge plate with free edges when performing a hydroelastic analysis under the action of waves. The analysis consists of separating the hydrodynamic analysis from the dynamic response analysis of the VLFS. The deflection of the plate is decomposed into vibration modes where as many higher modes as possible should be used to capture the actual deflection shapes and the stresses. It is generally accepted that finite element method and the Ritz-type energy method fail to model zones with steep gradients which are encountered in, for instance, the stress resultants near the free edges of plates [Journal of Engineering Mechanics 1983;109(2):537–56]. Moreover, the natural boundary conditions are not satisfied completely because they are not enforced a priori [International Journal of Solids and Structures 2001;38:6525–58, Journal of Computational Structural Engineering 2001;1(1):49–57, Journal of Structural Engineering ASCE 2002;128(2):249–57, Computers and Structures 2002:80(2):145–54]. Exact solutions for frequencies, mode shapes and modal stress resultants are thus very important as they provide valuable benchmarks for assessing the convergence, accuracy and validity of numerical results obtained using the finite element method. To this end, we present the exact vibration results for stepped circular plates with free edges. When employed in a hydroelastic analysis, these exact vibration solutions yield accurate deflections and stress resultants (stresses) for circular VLFSs with stepped drafts.