Keywords: free vibration, circular and annular plates, higher-order plate theories, variable-kinematic Ritz method.

Modelling approaches for free vibration analysis of plate-like structures range from fully three-dimensional models, without any simplifying assumption on the kinematics of deformation, to traditional plate theories, such as classical plate theory (CPT) and first-order shear deformation theory (FSDT), based on a reduction of the three-dimensional problem to simple and economical two-dimensional models.

Many attempts lying in the middle have appeared in the recent literature. They fall into the category of so-called refined or higher-order plate theories, where the conventional kinematics of FSDT is enriched with various higher-order terms such as power series expansion of the thickness coordinate. Such refined theories preserve the two-dimensioanl nature of the model and avoid the complexity and computational inefficiency of three-dimensional elasticity solutions.

A novel Ritz-based formulation based on an entire class of two-dimensional higher-order theories is presented here to accurately compute eigenvalues of thick and thin isotropic circular and annular plates with arbitrary boundary conditions. The method is capable of handling in an unified way arbitrary refinements of classical theories, without the need of a new modelling effort each time, thus giving the ability to easily perform comparisons of different two-dimensional theories of increasing complexity. Considering the circumferential symmetry of circular plates, the present formulation is also computationally efficient since only a single series of trial functions in the radial direction are required. In addition, relying on a global Ritz approximation, the method has a high spectral accuracy and converges faster than local methods.

The formulation relies on a suitable expansion of invariant kernels of the mass and stiffness matrix. The invariance is to be intended with respect to both the order of the theory and the type of Ritz trial functions. Upper-bound frequency values are presented using products of boundary-compliant functions and Chebyshev polynomials. It is shown that the method exhibits good convergence properties and a high numerical stability. Increasing accuracy towards three-dimensional values in terms of frequency parameters can be obtained with theory refinement. Further, it is found that kinematic plate models of lower order are more sensitive to the thickness-to-radius ratio, whereas accuracy is substantially independent of plate thickness when a highly refined theory is adopted.

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