Keywords: vibration, Mindlin plates, modal stress-resultants, analytical methods, Ritz method, least squares curve fitting.

In the hydroelastic analysis of pontoon-type Very Large Floating Structures (VLFS), such structures are usually modelled as plates with free edges. The coupled fluid- structure interaction problem may be solved by firstly decomposing the unknown deflection of the plate into modal functions associated with a freely vibrating plate in air. The second step involves substituting the modal functions into the hydrodynamic equations and solving the boundary value problem. The modal amplitudes of the set of equations of motion obtained are then back-substituted into the modal functions and the stress-resultant functions for the actual deflections and stress-resultants of the VLFS under wave forces.

When performing vibration analyses of plates with free edges using either the Ritz method or the finite element method, very accurate natural frequencies of vibrations and modal deflections can be obtained. However, it was observed that the modal stress-resultants, especially the shear forces and the twisting moments, do not quite satisfy the natural boundary conditions despite achieving excellent convergence of the natural frequencies. This is because (a) the methods only stipulate the satisfaction of the geometric boundary conditions at the outset; relying on the minimization of the Lagrangian to satisfy the natural boundary conditions, and (b) the presence of very steep gradients in the stress-resultant distributions (shear forces and twisting moments) near the free edges. Many analysts do not recognize the violations of the natural boundary conditions because they expect the stress-resultants to vanish at the free edges. Furthermore, "oscillations" of the stress- resultant values were observed. In view of the fact that the accurate prediction of the hydroelastic responses of the floating structures depend on the satisfaction of the required boundary conditions and accurate distributions of the modal stress- resultants, a study that aims to address the aforementioned problems would be of a great value.

This paper highlights the problems encountered in computing accurate stress resultants when using the Ritz method for plates with free edges and it also presents some remedies for solving the problems. In the study, rectangular plates with some free edges and circular plates with a free boundary were considered. An analytical method was also employed to obtain the solutions. The benchmark exact solutions were used to assess the validity, convergence and accuracy of the Ritz (numerical) results. It will be shown herein that the modal stress resultant distributions obtained by the Ritz method do not satisfy the natural boundary conditions as well as they contain oscillations.

In order to address the aforementioned problems, we examine the use of penalty functionals in the total potential energy functional to enforce the natural boundary conditions at these free edges. It is found that the inclusion of the penalty functionals for bending moment and shear force has a positive effect in improving the accuracy of the modal stress resultants predicted by the Ritz method. On the other hand, the penalty functional for the twisting moment stiffens the plate artificially and causes large oscillations of the modal stress resultants, making the solutions more erroneous. Thus the preferred Ritz scheme in calculating the modal stress resultants is the one with penalty functionals for both shear force and bending moment.

Based on the results obtained in the vibration analyses of rectangular and circular plates, it is found that the peak values of the Ritz stress resultants are very accurate when oscillations of stress resultants are not present. In the case of stress resultants with oscillations, a least-squares curve fitting technique is proposed to smooth out the results, thereby remedying the oscillating problem. But it was still difficult to handle the excessive oscillations of the Ritz shear force near the central portion of the freely vibrating circular plate. This problem requires the attention of researchers in the future development of the Ritz method for the analysis of circular plates.

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