Semi-Analytical Analysis of Thin Plate Deflections Using the Clebsch–Ritz Method

Plates of various geometries and materials are widely employed in structural and mechanical engineering, where accurate prediction of their deformation behavior is crucial for design and safety. This study presents a semi-analytical approach for determining the deflection of thin rectangular plates resting on an elastic foundation with free boundary conditions. The proposed method integrates Clebsch’s analytical solution of the plate deformation equation with the Ritz energy method to compute the unknown coefficients governing the displacement field. By combining the strain energy of deformation with the external work of the applied load, the governing variational equations are established and solved to determine the deformation parameters. The developed formulation allows direct evaluation of the plate deflection at any point for arbitrary loading conditions. Numerical implementation using *Mathematica* validates the accuracy of the model and confirms its efficiency in representing the physical behavior of plates on elastic media. The proposed Clebsch–Ritz approach offers a reliable and adaptable framework for analyzing static deformation problems in structural and mechanical applications.