Buckling of Plates with Arbitrary Geometric Configurations via the Discrete Ritz Method

A discrete Ritz method (DRM) is presented for buckling analysis of plates with arbitrary geometric configurations employing the concept of representing structural energy with global variable stiffness. In the DRM approach, a minimum rectangular domain covering the plate is generated, and openings are created within this domain based on the geometric boundaries of the plate. By applying the Gauss-Legendre quadrature rule and generating sufficient Gaussian points within the rectangular domain, along with assigning zero stiffness and thickness for the Gaussian points in the cutouts, the energy of plates with arbitrary geometric configurations can be numerically simulated. This methodology transforms the rectangular domain into a discrete model of a variable stiffness system, enabling the prediction of the deformation of plates of any shape. Furthermore, a prebuckling in-plane stress field is calculated and integrated into the eigenvalue buckling analysis of plates. The DRM formulation ensures that the energy functionals and computation procedures remain standardized and unaffected by variations in plate geometry, thus overcoming the limitations of traditional Ritz methods, which cannot be applied to complex geometric domains. To validate the efficacy of DRM, five numerical examples of plates with distinct geometries are adopted, and the results are compared with those from the literature and the finite element method. Numerical results show that DRM is capable of handling of plates with complex geometries and distinct boundary conditions, and DRM results are almost consistent with those of existing literature, indicating the feasibility and stability of DRM.