Thin rectangular plates are widely used in civil, mechanical, and aeronautical engineering, where accurate prediction of buckling behaviour is critical for structural reliability. While most classical analyses focus on small-deflection conditions, such approaches neglect the nonlinear effects induced by large out-of-plane displacements. This study aims to apply the new general mathematical model for biaxial buckling of a thin isotropic plate with large deflection to formulate new specific equations for six plate boundary conditions. The nonlinear buckling behaviour is investigated for plates with boundary conditions CCCC (clamped - clamped - clamped - clamped), CSSS (clamped - simply supported - simply supported - simply supported), CSCS (clamped - simply supported - clamped - simply supported), CCSS (clamped - clamped - simply supported - simply supported), CCCS (clamped - clamped - clamped - simply supported) and SSSS (simply supported all-round)) subjected to biaxial compressive loads. The new specific equations of this work allow for the evaluation of buckling load coefficients across varying aspect ratios, biaxial buckling ratios (n), and deflection-to-thickness ratios (w/t). The results obtained reveal that the biaxial buckling load coefficient (and load) decreases with increasing ‘n’ but increases with an increase in w/t. This highlights the combined influence of loading distribution and geometric nonlinearity. Deducing from the new biaxial equations for the uniaxial loading case for the purpose of comparison in square plates, the large-deflection buckling coefficients compared showed negligible deviation (~0%) from prior works, validating the proposed equations’ accuracy. Comparative analysis on values of buckling and postbuckling loads of CCCS plates against existing work under biaxial loading shows percentage differences between 0% and 5.56%, with the present results consistently upper bound but within acceptable engineering tolerances. The further comparison of biaxial buckling and post-buckling load coefficients of Sthe SSS plate with existing works has also shown the adequacy of this new model. The findings demonstrate that the proposed linear and nonlinear buckling equations not only align with established studies but also offer enhanced predictive capability for large deflection analysis of thin isotropic rectangular plates.

Thin rectangular plates, due to their small thickness relative to length and width and their high strength-to-weight ratio, are widely used in structural elements such as ship hulls, bridge decks, and aircraft wings. They are prone to nonlinear buckling under compressive forces, especially under biaxial in-plane compressive loading with large displacements, where linear theories often fail and membrane stresses complicate analysis. This study aimed to formulate a general mathematical equation for buckling analysis of thin rectangular isotropic plates with large displacements subject to biaxial in-plane forces using the Ritz potential energy functional method, and incorporates both geometric and material nonlinearities. Based on the formulated general equation, a specific equation for an all-round simply supported (SSSS) plate was developed using polynomial displacement shape function to determine the stiffness characteristics. Numerical values for critical buckling and post-buckling loads under biaxial compression for a square plate case were obtained. To validate these results, a comparison with values in the literature was made and the results show high consistency. The uniaxial buckling deviations ranged 0.047–0.10%, while undeformed biaxial buckling coefficients across varying aspect ratios and loading ratios (n = Ny/Nx) showed near-zero differences. From the two studies used for comparison, the maximum deviation is 24.42% and the minimum deviation is 1.12%. This indicates that the new model is adequate. Also, the adequacy of this new equation can be judged based on the simplicity of the formulation, and the closed agreement of the obtained numerical results with established results in the literature. This research enhances theoretical understanding of nonlinear buckling in thin plates and offers practical insights for improving structural reliability and efficiency in civil, mechanical, aerospace, and marine engineering. Therefore, the conclusion is that the model is suitable for buckling and post-buckling analysis of thin rectangular isotropic plates.