Analytic Buckling Solution of Composite Classical Rectangular Thin Plates Using Galerkin Variational Approach

Abstract

This paper focuses on investigating the mechanical buckling behavior of composite and orthotropic classical rectangular plates using the Galerkin theory. By utilizing the classical plate theory of elasticity, the governing equations for the functionally graded plate material subjected to a uniaxially distributed load along the x and y axes are derived based on the principle of work energy. Consequently, the displacement function is determined as the solution to a differential equation that satisfies specific boundary conditions. The equilibrium expression derived from the work-energy equation is then minimized and analytically solved to determine the stresses and critical buckling loads of an orthotropic composite rectangular plate. The critical buckling loads of all edges, including clamped and simply supported orthotropic and isotropic thin rectangular plates with aspect ratios ranging from 0.1 to 3, are calculated and compared with existing literature to assess the stiffness of isotropic and orthotropic plates. The findings reveal that the critical buckling load of a clamped plate decreases as the aspect ratio increases from 0.1 to 1 but increases for aspect ratios between 1 and 3. The average percentage variation of critical buckling load values obtained from previous studies using orthotropic and isotropic (classical lamination plate theory [CLPT]) materials is found to be 1.93% and 3.65%, respectively. Additionally, it is observed that the critical buckling load of clamped edges in isotropic plates is lower than that of orthotropic plates at lower aspect ratios (0.4–1.0) but higher at higher aspect ratios (above 1.0). The results indicate that plates with all-round clamped boundaries exhibit higher buckling loads due to their effective fixation at all boundaries, which provides greater resistance to deformation and buckling compared to plates with all edges simply supported, where there is more flexibility for deformation under loads. This highlights the influence of boundary conditions on the critical buckling load of the plate. Furthermore, a significant percentage difference is noted when comparing approximate solutions, while a low percentage difference of 0.43% with the author’s derived displacement function indicates close agreement, demonstrating the high level of accuracy and reliability of the proposed model in predicting buckling load of plates. Hence, it can be recommended for stability analysis, especially where the material properties need to be controlled in a specific direction to optimize performance.