This study examines the buckling behavior of thick composite plates with varying composite lay-up arrangements. The equations used in this study were derived from the Sinusoidal Shear Deformation Theory (SSDT) based on thick composite plates. A compressive load was applied to the composite plate along its axis, and the equations of motion for the system were obtained using the energy method and Hamilton’s principle. These equations were solved using the Generalized Differential Quadrature Method (GDQM) under clamped Boundary Conditions (BCs). Through this process, the Critical Buckling Load (CBL) was determined. The Chebyshev-Gauss–Lobatto function was used to determine the point distribution and spacing. The study examines the impact of various parameters, including the fiber lay-up arrangement, the number of layers, the elastic modulus ratio, and the aspect ratio, on the buckling behavior of composite plates. The results show that changing the angle of the fibers’ arrangement for the [θ/θ] lay-up arrangement of fibers in the composite plate led to a decrease in the CBL from 0º to 15º, followed by an increase from 15º to 45º. Moreover, the research shows that enhancing the anisotropy of materials for thin plates had a more significant impact on the CBL as compared to plates with relatively greater thickness.