Ritz-based analytical solution for static, free vibration, and buckling analyses of porous two-directional functionally graded beams

In this study, the static, free vibration, and buckling analyses of porous two-directional functionally graded (2D FG) beams were carried out using a Ritz-based analytical solution method based on the parabolic shear deformation theory. The choice of trial functions used in the Ritz method distinguishes different approaches and determines the accuracy of the analytical solution. In this work, polynomial series functions suitable for various boundary conditions were proposed. It was also assumed that the material properties of the beam vary along both the thickness (z-axis) and length (x-axis) directions according to a power-law distribution. The beam cross-section was considered with three different types of porosity distributions. The equations of motion were derived using Lagrange's principle. The analysis results were obtained in a non-dimensional form depending on various boundary conditions, grading indices (px, pz), porosity coefficient (e), porosity distribution, and slenderness ratios (L/h). First, the suitability of the proposed polynomial series functions for this problem was verified by comparison with the literature, and the obtained results were found to be in good agreement with existing studies. Then, the static, free vibration, and buckling analyses of porous two-directional functionally graded beams were performed in detail, and the results obtained with the proposed method were presented through tables and graphs.