Free Vibration and Buckling Analysis of Two Directional Functionally Graded Beams Using a Four-Unknown Shear and Normal Deformable Beam Theory

Abstract

This study presents the free vibration and buckling behavior of two directional (2D) functionally graded beams (FGBs) under arbitrary boundary conditions (BCs) for the first time. A four-known shear and normal deformation (Quasi-3D) theory where the axial and transverse displacements are assumed to be cubic and parabolic variation through the beam depth is employed based on the framework of the Ritz formulation. The equations of motion are derived from Lagrange’s equations. The developed formulation is validated by solving a homogeneous beam problem and considering different aspect ratios and boundary conditions. The obtained numerical results in terms of dimensionless fundamental frequencies and dimensionless first critical buckling loads are compared with the results from previous studies for convergence studies. The material properties of the studied problems are assumed to vary along both longitudinal and thickness directions according to the power-law distribution. The axial, bending, shear and normal displacements are expressed in polynomial forms with the auxiliary functions which are necessary to satisfy the boundary conditions. The effects of shear deformation, thickness stretching, material distribution, aspect ratios and boundary conditions on the free vibration frequencies and critical buckling loads of the 2D-FGBs are investigated.