A Ritz solution for comprehensive analysis of functionally graded beams with various boundary conditions

The present study applies a Ritz solution that employs Laguerre polynomials to comprehensively analyse functionally graded beams characterised by varying material properties across their thickness. The material distribution within the beams follows a power-law relationship, indicating a continuous variation in the material distribution throughout the beams. Furthermore, the beam displacement is established using the higher-order shear deformation theory, and the governing equation is formulated based on the Lagrange equation. Various typical boundary conditions, including clamped-clamped, clamped-simply supported, clamped-free, and simply-supported, are considered. The proposed method is validated through numerical simulations, shedding light on the influence of boundary conditions, power-law exponents, and slenderness on the buckling, free vibration, and bending behaviours of functionally graded beams.