Abstract
The aim of this study is to develop and verify a variationally consistent finite element formulation for the bending and free vibration analysis of functionally graded (FG) beams resting on a physically derived two-parameter elastic foundation.The specific objective is to extend the previously developed analytical foundation model and simplified finite element formulation for homogeneous beams to metal-ceramic FG beams while preserving the mechanically based determination of the effective foundation parameters.To ensure the internal consistency of the model, two refinements are introduced.First, the bending stiffness is determined by accounting for the full distribution of Young's modulus across the beam thickness relative to the actual neutral axis, rather than using a