The present article deals with the free vibration analysis of three-dimensional metallic and functionally graded beams with arbitrary boundary conditions. The investigation is carried out by using refined variable-kinematics quasi-3D beam theories hierarchically generated by using the method of power series expansion of displacement components. Each displacement variable, in the displacement field, can be expanded at any desired order independently from the others and regarding to the results accuracy and the computational cost. The weak-form of the governing equations is derived via the Principle of the Virtual Displacements (PVD), while the Ritz method is used as solution technique. Algebraic Ritz functions, orthogonalised by using the Gram–Schmidt process, are employed in the analysis. Convergence and accuracy of the proposed formulation have been thoroughly examined. A comprehensive assessment of the developed beam models, for various boundary conditions, is also provided. The effect of significant parameters such as length-to-thickness ratio (slenderness ratio), volume fraction index and material properties, on the natural frequencies and mode shapes, is discussed
