A discrete model for free vibration of 3D-FGM Timoshenko beams under various boundary conditions

Spatial variations in stiffness and mass density make the free-vibration behavior of tridirectionally functionally graded material (3D-FGM) beams highly sensitive to material gradation, while computationally efficient and accurate models for broad parametric studies remain limited. In this work, a discrete Timoshenko-beam formulation is developed for free-vibration analysis of 3D-FGM beams by representing the structure as an $$N$$ -DOF spring–mass system, where bending is modeled through rotational springs and transverse shear is captured through shear springs of stiffness $$K_i^{s}$$ . Using Lagrange’s formalism, the governing equations are derived in compact matrix form and reduced to a generalized eigenvalue problem to compute natural frequencies and mode shapes. The proposed model is validated against benchmark results for homogeneous Timoshenko beams, transversely graded FGM beams, and porous bidirectional functionally graded (2D-FGM) beams, showing good agreement in benchmark cases; for $$N=1000$$ , the maximum absolute percentage error is $$1.64\%$$ for the porous 2D-FGM Type 1 case, and the runtime is about $$1.5$$ s in Matlab. The parametric study shows that vibration response is influenced by slenderness ratio, gradation indices, and boundary conditions: increasing slenderness generally increases frequency parameters, gradation indices modify the stiffness–inertia balance and modal characteristics, and clamped configurations yield higher frequencies than less constrained supports. These results indicate that the proposed discrete framework provides an efficient tool for free-vibration analysis of 3D-FGM beam structures while accounting for shear deformation.