Evaluation of natural frequencies with two-node finite element in Timoshenko beam theory using quintic transverse displacement interpolation

Accurate determination of natural frequencies is essential for ensuring the dynamic stability of structures across a wide range of fields, including structural, aeronautical, and mechanical engineering. Resonance, which occurs when external dynamic forces match a structure’s natural frequency, can cause significant damage or failure. The finite element method is widely used for such analyses owing to its matrix-based formulation and suitability for computational implementation. This study presents a finite element framework for computing translational, rotational, and curvature-based natural frequencies of a prismatic beam, using a quintic interpolation function for transverse displacement based on Timoshenko beam theory. While linear interpolation is adequate for axial displacements, for transverse displacements, due to their sinusoidal behavior under loading, require higher-degree interpolation for accurate modeling. A quintic polynomial introduces six degrees of freedom per element in any local transverse axis direction, compatible with two-node configuration. This approach facilitates consistent assembly of global matrices and the formulation allows flexible mesh configurations and supports efficient matrix-based numerical implementation.