Wave Motion in Elastic Structures - Timoshenko Theory

The need to ensure the high performance of mechanical structures under extreme operating conditions has motivated extensive research in the field of Structural Health Monitoring (SHM). SHM aims to detect damage at early stages in order to prevent structural failures and is widely applied to high-risk engineering systems such as civil infrastructure, aeronautical and offshore structures, pipelines in the petrochemical industry, and amusement vehicles such as roller coasters. Wave propagation analysis, particularly at high frequencies, is an important approach for monitoring these structural systems. In this context, the present work investigates wave modes in a beam through the analysis of wavenumbers and dispersion curves. While the Euler–Bernoulli beam theory assumes that cross-sections remain plane and perpendicular to the neutral axis, neglecting rotary inertia and shear deformation, the Timoshenko beam theory provides a more accurate description at higher frequencies or shorter wavelengths by accounting for these effects. The main objective of this work is to derive the governing equations of the Timoshenko beam theory, including the solution for the wavenumber from the dispersion equation. The results present the dispersion curves and a schematic representation of the wave modes in a Timoshenko beam.