Dynamics of heavy beams: Closed-form vibrations of gravity-loaded Rayleigh–Timoshenko columns

Abstract

Abstract We present the first full analytical solution to the mode-shape equation of the gravity-loaded Rayleigh-Timoshenko beam, thus describing in closed form the vibration modes of beam–columns, bending cables, and stiff marine risers under self-weight. The modes are given in terms of a family of complex “Hankel-Airy” functions, along with their asymptotic expansions, orthogonality relations and series representations. It is shown that the new solutions generalize multiple approximate and asymptotic solutions in the literature and that their numerical approximations closely match past numerical and experimental results as well as models built using the ANSYS software package.