Exact analytical solution to the 3D Navier-Lame equation for a curved beam of constant curvature subject to arbitrary dynamic loading

Abstract

This paper presents an exact analytical solution to the 3D transient dynamics of a linear elastic, isotropic homogeneous, curved beam, with uniform rectangular cross-section. The solution technique uses vector identities to decouple the governing equations. The decoupled equations are then solved by method of separation of variables. The solutions to the decoupled equations can be recombined to form a new equation. Solving this new equation yields the displacement field. To demonstrate the capabilities of the proposed solution technique, a generic case study was modeled and computed. A curved beam is subjected to a longitudinal impulse loading and the transient displacement field is calculated. This solution technique is valid for any curvature so long as the curvature is the same throughout the beam. This includes the limiting case where the inner radius of the beam goes to zero and the curved beam becomes a section of a disk. The presented solution results were compared with an approximate solution from the literature and experimental data from the literature. The solution is also compared with an explicit FEM solution conducted by the Authors. The presented solution agrees with the results from the literature and from the FEM solutions conducted by the Authors. This paper demonstrates that the accuracy and robustness of the proposed solution technique meets the needs of many potential applications.