A new finite element formulation for free vibrations of planar curved beams

This study aims to derive a new finite element formulation for in-plane free vibrations of curved beams with arbitrary curvature, and cross-section variation. The stiffness matrix presented in this study are obtained from the exact solution of the static problem, considering the effects of axial extension, shear deformation. Using the exact solution for point loads, a consistent mass matrix is obtained, considering the effect of rotatory inertia. Numerous examples, related to the free vibrations of planar curved beams are solved to validate the presented approach. It is proved that presented formulation does not suffer from any locking phenomena. Circular beams with varying cross-section are investigated by assembling uniform elements. Parabolic, elliptic, and sinusoidal beams are examined by both using variable curvature elements, and assembling circular beam elements. This new formulation is thought to be an effective tool in structural analysis of curved beams.