Benchmarks for higher-order modes evaluation in the free vibration response of open thin-walled beams due to the cross-sectional deformations

Highly flexible thin-walled beams with complex open cross-sections are sensitive to torsional and warping effects. The analysis of higher-order vibration modes in these structures needs more accurate and precise methods in order to achieve reliable results and detect the cross-sectional deformations in the structures’ free vibration response. This paper analyzes higher vibration modes in a series of thin-walled beams, which were proposed by Chen as benchmark problems. These are all open-section thin-walled beams with complex geometries. Global vibration modes, such as bending and torsion, related to the rigid cross-sectional deformations can be detected via classical and shear refined theories. However, cross-sectional deformations appear at higher frequencies, and these modes are mixed with the global ones. To highlight this fact, this paper compares classical beam theories with refined ones based on the Carrera Unified Formulation (CUF) and the shell results using the commercial finite element (FE) software and the data available from the literature. The CUF FEs based on the power of cross-sectional deformation coordinates (x, z) and those based on the Lagrangian polynomials are implemented and compared using Modal Assurance Criterion. A number of interesting conclusions are drawn about the effectiveness of classical and CUF-based results. The need for models capable of detecting cross-sectional deformations is outlined. In fact, many modes are lost by classical beam theories; on the other hand, they show rigid cross-section modes that do not really exist. This fact is also confirmed by the shell models, which are more expensive in terms of computational costs regarding the efficient CUF ones proposed here.