Torsional warping eigenmodes of FGM beams with longitudinally varying material properties

In this paper, the influence of torsional warping of thin-walled cross-sections of twisted Functionally Graded Material (FGM) beams with a longitudinal polynomial variation of the material properties on their eigenvibrations is investigated, considering the secondary deformations due to the angle of twist. The transfer relations needed for the transfer matrix method are derived. Based on them, the local finite element equations of the twisted FGM beam are established. The warping part of the first derivative of the twist angle, caused by the bimoment, is considered as an additional degree of freedom at the beam nodes. The focus of the numerical investigation, with and without consideration of the Deformation due to the Secondary Torsional Moment (STMDE), is on modal analysis of straight cantilever FGM beams with doubly symmetric open and closed cross-sections. The influence of the longitudinal variation of the material properties and the secondary torsion moment on the eigenfrequencies is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid and warping beam finite elements.