Numerical free vibration analysis of homogeneous or composite beam using a refined beam theory built on Saint Venant’s solution

Abstract

Free vibration problem of an arbitrary cross-sectional homogeneous or composite beam is investigated using a refined 1D beam theory (RBT). This theory includes a set of 3D displacement modes of the cross-section (CS) which reflects its mechanical behavior: the main part of these sectional modes is extracted from the 3D Saint Venant’s solution and another part is related to the CS dynamic behavior. These sectional modes, which are first derived from a CS analysis, lead to a consistent 1D beam model which really fits the section nature (shape and materials), and hence the beam problem.The numerical strategy to apply such general approach, is based on a first set of CS problems solved by 2D-FEM computations to get the sectional modes, and then the dynamic beam problem is solved by 1D-FEM computation according to RBT displacement model to provide (in fine) the first natural frequencies and 3D vibration mode shapes of the beam. To do so and in order to easily apply such method, a user friendly Matlab numerical tool named CSB (Cross-Section and Beam analysis) has been developed.To illustrate the capabilities and the accuracy of the method to catch the main 3D-effects, such as elastic/inertial coupling effects and 3D local/global mode shapes, a significant set of beam cross-section configurations with isotropic and anisotropic materials are analyzed. The first ten natural frequencies and 3D mode shapes are systematically compared to those obtained by full 3D-FEM computations, and some of them to literature.