An analytical approach for the cross-sectional analysis of generalised beam theory

This paper presents an analytical approach for the cross-sectional analysis carried out within the framework of the generalised beam theory. With this method, the in-plane deformations of the conventional modes are determined as the dynamic modes of an unconstrained planar frame, from which the corresponding warping distributions are post-processed. The shape functions adopted for the transverse displacements of the stiffness elements used for the dynamic analysis are derived based on the analytical solutions of the beam problem. In the case of the axial displacements, two sets of shape functions are considered, which correspond to a linear polynomial representation as well as to the analytical dynamic solution. The differences between these two approximations are discussed in relation to the condition of inextensibility applied within the generalised beam theory to each plate segment forming the cross-section. With the proposed procedure, the conventional modes, which include the distortional, shear and local modes, do not need to be calculated separately, as carried out with the classical generalised beam theory, and the approach is applicable to open, partially closed and closed sections. Finally, the accuracy of the proposed approach is outlined by means of two numerical case studies.