On the relationship of the shear deformable Generalized Beam Theory with classical and non-classical theories

The possibility to establish clear relationships between the results of the Generalized Beam Theory (GBT) and those of the classical beam theories is a crucial issue for a correct theoretical positioning of the GBT within the other existing beam theories as well as for the application of the GBT in the current engineering practice. With this in mind, the recovery of classical and non-classical beam theories within the framework of the GBT is presented in this paper. To this purpose, a new formulation of the GBT with shear deformation is conceived. Particularly, the formulation recently proposed by the authors is here modified by introducing new definitions of the kinematic parameters and of the generalized deformations, and extended to the dynamic case. Firstly, it is shown that a suitable choice of the flexural deformation modes allows recovering the Vlasov beam theory, both with and without shear deformation. Also, the analytical solution of the non-uniform torsion problem with shear deformation is given. Then, the recovery of the Capurso beam theory using the nonlinear warping deformation modes is illustrated.