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A geometrically nonlinear Generalized Beam Theory is formulated and the results in the framework of buckling analyses are discussed. The geometrically nonlinear model is recovered reusing the model available in the linear context. This generalization to the nonlinear context is obtained exploiting the corotational based method called Implicit Corotational Method, starting from a mixed energy description of the continuum in terms of nonlinear Biot stress/strain tensors and using convenient assumptions for the linear stress/strain tensors. Once obtained, the nonlinear model has been implemented using a flexibility based finite element. The results of buckling analyses for different and complex beam cross-section geometries, emphasizing distortional and local buckling behaviors, are presented and comparisons with finite element shell models made.

The paper deals with geometrically nonlinear finite element analysis of folded-plate and shell structures. A Koiter asymptotic approach is proposed, based on the reuse of a linear element in the nonlinear context through a corotational formulation.The corotational approach represents a simple and effective way to satisfy the basic requirement of Koiter analysis, i.e. full objectivity in the finite element modeling. In fact, starting simply from a suitable linear finite element and implementing the corotational algebra proposed in Garcea et al. (2009), Zagari (2009) lead to objective explicit expressions for the first four variations of the strain energy which are needed by asymptotic analysis.The shell element used here is the flat shell quadrangular element with 4 nodes and 6 dofs per node proposed in Madeo et al. (2012) and called MISS-4: a mixed element, based on the Reissner–Mindlin plate theory, with an Allman-like quadratic interpolation for displacements and an equilibrated isostatic interpolation for the stress resultants. The element is free from locking and spurious zero-energy modes, so it appears a suitable candidate for nonlinear corotational analysis.The results of the numerical validation show the effectiveness and accuracy of the proposed approach, and its excellent overall robustness for both mono- and multi-modal buckling problems, also in the presence of strong nonlinear pre-critical behavior.

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.