A geometrically nonlinear shell theory for thin-walled tubes and beams subjected to large displacements and cross-section deformation

Thin-walled tubular shells, encompassing thin-walled beams and tubes, are characterized by their slenderness, making them susceptible to significant geometric nonlinearities under load. Accurately predicting their structural behaviour requires advanced analytical frameworks. This paper presents a geometrically nonlinear shell theory specifically developed for thin-walled tubular shells with both open and closed sections, capable of capturing their nonlinear response when subjected to large displacements and rotations, and small strains. The proposed formulation is based on intrinsic geometry and a moving frames approach, deriving the shell model from the 3D continuum to describe deformations. In this way deformations are described in their most general form, valid for large displacements and rotations. By incorporating cross-sectional deformations — including in-plane and out-of-plane warping — the model bridges traditional distinctions between beam and shell theories, enabling the analysis of intermediate structural behavior. Emphasizing both mathematical rigor and engineering applicability, this paper validates the proposed model through numerical simulations and comparative finite element analyses. Two case studies are analysed: a circular tube undergoing bending ovalization, being a classical benchmark, and an open-section tube having a U-shaped profile that experiences significant in-plane warping. These examples demonstrate the model's effectiveness to accurately capture large cross-sectional deformations in both open and closed thin-walled structures.