Identify the unloading branch in theoretical post-buckling analysis under compression with pre-torsion

Determine which bifurcation branch of the post-buckling solutions obtained via the Donnell shell theory solved with the Galerkin method corresponds to the physical unloading equilibrium path of clamped–clamped thin cylindrical shells subjected to axial compression with a specified pre-torsion, where branches are indexed by the circumferential wavenumber N.

Background

The paper develops a Donnell shell theory framework solved by the Galerkin method to analyze buckling and post-buckling behavior of clamped–clamped cylindrical shells under combined torsional and axial loads, and validates predictions with finite element analyses (FEA) and experiments.

In the compression-with-pre-torsion case, the theoretical model produces multiple bifurcation branches after buckling that correspond to different circumferential wavenumbers N. While loading paths can be characterized, the authors explicitly state they cannot determine which theoretical branch corresponds to the physical unloading path, and therefore report unloading results only from FEA. Resolving this branch-selection ambiguity would complete the theoretical description of the hysteretic unloading behavior in this setting.