  The structural analysis of ultra-lightweight flexible shells and membranes
may require the adoption of complex nonlinear strain-displacement relations.
These may be approximated and simplified in some circumstances, e.g., in the
case of moderately large displacements and rotations, in some others may be
not. In this paper, the effectiveness of various geometrically nonlinear strain
approximations such as the von Karman strains is investigated by making use of
refined shell formulations based on the Carrera Unified Formulation (CUF).
Furthermore, geometrical nonlinear equations are written in a total Lagrangian
framework and solved with an opportune Newton-Raphson method. Test cases
include the study of shells subjected to pinched loadings, combined flexure and
compression, and post-buckling including snap-through problems. It is
demonstrated that full geometrically nonlinear analysis accounting for full
Green-Lagrange strains shall be performed whenever displacements are higher
than the order of magnitude of the thickness and if compressive loads are
applied.
