- A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain-displacement relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Krmn type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Krmn type for vibration amplitudes of about two times the shell thickness for the studied case. 2010 Elsevier Ltd. All rights reserved.