Non-linear normal modes (NNMs) are used for deriving in a systematic manner the type of non-linearity of circular plates and shallow spherical shells with a free edge. A special attention is paid to geometric imperfections, that are unavoidable in real systems. It is quantitatively shown, for a number of different axisymmetric and asymmetric imperfections, how the hardening type non-linearity, that is typical of flat plate displaying only cubic non-linearity, is turned to a softening type behaviour for an imperfection amplitude being a fraction of the plate thickness. The role of 2:1 internal resonance in this process is underlined. When damping is included in the calculation, it is found that the softening behaviour is generally favoured, but its effect remains limited.