A theoretical analysis, based on plastic shell theory and utilizing extremum and variational principles combined with numerical approaches, has been conducted to investigate the quasi-static flexural behaviour of mild steel circular pipes. Attention is focused on the hardening-softening relations between the bending moment and the curvature of pipes which, according to observations in pipe-whip experiments, governs the deformation process and can lead to localization. As the study is concerned with free pipe-whip behaviour of the high pressure piping system, in the theoretical analysis the circular pipe is assumed to be subjected to a bending moment as well as a relatively small tension force in the axial direction. In addition to numerical examples for various cases, a quasi-static four-point bending test was carried out to verify the hardening-softening moment vs curvature relation of pipes predicted by the present theory in the case of pure bending. A simpler, approximate procedure which predicts the moment-curvature relationship with reasonable accuracy is also described.