Many thin cylindrical shells are used in structural applications in which the dominant loading condition is global bending. Key examples include chimneys, tubular piles, wind generation towers and tall silos. Their thickness lies in a tricky range which is extremely thin for the structural tube community and very thick for the shell buckling community. The buckling strength of these structures is dominated by extensive plasticity, but the fully plastic state is usually far from being attained. This paper explores the buckling strength of imperfect thin cylindrical shells under global bending in the elastic-plastic range. The capacity curves of the new Eurocode EN 1993-1-6 (2007) are used to match the final results. The results show that the capacity curves can capture this buckling behavior accurately and safely for different types of material models. It is assumed that the shell is held circular by rings or boundaries at reasonable intervals, effectively restraining ovalisation. It is hoped that these results will make a useful contribution towards resolving the misunderstandings and controversy that has been evident in this field in recent years.