This paper provides approximate J estimates for off-centred, circumferential through-wall cracks in cylinders under bending and under combined tension and bending. The proposed method is based on the reference stress approach, where the dependence of elastic and plastic influence functions of J on the cylinder/crack geometry, the off-centred angle and strain hardening is minimised through the use of a proper normalising load. Based on published limited FE results for off-centred, circumferential through-wall cracks under bending, such normalising load is found, based on which the reference stress based J estimates are proposed for more general cases, such as for a different cylinder geometry and for combined loading. Comparison of the estimated J with extensive FE J results shows overall good agreements for different crack/cylinder geometries and for combined tension and bending, which provides sufficient confidence in the use of the proposed method to fracture mechanics analyses of off-centred circumferential cracks. Furthermore, the proposed method is simple to use, giving significant merits in practice.Copyright © 2003 by ASME