A round pipe with an external longitudinal surface flaw is assumed to be subjected to different types of elementary hoop stress distributions acting perpendicular to the crack faces. The stress intensity factors (SIFs) along the crack front for such elementary cases are determined through the three-dimensional finite element method. Then these results are used to compute approximate values of SIFs in the case of complex loadings (such as internal pressure, thermal stresses, residual stresses due to plasticization, etc.), by employing both the superposition principle and the power series expansions of the actual hoop stresses. Thick and thin-walled pipes are examined, and some results are compared with those presented by other authors.