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A portion of a flawed thin-walled shell with an elliptical-arc external surface flaw located in a straight zone (pipe) or in a joint zone (elbow) is analysed to evaluate stress field and fatigue life. Such a portion is assumed to be a part of a shell of revolution, described by two principal curvature radii (R1 and R2). By employing the superposition principle and the power series expansion of the actual stresses, an approximated stress-intensity factor (SIF) expression can be determined for different actual loading conditions. In the present paper, the SIFs (weight functions) for five elementary stress distributions are determined through a FE analysis, by varying the relative curvature radius 1 2 r = R / R of the shell from 0 to infinity. Then, the SIFs for cylindrical and spherical shells under various loading conditions are computed through the above weight functions. Finally, a numerical simulation is carried out to predict the crack growth under cyclic internal pressure with constant amplitude. Some results are compared with those determined by other authors.