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This paper is concerned with the integrity assessment of defective pressurized pipelines and pressure vessels by means of diverse methods comparatively, for two kinds of failure processes: (I) plastic collapse under internal pressure monotonically increasing in time; (II) incremental collapse (or more generally lack of shakedown, “inadaptation”) under fluctuating internal pressure. The defects considered are part-through slots of various geometrical configurations. The analysis methods employed and comparatively discussed are as follows: for situation (I), empirical formulae of current industrial use, rigid-plastic limit analysis, elastic-plastic time-stepping
computations by a commercial nonlinear code; for situation (II), a direct method based on classical shakedown theory and evolutive computations like for (I). The main assumptions are Mises perfect elastoplasticity and small deformations. As the main conclusion, the kinematic direct methods of limit and shakedown analysis in a finite element setting turn out to represent a cost-effective and reliable tool for integrity assessment in the present context.