An accurate method for the analysis of remaining life for pressure vessels already in service would certainly help operation managers facing the question of when to replace mayor pieces of processing equipment. Operating companies would also realize the consequent savings derived from knowing in advance the probability of failure, or from safely extend the operating life of facilities. In this paper the first steps are undertaken towards a probabilistic analysis model for predicting the reliability of cracked pressure vessels as a function of applied operating loads using fracture mechanics. The analysis takes into account the variability in material properties, uncertainty in operating loads, continuing degradation of structural residual strength due to crack initiation and propagation and the combined effect of existing crack and newly initiating crack. The information gathered from inspections and non-destructive tests can be incorporated, together with the stresses caused by the operating loads, into the calculations. A numerical example is provided to demonstrate the methodology.


The development of cracks due to fatigue is one of the main causes of interruption of service for pressure vessels. Cracked pressure vessels would generally be out of service and receive appropriate repair. However, sometimes the vessels need to continue to service until the maintenance or repair condition becomes available. In addition, leaving the cracked pressure vessel in service, provided that it maintains acceptable reliability and safety level, has great economical benefit. Because of these reasons, analysis and assessment of the reliability and safety of cracked pressure vessels becomes an increasingly interesting topic for the petroleum and gas industries.

Traditional analysis for the reliability of cracked pressure vessels are based on classical fracture mechanics methods, such as the application of Paris-Erdogan formulation[1,2], and focus generally on single crack initiation and propagation using deterministic analysis. This deterministic technique does not take into consideration the variability and uncertainty of material properties and working loads that, as measurements show, demonstrate significant variation.

In this paper a new reliability analysis model is proposed for cracked pressure vessels based on probabilistic fracture mechanics technique. This new model takes into account an extended set of variables such as:

  • variability in material properties;

  • uncertainty in working loads;

  • continuing degradation of structural residual strength due to crack initiation and propagation and

  • the combined effect of existing and newly initiating cracks.

Compared to traditional analysis models based on deterministic analysis techniques, the probabilistic model it is expected to provide a more accurate prediction for the reliability and the safety of cracked pressure vessels. This increased accuracy will improve the decision making process and should translate into significant savings in inspection, repair and replacement expenses for cracked pressure vessels.

Reliability Analysis Model

The basic procedure for calculation of reliability of cracked pressure vessels in the proposed analysis model is as follows:

1) identify the probability of fracture failure at loading cycle N due to the crack initiating at loading cycle Nd (Nd < N);

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