G. Duvenhage: University of Pretoria, Pretoria, 0002 and Engineering (Mechanical) Division, Sasol Synthetic Fuels, P0 Box 600, Secunda 2302, South Africa


J. Wannenburg: University of Pretoria, Pretoria, 0002, and Laboratory for Advanced Engineering (Pty) Ltd, P0 Box 30536 Pretoria 0132, South Africa



First published in: Int.J.Fatigue Vol. 17, No. 7, pp477 – 483, 1995.



Keywords: Remaining life analysis, Pressure vessel, Fracture mechanics





Fracture mechanics has evolved into an engineering tool able to solve problems related to the safety of structures containing defects. In practice, however, a wide choice of parallel methods as well as unsolved discrepancies in the theory requires engineering judgment and pragmatism. In this paper a practical case study is presented, concerning a maintenance strategy assessment performed on a welded pressure vessel. The pressure vessel is analyzed in terms of leak-before-break arguments, proof test logic, fatigue initiation life, defect initiation and unstable growth critical sizes, fatigue crack propagation that leads to fracture or leakage, as well as probabilistic methods based on defect size distributions and NDE detection probabilities. The paper deals with some background to the theories employed, the different steps of the assessment methodology, as well as with the interpretation of the result to design a practical and safe maintenance strategy.





The continuous use of process equipment well beyond its intended lifespan and with the minimum downtime is of growing importance, owing to economic constraints and environmental concerns. It is the responsibility of fracture mechanics analysts to assess the risk involved with the operating of equipment and to propose effective fracture control measures.


In past years fracture mechanics and fatigue analysis techniques have evolved into efficient engineering tools, which can be used to assess fitness for purpose of structures. Various different techniques exist and new ones are proposed continually and need continued improvement to ensure it stays valid well into the future. In this paper these concepts are applied to a case study, demonstrating the uses and associated problems of fracture mechanics.


Table 1 summarizes the various fitness-for-purpose concepts and the related arguments that may have relevance to the typical pressure vessel case study. These concepts have all been evaluated in terms of the present case study. The logical methodology followed during the assessment is depicted in Figure 1. This methodology is also maintained for the ordering of this paper.




Analysis Required


Demonstrate that critical surface defects will not become critical through-thickness defects when fracturing leading to leaking will occur before catastrophic failure.

Fracture analysis for through-thickness defects.

Proof test logic

Demonstrate that the largest defect that can exist after a proof test will not propagate to critical size at work pressure before next proof test.

Fracture analysis at proof test and working pressures, as well as propagation analysis.

Fatigue life

Demonstrate that calculated fatigue life of structure is far in excess of required vessel life.

Fatigue (stress-life) analysis.

Critical defect sizes

Demonstrate that calculated critical defect sizes are larger than what can safely be detected with NDE techniques. To be used with scheduled inspections.

Fracture analysis. (initiation and stable growth)

Damage tolerance

Demonstrate the sufficient propagation life (from NDE-detectable size to critical size) exists such that a safe number of inspections can be scheduled during propagation life.

Fracture analysis and propagation analysis.

Probabilistic methods

Demonstrate that the probability of failure associated with any of the above concepts is acceptable.

Statistical analysis of defect sizes and detection probabilities.

Table 1: Fitness-for-purpose concepts and related arguments

Text Box:  
Figure 1: Assessment Methodology




The pressure vessel involved in this case study is utilized in a pressure swing process. The purpose of the process is to purify a contaminated hydrogen-rich feed stream for re-usage in a reactor. Contaminants are removed by means of an adsorbent placed in the vessel. Adsorption of contaminant occurs at high pressure and the adsorbent is then cleaned at a low pressure. The pressure cycle is between 110 kPa and 2290 kPa and is repeated every 15 minutes. The operating temperature is 320C.


The vessel was manufactured from a low-carbon pressure vessel steel and was fully post-weld heat treated.


The system has been operating for 12 years, and after recent failures of similar hydrogen-containing vessels it was decided to increase the inspection frequency. At the same time a project was initiated to perform a detailed integrity assess­ment based on state-of-the-art fracture mechanics principles, to arrive at a safe and optimal maintenance strategy.







Two types of stress need to be determined. Primary stresses, which are caused by actual loading conditions such as internal pressure and weight, are calculated by means of finite element analysis. Figure 2 shows a model of the vessel used in the finite element analysis. Secondary stresses refer to the residual stresses caused by the welding. Direct determination and measurement of these stresses are nearly impossible, and therefore literature was used to assess the levels of these stresses parallel and transverse to the weld.

Text Box:  
Figure 2: Finite Element Model of Pressure Vessel


Stress-strain properties


The material's stress-strain properties are used to generate a material-specific failure assessment diagram, whereas the yield and ultimate stresses are needed in the assessment of limit loads and residual stresses:


s y = 360 MPa

s u = 504 MPa


The stress-strain properties are represented by a Ramberg-Osgood relation of the following form:






Toughness properties


Text Box:  
Figure 3: Unloading Compliance toughness Test Record
Assessment documents require toughness to be expressed in terms of the plane strain value K Ic . Owing to the thickness limitations imposed it can be difficult to do a valid test, as the specimens become too large. This problem can be alleviated in two ways. One is to do the test on substandard specimens and to adapt the result for thicker specimens. Another method is to convert J data into KIc. J data are less stringent on the thickness limitations and smaller specimens could therefore be used. This method is recognized to give a good representation of the toughness data.

Text Box:  
Figure 4: J-R curve

Specimens were prepared so as to measure accurately the toughness properties of the weld material and the heat affected zone. The J - Da curve was measured by a compliance technique utilizing a clip gauge to measure crack mouth opening displacement. Figure 3 shows the load vs crack mouth opening displacement of the test. Crack extension is calculated from the slopes of the partial unloading lines. The value of these slopes is known as the compliance value. Data manipulation was done as outlined in ref. 6. Owing to the small specimen size, a linear representation of the J - Da curve was obtained, as shown in Figure 4. From the J - Da line the initiation toughness was obtained. These elastic-plastic toughness data were then converted to linear elastic data. The initiation toughness as well as the increase in toughness due to ductile tearing is shown in the following:




KIC = 233 MPa m½                                                                                                      (3)


JC(da) = 238.1 + 681.7(da-0.02)                                                                                 (4)




Crack propagation rates


The measurement of crack growth rates, especially where environmental conditions could influence the results, is heavily influenced by loading rate and test time. It is a known fact that a hydrogen environment would increase the crack growth rate substantially and should therefore be anticipated.


A modified Forman equation was chosen to represent the crack growth data. This allowed for the incorporation of a stress ratio in the analysis. Experimental data from literature were used in the calculation of constants.


The crack propagation rate is represented by:




where da/dN is measured in mm cycle -1 and DK in MPa m ½.


NDE results


Thorough NDE examination as part of statutory inspections has been carried out on the vessel over a number of years. Some 140 defects have been detected, with sizes of up to 120 mm length and 3 mm in depth. Unfortunately, only the maximum lengths and depths of defects found during each inspection were recorded.





Fatigue life analysis


A fatigue life analysis based on the stress life approach was followed to obtain an estimate of the expected life of the vessel. The method used is outlined in ref. 4 and is considered as an alternative to BS 5500 appendix C.


The fatigue curves, or S-N curves, are two deviations below the mean, representing a 97.7% probability of survival. Comparisons of these S-N curves and fatigue test data obtained from cyclic pressure tests on welded vessels indicate that they are conservative for pressure vessels, but not excessively so.


The minimum life obtained for any joint was 32.2 years. This did not include effects such as overloading, stress relief, environment and welding processes, and it could therefore be judged as a conservative estimate of the vessel life.


Critical flaw size analyses


Low-carbon pressure vessel steel as used in the manufacturing of this vessel behaves in a ductile manner. This would require an elastic-plastic analysis, which accounts for the plasticity associated with ductile fracture. A linear-elastic analysis in this case would be unconservative. Any fracture analysis is, however, just as good as the quality of the input data, and the non-availability of geometry factors for an elastic-plastic analysis restricts its use.


Text Box:  
Figure 5: Failure Assessment Diagram
This problem is overcome by the now well-known two criteria approach, in which failure is predicted as a function of brittle fracture and plastic collapse. This concept is graphically presented in a failure assessment diagram, as shown in Figure 5.


The two criteria are represented on the failure assessment diagram as a normalized crack driving force KR = K/KC on the vertical axis, with a brittle failure limit at KR = 1; and a normalized stress SR = s/sy on the horizontal axis, with a plastic collapse limit at SR = 1. Assessment points falling inside the diagram are considered safe.


In regions of high stress, near the yield limit, plasticity prevails, and the horizontal line (i.e. linear elastic fracture mechanics) is not valid. Models have been developed in which the fracture line in regions of high stresses has been modified to account for plasticity. The limit for yield failure has also been increased to allow for work-hardening in high-stress areas. With the use of the failure assessment diagram it is therefore possible and valid to conduct an elastic-plastic analysis without the elastic-plastic geometry factors and without being overly conservative.


Three recognized methods exist for the calculation of critical defect sizes: the CEGB R6 method, British Standard Institution PD-6493 document, and the EPRI elastic-plastic analysis methodology.


The R6 and PD-6493 methods use elastic geometry factors in the analysis. Plastic effects are incorporated by employing a failure assessment diagram. The EPRI method is a full elastic-plastic analysis using elastic-plastic geometry factors and material properties. The application of this method is limited owing to the unavailability of geometry factors.


Through-thickness flaws: An analysis was performed using CRACKWISETM to determine the critical through-thickness defect sizes. The analysis performed corresponds to a PD-6493 level 2 analysis. The minimum through-thickness flaw was calculated to be 410 mm. This length is approximately 12 times the average thickness of the vessel.


Surface flaws: An analysis was performed to calculate the critical surface flaw dimensions. The results are represented as a graph of critical crack depth vs length. Flaws on the inside and the outside of the vessel were considered.


The critical defect analysis was conducted based on the CEGB R6 methodology. A category 3 analysis employing a material-specific failure assessment curve (level 2 in R6) was used. The limit crack sizes for initiation and stable crack growth were calculated. A graphical representation of the analysis process is shown in Figure 6.


Figure 7 shows the combined results of tolerable flaw sizes for all the different positions and flaw orientations considered on the vessel, based on initiation toughness at working pressure.



Axial flaws in the circumferential weld were found to be the smallest allowable cracks. As it was considered impractical to impose a maintenance strategy that differentiates between different flaw locations or orientations, it was decided to use these calculated values as the minimum allowable for the rest of the vessel. Figure 8 shows the graph for critical depth vs length for initiation and stable growth analysis performed on axial cracks in the circumferential welds.


Crack propagation analysis


Various scenarios exist in which cracks can grow to produce cracks that could cause failure. The calculated critical flaw sizes showed that large cracks could exist without causing fracture failure. It was therefore decided to base the crack propagation analysis on cracks that could propagate to produce a leak. Leakage of hydrogen could cause a fire hazard and would therefore also constitute a failure. It would therefore be appropriate to determine the time that a known flaw will take to start leaking. A conservative approach would therefore be to analyse the propagation of a small circular flaw until breakthrough.


Crack propagation analyses are very dependent on the initial crack sizes. It was therefore decided to do a reverse calculation, in which the end size is known and the initial size is calculated as a function of the period in service, assuming that the defect would be semicircular throughout its life. The semicircular assumption would be conservative in the practical situation where the surface breaking length is measured and the depth is assumed to be half of this length. Figure 9 shows the crack length as a function of time to produce a leak.


These crack growth calculations should be conservative, as they do not account for the crack retardation that occurs after an overload, as in the case of a pressure test, and the worst case stress situations were used.




Probabilistic analysis


For the purpose of a probabilistic damage tolerance argument to be discussed later, it was necessary to obtain a mathematical description of the statistical distribution of defect sizes that can be expected in the vessel. It has been found in previous studies6 that a log-normal distribution may be used for this purpose:




with y = ln x being normally distributed (mean = m y , variance = sy2), x = defect size (either depth or length).


By definition:




Text Box:  
Figure 10: Log-Normal Distribution of Defect Length
The mean and variance of the natural logarithms of both the lengths and depths of defects found during inspections should be calculated. As these data were not available for the present case, a distribution function was used that was obtained from inspections done on similar welds on similar thickness material. The log-normal probability density function for defect lengths assumed to be valid for this case is shown in Figure 10.


The cumulative probability function is shown in Figure 11. From this graph it is possible to read off the total fraction of defects that will be smaller than a certain length. (e.g. 92% defects will be smaller than 40 mm in length)


Text Box:  
Figure 11: Probability Function for Defect Length




Results for various analyses have been presented. These results were subsequently applied to different possible fitness for purpose arguments.


Fatigue life


The results of the fatigue life analysis indicated that the vessel may potentially fail due to a fatigue mechanism before the intended end of the design life. This precluded the possibility of demonstrating fitness for purpose for the total vessel life by ensuring a defect-free vessel through a once-only inspection. Other fitness-for-purpose arguments are therefore required.


Proof test logic


Proof test logic implies the calculation of the maximum time between inspections, based on the time it would take for the largest crack that could exist after a proof test to propagate to a critical crack size at working pressure, causing failure of the vessel.

Text Box:  
Figure 12: Comparison of Fatigue Crack Growth Rates.

A leak at proof test would be considered as failure of the vessel; therefore the calculation is performed for surface flaws only.


The crack propagation analysis is done using geometry relations that address growth in the depth and length directions separately. A piecewise linear integration is used to assess the crack propagation.


The time to failure for surface flaws ranged from 17 days (1647 cycles) for shallow cracks to 12 days (1080 cycles) for deep cracks. Figure 12 shows the growth of two cracks from critical at proof test to critical at working pressure.


The results of this argument showed that a pressure test will not be valid for demonstrating fitness for purpose for the prolonged use of the vessel.




The results of the critical defect size analyses indicated that the vessel would leak before catastrophic failure (the critical size of a through-thickness defect was 410 mm, whereas a surface flaw 410 mm long would have to be 17 mm deep before breakthrough would occur) but, because the leaking of hydrogen from the vessel could not be permitted, leak-before-break could not be considered as a valid fitness-for-purpose concept.


Critical flaw sizes


The critical flaw size results indicated that it is safe to operate the vessel with defects of easily detectable sizes. This makes it viable to employ damage tolerance principles to demonstrate and maintain fitness for purpose for the vessel.


Damage tolerance


It was concluded that the failure mechanism to be guarded against would be the propagation of a small surface flaw until leakage occurs. Inspections should therefore be scheduled such that flaws that are large enough to propagate to leakage before the next inspection would be detected and repaired during each inspection Figure 9 provides the information necessary for this decision.


A deterministic argument in this respect would be based on a decision as to what size crack would be large enough not to be missed during inspection. It could be argued that a crack with a length of 25 mm would always be detected, and from Figure 9, inspections every 20 months would be prescribed.


With a probabilistic argument, it would be possible to calculate confidence levels for such a decision. From Figure 11 it may be seen that 80% of all defects will be smaller than 25 mm. This fact, together with an estimation of detection probabilities, would make it possible to calculate the probability that a crack that will grow to leakage before the next inspection will remain in the vessel after an inspection.


Owing to the lack of sufficient information, a detailed probabilistic analysis could not be performed in detail for the present case. However, from the literature and the estimated distribution of defect lengths assumed above, the following calculation may be performed.


The probability that, if a defect exists, it will grow to leakage before the next inspection and remain in the structure after inspection and repair, is calculated as


Pfd = P(ac < a) (1 - Pd)                                                                                                (9)


with Pd = probability that a defect will be detected by NDE.


The probability that the vessel contains a critical defect after inspection and repair is given by:


Pf = 1 - (1 - Pfd)N                                                                                                         (10)


with N = number of defects in the vessel before inspection.


P(ac < a) can be obtained from Figures 9 and 11.


For a length of 25 mm (corresponding to the time to the next inspection of 20 months from Figure 9) the probability for a larger crack to exist will be 20% (from Figure 11).


A typical value for Pd cited in the literature is 90%. Some 4-6 defects have been detected on average per inspection, which gives an indication of the value to be used for N.


Using these values, a probability of failure (leakage) could be estimated as 11%. Decreasing the time to inspection to 5 months (corresponding to a crack length of 35 mm from Figure 9) would change this to 6%.


These results seem to be unacceptable. However, there are two important factors that make these calculations overly conservative. First, the assumption that a detected flaw would be semicircular implies very conservative growth rates. To obtain more realistic results would, however, require knowledge of the two-dimensional size distributions of defects to be expected in the vessel, which is currently not available.


Second, it is a known fact that detection probabilities are a function of defect size (a larger defect would be more easily detected than a smaller defect), which has not been taken into account, again owing to the lack of information on which to base more realistic estimates.





This paper presented a practical case study to demonstrate the use of various fracture mechanics analysis techniques, arguments and methods, to develop a maintenance strategy for a welded pressure vessel.


Material toughness properties were determined using the unloading compliance technique, yielding J - Da curves. These tests are difficult to execute, and confusion reigns in the literature concerning the interpretation of the results, but it was demonstrated that the R6 interpretation of these results yields useful engineering values for toughness.


Critical defect size analyses were performed according to the R6 method and the PD-6493 method. The analyses demonstrated that realistic results could only be obtained by performing an elastic-plastic analysis, incorporating stable crack growth. Fatigue crack propagation analyses were performed, which demonstrated the importance of engineering pragmatism concerning the choice of initial crack size.


A leak-before-break situation was shown to exist for the vessel, but fitness for purpose could not be        proven using this result, as leaking may cause an explosion. Proof test logic was followed, which also could not demonstrate fitness for purpose.


It was found that the vessel may be maintained using damage tolerance principles. The limiting scenario that would govern the inspection schedules was found to be the propagation of surface cracks through-thickness to cause leakage. Inspections scheduled for every 20 months would require the detection and repair of all defects longer than 25 mm.


A probabilistic assessment of the risk of leakage, should the above schedules be maintained, yielded very conservative results. It would be possible to obtain more realistic results, should more complete inspection results be recorded in future.





1) ‘Guidance on methods for assessing the acceptability for flaws in fusion welded structures', PD 6493, British Standards Institution, 1991

2) Kumar, V., German, M.D. and Shih, C.F. 'An Engineering Approach for Elastic Plastic Fracture Analysis'. EPRI Internal Report NP 1931,1981, Palo Alto, California.

3) Mime, I., Ainsworth, R.A et al. Int. J. Pressure Vessels Piping 1988, 32, 105

4) Assessment of vessels subject to fatigue: alternative approach to method in appendix C, Enquiry Case 5500/79: May 1986 to BS 5500:1988, British Standards Institution.

5) Duvenhage, G.H. Fitness-for-purpose assessment of a pressure vessel subjected to cyclic loads based on fracture mechanics methodologies, Masters Degree Thesis, University of Pretoria, 1994.

6) Wannenburg, J.A. Fracture mechanics methodology for the integrity assessment of welded structures, Masters Degree Thesis, University of Pretoria, 1992.

7) Broek, D. 'The Practical Use of Fracture Mechanics', Kluwer Academic Press, Dordrecht 1989.