THE USE OF PROBABILITY THEORY IN FRACTURE MECHANICS – A CASE STUDY
J. Wannenburg: University of Pretoria, Pretoria 0002, South Africa; and Laboratory for Advanced Engineering (Pty) Ltd, P0 Box 30536, Sunnyside, Pretoria 0132, South Africa
G.C. Klintworth: University of Pretoria, Pretoria 0002, South Africa; and Mecaic (Pty) Ltd, P0 Box 7958, Hernopsemmer 0046, South Africa
A. D. Raath: University of Pretoria, Pretoria 0002, South Africa; and Laboratory for Advanced Engineering (Pty) Ltd, P0 Box 30536, Sunnyside, Pretoria 0132, South Africa
Keywords: Probability theory, Fracture mechanics, Pipeline, Probability of failure, Case study, Statistics, Non-Destructive Inspection
A probabilistic approach to fracture mechanics in the form of a typical case study is described wherein the integrity of a high-pressure, water pipeline is assessed. An analysis methodology is discussed incorporating the probability density functions of defect sizes, the statistics of defect occurrences and the statistical distribution of material properties. This method enables the analyst to supply a very simple assessment of safety, based on the probability of failure, (a single number) which may be compared to accepted industrial standards. (e.g. 10-6 for nuclear applications) It is argued that this method often offers the only way to scientifically and economically assess the integrity of fracture prone structures.
Failure of a stressed member may occur when an inherent crack propagates in an unstable manner to cause separation. Such failure is termed fracture failure and has become increasingly important in modern engineering due to the use of stronger materials in optimal designs. This is especially true for welded constructions where the use of stronger materials implies higher operating stresses, which, together with the fact that the occurrence of welding defects is usually unavoidable, causes such designs to be prone to fracture failure.
The task of a fracture mechanics analyst is to provide information (normally a safety assessment) on which a fracture control plan for a particular component or structure may be based. In the design phase the fracture control plan involves materials selection, choice of operating stress levels, manufacturing method and planned inspection and repair schedules. This responsibility is however often ignored in the design phase and, when the problem becomes apparent during operation, any viable fracture control measure becomes extremely costly.
In engineering, a safety assessment is always required to be both conservative as well as sufficiently accurate. This aspect is illustrated in Fig. 1. A simplified analysis (to the left on the horizontal axis) would imply that conservative assumptions are made wherever uncertainties exist, therefore yielding a conservative assessment of safety (low on the vertical axis). If the level of safety thus obtained would imply costly fracture control measures such as replacement or repair, the option remains to perform a more detailed analysis which could lead to less conservative results. Performing a more detailed analysis could therefore be seen as an option for fracture control although the true safety level is not altered.
It is obviously not known whether a more detailed analysis would in fact result in an appreciably higher estimated level of safety since the true safety level is unknown. A simplified analysis should however always result in a more conservative assessment than a more detailed analysis. If this is not the case then non-conservative assumptions had been made, implying an invalid assessment.
It is well accepted that certain parameters involved in a fracture mechanics analysis are probabilistically distributed variables. Material properties always exhibit scatter, defect occurrences and sizes are statistically variable and loading may also be random or pseudo-random. When required to supply a deterministic assessment of safety, the analyst is forced, according to the philosophy described above, to assume the lowest values of various variables.
This often leads to very conservative results. A deterministic analysis of any welded structure could, in fact, often conclude that the structure is unsafe since there may often be a possibility that a welding defect exists which is larger than the critical size. This fact is usually ignored and ambiguous assumptions are made on a ‘gut feel’ basis to prove or disprove integrity.
A probabilistic approach, in the opinion of the present author, therefore often offers the only way with which to incorporate sufficient detail into an analysis to be able to furnish a realistic assessment of safety. The purpose of such an assessment is then to supply a single number, namely, the “probability of failure”, to decision-makers. Even if this number turns out to be very small, it should not be the responsibility of a fracture mechanics analyst to decide whether it is negligible or not. Such a decision should be made on a management level where a simple calculation, namely, the probability of failure multiplied by the consequential cost of failure, should indicate how money should be spent on further fracture control measures.
Probabilistic fracture mechanics has mainly been used in the nuclear energy field where it has been recognized that it is unacceptable to base a safety assessment on unscientific assumptions. In the present paper, a case study is presented to illustrate the use of probabilistic fracture mechanics in a non-nuclear application.
Assume a high-pressure water pipeline with a diameter of 6 m and a wall-thickness of 30mm. The pipe is made up from cans which are welded together with circumferential manual arc butt welds on site. Each can is made up from four rings, each ring consisting of three rolled plates welded together with axial machine welds. The rings are welded together with circumferential machine welds. The configuration is shown in Fig. 2. Note that the rings are rotated so that the axial welds do not fall in line.
The pipe is manufactured from a quenched and tempered steel with a yield strength of 700MPa. The design was based on a simple yielding criterion without consideration to fracture mechanics. The concern regarding the integrity of the pipeline developed when welding defects were found during inspection of the manual arc welds joining the prefabricated cans of the pipeline. These defects developed due to hydrogen cracking, which occurred due to insufficient preheating in a damp environment on site, together with the presence of welding induced residual stresses.
The pipeline is mainly stressed by internal water pressure caused by a static head. Superimposed on the static head pressure are pressure fluctuations caused by operational mode changes. Fatigue crack propagation therefore had to be considered as a potential complicating aspect. In addition to this, the welds were not stress relieved and welding residual stresses also have to be considered during the analyses.
The consequence of failure of the pipeline would be flooding of costly plant as well as possible loss of life. The logic followed during the integrity assessment of the pipeline is diagrammatically depicted in Fig. 3.
NON DESTRUCTIVE INSPECTION (NDI) RESULTS
As shown in Fig. 3, the first step was to conduct an extensive NDI exercise, involving magnetic particle as well as ultrasonic inspection techniques. This was done after a preliminary analysis had indicated that welding cracks would indeed have an influence in decreasing the integrity of the pipeline. The aim of the NDI exercise was to determine the extent of occurrences as well as the sizes of the welding defects. Without this information, no analyses would have been possible. In excess of 1000 m of weld were inspected.
The inspection exposed more than 500 defects. All were situated in the circumferential manual arc welds. A little less than half of these defects were found to be orientated in the axial direction, as shown in Fig. 4. The defects were all surface breaking or near surface breaking.
An analysis was conducted according to the procedures stipulated in the British Standards Institution document PD 6493,1 in order to determine tolerable defect sizes. The results of this analysis are shown in Fig. 5, depicting a tolerable defect depth versus length graph, any defect lying below this curve being safe. Based on these results, as well as the fact that probable fatigue crack growth of the defects was not quantified at that stage, it was decided to repair all defects found during the inspection. The repair exercise represented the first excursion down the assessment logic flow diagram depicted in Fig. 3.
The defects were sized during the inspection and repair exercise according to their length and depth dimensions. It is proposed6 that a lognormal probability density function could be fitted to defect size results, or
with x = defect size related parameter (x = length/l0 and x = depth/2); = variance; and = standard deviation.
The lognormal distribution function achieved a remarkably good fit to the size data, as can be observed in Fig. 6. The defect length and depth dimensions are shown in the upper and lower graphs respectively. The ‘after fatigue growth’ curves are discussed later.
As shown in Fig. 3, the next step was to assess the post-repair integrity of the pipeline. The PD 6493 method used to establish tolerable defect sizes for the repair program is inherently very conservative due to reasons beyond the scope of this paper. It was therefore decided to perform a more detailed analysis to obtain the best estimate of critical defect sizes. A J-integral based analysis method, proposed by the British Central Electricity Generation Board (CEBG) in their document known as R6,2 was subsequently employed.
This necessitated a material testing exercise to determine the material toughness properties relevant to the R6 method. The method requires the initiation toughness in terms of J (=JIC) or K (= KIC), as well as the crack growth resistance or Jr -curve, to be quantified (commonly used fracture mechanics notation has been used). Fracture tests on three-point bend specimens were performed according to the procedures proposed in R6. These tests were performed on welded as well as base metal specimens.
The unloading compliance technique7 was used to construct the J - Da curves. According to this technique, the stable crack extension is measured by partially unloading the specimen at various clipgauge displacement levels. The slopes of the unloading curves are then a measure of the change in compliance of the specimen due to stable crack growth. A typical test record is shown in Fig. 7.
As mentioned earlier, fracture toughness data will always exhibit scatter. In order to quantify the distribution of fracture toughness data, a sufficient number of specimens need to be tested to represent an adequate statistical sample. Initiation fracture toughness data may be described using a Weibull probability density function.3
In the present case, a limited number of specimens were available. A conservative engineering approach therefore had to be assumed by choosing the lowerbound toughness values obtained in the tests. These results are listed below:
Weld metal: J0.2 = 46 KJ m2 dJ/da = 70 MPa
Base metal: J0.2 = 119 KJ m2 dJ/da = 216 MPa
Initiation is assumed to commence after 0.2% apparent crack extension.
The Jr - curve is approximated2 in R6, by:
CRITICAL DEFECT SIZE ANALYSIS
The CEGB R6 method is a descendant of the Two Criteria method. These criteria are LEFM and plastic collapse and are represented on a Failure Assessment Diagram (FAD) as a normalized crack driving force, Kr = K/KIC on the vertical axis with a brittle fracture limit at Kr = 1(K = KIC) and a normalized stress on the horizontal axis with a plastic collapse failure limit at Lr = 1 (s = syield). Assessment points falling inside the diagram (shown in Fig. 8) are considered safe. In R6 this concept is refined by using EPFM to interpolate between the two extremes, as shown by the dashed line in Fig. 8. This curve then represents the R6 assessment line.
The philosophy behind this assessment line and how the R6 method relates to the J-integral approach in elastic-plastic fracture mechanics is described in the following paragraphs.
In the linear elastic regime, the stress intensity factor, K, is used as a measure for the crack driving force and the failure criterion is simply:
K ³ KIC (3)
K is a function of geometry, stress and crack size, or
with Y = geometry correction factor; s = applied stress; and a = crack size.
Equation (3) may also be written in terms of fracture energy parameters, or:
G > GIC (5)
The expression for G may be written in more general terms (using ):
It has been shown4 that eqn (6) is also valid for non-linear elastic material behaviour (¹ s/E) In the non-linear case, the geometry factor Y would be different to Y in the linear case and is therefore given a new name:
It has also been established4 that Gnon-linear is equivalent to the value of the well-known J-integral, or
The failure criterion then becomes:
J ³ JIC (9)
Equation (9) may then be manipulated by dividing with G (linear), inverting and taking the square root, or
As was mentioned earlier, the vertical axis of the R6 FAD represents a normalized crack driving force:
and from eqns (5a) and (9a) it is clear that
Evaluation of the left-hand side of eqn (10) therefore only requires the linear elastic crack driving force (K) and the fracture toughness (KIC) to be known. The right-hand side of eqn. (10) is denoted Krf in R6, therefore establishing the R6 failure criterion:
Kr ³ Krf (13)
Krf as a function of a normalized applied stress parameter (Lr = s/syield) then represents the R6 failure assessment line. Evaluation of Krf requires the elastic-plastic crack driving force (J) to be known. Using eqns. (5a) and (8) Krf may then be written as:
Krf therefore a function of geometry factors, the material stress-strain relationship ( = f(s)), as well as the applied stress. Finite element analysis results for different geometries and different materials have shown8 that Krf is only weakly geometry dependent and that a lower bound universal failure assessment line could therefore be established which would be applicable to materials not exhibiting a high initial work-hardening rate:
Krf = (1 – 0.14 Lr2)(0.3 + 0.7 exp(-0.65 Lr6)) (15)
with Lr = applied stress/collapse stress; and Krf = critical value for Kr.
Residual stresses are accounted for by adding the crack driving force due to residual stresses (Ks) to the crack driving force due to externally applied stresses (Kp). Residual stresses are assumed not to contribute towards plastic collapse and therefore have no influence on the Lr coordinate. A plasticity correction factor (r) is then added to the Kr coordinate to take into account plasticity at the crack tip caused by residual stresses:
Kr = (Ks + Kp) / KIC + r (16)
Having broadly described the principles behind R6, it can be applied to the present case study.
The maximum externally applied stress in the pipeline is caused by a maximum internal pressure peak of 3.8 MPa:
shoop = Pr/t (17)
= 3.8 x 3/0.03
= 380 MPa
A simplified version of the well known Newman & Raju solution for elliptical surface cracks subjected to uniform tensile loading9 was used to calculate Kp:
with a = defect depth; c = defect length/2; and t = wall-thickness.
A parabolic through-thickness residual stress profile, as shown in Fig. 9, was assumed. A weight function technique10 was used to calculate Ks.
An initiation analysis was performed to determine the critical crack dimensions at which initiation will take place. The results of this analysis are shown in Fig. 10 for an axial defect in weld metal, giving the relationship between defect depth and length for the conditions of initiation and stable growth instability (discussed later) respectively.
The influence of the residual stress profile on the initiation condition can clearly be seen.
A stable crack growth analysis was subsequently performed to determine the critical defect dimensions at which instability would occur. A locus of assessment points was obtained for different initial crack depths and postulated crack growth increments. Instability would occur when a locus falls in its entirety outside the FAD and touches at one point only, or
Kr(a + Da) = Krf (19)
This procedure is illustrated in Fig. 11 which depicts a portion of the FAD shown in Fig. 8. The assessment line in Fig. 11 is described by eqn (15). The crack depth is incrementally increased until the initiation condition is reached (ai in Fig. 11). A crack growth increment is then added to this initial depth and a new assessment coordinate is calculated. Due to the rising fracture toughness, the curve drops below the assessment line, implying arrest. The initial depth is then incrementally increased and the procedure repeated. The critical initial crack depth (acr) is found when the locus of stable crack growth assessment coordinates does not intersect the assessment line.
The result of the stable crack growth analysis is shown in Fig. 10 for an axial defect in weld metal.
The critical crack size analyses discussed above were based on the most severe loading conditions expected to occur during the life of the pipeline. The fatigue crack growth of defects due to variable loading during the operational life were however not taken into account.
A fatigue crack growth analysis was subsequently performed. Material tests were performed to determine the fatigue crack growth properties of the weld metal based on the Paris crack growth rate expression:
with DK in MPaÖm and da/dN in m/cycle.
It was assumed that the growth rate would double in water:
A fatigue crack growth analysis was performed for each defect found during the inspection exercise to determine estimated end-of-life size probability density functions for both length and depth. The analyses were performed by calculating the incremental crack growth in both the length and depth directions for each sequential cycle and then updating the aspect ratio before continuing to the next cycle. The sequence of cycles and stress ranges for each cycle expected during the life of the pipeline were accurately simulated.
Typical crack growth curves for defect depth and length are shown in Fig. 12. The end-of-life PDFs for length and depths are shown in Fig. 6.
Figure 12: Fatigue crack growth curves. (a) Length; (b) depth.
It became obvious that the integrity of the pipeline could not be deterministically demonstrated since there was always a possibility that a critical defect had been missed during the inspection and repair exercise. It was therefore necessary to quantify this risk.
The greatest risk involved axially oriented defects which are situated in circumferential welds but which are aligned with axial welds. Axially orientated defects not aligned with axial welds would propagate into the very much tougher base metal and would subsequently arrest. Circumferentially orientated defects are subjected to a fraction of the hoop stress and would not become critical.
The probability that, if such a defect exists and was missed during the inspection and repair program, it would be critical before or at the end of the design life of the pipeline was quantified as follows:
Pd = P(size > critical size)
with l = length; lc = critical length; d = depth; and dc = critical depth.
The stable crack growth critical size curve shown in Fig. 10, establishes the relationship between critical depth and critical length: lc = f(dc).
The above function was approximated by a straight line fit of the actual curve with a vertical cut-off at l = 62 mm and a horizontal cut-off at d = 2.5 mm. When a surface defect fails it must be re-categorized as a through-thickness crack and should then be re-analyzed. The critical length of a through-thickness crack was calculated as 50 mm. This implies that, in this case, any surface defect that becomes critical would propagate in the thickness direction until it breaks the surface and would then further propagate in an unstable manner in the length direction to split open the pipe. The approximated function is shown in Fig. 13.
Equation (15) was then evaluated as follows:
The above expression was solved numerically and yielded the following result:
Pd = 2.2 x 10-2
The probability of failure of the pipeline was then calculated as follows:
Pf = 1 – (1 – Pd)n (25)
with n = number of defects remaining in the pipeline after the inspection program.
The number of defects remaining in the pipeline was estimated as follows. Approximately 230 axially orientated defects were found during the inspection. All these defects were repaired. Typical detection probabilities cited in the literature3,6 range from 0.6 (poor inspection) to 0.95 (good inspection). The probability would also be a function of defect size.
A detection probability of 0.95 was assumed in the present case, independent of crack size. This implied that a probable number of 11.9 defects had been missed. Of these, only a fraction would be aligned with axial welds. This fraction was quantified as the ratio of six times the width of an axial weld (there are six axial welds connected to each circumferential weld), to the circumference of the pipe, or fraction = 6 x 20/(2p x 3000) = 0.007.
The probability of failure was then calculated:
Pf = 1 – (1 – 2.2x10-2)0.007x11.9
= 1.87 x 10-3
The result of the probabilistic analysis indicated that the probability of failure was unacceptable. A probability of failure of 5 X 10-4 has been proposed as a maximum tolerable value.3 In nuclear applications a value of 10-6 is usually required. In the present ease, a probable cost of failure was calculated:
Probable cost of failure = 1.87 x 10-3 x $500 000 000
= $1 000 000
Analyses were thereafter performed to determine which fracture control option would yield the best results (lower the probability of failure in the most cost-effective way). One of the greatest advantages of the probabilistic approach is that the sensitivity of the risk of failure (which is the single most important result of any failure analysis) to any influencing parameter may easily be quantified.
The final solution was to embed the pipeline in concrete, thereby lowering the stresses to such an extent that the probability of failure became acceptable.
The interaction between fracture mechanics analyses and NDI results has traditionally been addressed by assuming a critical defect size based on the NDI techniques’ size limitations. The fact however remains that detection probabilities are operator dependent and that there will always be a possibility that a defect larger than this size would be missed. It is senseless to base a detailed and costly analysis on such a ‘gut-feel’ baseline assumption.
Probabilistic fracture mechanics therefore often offers the only scientific way to assess the integrity of fracture prone structures.
It seems appropriate that research effort is expended on this subject. Some results on statistical distributions of fracture initiation toughness data have been published, very few however on Jr data. The size dependence of NDI detection probabilities is also a subject that requires attention.
1. British Standards Institution, Guidance on some methods for the derivation of acceptance levels for defects in fusion welded joints. PD 6493, 1980.
2. Milne, I., Ainsworth, R. A., Dowling, A. R. & Steward, A. T., Assessment of the integrity of structures containing defects. R/H/R6-Rev.3, 1986.
3. Mudge, P. J. & Williams, S., Statistical aspects of defect sizing using ultrasonics, 1986.
4. Broek, D., The Practical Use of Fracture Mechanics. Kluwer Academic Publishers, 1989, pp.90-1.
5. Takashima, H., Mimaki, T. & Hagiwara, Y., Reliability analysis of spherical tank by Monte Carlo simulation. In The Mechanism of Fracture, ed. V. S. Goel. America Society of Metals, 1986, pp.537-46.
6. Carlsson, J., Probabilistic fracture mechanics. In Advances in Elasto-Plastic Fracture Mechanics, ed. L. H. Larsson. Applied Science Publishers, 1980, pp.418-19.
7. Faucher, B. & Tyson, W. R., A comparison crack-mouth opening and load-line displacement for J-integral evaluation using bend specimens. Elastic-plastic fracture test methods. ASTM STP, 856 (1983).
8. Turner, C. E., Methods for post-yield fracture safety assessment. In Post-yield Fracture Mechanics, Elsevier Science Publishers, 1984, pp.326-8.
9. Newman, J. C. & Raju, J. S. Eng. Frac. Mech., 15 (1981) 185-92.
10. Mattheck, C., Munz, D. & Stamm, H., Stress intensity factor for semi-elliptical surface cracks loaded by stress gradients. Eng. Frac. Mech.,18(3) (1983) 633-41.