- Email: [email protected]
The extrapolation of creep rupture data by PD6605 – An independent case study
International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
The extrapolation of creep rupture data by PD6605 e An independent case study J. Bolton* 65 Fisher Avenue, Rugby, Warks CV22 5HW, United Kingdom
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 January 2010 Received in revised form 6 February 2011 Accepted 17 March 2011
The worked example presented in BSI document PD6605-1:1998, to illustrate the selection, validation and extrapolation of a creep rupture model using statistical analysis, was independently examined. Alternative rupture models were formulated and analysed by the same statistical methods, and were shown to represent the test data more accurately than the original model. Median rupture lives extrapolated from the original and alternative models were found to diverge widely under some conditions of practical interest. The tests prescribed in PD6605 and employed to validate the original model were applied to the better of the alternative models. But the tests were unable to discriminate between the two, demonstrating that these tests fail to ensure reliability in extrapolation. The difficulties of determining when a model is sufficiently reliable for use in extrapolation are discussed and some proposals are made. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Creep rupture Extrapolation Statistical modelling
1. Introduction The prohibitive cost of long-term rupture testing for creepresistant steels and other engineering alloys has led to the development of numerous procedures whereby data for limited test times are extrapolated to commercially useful lifetimes. Such procedures first represent the available data by a graphical or mathematical model that reflects the observed dependence of mean rupture life upon stress and temperature, then extrapolate that model assuming that it is a reliable predictor of behaviour over somewhat longer times. There is no guarantee that this will be so, since, for example, changes in microstructure might precipitate an unanticipated change in behaviour. However, the prediction is presumed to be reliable provided that there is no evidence to the contrary, i.e. provided that the following conditions are met: (i) The model accurately follows the variation of rupture time with stress and temperature displayed by the test results for typical casts, (ii) The model is well behaved over the extrapolated interval, exhibiting no changes in curvature that appear to originate only in the mathematical formulation of the model, (iii) There is no separate body of data for a similar alloy that contradicts the extrapolated trend of the model, such as the analyst should also take into account.
* Tel.: þ44 (0) 1788 565366. E-mail address: [email protected]. 0308-0161/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2011.03.001
This paper discusses the assessment of rupture models for a single dataset, without subsequent adjustments based on data for any similar alloy, and incorporating no deliberate conservatism. In this context, judgement of the reliability of extrapolation depends only upon the model’s accuracy in representing the data and its orderly behaviour over the extrapolated interval. One such extrapolation method, which has been widely promoted in Europe and the United States, is presented in British Standard document PD6605-1:1998 [1]. This document presents a statistical procedure for finding an optimised mathematical model of the dependence of median rupture life on stress and temperature. It includes recommended tests for verifying the reliability of that model, so that a model based on data terminating at, for example, 100,000 h may be extrapolated to two or three times that duration with some confidence. The statistical method at the core of this procedure is described in Ref. [2]. It consists of iterating the numerical coefficients of each of a series of candidate equations, first to optimise their coefficients and then to determine which has the greatest likelihood of being an accurate model. A library of candidate equations for established rupture models is recommended in Ref. [1], and the user may introduce other equations. The statistical method employed to calculate numerical values of Likelihood or Deviance for each model is recapitulated here in Appendix 1. A recent and generally favourable review of the PD6605 procedure has been published as Ref. [3]. Appendix C of PD6605 presents a worked example to illustrate the recommended analysis and validity tests. A preferred model, from the standard library, is established and validated, and extrapolated median rupture properties are then determined. It is
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
this case that has been independently analysed in this study. Alternative models of the same data are presented here, formulated to follow the shapes of LogStress-LogTime curves displayed by individual casts more closely, with numerical coefficients optimised by the same statistical methods. The models are compared both graphically and by the calculation of statistical Deviance, and the alternative models are shown to be more accurate. The overestimation of extrapolated lives by PD6605 is then determined relative to the implicitly more reliable alternatives. The tests prescribed by PD6605 to ensure model reliability are discussed, and are found to be ineffective in discriminating between the models considered. Criteria for the assessment of model reliability are discussed, leading to the conclusion that there is no adequate set of quantitative criteria. Visual assessment by a vigilant analyst is indispensable to the selection of a model of sufficient reliability to warrant extrapolation. Suggestions are made to facilitate the selection process. 2. Data source and model construction The dataset used for this study was that of the worked example in PD6605 for a 12Cr alloy steel. This consists of rupture data for eighteen casts tested, to varying extents, over five temperatures between 450 C and 600 C. The test population included 248 tests, 217 of which had failed and 31 of which had remained unbroken. The data are re-plotted in the figures of this paper together with the median curves for Model MH03, determined by PD6605 to be the best of its library and to pass all of its specified tests. The alternative models, designated the ADA and SDS models, were formulated for this investigation in order to achieve a closer match to the visible trends in the data. The designations MH03, ADA and SDS are used throughout this paper to signify either the mathematical forms of the models or their exact forms, including optimised sets of numerical coefficients, according to context. The ADA (Arc-Dip-Arc) and SDS (Slope-Dip-Slope) models were designed to duplicate the pronounced inflections evident in LogStress versus LogTime plots of the data. Their mathematical forms are quite different, as described in Appendix 2, but both define an inflected transition between an upper and lower curve of LogStress
159
versus LogTime with a varying, but always negative, gradient. These models are incapable of paradoxical reversals when extrapolated, whereby stress increases with increasing rupture time or rupture time decreases with decreasing stress, as may occur with models formulated without deliberate design of the resultant curve shapes. 3. Comparison of models to data Fig. 1 shows the rupture test data and competing models at temperatures of 600 C and 500 C, and Fig. 2 shows the same comparison at 550 C and 475 C, separating the data at adjacent temperatures that would otherwise overlap. The very few data at 450 C are not shown, but were included in the statistical analysis. Each individual cast is distinguished at each temperature. Casts considered typical, being of medium strength and having been tested at more than one temperature, are shown more prominently so that the typical data trends stand out against the general scatter. At 600 C (Fig. 1) data for typical casts can be seen to define the shape of the LogStresseLogTime curve very consistently. These casts exhibit a severe inflection, starting at about 3000 h and apparently nearing completion at about 30,000 h, and other casts display a consistent behaviour. The MH03 model is visibly too smooth in relation to the individual cast data, showing an insufficiently sharp inflection; whereas the ADA and SDS models follow the individual cast data more closely over the whole range of test times, with the ADA model displaying a somewhat exaggerated inflection. The numerical coefficients of these models were derived by Maximum Likelihood analysis of the data at all temperatures, as described later. Hence none is necessarily the best fit that might be obtained by analysis of the data at 600 C alone. The importance of establishing an accurate model at this temperature is that it provides a guide to the curve shapes at lower temperatures, where the data extend less far into the inflection. In the absence of evidence to the contrary, it is to be expected that a similar characteristic shape will appear after progressively longer times at progressively lower temperatures. Typical casts at 550 C (Fig. 2) display a consistent trend, with clear evidence of an inflection similar to that at 600 C, but the data terminate well before completion of the inflection. It is again evident that the MH03 model is too smooth, lacking a sufficiently
Stress, MPa
1000
100
C14 500C Rupt
C22 500C Rupt
C22 500C Unbr
C24 500C Rupt
C25 500C Rupt
C26 500C Rupt
C27 500C Rupt
C27 500C Unbr
C74 500C Rupt
C74 500C Unbr
C75 500C Rupt
C75 500C Unbr
C76 500C Rupt
C76 500C Unbr
MH03 500C
ADA 500C
SDS 500C
C29 600C Rupt
C29 600C Unbr
C30 600C Rupt
C30 600C Unbr
C31 600C Rupt
C32 600C Rupt
C32 600C Unbr
C33 600C Rupt
C34 600C Rupt
C34 600C Unbr
MH03 600C
ADA 600C
SDS 600C
100
1000
10000
100000
Time, h
Fig. 1. 12Cr steel rupture data and models at 600 and 500 C.
10 1000000
160
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Stress, MPa
1000
100 C29 475C Rupt
C30 475C Rupt
C31 475C Rupt
C32 475C Unbr
C33 475C Rupt
C33 475C Unbr
C34 475C Rupt
MH03 475C
ADA 475C
SDS 475C
C14 550C Rupt
C22 550C Rupt
C22 550C Unbr
C24 550C Rupt
C25 550C Rupt
C25 550C Unbr
C26 550C Rupt
C26 550C Unbr
C27 550C Rupt
C29 550C Rupt
C29 550C Unbr
C30 550C Rupt
C30 550C Unbr
C31 550C Rupt
C32 550C Rupt
C33 550C Rupt
C33 550C Unbr
C34 550C Rupt
C34 550C Unbr
C71 550C Rupt
C72 550C Rupt
C73 550C Rupt
C74 550C Rupt
C75 550C Rupt
C76 550C Rupt
MH03 550C
ADA 550C
SDS 550C
100
1000
C32 475C Rupt
10000
10 1000000
100000
Time, h
Fig. 2. 12Cr steel rupture data and models at 550 and 475 C.
sharp inflection. Typical gradients at 30,000 to 60,000 h are visibly closer to the SDS model than to either of the other models. The differences between the models are sufficient at 550 C to be of practical importance. A median rupture life of 100,000 h according to the MH03 model is an overestimate, relative to the SDS model, of about 50%. At 500 C (Fig. 1) the three models are in close agreement up to about 60,000 hours. Beyond that time the MH03 model is clearly too high relative to the scatter of data. The data exhibit the early stages of inflections, but only for the weakest cast (C22) is a welldeveloped inflection visible. Extrapolation of the median curve depends heavily upon projecting trends established at 550 C and 600 C, for which the SDS model appears the most reliable. At 500 C the discrepancies between the models at times of practical importance are large. Rupture lives of 200,000 to 300,000 h according to the MH03 model are overestimates, relative to the SDS model, of 50% to 100%. At 475 C (Fig. 2) the data for typical casts are consistent with all three models. Extrapolation into the inflected portion of the rupture curve is totally dependent upon model accuracy at higher temperatures, which favours the SDS model. Discrepancies between the two models at 475 C and at times of practical importance are significant. Rupture lives of 200,000 to 300,000 h according to the MH03 model are overestimates, relative to the SDS model, of up to 50%.
4. Likelihood and deviance calculations In the statistical analysis discussed here, each mathematical model incorporates a “systematic” component and a “random” component. The systematic component is an equation that describes the median time to rupture in terms of stress and temperature, incorporating a number of numerical coefficients as in Appendix 2. The random component is an equation that defines the distribution of rupture times about the median, incorporating additional coefficients as in Appendix 1. An optimum set of coefficients for both components is obtained by iteration of their values to obtain the maximum overall Likelihood. The overall Likelihood is the product of all individual
Likelihoods for each failed and unfailed test, determined by statistical analysis as described in Refs. [1,2] and recapitulated in Appendix 1. Optimised rupture models are conveniently compared in terms of Deviance, which is related to Likelihood as in Appendix 1. The model that corresponds most accurately to the data is that with the lowest value of Deviance. Values of Deviance for the models considered are shown in Table 1. The models are fully defined in Appendices 1 and 2 and assume Weibull two-parameter distributions for the random component, with heteroscedasticity. The lower Deviance of the SDS model confirms what is visually evident from Figs. 1 and 2, that this model is a significantly better fit to the data than the other two. Part of this difference is required to justify the inclusion of additional terms, as discussed in Appendix 2, but the requirement amounts to only a small fraction of the improvement achieved. The SDS model is therefore much more likely to be a reliable basis for extrapolation, provided that there is no separate evidence to the contrary. The difference in total Deviance, DD, between any two models in Table 1 can be expressed in terms of relative Likelihood. If one or other model is postulated to be an exact model of the median and distribution of the data, the Likelihood of the set of test results being observed is greater by expðDD=2Þ for the model of lower Deviance. Thus the Likelihood of the ADA model is 9.0 times greater than for the MH03 model, and over a population of 248 tests the Likelihood is 0.9% greater per test. The Likelihood of the SDS model is 2.3 107 times greater than for the MH03 model and is 7.1% greater per test. The lower Deviances of the ADA and SDS models show that they are progressively more accurate representations of the test data, implying that they are increasingly more reliable for extrapolation - subject to appropriate tests of validity. Table 1 Calculated Deviances for the SDS, ADA and MH03 models. Test population
Deviance
Category
Number
MH03 model
ADA model
SDS model
Failed Unfailed Total
217 31 248
4393.7 25.6 4419.3
4358.7 56.2 4414.9
4359.3 26.1 4385.4
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
161
It is remarkable that the MH03 and ADA models, which differ little in terms of total Deviance or average Likelihood, give widely divergent results when extrapolated. At a stress of 200 MPa and a temperature of 500 C, their median rupture lives are 287,000 h and 111,000 h respectively. The wide divergence associated with only a small difference in Deviance suggests that neither model is reliable. The ADA model is not discussed further here because the significantly lower Deviance of the SDS model shows the latter to be more accurate. There is a minor distinction to be made between the MH03 model for which results are given in Table 1 and the corresponding model (MH03a) of PD6605. They are identical in terms of the mathematical form of the rupture equation (Appendix 2) but differ slightly in other respects. In order to ensure strict comparability with the ADA and SDS models, the MH03 model coefficients were re-optimised for exactly the same dataset and with the same form of heteroscedasticity relationship (Appendix 1), resulting in the Deviance values in Table 1. The total Deviance of the MH03 model is slightly higher than the Deviance of 4417.7 reported for the MH03a (PD6605) model, as a result of using the simpler heteroscedasticity relation discussed in Appendix 2. However, the differences between median rupture lives calculated from the MH03 and MH03a models are insignificantly small, and are barely visible on the scale of Figs. 1 and 2. Rupture lives, Deviances, etc quoted in this paper are for the MH03 model unless otherwise stated, but for practical purposes the MH03 and MH03a models are identical, so comparisons to the SDS model apply equally to both.
Table 2 Relative errors in MH03 model extrapolated lives.
5. Prescribed validity tests
7. Discussion of model validity criteria
A series of validity tests is prescribed in PD6605 with the purpose of ensuring that a model that fits the data acceptably well is sufficiently reliable to warrant extrapolation. Some specify selfevident requirements, such as,
The failure of the PD6605 procedure in the case examined is that its validity tests did not reject the MH03a model, thus failing to compel the original analyst to seek a more accurate alternative. The flaw in the procedure is its default assumption that a model that satisfies its prescribed tests is a reliable basis for extrapolation. Failure to satisfy these tests may be a sufficient reason for rejection, but the converse is clearly untrue; success in satisfying these tests is not a sufficient reason for acceptance. The prime consideration for validation of a model initially selected for its low Deviance is that its rupture curves should visibly duplicate the main trends evident in a graphical presentation of the test data. Figs. 1 and 2 show that the MH03, ADA and SDS models all satisfy this condition in a broad sense, although they differ in their degree of precision. The next consideration is that the model should extrapolate in a stable fashion to well beyond the usual extrapolation interval of two to three times, e.g. to six times the test programme duration. The curves should display no changes in curvature that appear to originate in the mathematical formulation of the model rather than in the visible trends of the data. Figs. 1 and 2 show that the MH03, ADA and SDS models all satisfy this requirement over at least a sixfold extrapolation in time. A further consideration is that the model should produce consistent extrapolations, exhibiting only moderate sensitivity to the relatively sparse data at the longest times. This can be verified by comparative extrapolations from a model based upon a dataset masked at half the eventual test programme duration. For the case examined the nominal duration, bounding 95% of all tests, is 80,000 h. The dataset was masked at 40,000 h by altering each test record at over 40,000 h, whether ruptured or unbroken, to show instead the former state of unbroken at 40,000 hours. The coefficients of the MH03 and SDS models were then re-optimised for minimum Deviance. Rupture lives were re-calculated at the stress corresponding, according to the initial models, to rupture at 160,000 h. The results are given in Table 3.
a) Visual confirmation that the model conforms to the major trends of stress and temperature dependence exhibited by the data, b) Visual confirmation that the model is well behaved in extrapolation to well beyond the intended extrapolation interval, exhibiting no paradoxical changes in gradient. It can be seen from Figs. 1 and 2 that all three models broadly satisfy those requirements. Additional requirements stipulate, c) Limits to the gradient of and scatter about a linear regression of Log(model rupture time) with respect to Log(test rupture time), d) Sensitivity of extrapolation to the random removal of a fraction of the data, e) Sensitivity of extrapolation to especially influential casts. All of the prescribed tests were satisfied by the original MH03a model, relieving the analyst of any obligation to search for a better model, and resulting in its endorsement as sufficiently reliable for extrapolation. However, the SDS model also passes these validity tests, which creates a contradiction: two models that result in gross differences in extrapolated rupture life cannot both be valid. It follows that the prescribed tests are not sufficiently discriminating to reject all unreliable models. 6. Extrapolated rupture lives Median rupture lives extrapolated from the SDS and MH03 models at the same stress are shown in Table 2 at temperatures of practical
Temperature, C
Median rupture life, kh
MH03 relative error
MH03 model
SDS model
550
300 200 100
253 150 67
þ19% þ33% þ49%
525
300 200 100
161 109 69
þ86% þ83% þ45%
500
300 200 100
152 123 86
þ97% þ63% þ16%
interest. The corresponding lives predicted by the original MH03a (PD6605) model are within 3% either way of the values given for the MH03 model. Relative errors listed in the final column are relative to the SDS model, which appears to be the more reliable of the two. By comparison to the margins conventionally employed in, for example, the design of power plant, overestimates of up to 30% in extrapolated life are unlikely to impinge upon plant safety. But overestimates of 50% or more introduce some increasing risk that expected design lives might not be attained, and that service failures might occur. Design use of the values recommended in PD6605, or similarly derived for an alloy with similar characteristics, is therefore potentially hazardous.
162
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Particularly at 600 C where the inflection in the data is most fully defined, the model curve is distinctly smoother than the data trend suggests. At 550 C, where the inflection is less well defined, it is nonetheless evident that the MH03 model does not match the typical gradients displayed by the data. And there is further evidence of this at 500 C. A vigilant analyst should recognise these observations as indicative of an unsatisfactory model, and should pursue an alternative model that displays a more pronounced inflection. The ADA model better duplicates the sharpness of the inflection but appears to somewhat exaggerate the effect. b) The SDS model appears in these figures to provide a very close approximation to the data trends, particularly at 600 C where the inflection in the data is most fully developed. A vigilant analyst should recognise also that, e.g. at 550 C, the model succeeds in passing through the median of rupture times at a given stress rather than through the median of rupture stresses at a given time, which can be the source of visual confusion for inflected curves. c) The most difficult criterion of judgement is the likelihood of devising a model that duplicates the visible data trends still better than the SDS model; and this is ultimately a qualitative and subjective judgement. Only if that likelihood seems remote is it fair to conclude that the SDS model is reliable for extrapolation, in the sense that it is as reliable as any that is likely to be found. Judging this from Figs. 1 and 2 is made difficult by the amount of scatter between casts, and it is helpful to re-plot the data as in Fig. 3. In this figure data are re-plotted in terms of equivalent time at a single, chosen temperature, test failure times being multiplied by the ratio of model rupture times at the chosen and actual temperatures. Scatter is suppressed by plotting only the median rupture time for each sub-population at a common stress and temperature. The gradation in symbol size indicates the sub-population size, which varies from one to fifteen. Judged on the basis of coincidence of the model with data for the larger sub-populations, and continuity of the data trend from one temperature to another, the SDS model appears (to the writer) to be unlikely to be surpassed by devising further models.
Table 3 Extrapolated median rupture lives for a dataset masked at half-range. Extrapolated median rupture lives for masked dataset, kh at stresses for an initial extrapolated life of 160 kh Temperature, C MH03 masked MH03 change, % SDS masked SDS change, %
600 168.0 5.0 176.0 10.0
575 167.5 4.7 174.1 8.8
550 166.9 4.3 171.4 7.1
525 166.1 3.8 168.0 5.0
500 165.4 3.4 163.6 2.3
475 165.0 3.1 160.4 0.2
It is suggested that changes within 20% either way are acceptable, and hence that both the MH03 and SDS models are satisfactory as regards consistency of extrapolation. None of the criteria discussed above leads to outright rejection of either the MH03 or SDS model, so the SDS model must be judged the more reliable of the two on the basis of its lower Deviance. But this judgement is only relative e because a further model might be devised that is still more accurate than the SDS model. Hence it cannot be concluded that the SDS model is sufficiently reliable for extrapolation. That conclusion could be reached only if there were some absolute criterion of accuracy, such as whether model Deviance is within an acceptable limit. But Deviance reflects both the divergence between model and data and the scatter between individual casts, and scatter may be the predominant factor. So a criterion based upon Deviance is not practical. There is no quantitative criterion of accuracy, so the evaluation of a series of progressively better models, i.e. models of progressively lower Deviance that satisfy appropriate validity criteria, has no numerically definable conclusion. There is no end point at which an analyst, relying upon the criteria of PD6605 or those discussed above, could conclude that a still better model might not produce significantly different extrapolated rupture lives. The only practical solution that presents itself is to apply the first criterion above, of accurate duplication of the trends evident in the data, with the utmost vigilance. Thus, a) The MH03 model appears, in Figs. 1 and 2, to provide a reasonable approximation to the data trends e but careful visual examination reveals that it is far from perfect.
Stress, MPa
1000
100
10 1.E-01
450C, pop.2
450C, pop.3
475C, pop.1
475C, pop.2
475C, pop.3
475C, pop.4-5
475C, pop.6-8
500C, pop.1
500C, pop.2
500C, pop.3
500C, pop.6-8
550C, pop.1
550C, pop.2
550C, pop.3
550C, pop.4-5
550C, pop.6-8
550C, pop.9-11
550C, pop.12-15
600C, pop.1
600C, pop.4-5
600C, pop.6-8
Model
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
o
Equivalent (SDS) time at 525 C, h Fig. 3. Median rupture times for all test sub-populations standardised at 525 C by SDS model.
1.E+07
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Thus the key challenges facing the data analyst are to recognise when it is feasible that a better model might be devised, and then to devise it. If the analyst lacks the mathematical skills to devise a better model, then the challenge is to recognise when more expert assistance is required. The plethora of validity tests in PD6605 is inimical to these key judgements because they present the semblance of a sufficient proof of model reliability. On the basis of these tests, the original analysis endorsed a model that is unconvincing when tested by conventional, visual comparison to the data on a LogStresseLogTime plot, as in Figs. 1 and 2. 8. Conclusion Comparison of the rupture model recommended in PD6605 for a 12Cr alloy steel to an alternative model revealed a contradiction. Both models conformed to the validity criteria prescribed in PD6605, but there were gross discrepancies between their extrapolated lives for the same conditions of stress and temperature. The PD6605 recommended model predicted median rupture lives up to twice as long as the alternative model. The alternative model was found to be the statistically better fit to the data and hence, with no evidence from within the data to support any contrary conclusion, to be the more reliable basis for extrapolation. It follows that the PD6605 validation tests are not sufficiently discriminating. The ineffectiveness of these tests is a fundamental deficiency in PD6605, because they are the means of prompting the analyst to seek a more suitable model. In the absence of effective, quantitative tests of validity, the most discriminating means of verifying a model is the conventional, visual comparison of model to test data on a LogStresseLogTime plot. Any visible departure of a candidate model from the typical shapes exhibited by individual casts should be regarded as indicating that the model is unsatisfactory. Judgement based upon this comparison is subjective, but appears in the case examined to be effective when applied sufficiently critically. The demonstrated unreliability of PD6605 indicates that it is insufficiently developed to warrant the authority of a British Standard. The British Standards Institution was alerted to this problem in November 2009 and is presently, in April 2011, considering whether to constitute a panel to revise the procedure or to withdraw it. Acknowledgement The writer is indebted to Mr Derek Bolton for advice on the mathematics of the statistical analysis, and for providing independent corroboration of numerical results by means of a Cþþ program. Appendix 1. Calculation of likelihood A. Dependence of likelihood on the systematic component of the rupture equation Any candidate equation for the dependence of mean rupture time on applied stress and temperature (the systematic component of the rupture model) contains a number of numerical coefficients. For any trial set of numerical coefficients, the discrepancy between each observed failure time (tri) and the model expected failure time (tmi) may be expressed as tri/tmi. The distribution of all such discrepancies may then be expressed as a cumulative probability,
Prftr =tm < zg ¼ f ðzÞ Function f(z) is assumed to be of a convenient mathematical type, usually a Weibull or a LogLogistic distribution of tr/tm, as discussed later.
163
For any trial set of coefficients, the Likelihood (LFi) of an individual test failure time tri is provisionally, or finally, calculated as,
LFi ¼
dPrftr =tm < tri =tmi g dtr
This is not the Probability of the individual discrepancy. It is the gradient, or Probability Density, which is directly proportional to the probability of failure within a small interval either side of the observed rupture time. Likelihood has the dimensions of inverse time. The combined Likelihood of all the observed failure times, for a population NF of failed tests, is the product of the individual Likelihoods,
LF ¼
NF Y
LFi
i¼1
Evaluation of LF is a means of testing one set of coefficients against another set, the set with the higher Likelihood being preferred. Thus a Maximum Likelihood may be obtained by trial optimisation of the coefficients. A typical set of rupture data contains a number of unbroken test results, which can be used in a similar way. The Likelihood of individual survival time tui is,
LUi ¼ 1 Prftui =tmi < zg This is the Probability of the individual survival, which is dimensionless, and the combined Likelihood of a population NU of survivals is the product,
LU ¼
NU Y
LUi
i¼1
Hence the combined Likelihood of both the observed failures and observed survivals is the product,
L ¼ LF $LU A preferred rupture model is found by optimising the values of coefficients to give the Maximum Likelihood L, including both failed tests and unbroken tests. Candidate rupture models are compared in PD6605 in terms of Deviance, D, which is related to Likelihood, L, by the equation,
D ¼ 2lnðLÞ Strictly, Deviance is given by the difference 2½lnðLsat Þ lnðLÞ, where Lsat is the Likelihood of the Saturated Model e a model having a number of terms equal to the number of test results. But since competing models are ranked on the basis of their difference in Deviance, the omission of this common term is unimportant. C. Dependence of Likelihood on the random component of the rupture equation The standard mathematical distributions of time discrepancy considered in PD6605 and in this paper are the Weibull and LogLogistic distributions. Note that, throughout this paper, the term “mean rupture time” is used in accordance with engineering usage. However, the statistical analysis of discrepancies tr/tm is actually based on the median (middle) value rather than the mathematical mean (arithmetic or logarithmic average) value, and the median value is taken to be 1. For a Weibull (two-parameter) distribution of tr/tm the Cumulative Probability and Probability Density are:
164
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SD ¼ b Gð1 þ 2=aÞ G2 ð1 þ 1=aÞ
Prftr =tm < tri =tmi g ¼ 1 eðtri =btmi Þ a a tri a1 eðtri =btmi Þ d Prftr =tm < tri =tmi g ¼ b btmi tmi dtr where b ¼ elnð1=ln2Þ=a , as required for Median ðtr =tm Þ ¼ 1, and a is a coefficient determined by trial to result in Maximum Likelihood. For a Log-Logistic distribution of tr/tm the Cumulative Probability and Probability Density are, for Median (tr/tm) ¼ 1:
Prftr =tm < tri =tmi g ¼
1 1 þ ðtri =tmi Þ1=b
d Prftr =tm < tri =tmi g ¼ dtr
ðtri =tmi Þðbþ1Þ=b 2 btmi 1 þ ðtri =tmi Þ1=b
a ¼ expðD0 þ D1 T þ D2 log sÞ
where b is a coefficient determined for Maximum Likelihood. PD6605 recommends the Weibull distribution as a default option, although this is not fully justified unless the Maximum Likelihood found for this distribution is greater than for the LogLogistic distribution. In the case studied slightly greater values of Likelihood were obtained for the Weibull distribution, and only figures for that case are quoted. Values of the coefficient a, for the case studied, are given in Appendix 2. If the values of a (or b) above are constant, then the scatter in tr/tm (or the Standard Deviation of tr/tm) is uniform and the distribution is “homoscedastic”. However it is common, as in the case studied, to find that the distribution is “heteroscedastic” e the scatter displays a variation with stress, or temperature. Heteroscedasticity is incorporated in the random component of the rupture model by defining a (or b) as a function of stress, s, and temperature, T. In the case studied the relationship chosen to represent the observed trends in Standard Deviation was,
a ¼ D0 þ D1 s This equation has no temperature term because calculated Likelihoods were found to be insensitive to the inclusion of such a term. The suitability of this relationship is shown in Fig. 4, where Standard Deviations of tri =tmi for the SDS model, for sub-populations of at least four failed tests at a common stress (or narrow range of stress) and common temperature, are plotted against stress. Standard Deviation values for the model Weibull distribution were calculated from,
This equation was not used in the present study because it predicts that Standard Deviation diminishes rapidly, and improbably, at low stresses below the range tested. It also includes a temperature term, which was found redundant in the statistical analysis of the MH03, ADA and SDS models. The inclusion of heteroscedasticity has little effect upon determination of the optimum coefficients of the median rupture equation, and hence little effect upon extrapolated median rupture lives. It is important only in the context of estimating extrapolated rupture lives at a high confidence level, e.g. at 95% confidence, which is not discussed here. Appendix 2. Details of rupture models A. MH03 model The MH03 model is a model of the MansoneHaferd type, and has the following equation for median rupture life,
o n lnðtm Þ ¼ B0 þ B1 þ B2 log s þ B3 ðlog sÞ2 þB4 ðlog sÞ3 T where s ¼ Test stress, MPa; T ¼ Test temperature, K; and B0 ; //B4 are numerical coefficients. In this form the equation includes logarithmic terms both to base e and base10. For the MH03a model, determined by Maximum Likelihood calculation for the subject data set as reported in PD6605, the numerical coefficients of the systematic component of this model were,
B0 ¼ 47:409; B1 ¼ 0:099122; B2 ¼ 0:19674;
10
Standard Deviation of tr/tm (SDS Model)
where G is the Gamma function. The scatter in Standard Deviation for the test results is large, which is not surprising in view of the small sub-population sizes, but Fig. 4 nonetheless supports the dependence on stress and independence of temperature deduced from Likelihood calculations. The model Standard Deviation diminishes to a steady value at low stresses. Values of coefficients D for the case studied are given in Appendix 2. The heteroscedasticity relationship recommended in PD6605 and used for the MH03a model was:
B3 ¼ 0:09328; B4 ¼ 0:015216
Data 475C
And the random coefficients defining the Weibull a parameter, as a ¼ expðD0 þ D1 T þ D2 log sÞ, were,
Data 500C Data 550C
D0 ¼ 3:311; D1 ¼ 0:0003944; D2 ¼ 1:3567
Data 600C SDS Random Model
The model was selected in PD6605 as the best of its library of models, resulting in a lower Deviance than any other model considered that satisfied the prescribed validity tests. For the MH03 model established for this study, which generates rupture lives for any test condition almost identical to those for the MH03a model, the systematic coefficients were,
1
B0 ¼ 47:58; B1 ¼ 0:098865; B2 ¼ 0:19674; B3 ¼ 0:09328; B4 ¼ 0:015212
0.1 0
100
200
300
400
500
600
Stress, MPa Fig. 4. Heteroscedasticity for sub-populations (min. 4 tests) at a common stress and temperature.
And the random coefficients defining the Weibull a parameter, as a ¼ D0 þ D1 s, were
D0 ¼ 2:30; D1 ¼ 0:002795
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
B. ADA model
Table 4 Deviance reductions justifying additional coefficients.
The ADA model was developed for the present study and its structure is described in general terms in Section 2. In mathematical terms it is described by the following equation,
ln tm
165
¼ Vln V B0 expðDT=B1 Þ 1 þ ð1 VÞln V B2 expðDT=B1 Þ 1
Model
No. coeff’ts
Total deviance
Deviance reductions relative to MH03 (or to ADA) model Reduction
where V ¼ fs=ðB3 expðDT=B4 ÞÞgB5 ; s ¼ test stress, MPa; DT ¼ 600
-Test temperature, C; and B0 ; .B5 are numerical coefficients. The systematic coefficients of this model for the subject dataset, determined by Maximum Likelihood calculation, were,
B0 ¼ 17:66; B1 ¼ 566:8; B2 ¼ 2:827; B3 ¼ 396:6; B4 ¼ 185:7; B5 ¼ 1:704;
MH03 MH03a ADA SDS
7 8 8 9
4419.3 4417.7 4414.9 4385.4
0 1.6 4.4 33.9
0 29.5
PD6605 criterion
Akaike criterion
2.7 2.7 5.4
2.1 2.1 4.3
2.7
2.2
PD6605 imposes a criterion for the acceptance of Model B (containing kB parameters) and rejection of Model A (containing kA parameters) in the form:
And the numerical coefficients of the Weibull distribution were,
Deviance ðModel AÞ Deviance ðModel BÞ > 2:7ðkB kA Þ
D0 ¼ 2:32; D1 ¼ 0:00270
A more suitable criterion is the Akaike Information Criterion [4] which requires a reduction in Deviance of,
C. SDS model
2kB ½1 þ ðkB þ 1Þ=ðN kB 1Þ 2kA ½1 þ ðkA þ 1Þ=ðN kA 1Þ
The SDS model was also developed for the present study and its structure is described in general terms in Section 2. In mathematical terms it is described by the following equation,
where N ¼ Number of test results (248). Table 4 shows the Deviance reductions for each of the models discussed and the requirements of the above criteria. Figures are given relative to the MH03 model and, in italics, relative to the ADA model. The reduction in Deviance between the MH03 and MH03a models is seen to be insufficient to justify the additional (random) coefficient of the latter. This confirms the redundancy of a temperature term in the heteroscedasticity equation evident from the statistical analysis of the MH03 model. The progressively higher numbers of (systematic) coefficients in the ADA and SDS models achieved reductions in Deviance that, in each case, satisfied these criteria.
ln tm ¼ B0 þ B1 ln S þ B2 S þ B3 DT þ B4 = 1 þ ðS=B5 Þ2 where S ¼ seDT=B6 ; s ¼ Test stress, MPa; DT ¼ 600 -test temperature, C; and B0 ; .B6 are numerical coefficients. The numerical coefficients of this model for the subject 12Cr dataset, determined by Maximum Likelihood calculation, were,
B0 ¼ 48:788; B1 ¼ 2:6009; B2 ¼ 0:065587; B3 ¼ 0:02506; B4 ¼ 24:57; B5 ¼ 219:45; B6 ¼ 235:66 And the random coefficients defining the Weibull a parameter, as a ¼ D0 þ D1 s, were
D0 ¼ 2:543; D1 ¼ 0:003215 D. Number of model coefficients The MH03 model includes seven (systematic and random) numerical coefficients compared to eight for the original MH03a model, eight for the ADA model and nine for the SDS model.
References [1] British Standards Institute. Published Document 6605-1:1998, Guidance on methodology for assessment of stress-rupture data e Part 1: Procedure for derivation of strength values, 15 Dec 1998. [2] Davies RB, Hales R, Harman JC, Holdsworth SR. Statistical modelling of creep rupture data. Journal of Engineering Materials Technology (ASME) 1999;121(3): 264e71. [3] Bullough CK, Norman M. The PD6605 creep rupture assessment procedure e an appraisal of its application ten years on, Creep and fracture in high temperature components, 2nd ECCC Creep conference, Zurich, 21e23 Apr 2009. [4] Akaike, Hirotogu. A new look at the statistical model identification. IEEE Trans Automatic Control 1974;19(6):716e23.
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
The extrapolation of creep rupture data by PD6605 e An independent case study J. Bolton* 65 Fisher Avenue, Rugby, Warks CV22 5HW, United Kingdom
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 January 2010 Received in revised form 6 February 2011 Accepted 17 March 2011
The worked example presented in BSI document PD6605-1:1998, to illustrate the selection, validation and extrapolation of a creep rupture model using statistical analysis, was independently examined. Alternative rupture models were formulated and analysed by the same statistical methods, and were shown to represent the test data more accurately than the original model. Median rupture lives extrapolated from the original and alternative models were found to diverge widely under some conditions of practical interest. The tests prescribed in PD6605 and employed to validate the original model were applied to the better of the alternative models. But the tests were unable to discriminate between the two, demonstrating that these tests fail to ensure reliability in extrapolation. The difficulties of determining when a model is sufficiently reliable for use in extrapolation are discussed and some proposals are made. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Creep rupture Extrapolation Statistical modelling
1. Introduction The prohibitive cost of long-term rupture testing for creepresistant steels and other engineering alloys has led to the development of numerous procedures whereby data for limited test times are extrapolated to commercially useful lifetimes. Such procedures first represent the available data by a graphical or mathematical model that reflects the observed dependence of mean rupture life upon stress and temperature, then extrapolate that model assuming that it is a reliable predictor of behaviour over somewhat longer times. There is no guarantee that this will be so, since, for example, changes in microstructure might precipitate an unanticipated change in behaviour. However, the prediction is presumed to be reliable provided that there is no evidence to the contrary, i.e. provided that the following conditions are met: (i) The model accurately follows the variation of rupture time with stress and temperature displayed by the test results for typical casts, (ii) The model is well behaved over the extrapolated interval, exhibiting no changes in curvature that appear to originate only in the mathematical formulation of the model, (iii) There is no separate body of data for a similar alloy that contradicts the extrapolated trend of the model, such as the analyst should also take into account.
* Tel.: þ44 (0) 1788 565366. E-mail address: [email protected]. 0308-0161/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2011.03.001
This paper discusses the assessment of rupture models for a single dataset, without subsequent adjustments based on data for any similar alloy, and incorporating no deliberate conservatism. In this context, judgement of the reliability of extrapolation depends only upon the model’s accuracy in representing the data and its orderly behaviour over the extrapolated interval. One such extrapolation method, which has been widely promoted in Europe and the United States, is presented in British Standard document PD6605-1:1998 [1]. This document presents a statistical procedure for finding an optimised mathematical model of the dependence of median rupture life on stress and temperature. It includes recommended tests for verifying the reliability of that model, so that a model based on data terminating at, for example, 100,000 h may be extrapolated to two or three times that duration with some confidence. The statistical method at the core of this procedure is described in Ref. [2]. It consists of iterating the numerical coefficients of each of a series of candidate equations, first to optimise their coefficients and then to determine which has the greatest likelihood of being an accurate model. A library of candidate equations for established rupture models is recommended in Ref. [1], and the user may introduce other equations. The statistical method employed to calculate numerical values of Likelihood or Deviance for each model is recapitulated here in Appendix 1. A recent and generally favourable review of the PD6605 procedure has been published as Ref. [3]. Appendix C of PD6605 presents a worked example to illustrate the recommended analysis and validity tests. A preferred model, from the standard library, is established and validated, and extrapolated median rupture properties are then determined. It is
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
this case that has been independently analysed in this study. Alternative models of the same data are presented here, formulated to follow the shapes of LogStress-LogTime curves displayed by individual casts more closely, with numerical coefficients optimised by the same statistical methods. The models are compared both graphically and by the calculation of statistical Deviance, and the alternative models are shown to be more accurate. The overestimation of extrapolated lives by PD6605 is then determined relative to the implicitly more reliable alternatives. The tests prescribed by PD6605 to ensure model reliability are discussed, and are found to be ineffective in discriminating between the models considered. Criteria for the assessment of model reliability are discussed, leading to the conclusion that there is no adequate set of quantitative criteria. Visual assessment by a vigilant analyst is indispensable to the selection of a model of sufficient reliability to warrant extrapolation. Suggestions are made to facilitate the selection process. 2. Data source and model construction The dataset used for this study was that of the worked example in PD6605 for a 12Cr alloy steel. This consists of rupture data for eighteen casts tested, to varying extents, over five temperatures between 450 C and 600 C. The test population included 248 tests, 217 of which had failed and 31 of which had remained unbroken. The data are re-plotted in the figures of this paper together with the median curves for Model MH03, determined by PD6605 to be the best of its library and to pass all of its specified tests. The alternative models, designated the ADA and SDS models, were formulated for this investigation in order to achieve a closer match to the visible trends in the data. The designations MH03, ADA and SDS are used throughout this paper to signify either the mathematical forms of the models or their exact forms, including optimised sets of numerical coefficients, according to context. The ADA (Arc-Dip-Arc) and SDS (Slope-Dip-Slope) models were designed to duplicate the pronounced inflections evident in LogStress versus LogTime plots of the data. Their mathematical forms are quite different, as described in Appendix 2, but both define an inflected transition between an upper and lower curve of LogStress
159
versus LogTime with a varying, but always negative, gradient. These models are incapable of paradoxical reversals when extrapolated, whereby stress increases with increasing rupture time or rupture time decreases with decreasing stress, as may occur with models formulated without deliberate design of the resultant curve shapes. 3. Comparison of models to data Fig. 1 shows the rupture test data and competing models at temperatures of 600 C and 500 C, and Fig. 2 shows the same comparison at 550 C and 475 C, separating the data at adjacent temperatures that would otherwise overlap. The very few data at 450 C are not shown, but were included in the statistical analysis. Each individual cast is distinguished at each temperature. Casts considered typical, being of medium strength and having been tested at more than one temperature, are shown more prominently so that the typical data trends stand out against the general scatter. At 600 C (Fig. 1) data for typical casts can be seen to define the shape of the LogStresseLogTime curve very consistently. These casts exhibit a severe inflection, starting at about 3000 h and apparently nearing completion at about 30,000 h, and other casts display a consistent behaviour. The MH03 model is visibly too smooth in relation to the individual cast data, showing an insufficiently sharp inflection; whereas the ADA and SDS models follow the individual cast data more closely over the whole range of test times, with the ADA model displaying a somewhat exaggerated inflection. The numerical coefficients of these models were derived by Maximum Likelihood analysis of the data at all temperatures, as described later. Hence none is necessarily the best fit that might be obtained by analysis of the data at 600 C alone. The importance of establishing an accurate model at this temperature is that it provides a guide to the curve shapes at lower temperatures, where the data extend less far into the inflection. In the absence of evidence to the contrary, it is to be expected that a similar characteristic shape will appear after progressively longer times at progressively lower temperatures. Typical casts at 550 C (Fig. 2) display a consistent trend, with clear evidence of an inflection similar to that at 600 C, but the data terminate well before completion of the inflection. It is again evident that the MH03 model is too smooth, lacking a sufficiently
Stress, MPa
1000
100
C14 500C Rupt
C22 500C Rupt
C22 500C Unbr
C24 500C Rupt
C25 500C Rupt
C26 500C Rupt
C27 500C Rupt
C27 500C Unbr
C74 500C Rupt
C74 500C Unbr
C75 500C Rupt
C75 500C Unbr
C76 500C Rupt
C76 500C Unbr
MH03 500C
ADA 500C
SDS 500C
C29 600C Rupt
C29 600C Unbr
C30 600C Rupt
C30 600C Unbr
C31 600C Rupt
C32 600C Rupt
C32 600C Unbr
C33 600C Rupt
C34 600C Rupt
C34 600C Unbr
MH03 600C
ADA 600C
SDS 600C
100
1000
10000
100000
Time, h
Fig. 1. 12Cr steel rupture data and models at 600 and 500 C.
10 1000000
160
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Stress, MPa
1000
100 C29 475C Rupt
C30 475C Rupt
C31 475C Rupt
C32 475C Unbr
C33 475C Rupt
C33 475C Unbr
C34 475C Rupt
MH03 475C
ADA 475C
SDS 475C
C14 550C Rupt
C22 550C Rupt
C22 550C Unbr
C24 550C Rupt
C25 550C Rupt
C25 550C Unbr
C26 550C Rupt
C26 550C Unbr
C27 550C Rupt
C29 550C Rupt
C29 550C Unbr
C30 550C Rupt
C30 550C Unbr
C31 550C Rupt
C32 550C Rupt
C33 550C Rupt
C33 550C Unbr
C34 550C Rupt
C34 550C Unbr
C71 550C Rupt
C72 550C Rupt
C73 550C Rupt
C74 550C Rupt
C75 550C Rupt
C76 550C Rupt
MH03 550C
ADA 550C
SDS 550C
100
1000
C32 475C Rupt
10000
10 1000000
100000
Time, h
Fig. 2. 12Cr steel rupture data and models at 550 and 475 C.
sharp inflection. Typical gradients at 30,000 to 60,000 h are visibly closer to the SDS model than to either of the other models. The differences between the models are sufficient at 550 C to be of practical importance. A median rupture life of 100,000 h according to the MH03 model is an overestimate, relative to the SDS model, of about 50%. At 500 C (Fig. 1) the three models are in close agreement up to about 60,000 hours. Beyond that time the MH03 model is clearly too high relative to the scatter of data. The data exhibit the early stages of inflections, but only for the weakest cast (C22) is a welldeveloped inflection visible. Extrapolation of the median curve depends heavily upon projecting trends established at 550 C and 600 C, for which the SDS model appears the most reliable. At 500 C the discrepancies between the models at times of practical importance are large. Rupture lives of 200,000 to 300,000 h according to the MH03 model are overestimates, relative to the SDS model, of 50% to 100%. At 475 C (Fig. 2) the data for typical casts are consistent with all three models. Extrapolation into the inflected portion of the rupture curve is totally dependent upon model accuracy at higher temperatures, which favours the SDS model. Discrepancies between the two models at 475 C and at times of practical importance are significant. Rupture lives of 200,000 to 300,000 h according to the MH03 model are overestimates, relative to the SDS model, of up to 50%.
4. Likelihood and deviance calculations In the statistical analysis discussed here, each mathematical model incorporates a “systematic” component and a “random” component. The systematic component is an equation that describes the median time to rupture in terms of stress and temperature, incorporating a number of numerical coefficients as in Appendix 2. The random component is an equation that defines the distribution of rupture times about the median, incorporating additional coefficients as in Appendix 1. An optimum set of coefficients for both components is obtained by iteration of their values to obtain the maximum overall Likelihood. The overall Likelihood is the product of all individual
Likelihoods for each failed and unfailed test, determined by statistical analysis as described in Refs. [1,2] and recapitulated in Appendix 1. Optimised rupture models are conveniently compared in terms of Deviance, which is related to Likelihood as in Appendix 1. The model that corresponds most accurately to the data is that with the lowest value of Deviance. Values of Deviance for the models considered are shown in Table 1. The models are fully defined in Appendices 1 and 2 and assume Weibull two-parameter distributions for the random component, with heteroscedasticity. The lower Deviance of the SDS model confirms what is visually evident from Figs. 1 and 2, that this model is a significantly better fit to the data than the other two. Part of this difference is required to justify the inclusion of additional terms, as discussed in Appendix 2, but the requirement amounts to only a small fraction of the improvement achieved. The SDS model is therefore much more likely to be a reliable basis for extrapolation, provided that there is no separate evidence to the contrary. The difference in total Deviance, DD, between any two models in Table 1 can be expressed in terms of relative Likelihood. If one or other model is postulated to be an exact model of the median and distribution of the data, the Likelihood of the set of test results being observed is greater by expðDD=2Þ for the model of lower Deviance. Thus the Likelihood of the ADA model is 9.0 times greater than for the MH03 model, and over a population of 248 tests the Likelihood is 0.9% greater per test. The Likelihood of the SDS model is 2.3 107 times greater than for the MH03 model and is 7.1% greater per test. The lower Deviances of the ADA and SDS models show that they are progressively more accurate representations of the test data, implying that they are increasingly more reliable for extrapolation - subject to appropriate tests of validity. Table 1 Calculated Deviances for the SDS, ADA and MH03 models. Test population
Deviance
Category
Number
MH03 model
ADA model
SDS model
Failed Unfailed Total
217 31 248
4393.7 25.6 4419.3
4358.7 56.2 4414.9
4359.3 26.1 4385.4
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
161
It is remarkable that the MH03 and ADA models, which differ little in terms of total Deviance or average Likelihood, give widely divergent results when extrapolated. At a stress of 200 MPa and a temperature of 500 C, their median rupture lives are 287,000 h and 111,000 h respectively. The wide divergence associated with only a small difference in Deviance suggests that neither model is reliable. The ADA model is not discussed further here because the significantly lower Deviance of the SDS model shows the latter to be more accurate. There is a minor distinction to be made between the MH03 model for which results are given in Table 1 and the corresponding model (MH03a) of PD6605. They are identical in terms of the mathematical form of the rupture equation (Appendix 2) but differ slightly in other respects. In order to ensure strict comparability with the ADA and SDS models, the MH03 model coefficients were re-optimised for exactly the same dataset and with the same form of heteroscedasticity relationship (Appendix 1), resulting in the Deviance values in Table 1. The total Deviance of the MH03 model is slightly higher than the Deviance of 4417.7 reported for the MH03a (PD6605) model, as a result of using the simpler heteroscedasticity relation discussed in Appendix 2. However, the differences between median rupture lives calculated from the MH03 and MH03a models are insignificantly small, and are barely visible on the scale of Figs. 1 and 2. Rupture lives, Deviances, etc quoted in this paper are for the MH03 model unless otherwise stated, but for practical purposes the MH03 and MH03a models are identical, so comparisons to the SDS model apply equally to both.
Table 2 Relative errors in MH03 model extrapolated lives.
5. Prescribed validity tests
7. Discussion of model validity criteria
A series of validity tests is prescribed in PD6605 with the purpose of ensuring that a model that fits the data acceptably well is sufficiently reliable to warrant extrapolation. Some specify selfevident requirements, such as,
The failure of the PD6605 procedure in the case examined is that its validity tests did not reject the MH03a model, thus failing to compel the original analyst to seek a more accurate alternative. The flaw in the procedure is its default assumption that a model that satisfies its prescribed tests is a reliable basis for extrapolation. Failure to satisfy these tests may be a sufficient reason for rejection, but the converse is clearly untrue; success in satisfying these tests is not a sufficient reason for acceptance. The prime consideration for validation of a model initially selected for its low Deviance is that its rupture curves should visibly duplicate the main trends evident in a graphical presentation of the test data. Figs. 1 and 2 show that the MH03, ADA and SDS models all satisfy this condition in a broad sense, although they differ in their degree of precision. The next consideration is that the model should extrapolate in a stable fashion to well beyond the usual extrapolation interval of two to three times, e.g. to six times the test programme duration. The curves should display no changes in curvature that appear to originate in the mathematical formulation of the model rather than in the visible trends of the data. Figs. 1 and 2 show that the MH03, ADA and SDS models all satisfy this requirement over at least a sixfold extrapolation in time. A further consideration is that the model should produce consistent extrapolations, exhibiting only moderate sensitivity to the relatively sparse data at the longest times. This can be verified by comparative extrapolations from a model based upon a dataset masked at half the eventual test programme duration. For the case examined the nominal duration, bounding 95% of all tests, is 80,000 h. The dataset was masked at 40,000 h by altering each test record at over 40,000 h, whether ruptured or unbroken, to show instead the former state of unbroken at 40,000 hours. The coefficients of the MH03 and SDS models were then re-optimised for minimum Deviance. Rupture lives were re-calculated at the stress corresponding, according to the initial models, to rupture at 160,000 h. The results are given in Table 3.
a) Visual confirmation that the model conforms to the major trends of stress and temperature dependence exhibited by the data, b) Visual confirmation that the model is well behaved in extrapolation to well beyond the intended extrapolation interval, exhibiting no paradoxical changes in gradient. It can be seen from Figs. 1 and 2 that all three models broadly satisfy those requirements. Additional requirements stipulate, c) Limits to the gradient of and scatter about a linear regression of Log(model rupture time) with respect to Log(test rupture time), d) Sensitivity of extrapolation to the random removal of a fraction of the data, e) Sensitivity of extrapolation to especially influential casts. All of the prescribed tests were satisfied by the original MH03a model, relieving the analyst of any obligation to search for a better model, and resulting in its endorsement as sufficiently reliable for extrapolation. However, the SDS model also passes these validity tests, which creates a contradiction: two models that result in gross differences in extrapolated rupture life cannot both be valid. It follows that the prescribed tests are not sufficiently discriminating to reject all unreliable models. 6. Extrapolated rupture lives Median rupture lives extrapolated from the SDS and MH03 models at the same stress are shown in Table 2 at temperatures of practical
Temperature, C
Median rupture life, kh
MH03 relative error
MH03 model
SDS model
550
300 200 100
253 150 67
þ19% þ33% þ49%
525
300 200 100
161 109 69
þ86% þ83% þ45%
500
300 200 100
152 123 86
þ97% þ63% þ16%
interest. The corresponding lives predicted by the original MH03a (PD6605) model are within 3% either way of the values given for the MH03 model. Relative errors listed in the final column are relative to the SDS model, which appears to be the more reliable of the two. By comparison to the margins conventionally employed in, for example, the design of power plant, overestimates of up to 30% in extrapolated life are unlikely to impinge upon plant safety. But overestimates of 50% or more introduce some increasing risk that expected design lives might not be attained, and that service failures might occur. Design use of the values recommended in PD6605, or similarly derived for an alloy with similar characteristics, is therefore potentially hazardous.
162
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Particularly at 600 C where the inflection in the data is most fully defined, the model curve is distinctly smoother than the data trend suggests. At 550 C, where the inflection is less well defined, it is nonetheless evident that the MH03 model does not match the typical gradients displayed by the data. And there is further evidence of this at 500 C. A vigilant analyst should recognise these observations as indicative of an unsatisfactory model, and should pursue an alternative model that displays a more pronounced inflection. The ADA model better duplicates the sharpness of the inflection but appears to somewhat exaggerate the effect. b) The SDS model appears in these figures to provide a very close approximation to the data trends, particularly at 600 C where the inflection in the data is most fully developed. A vigilant analyst should recognise also that, e.g. at 550 C, the model succeeds in passing through the median of rupture times at a given stress rather than through the median of rupture stresses at a given time, which can be the source of visual confusion for inflected curves. c) The most difficult criterion of judgement is the likelihood of devising a model that duplicates the visible data trends still better than the SDS model; and this is ultimately a qualitative and subjective judgement. Only if that likelihood seems remote is it fair to conclude that the SDS model is reliable for extrapolation, in the sense that it is as reliable as any that is likely to be found. Judging this from Figs. 1 and 2 is made difficult by the amount of scatter between casts, and it is helpful to re-plot the data as in Fig. 3. In this figure data are re-plotted in terms of equivalent time at a single, chosen temperature, test failure times being multiplied by the ratio of model rupture times at the chosen and actual temperatures. Scatter is suppressed by plotting only the median rupture time for each sub-population at a common stress and temperature. The gradation in symbol size indicates the sub-population size, which varies from one to fifteen. Judged on the basis of coincidence of the model with data for the larger sub-populations, and continuity of the data trend from one temperature to another, the SDS model appears (to the writer) to be unlikely to be surpassed by devising further models.
Table 3 Extrapolated median rupture lives for a dataset masked at half-range. Extrapolated median rupture lives for masked dataset, kh at stresses for an initial extrapolated life of 160 kh Temperature, C MH03 masked MH03 change, % SDS masked SDS change, %
600 168.0 5.0 176.0 10.0
575 167.5 4.7 174.1 8.8
550 166.9 4.3 171.4 7.1
525 166.1 3.8 168.0 5.0
500 165.4 3.4 163.6 2.3
475 165.0 3.1 160.4 0.2
It is suggested that changes within 20% either way are acceptable, and hence that both the MH03 and SDS models are satisfactory as regards consistency of extrapolation. None of the criteria discussed above leads to outright rejection of either the MH03 or SDS model, so the SDS model must be judged the more reliable of the two on the basis of its lower Deviance. But this judgement is only relative e because a further model might be devised that is still more accurate than the SDS model. Hence it cannot be concluded that the SDS model is sufficiently reliable for extrapolation. That conclusion could be reached only if there were some absolute criterion of accuracy, such as whether model Deviance is within an acceptable limit. But Deviance reflects both the divergence between model and data and the scatter between individual casts, and scatter may be the predominant factor. So a criterion based upon Deviance is not practical. There is no quantitative criterion of accuracy, so the evaluation of a series of progressively better models, i.e. models of progressively lower Deviance that satisfy appropriate validity criteria, has no numerically definable conclusion. There is no end point at which an analyst, relying upon the criteria of PD6605 or those discussed above, could conclude that a still better model might not produce significantly different extrapolated rupture lives. The only practical solution that presents itself is to apply the first criterion above, of accurate duplication of the trends evident in the data, with the utmost vigilance. Thus, a) The MH03 model appears, in Figs. 1 and 2, to provide a reasonable approximation to the data trends e but careful visual examination reveals that it is far from perfect.
Stress, MPa
1000
100
10 1.E-01
450C, pop.2
450C, pop.3
475C, pop.1
475C, pop.2
475C, pop.3
475C, pop.4-5
475C, pop.6-8
500C, pop.1
500C, pop.2
500C, pop.3
500C, pop.6-8
550C, pop.1
550C, pop.2
550C, pop.3
550C, pop.4-5
550C, pop.6-8
550C, pop.9-11
550C, pop.12-15
600C, pop.1
600C, pop.4-5
600C, pop.6-8
Model
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
o
Equivalent (SDS) time at 525 C, h Fig. 3. Median rupture times for all test sub-populations standardised at 525 C by SDS model.
1.E+07
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
Thus the key challenges facing the data analyst are to recognise when it is feasible that a better model might be devised, and then to devise it. If the analyst lacks the mathematical skills to devise a better model, then the challenge is to recognise when more expert assistance is required. The plethora of validity tests in PD6605 is inimical to these key judgements because they present the semblance of a sufficient proof of model reliability. On the basis of these tests, the original analysis endorsed a model that is unconvincing when tested by conventional, visual comparison to the data on a LogStresseLogTime plot, as in Figs. 1 and 2. 8. Conclusion Comparison of the rupture model recommended in PD6605 for a 12Cr alloy steel to an alternative model revealed a contradiction. Both models conformed to the validity criteria prescribed in PD6605, but there were gross discrepancies between their extrapolated lives for the same conditions of stress and temperature. The PD6605 recommended model predicted median rupture lives up to twice as long as the alternative model. The alternative model was found to be the statistically better fit to the data and hence, with no evidence from within the data to support any contrary conclusion, to be the more reliable basis for extrapolation. It follows that the PD6605 validation tests are not sufficiently discriminating. The ineffectiveness of these tests is a fundamental deficiency in PD6605, because they are the means of prompting the analyst to seek a more suitable model. In the absence of effective, quantitative tests of validity, the most discriminating means of verifying a model is the conventional, visual comparison of model to test data on a LogStresseLogTime plot. Any visible departure of a candidate model from the typical shapes exhibited by individual casts should be regarded as indicating that the model is unsatisfactory. Judgement based upon this comparison is subjective, but appears in the case examined to be effective when applied sufficiently critically. The demonstrated unreliability of PD6605 indicates that it is insufficiently developed to warrant the authority of a British Standard. The British Standards Institution was alerted to this problem in November 2009 and is presently, in April 2011, considering whether to constitute a panel to revise the procedure or to withdraw it. Acknowledgement The writer is indebted to Mr Derek Bolton for advice on the mathematics of the statistical analysis, and for providing independent corroboration of numerical results by means of a Cþþ program. Appendix 1. Calculation of likelihood A. Dependence of likelihood on the systematic component of the rupture equation Any candidate equation for the dependence of mean rupture time on applied stress and temperature (the systematic component of the rupture model) contains a number of numerical coefficients. For any trial set of numerical coefficients, the discrepancy between each observed failure time (tri) and the model expected failure time (tmi) may be expressed as tri/tmi. The distribution of all such discrepancies may then be expressed as a cumulative probability,
Prftr =tm < zg ¼ f ðzÞ Function f(z) is assumed to be of a convenient mathematical type, usually a Weibull or a LogLogistic distribution of tr/tm, as discussed later.
163
For any trial set of coefficients, the Likelihood (LFi) of an individual test failure time tri is provisionally, or finally, calculated as,
LFi ¼
dPrftr =tm < tri =tmi g dtr
This is not the Probability of the individual discrepancy. It is the gradient, or Probability Density, which is directly proportional to the probability of failure within a small interval either side of the observed rupture time. Likelihood has the dimensions of inverse time. The combined Likelihood of all the observed failure times, for a population NF of failed tests, is the product of the individual Likelihoods,
LF ¼
NF Y
LFi
i¼1
Evaluation of LF is a means of testing one set of coefficients against another set, the set with the higher Likelihood being preferred. Thus a Maximum Likelihood may be obtained by trial optimisation of the coefficients. A typical set of rupture data contains a number of unbroken test results, which can be used in a similar way. The Likelihood of individual survival time tui is,
LUi ¼ 1 Prftui =tmi < zg This is the Probability of the individual survival, which is dimensionless, and the combined Likelihood of a population NU of survivals is the product,
LU ¼
NU Y
LUi
i¼1
Hence the combined Likelihood of both the observed failures and observed survivals is the product,
L ¼ LF $LU A preferred rupture model is found by optimising the values of coefficients to give the Maximum Likelihood L, including both failed tests and unbroken tests. Candidate rupture models are compared in PD6605 in terms of Deviance, D, which is related to Likelihood, L, by the equation,
D ¼ 2lnðLÞ Strictly, Deviance is given by the difference 2½lnðLsat Þ lnðLÞ, where Lsat is the Likelihood of the Saturated Model e a model having a number of terms equal to the number of test results. But since competing models are ranked on the basis of their difference in Deviance, the omission of this common term is unimportant. C. Dependence of Likelihood on the random component of the rupture equation The standard mathematical distributions of time discrepancy considered in PD6605 and in this paper are the Weibull and LogLogistic distributions. Note that, throughout this paper, the term “mean rupture time” is used in accordance with engineering usage. However, the statistical analysis of discrepancies tr/tm is actually based on the median (middle) value rather than the mathematical mean (arithmetic or logarithmic average) value, and the median value is taken to be 1. For a Weibull (two-parameter) distribution of tr/tm the Cumulative Probability and Probability Density are:
164
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SD ¼ b Gð1 þ 2=aÞ G2 ð1 þ 1=aÞ
Prftr =tm < tri =tmi g ¼ 1 eðtri =btmi Þ a a tri a1 eðtri =btmi Þ d Prftr =tm < tri =tmi g ¼ b btmi tmi dtr where b ¼ elnð1=ln2Þ=a , as required for Median ðtr =tm Þ ¼ 1, and a is a coefficient determined by trial to result in Maximum Likelihood. For a Log-Logistic distribution of tr/tm the Cumulative Probability and Probability Density are, for Median (tr/tm) ¼ 1:
Prftr =tm < tri =tmi g ¼
1 1 þ ðtri =tmi Þ1=b
d Prftr =tm < tri =tmi g ¼ dtr
ðtri =tmi Þðbþ1Þ=b 2 btmi 1 þ ðtri =tmi Þ1=b
a ¼ expðD0 þ D1 T þ D2 log sÞ
where b is a coefficient determined for Maximum Likelihood. PD6605 recommends the Weibull distribution as a default option, although this is not fully justified unless the Maximum Likelihood found for this distribution is greater than for the LogLogistic distribution. In the case studied slightly greater values of Likelihood were obtained for the Weibull distribution, and only figures for that case are quoted. Values of the coefficient a, for the case studied, are given in Appendix 2. If the values of a (or b) above are constant, then the scatter in tr/tm (or the Standard Deviation of tr/tm) is uniform and the distribution is “homoscedastic”. However it is common, as in the case studied, to find that the distribution is “heteroscedastic” e the scatter displays a variation with stress, or temperature. Heteroscedasticity is incorporated in the random component of the rupture model by defining a (or b) as a function of stress, s, and temperature, T. In the case studied the relationship chosen to represent the observed trends in Standard Deviation was,
a ¼ D0 þ D1 s This equation has no temperature term because calculated Likelihoods were found to be insensitive to the inclusion of such a term. The suitability of this relationship is shown in Fig. 4, where Standard Deviations of tri =tmi for the SDS model, for sub-populations of at least four failed tests at a common stress (or narrow range of stress) and common temperature, are plotted against stress. Standard Deviation values for the model Weibull distribution were calculated from,
This equation was not used in the present study because it predicts that Standard Deviation diminishes rapidly, and improbably, at low stresses below the range tested. It also includes a temperature term, which was found redundant in the statistical analysis of the MH03, ADA and SDS models. The inclusion of heteroscedasticity has little effect upon determination of the optimum coefficients of the median rupture equation, and hence little effect upon extrapolated median rupture lives. It is important only in the context of estimating extrapolated rupture lives at a high confidence level, e.g. at 95% confidence, which is not discussed here. Appendix 2. Details of rupture models A. MH03 model The MH03 model is a model of the MansoneHaferd type, and has the following equation for median rupture life,
o n lnðtm Þ ¼ B0 þ B1 þ B2 log s þ B3 ðlog sÞ2 þB4 ðlog sÞ3 T where s ¼ Test stress, MPa; T ¼ Test temperature, K; and B0 ; //B4 are numerical coefficients. In this form the equation includes logarithmic terms both to base e and base10. For the MH03a model, determined by Maximum Likelihood calculation for the subject data set as reported in PD6605, the numerical coefficients of the systematic component of this model were,
B0 ¼ 47:409; B1 ¼ 0:099122; B2 ¼ 0:19674;
10
Standard Deviation of tr/tm (SDS Model)
where G is the Gamma function. The scatter in Standard Deviation for the test results is large, which is not surprising in view of the small sub-population sizes, but Fig. 4 nonetheless supports the dependence on stress and independence of temperature deduced from Likelihood calculations. The model Standard Deviation diminishes to a steady value at low stresses. Values of coefficients D for the case studied are given in Appendix 2. The heteroscedasticity relationship recommended in PD6605 and used for the MH03a model was:
B3 ¼ 0:09328; B4 ¼ 0:015216
Data 475C
And the random coefficients defining the Weibull a parameter, as a ¼ expðD0 þ D1 T þ D2 log sÞ, were,
Data 500C Data 550C
D0 ¼ 3:311; D1 ¼ 0:0003944; D2 ¼ 1:3567
Data 600C SDS Random Model
The model was selected in PD6605 as the best of its library of models, resulting in a lower Deviance than any other model considered that satisfied the prescribed validity tests. For the MH03 model established for this study, which generates rupture lives for any test condition almost identical to those for the MH03a model, the systematic coefficients were,
1
B0 ¼ 47:58; B1 ¼ 0:098865; B2 ¼ 0:19674; B3 ¼ 0:09328; B4 ¼ 0:015212
0.1 0
100
200
300
400
500
600
Stress, MPa Fig. 4. Heteroscedasticity for sub-populations (min. 4 tests) at a common stress and temperature.
And the random coefficients defining the Weibull a parameter, as a ¼ D0 þ D1 s, were
D0 ¼ 2:30; D1 ¼ 0:002795
J. Bolton / International Journal of Pressure Vessels and Piping 88 (2011) 158e165
B. ADA model
Table 4 Deviance reductions justifying additional coefficients.
The ADA model was developed for the present study and its structure is described in general terms in Section 2. In mathematical terms it is described by the following equation,
ln tm
165
¼ Vln V B0 expðDT=B1 Þ 1 þ ð1 VÞln V B2 expðDT=B1 Þ 1
Model
No. coeff’ts
Total deviance
Deviance reductions relative to MH03 (or to ADA) model Reduction
where V ¼ fs=ðB3 expðDT=B4 ÞÞgB5 ; s ¼ test stress, MPa; DT ¼ 600
-Test temperature, C; and B0 ; .B5 are numerical coefficients. The systematic coefficients of this model for the subject dataset, determined by Maximum Likelihood calculation, were,
B0 ¼ 17:66; B1 ¼ 566:8; B2 ¼ 2:827; B3 ¼ 396:6; B4 ¼ 185:7; B5 ¼ 1:704;
MH03 MH03a ADA SDS
7 8 8 9
4419.3 4417.7 4414.9 4385.4
0 1.6 4.4 33.9
0 29.5
PD6605 criterion
Akaike criterion
2.7 2.7 5.4
2.1 2.1 4.3
2.7
2.2
PD6605 imposes a criterion for the acceptance of Model B (containing kB parameters) and rejection of Model A (containing kA parameters) in the form:
And the numerical coefficients of the Weibull distribution were,
Deviance ðModel AÞ Deviance ðModel BÞ > 2:7ðkB kA Þ
D0 ¼ 2:32; D1 ¼ 0:00270
A more suitable criterion is the Akaike Information Criterion [4] which requires a reduction in Deviance of,
C. SDS model
2kB ½1 þ ðkB þ 1Þ=ðN kB 1Þ 2kA ½1 þ ðkA þ 1Þ=ðN kA 1Þ
The SDS model was also developed for the present study and its structure is described in general terms in Section 2. In mathematical terms it is described by the following equation,
where N ¼ Number of test results (248). Table 4 shows the Deviance reductions for each of the models discussed and the requirements of the above criteria. Figures are given relative to the MH03 model and, in italics, relative to the ADA model. The reduction in Deviance between the MH03 and MH03a models is seen to be insufficient to justify the additional (random) coefficient of the latter. This confirms the redundancy of a temperature term in the heteroscedasticity equation evident from the statistical analysis of the MH03 model. The progressively higher numbers of (systematic) coefficients in the ADA and SDS models achieved reductions in Deviance that, in each case, satisfied these criteria.
ln tm ¼ B0 þ B1 ln S þ B2 S þ B3 DT þ B4 = 1 þ ðS=B5 Þ2 where S ¼ seDT=B6 ; s ¼ Test stress, MPa; DT ¼ 600 -test temperature, C; and B0 ; .B6 are numerical coefficients. The numerical coefficients of this model for the subject 12Cr dataset, determined by Maximum Likelihood calculation, were,
B0 ¼ 48:788; B1 ¼ 2:6009; B2 ¼ 0:065587; B3 ¼ 0:02506; B4 ¼ 24:57; B5 ¼ 219:45; B6 ¼ 235:66 And the random coefficients defining the Weibull a parameter, as a ¼ D0 þ D1 s, were
D0 ¼ 2:543; D1 ¼ 0:003215 D. Number of model coefficients The MH03 model includes seven (systematic and random) numerical coefficients compared to eight for the original MH03a model, eight for the ADA model and nine for the SDS model.
References [1] British Standards Institute. Published Document 6605-1:1998, Guidance on methodology for assessment of stress-rupture data e Part 1: Procedure for derivation of strength values, 15 Dec 1998. [2] Davies RB, Hales R, Harman JC, Holdsworth SR. Statistical modelling of creep rupture data. Journal of Engineering Materials Technology (ASME) 1999;121(3): 264e71. [3] Bullough CK, Norman M. The PD6605 creep rupture assessment procedure e an appraisal of its application ten years on, Creep and fracture in high temperature components, 2nd ECCC Creep conference, Zurich, 21e23 Apr 2009. [4] Akaike, Hirotogu. A new look at the statistical model identification. IEEE Trans Automatic Control 1974;19(6):716e23.