1TWI, Granta Park, Great Abington, CB1 6AL, UK
2HSE, Rose Court, 2 Southwark Bridge, London SE1 9HS, UK
Paper presented at ECF 13, San Sebastian, Spain, 6 - 9 September 2000.
Abstract
The results of a comprehensive validation study of the BSI PD6493:1991 procedures were collated and processed in order to quantify the position of test failure points relative to the assessment line of the FAD. The results came mainly from wide plate tests with some vessel and pipe tests in several material types. Statistical analyses were carried out to derive probability distributions for the modelling uncertainty associated with the Level 2 FAD. Typical probabilistic fracture mechanics example calculations were then conducted to investigate the effect on failure probability estimates and the implications for partial safety factors given in BS 7910:1999 which were derived without considerations for modelling uncertainty in the FAD. The results show that failure probability estimates are reduced slightly, up to about an order of magnitude, by including modelling uncertainty in the fracture mechanics analyses. Regarding implications for the published partial safety factors (PSF's) in BS 7910:1999, the results suggest that the PSF's correspond to slightly lower failure probabilities than the specified target values. However, the reduction in failure probability by including modelling uncertainty is considered to be insufficient to warrant a change in partial safety factors in BS 7910:1999.
Introduction
Fracture mechanics procedures given in BSI PD6493:1991 [1] and BS7910:1999 [2] are based on the well known concept of the failure assessment diagram (FAD). It is also known from validation studies that assessment points falling outside the assessment lines in these diagrams do not necessarily represent failure. Therefore, there is a modelling uncertainty associated with the FAD's. The work described uses the results of large scale validation studies to estimate the uncertainty in the Level 2 FAD. Statistical analyses of the validation data is conducted to derive the modelling uncertainty which is then applied in example calculations to explore the effect on failure probability estimates. Also, the implications for partial safety factors - which were derived without consideration of modelling uncertainty - recommended in the BS 7910:1999 are discussed.
Wide plate and structural test data
In 1995, Challenger et al [3] reported the results of an extensive validation study of the PD6493:1991 fracture assessment procedures. The study was based on data generated over the years at TWI and other published data. It consisted mainly of wide plate test results covering several materials groups including pressure vessel and pipeline steels and aluminium alloys. The data also included some pressure vessel and pipe burst tests. Most of the results presented by Challenger for validating the Level 2 procedures were utilised in the present work. The details of the tests such as the material type, specimen geometry, tensile properties and fracture toughness (K mat or CTOD, δ mat) values, and the calculated fracture ratio K r (or √ δ r) and plastic collapse ratio S r were recorded on an EXCEL spreadsheet. In most cases, the same test data was analysed in terms of both K and CTOD [3] . Two sets of analyses were therefore carried out in the present work in terms of both parameters. The trend of results was very similar for both approaches, therefore only the K-based analysis is presented in this paper.
Uncertainty in failure assessment diagram
General
The validation data available was filtered to obtain the most appropriate data set for the analyses. Where the same test results were presented using different assumptions in the analysis, these were reviewed to decide on the appropriate set of results to include in this study. The data used in this analysis are shown along with the Level 2 FAD in Fig.1. Following screening of the data set, a total of 90 data test results was included in the analysis. These results are for ferritic steels and exclude results from stainless steels and aluminium alloys.
In order to quantify the modelling uncertainty in the FAD, a measure is required of the relationship between the assessment line and actual component failure point. Figure 2 illustrates the approaches adopted. For a failure point F, a measure of the uncertainty in prediction of the model can be obtained by comparing the radial distances R and r from the origin to the point and FAD respectively. In the classical 'safety factor' format, the ratio R/r could be used as the measure. An alternative measure of the uncertainty is also obtained from the difference R-r. This latter format is more appropriate for the TWI reliability software FORM and MONTE, so this formulation was used in this study to define modelling uncertainty, M u. Whilst Fig.2 illustrates the uncertainty measure with respect to one failure point, the procedure has to be applied to every data point available so that the general trend can be identified.
Variation of Modelling Uncertainty (M u) with FAD regions
An examination of the wide plate data (see Fig.1) shows the presence of a variation in behaviour with position in the FAD. Failure points are much closer to the FAD in the knee (elastic-plastic) region than in other regions. Also, the data points appear to be less scattered in the knee region. The variation of M u with the angle θ between the S r-axis and a line from the origin to the assessment point was examined and the results are shown in Fig.3, together with a mean regression line. The trend that emerges is that on average the margin of safety on the FAD is least in the middle (elastic-plastic) region, slightly higher in the 'plastic collapse' region and highest in the 'elastic fracture' region. However, there is a varying degree of scatter in the different regions. Statistical analysis was conducted on each set of data in order to evaluate the scatter quantitatively.
Statistical analysis of data
The data were divided into three equal sections or regions (see Fig.1). Statistical distributions were then fitted to the data falling in each region and also to all the data combined. Two data points fell inside the FAD thereby giving negative R-r values; so a constant was added to all calculated R-r values prior to distribution fitting. This modification amounted to introducing a location parameter or a threshold value corresponding to the minimum data value. The minimum calculated R-r value was -0.059 so a location parameter of -0.06 was used.
The statistical analyses were conducted using the MINITAB software package. This uses the maximum likelihood method to estimate parameters for normal, log e-normal, exponential and Weibull distributions. Goodness of fit is judged graphically in MINITAB by examining the probability plots. The choice being based on how close the points fall to the straight line, particularly in the tail regions of the distribution. The decision was further verified by using a TWI software DISTFIT which calculates the Kolmogorov-Smirnov statistic D* as a quantitative measure of goodness of fit. In all the analysis cases, selections based on both MINITAB and DISTFIT were in agreement.
A summary of the results of the statistical analyses is given in Table 1. This supports the trend observed in Fig.3. The order of the standard deviation values is similar to that of the mean values. The standard deviations are generally high implying a high degree of scatter.
Table 1 Results of statistical analysis of K-based validation data in terms of R-r
| FAD Region | Best-fit distr. | *Parameters | Moments |
|---|
| Location | Scale | Shape | Mean | Std. Dev |
|---|
Elastic
( θ =60-90°) |
Weibull |
-0.06 |
1.90 |
2.13 |
1.62 |
0.83 |
Elastic-plastic
( θ =30-60°) |
Weibull |
-0.06 |
0.55 |
1.08 |
0.47 |
0.49 |
Collapse
( θ =0-30°) |
Exponential |
-0.06 |
0.83 |
1.00 |
0.77 |
0.83 |
All
( θ =0-90°) |
Weibull |
-0.06 |
0.97 |
1.11 |
0.87 |
0.84 |
* The minimum (R-r) value was -0.054, so 0.06 was added to all values before calculating the scale and shape parameters; the location parameter was fixed at -0.06.
Use of modelling uncertainty in probabilistic analysis
General
The general definition of the Level 2 limit state function in the TWI software FORM and MONTE is may be stated as:
Z = X - Y [1]
where 'X' is the radial distance from the FAD origin to the failure assessment line and 'Y' is the distance from the FAD origin, along the same the same radial line to the assessment point. In the usual reliability analysis notation, Z > 0 is safe while Z ≤ 0 denotes the failure condition.
Extending this definition to include modelling uncertainty (M u = R-r) gives
Z = X + M u - Y [2]
Equation [2] is the failure condition implemented in the FORM/MONTE program. To analyse a case without allowance for modelling uncertainty M u is taken as a fixed value of 0 while the statistical distributions discussed in the preceding section are entered for M u in order to include this in the calculations.
Application to typical examples
Reliability calculations were carried out using input data selected to investigate the sensitivity of results in different regions of the FAD. In general, the input data are typical of welded components, but they were chosen such that deterministic assessments based on the mean property values lay in one of the three sections of the FAD, see Fig.1. Table 2 summarises the input data used for these analyses. It is noted that relatively high toughness values are selected in order to bias failure towards plastic collapse while lower values are used for fracture-dominated assessments. The reverse trend is noted with respect to tensile properties. In the elastic-plastic region, intermediate values of fracture toughness and tensile properties are used. A semi-elliptical surface flaw with a mean height of 2mm (normal distribution) and a length of 40mm (assumed fixed) was considered in the calculations. In general a coefficient of variation (cov) value of 10% was adopted for most variables. However, a slightly higher cov of 20% was assumed for fracture toughness and flaw height to reflect the fact that higher uncertainties tend to be associated with these two variables.
Table 2 Basic input data for probabilistic analyses
| FAD Region | 1 | 2 | 3 |
|---|
| Dominant failure mode |
Elastic fracture
( θ = 60-90°) |
Elastic-plastic
( θ = 30-60°) |
Plastic collapse
( θ = 0-30°) |
| Geometry |
|---|
| Thickness |
60 |
25 |
15 |
| Width |
1000 |
1000 |
100 |
| Stresses |
|---|
Primary membrane
Distribution
Mean
Cov |
Normal
90
0.10 |
Normal
250
0.10 |
Normal
240
0.10 |
Residual stress
Distribution
Mean
Cov |
Normal
379
0.10 |
Normal
379
0.10 |
Normal
180
0.10 |
| Material Properties |
|---|
Yield strength (N/mm 2)
Distribution
Mean
Cov |
Normal
550
0.10 |
Normal
450
0.10 |
Normal
300
0.10 |
Tensile strength (N/mm 2)
Distribution
Mean
Cov |
Normal
660
0.10 |
Normal
540
0.10 |
Normal
360
0.10 |
Fracture toughness, K mat (N/mm 3/2)
Distribution
Mean
Cov
Location parameter
Scale parameter
Shape parameter |
Weibull
2200
0.2
0.0
2375.9
5.83 |
Weibull
4000
0.2
0.0
4319.7
5.83 |
Weibull
7500
0.2
0.0
8099.5
5.83 |
For each case considered, three failure probability (P f) calculations were carried out using FORM. First, P f was evaluated without any allowance for modelling uncertainty (M u), then a second calculation included the uncertainty associated with the specific region. Finally, a third calculation is performed using M u derived from the combined data (including all regions). The results of the analyses are presented in Table 3. This shows a reduction in failure probability due to the inclusion of modelling uncertainty. However, the reduction in P f is in general small, typically less than one order of magnitude, except when using the appropriate M u for the elastic fracture region, where an order of magnitude reduction was obtained. For the elastic-plastic and plastic collapse regions, there was little difference between the results obtained from the region-specific uncertainty distribution and those based on a general weighted model covering all FAD regions.
Table 3 Effect of modelling uncertainty on failure probability estimates
| FAD Region | Failure probability, P f |
|---|
| No modelling uncertainty (M u =0) | *Region specific uncertainty | General uncertainty |
|---|
1
(Elastic fracture) |
5.94E-2 |
1.61E-3 |
1.53E-2 |
2
(Elastic-plastic) |
1.06E-2 |
4.40E-3 |
2.72E-3 |
3
(Plastic-collapse) |
2.35E-2 |
5.16E-3 |
3.73E-3 |
Overall, it seems reasonable to adopt the modelling uncertainty derived from the combined data (e.g. Weibull {location = -0.06, scale = 0.97, shape = 1.11}) in probabilistic analyses. This is appropriate because a region-specific model is only likely to give a significantly different result in only one region and also because it is not always known beforehand which failure mode is going to dominate.
Further probabilistic analyses were conducted to investigate the effect of modelling uncertainty (M u) over a wider range of failure probability estimates. This aspect of the work was dictated by the fact that the target failure probabilities considered in BSI 7910:1999 are in the range 2.3x10 -1 to 1x10 -5. The basic input data of Table 2 (elastic-region) were adopted in the analyses with the failure probability values varied in the required range by assuming a range of fracture toughness mean values. The general modelling uncertainty distribution (all regions, Table 1) was used in the analysis and the results are given in Table 4. Again this shows only a slight reduction in P f over the entire range of P f values considered and suggests a consistent effect of modelling uncertainty.
Table 4 Effect of M u over a wide range of failure probability estimates (Basic input data from Table 2, Region 1; variation with mean K)
*Mean fracture toughness, N/mm 3/2
Cov = 20% | Failure probability, P f |
|---|
No modelling uncertainty
(M u =0) | General uncertainty model |
|---|
1500
(Scale = 1620) |
3.90E-1 |
1.00E-1 |
2000
(scale = 2160) |
9.95E-2 |
2.55E-2 |
2200
(scale = 2376) |
5.94E-2 |
1.53E-2 |
3000
(scale = 3240) |
1.03E-2 |
2.74E-3 |
4000
(scale = 4320) |
1.95E-3 |
5.30E-4 |
5000
(scale = 5400) |
5.34E-4 |
9.69E-5 |
6000
(scale = 6480) |
1.85E-4 |
3.14E-5 |
7000
(scale = 7560) |
7.55E-5 |
1.48E-5 |
Cov is coefficient of variation
*Weibull distribution assumed with location parameter = 0 and shape parameter = 5.83
Discussion
General trend in results
The effect of including modelling uncertainty in the FAD is to reduce the failure probability estimates slightly, typically by less than one order of magnitude. It is thought that one reason for not having a more significant effect on the failure probability estimates is the large amount of scatter obtained in the validation tests. In effect, the benefit of adopting a limit state based on actual failure points lying predominantly outside the FAD is to a large effect cancelled out by the amount of scatter in these actual failures, see Table 1.
The present work was based on the Level 2 FAD of BSI PD6493:1991 because most of the results of the validation study used this assessment procedure. However, this FAD is not included in BS 7910:1999. There is therefore a need to extend the work to the new Level 2A FAD in BS 7910. However, it is considered that analyses based on the Level 2A FAD are likely to produce similar results to those obtained in the elastic region, but may be slightly different in the elastic-plastic and plastic collapse regions.
When conducting probabilistic fracture mechanics, it is considered appropriate to include modelling uncertainty in the probabilistic model. The general uncertainty model based on R-r (i.e. Weibull {location = -0.06, scale = 0.97, shape = 1.11} for the K-based approach) should be adequate for assessments to the Level 2 FAD. Where there is confidence regarding the dominant region of the FAD, then the appropriate model for that region may be used.
In the elastic-plastic region, the Level 2A FAD of BS 7910:1999 falls inside the PD6493:1991 Level 2 FAD used in the present work. It is therefore possible that failure probability estimates based on the BS 7910:1999 Level 2A FAD and including modelling uncertainty, would give much lower failure probability estimates in the elastic-plastic region, then those obtained in the present work.
Potential effects on BS 7910 target failure probabilities and PSF's
Modelling uncertainty (M u) on failure probability was found in this study to reduce the failure probability estimates only slightly. It is therefore likely that the current BS 7910 partial safety factors (PSF's) would, in most cases, not be significantly changed by including modelling uncertainty in the probabilistic fracture mechanics calculations. However, an examination of the current BS 7910:1999 PSF's suggests that these could be significantly different for small differences in target failure probabilities (e.g. 2.6 at P f = 7x10 -5 and 3.2 at P f = 1x10 -5 for K mat). Therefore any improvements in PSF's from consideration of M u are most likely for the lowest target failure probabilities.
Conclusions
Statistical analysis of an extensive database of large-scale validation data was carried out to derive the distribution of the modelling uncertainty associated with the Level 2 FAD. These were then incorporated into probabilistic analyses using typical example calculations. The following conclusions are drawn:
The margin of safety and hence the modelling uncertainty varies around the failure assessment diagram (FAD) with the highest margin in the elastic region and the minimum in the elastic-plastic region. However, the data is characterised by a large amount of scatter.
Modelling uncertainty for Level 2 K-based FAD in BSI PD6493:1999 can be described by a three parameter Weibull distribution (location = -0.06, scale = 0.97 and shape 1.11).
The reduction in failure probability by including modelling uncertainty is considered to be insufficient to warrant a change in partial safety factors in BS 7910:1999.
Acknowledgement
The work described in this paper was sponsored by the Health and safety Executive (HSE), United Kingdom, and the financial support is gratefully acknowledged. The authors also wish to thank Ms Ruth Sanderson of TWI who carried out much of the wide plate data processing and analysis.
References
- BSI PD6493:1991 (1991) Guidance on methods for assessing the acceptability of flaws in fusion welded structures. British Standards Institution, London.
- BS 7910:1999 (1999) Guide on methods for assessing the acceptability of flaws in fusion welded structures, British Standards Institution, London.
- Challenger, N.V., Phaal, R. and Garwood, S. J. (1995) Appraisal of PD 6493:1991 fracture assessment procedures Part I-III, TWI Report 512/1995.