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 andhighest 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 allcalculated 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 locationparameter 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 theanalysis 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
* 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.
| 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 |
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 analysisnotation, 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 suchthat 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-dominatedassessments. 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 of2mm
(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% wasassumed 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 theregion-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-specificmodel 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 (allregions, 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)
Cov is coefficient of variation
*Weibull distribution assumed with location parameter = 0 and shape
parameter = 5.83
*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 |
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 effecton 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
largeeffect 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 toextend 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 theelastic-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 FADand 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, notbe 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
smalldifferences 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 probabilisticanalyses 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 ischaracterised 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 outmuch 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.
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