Proceedings of the 18th International Conference on Environmental Degradation of Materials in Nuclear Power Systems – Water Reactors pp 1995-2011 | Cite as
A Statistical Analysis on Modeling Uncertainty Through Crack Initiation Tests
Abstract
Because a large time spread in most crack initiation tests makes it a daunting task to predict the initiation time of cracking, a probabilistic model, such as the Weibull distribution, has been usually employed to model it. In this case, although it might be anticipated to develop a more reliable cracking model under ideal cracking test conditions (e.g., large number of specimen, narrow censoring interval, etc.), it is not straightforward to quantitatively assess the effects of these experimental conditions on model estimation uncertainty . Therefore, we studied the effects of some key experimental conditions on estimation uncertainties of the Weibull parameters through the Monte Carlo simulations. Simulation results suggested that the estimated scale parameter would be more reliable than the estimated shape parameter from the tests. It was also shown that increasing the number of specimen would be more efficient to reduce the uncertainty of estimators than the more frequent censoring.
Keywords
Weibull distribution Estimation Monte carlo simulationNomenclature
- CDF
Cumulative distribution function
- \( F\left( \cdot \right) \)
Cumulative distribution function of Weibull distribution
- ECF
End cracking fraction
- \( \hat{\eta } \)
Estimator of Weibull scale parameter
- \( \hat{\beta } \)
Estimator of Weibull shape parameter
- EVD
Extreme value distribution
- EVDm
Extreme value distribution for minima
- GEVD
Generalized extreme value distribution
- iid
Independent and identically distributed
- \( s_{i} \)
Last censoring time of ith suspended specimen
- LCI
Length of censoring interval
- \( L\left( \cdot \right) \)
Likelihood function
- \( \mu \)
Location parameter of generalized extreme value distribution
- \( l\left( \cdot \right) \)
Log-likelihood function
- LB
Lower bound
- \( c_{{j_{L} }} \)
Lower bound time of censoring interval for jth cracking
- MLE
Maximum likelihood estimation
- C
Number of interval-censored cracked specimens
- S
Number of suspended specimens
Probability density function
- \( g\left( \cdot \right) \)
Probability density function of generalized extreme value distribution
- \( {\text{RE}}\left( \cdot \right) \)
Relative error
- \( {\text{RE}}_{50\% } \)
Relative error of median estimates
- \( {\text{RLCI}}_{90\% } \)
Relative length of 90% confidence interval
- RTD
Relative test duration
- \( \sigma \)
Scale parameter of generalized extreme value distribution
- \( \eta \)
Scale parameter of Weibull distribution
- \( \xi \)
Shape parameter of generalized extreme value distribution
- \( \beta \)
Shape parameter of Weibull distribution
- SCC
Stress corrosion cracking
- t
Time
- \( \eta_{\text{true}} \)
True Weibull scale parameter
- \( \beta_{\text{true}} \)
True Weibull shape parameter
- UB
Upper bound
- \( c_{{j_{U} }} \)
Upper bound time of censoring interval for jth cracking
- \( x \)
Variable of generalized extreme value distribution
Notes
Acknowledgements
This work was supported by the Nuclear Safety Research Program through the Korea Foundation of Nuclear Safety (KOFONS) granted financial resource from the Nuclear Safety and Security Commission (NSSC), Republic of Korea (No. 1403006), and was supported by “Human Resources Program in Energy Technology” of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), who granted the financial resources from the Ministry of Trade, Industry & Energy, Korea. (No. 20164010201000).
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