Structural reliability analysis with probability-boxes

1. Structural reliability analysis with probability-boxes Hao Zhang School of Civil Engineering, University of Sydney, NSW 2006, Australia Michael Beer Institute for Risk & Uncertainty, University of Liverpool, Liverpool, UK

2. Reliability assessment with limited dataA common scenario • Available data on structural strength and loads are typically limited. • Difficulty in identifying the distribution (type, parameters). • Competing probabilistic models. • Tail sensitivity. • Choice of probabilistic model is epistemic in nature.

3. Reliability assessment with limited dataOptions for solution • Bayesian approach • more subjective • high numerical effort • Imprecise probabilities • Probability box • Random set • Dempster-Shafer evidence theory

4. Presentation outline • Quasi interval Monte Carlo method • Different approaches for constructing P-boxes • Example

5. Monte Carlo method • Probability of failure, Pf , is estimated by • Inverse transform method rj : a sample of iid standard uniform random variates.

6. Interval Monte Carlo method When Fx( ) is unknown but bounded, interval samples can be generated Define then One has

7. Interval Monte Carlo method A lower and an upper bound for Pf can be estimated as Variance of direct interval Monte Carlo

8. Low-discrepancy sequences • Improvement of • sampling quality • convergence • numerical efficiency 2D scatter plots: (a) random sample; (b) Good lattice point; (c) Halton sequence; (d) Faure sequence.

9. Variance for interval quasi-Monte Carlo • A variance-type error estimate cannot be obtained directly because low-discrepancy sequences are deterministic. • An empirical variance estimate for interval quasi-Monte Carlo can be obtained by using randomized low-discrepancy sequence.

10. Presentation outline • Quasi interval Monte Carlo method • Different approaches for constructing P-boxes • Example

11. Construction of P-boxKolmogorov-Smirnov confidence limits Fn(x) = empirical cumulative frequency function Dnα= K-S critical value at significance level α with a sample size of n • Non-parametric. • The derived p-box has to be truncated.

12. Construction of P-boxChebyshev’s inequality If the knowledge of the first two moments (and the range) of the random variable is available, (one-sided or two-sided) Chebyshev inequality can be used. • Non-parametric. • Independent of sample size.

13. Construction of P-boxDistributions with interval parameters If the (unknown) statistical parameter (θ ) of the distribution varies in an interval • Parametric representation. • Confidence intervals on statistics provide a natural way to define interval bounds of the distribution parameters.

14. Construction of P-boxEnvelope of competing probability models When there are multiple candidate distribution models which cannot be distinguished by standard goodness-of-fit tests, Fi (x) = ith candidate CDF

15. Presentation outline • Quasi interval Monte Carlo method • Different approaches for constructing P-boxes • Example

16. Example Limit state: roof drift < 17.78 mm Roof drift is computed by (linear elastic) finite element analysis. 10-bar truss (after Nie and Ellingwood, 2005)

17. Example • The K-S limit and Chebyshev bound are truncated at 50 kN and 220 kN. • Type 1 Largest distribution with interval mean ([100.28, 125.69] kN, 95% confidence interval) • Five candidate distributions: T1 Largest, lognormal, Gamma, Normal, and Weibull, which all pass the K-S tests at a significance level of 5%.

18. Example

19. DiscussionK-S approach • K-S p-box yields a very wide reliability bound ([0, 0.246]). • The K-S wind load p-box itself is very wide, particularly in its upper tail. • K-S p-box has to be truncated at the tails. • The truncation points are often chosen arbitrarily. • The result may be influenced strongly by the truncation.

20. Discussion Chebyshev inequality • One-sided Chebyshev p-box yields a very wide reliability bound ([0, 0.103]). • It also has the truncation problem. • Chebyshev inequality is independent of the sample size. • Two sets of data, one with limited samples and a second with comprehensive samples, would lead to the same p-box if they have the same first 2 moments. • General conception: epistemic uncertainty can be reduced when the quality of data is refined.

21. Discussion Distribution with interval parameters • Pf varies between 0.0116 and 0.0266. • This interval bound clearly demonstrates the effect of small sample size on the calculated failure probability. • It appears that confidence intervals on distribution parameters is a reasonable way to define p-box, provided that the appropriate distribution form can be discerned.

22. Discussion Envelope of candidate distributions • Pf varies between 0.0006 and 0.0162. • The lower bound of Pf is contributed by the Weibull distribution. • If Weibull is discarded, the bounds of Pf will be [0.0032, 0.0162]. • These results highlight the sensitivity of the failure probability to the choice of the probabilistic model for the wind load.

23. Conclusions • Interval quasi-Monte Carlo method is efficient and its implementation is relatively straightforward. • A truss structure has been analysed. • Reliability bounds based on different wind load p-box models vary considerably. • Failure probabilities are controlled by the tails of the distributions.

24. Conclusions • Both K-S confidence limits and Chebyshev inequality have shown some practical difficulties to define p-boxes in the context of structural reliability analysis (tail sensitivity problem). • The most reasonable method to construct p-box for the purpose of reliability assessment seems to be their construction based on confidence intervals of statistics.