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. 2 Reliability assessment with limited data A 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. 3 Reliability assessment with limited data Options 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 4

5. Monte Carlo method  Probability of failure, P f, is estimated by  Inverse transform method r j : a sample of iid standard uniform random variates. 5

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

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

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

9. 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 10

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

12. Construction of P-box Chebyshev’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. 12

13. Construction of P-box Distributions 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. 13

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

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

16. Example 16 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 17  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 18

19. Discussion K-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. 19

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. 20

21. Discussion Distribution with interval parameters  P f 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. 21

22. Discussion Envelope of candidate distributions  P f varies between 0.0006 and 0.0162.  The lower bound of P f is contributed by the Weibull distribution.  If Weibull is discarded, the bounds of P f 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. 22

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. 23

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. 24