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Structural reliability analysis with probability-boxes

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### 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

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.

Reliability assessment with limited dataOptions for solution

- Bayesian approach
- more subjective
- high numerical effort
- Imprecise probabilities
- Probability box
- Random set
- Dempster-Shafer evidence theory

Presentation outline

- Quasi interval Monte Carlo method
- Different approaches for constructing P-boxes
- Example

Monte Carlo method

- Probability of failure, Pf , is estimated by
- Inverse transform method

rj : a sample of iid standard uniform random variates.

Interval Monte Carlo method

When Fx( ) is unknown but bounded, interval samples can be generated

Define

then

One has

Interval Monte Carlo method

A lower and an upper bound for Pf can be estimated as

Variance of direct interval Monte Carlo

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.

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.

Presentation outline

- Quasi interval Monte Carlo method
- Different approaches for constructing P-boxes
- Example

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.

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.

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.

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

Presentation outline

- Quasi interval Monte Carlo method
- Different approaches for constructing P-boxes
- Example

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)

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%.

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.

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.

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.

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.

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.

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.

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