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 data A common scenario.

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Structural reliability analysis with probability-boxes

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Structural reliability analysis with probability boxes

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 data a common scenario

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 data options for solution

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

Presentation outline

  • Quasi interval Monte Carlo method

  • Different approaches for constructing P-boxes

  • Example


Monte carlo method

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

Interval Monte Carlo method

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

Define

then

One has


Interval monte carlo method1

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

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

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 outline1

Presentation outline

  • Quasi interval Monte Carlo method

  • Different approaches for constructing P-boxes

  • Example


Construction of p box kolmogorov smirnov confidence limits

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 box chebyshev s inequality

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 box distributions with interval parameters

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 box envelope of competing probability models

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 outline2

Presentation outline

  • Quasi interval Monte Carlo method

  • Different approaches for constructing P-boxes

  • Example


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)


Example1

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%.


Example2

Example


Discussion k s approach

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

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

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

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

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.


Conclusions1

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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