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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

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%.
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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