Research Papers

A Robust Approach to Quantification of Margin and Uncertainty

[+] Author and Article Information
Daniel J. Segalman

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: segalman@egr.MSU.edu

Thomas L. Paez

Thomas Paez Consulting,
12605 Osito Ct NE,
Albuquerque, NM 87111
e-mail: tlpaez4444@gmail.com

Lara E. Bauman

Livermore, CA
e-mail: lara.bauman@gmail.com

Manuscript received October 7, 2015; final manuscript received February 26, 2017; published online April 4, 2017. Assoc. Editor: David Moorcroft.

J. Verif. Valid. Uncert 2(1), 011005 (Apr 04, 2017) (10 pages) Paper No: VVUQ-15-1045; doi: 10.1115/1.4036180 History: Received October 07, 2015; Revised February 26, 2017

A systematic approach to defining margin in a manner that incorporates statistical information and accommodates data uncertainty but does not require assumptions about specific forms of the tails of distributions is developed. A margin that is insensitive to the character of the tails of the relevant distributions (tail insensitive margin, TIM) is defined. This is complemented by the calculation of probability of failure (PoF) where the load distribution is augmented by a quantity equal to the TIM. This approach avoids some of the perplexing results common to traditional reliability theory where, on the basis of very small amounts of data, one is led to extraordinary claims of infinitesimal probability of failure. Additionally, this approach permits a more meaningful separation of statistical and engineering issues.

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Grahic Jump Location
Fig. 4

Probability of failure (PoF) calculated using multiple distributions fitted to the shifted load and strength data

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Fig. 5

A notional depiction of the translation of load realizations that causes the top 5% of the revised load to extend beyond the bottom 5% of the strengths. The abscissa represents load and strength data.

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Fig. 6

The load realizations translated by M95∕5, their approximating PDFs, the strength realizations, and their approximate strength PDFs

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Fig. 3

Multiple distributions fitted to the available load and strength realizations. The load and strength data are indicated by blue and red tick marks, respectively.

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Fig. 2

Histogram of 30 loads and 25 strengths. The histograms are each normalized to integrate to one.

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Fig. 1

An otherwise linear structure with nonlinear elements between nodes 5 and 21

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Fig. 8

The TIM/PFM margin and uncertainty approach embodied in Eq. (2.1) is suggested by the statistical statement about flood levels that have actually been observed (50-yr flood mark) and the distance (30 m) between rare, high flood levels and the base of the house shown

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Fig. 9

The TIM can be approximated using a combination of the empirical cumulative distribution function of load and a delta function approximation for the PDF of strength. The blue curve is the ECDF of load.

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Fig. 10

The probability of failure including margin (PFM) can be estimated by integrating the product of the KDE approximations for the PDF (fY) for strength and the complementary CDF for translated load (1 − FX(x − M)), where M is the tail independent margin (TIM). To put both plots in the same figure, the PDF of strength is normalized by its peak value.

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Fig. 12

KDE estimates for CDFs for load (left) and strength (right) from 1000 resamplings each. The low 20% load and the high 20% strength distributions are shown in thick blue and red lines, respectively. Abscissas are load (left) and strength (right), and ordinates are cumulative probabilities.

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Fig. 11

Bootstrap resampling is used to obtain 1000 other plausible sets of realizations of load and of strength. The low 20% load and the high 20% strength distributions are shown in thick blue and red lines, respectively. Abscissas are load (left) and strength (right), and ordinates are cumulative probabilities.

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Fig. 7

Contours of the joint PDF, fXfY (ellipses) and integration domains of Eq. (1.3) (lower, green cross-hatched region) and (2.1) (blue and green cross-hatched regions)



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