Contrasting the Statistical with the Mathematical Properties of NESSUS/FORM

Acknowledgement:
This work was sponsored in part by NASA Order No. C-80076A
Program Manager: Dr. Shantaram S. Pai (NASA GRC)

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The mathematical properties of the FORM (First Order Reliability Method) have been studied extensively by engineers for more than three decades. Focus has been on the non-linearity of the limit-state function (g-function) caused by the “transformation” of variables into Normals. Even when the variables are already Normal, the non-linearity of the g-function can sometimes be troublesome, and this too has been extensively studied and remedies such as SORM (Second Order RM) have been proposed. What has not been studied, however, is the statistical, as contrasted with the mathematical, behavior of FORM.Figure 1 illustrates the FORM paradigm simplified to only two dimensions, demand and capacity. The joint distribution of demand and capacity is “transformed” into an equivalent bivariate normal density. In this study the marginal distributions were defined to be normal to simplify the problem and to illustrate that the behavior is not a result of improper “transformation.” The g-function is where demand equals capacity so that it divides the space into fail and not-fail regions depending on whether demand exceeds capacity. The b factor then is the shortest distance from the center of the joint density to the g-function, in units of standard deviations.

The FORM/SORM Paridigm

Warning: The units are rubber-band yardsticks

This work contrasts the statistical and mathematical behavior using 10,000 nominally identical NESSUS/FORM problem evaluations, and observes that for a simple cantilever beam example the average expected number of failures is more than $$50 \times$$ the nominal value. Figure 2 presents the simple cantilever beam problem.