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For a simple failure analysis with one stress variable, there will be a time-variant failure barrier, <math>r(t)</math>, beyond which the system will fail. This simple case may have a deterministic solution, but for more complex systems, such as crack analysis of a large structure, there can be a very large number of variables, for instance, because of the large number of ways a crack can propagate. In many cases, it is infeasible to produce a deterministic solution even when the individual variables are all individually deterministic.<ref>Beck & Melchers, p. 2202.</ref> In this case, one defines a probabilistic failure barrier surface, <math> \mathbf R (t)</math>, over the [[vector space]] of the stress variables.<ref>Beck & Melchers, p. 2201.</ref>
If failure barrier crossings are assumed to comply with the [[Poisson counting process]], an expression for maximum probable failure can be developed for each stress variable. The overall probability of failure is obtained by averaging (that is, [[integral|integrating]]) over the entire variable vector space. FPI is a method of approximating this integral. The input to FPI is a time-variant expression, but the output is time-invariant, allowing it to be solved by [[first-order reliability method]] (FORM) or second-order reliability method (SORM).<ref>Beck & Melchers, p. 2201.</ref>
An FPI package is included as part of the core modules of the [[NASA]]-designed [[NESSUS Probabilistic Analysis Software|NESSUS]] software.<ref>Shah ''et al.'', p. 5.</ref> It was initially used to analyse risks and uncertainties concerning the [[Space Shuttle main engine]],<ref>Shah ''et al.'', p. 5.</ref> but is now used much more widely in a variety of industries.<ref>Riha ''et al.'', p. 3.</ref>
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