In many real systems the components’ strengths may be added together, rather than combined in parallel or series. Such an example is given in Chapter 5. This poses a challenge for analytical methods, as in general normal distributions and log normal distributions cannot be summed easily (except in limited cases such as independently distributed normal random variables). Therefore, in order to consider such systems in the imprecise probabilistic safety analysis framework, we resort to using the imprecise FORM approximations given in Qiu et al. [144].
In Chapter 5, Probabilistic Safety Analysis of a concrete containment was presented as part of a round robin international test exercise. Two experimental test cases (Sandia National Laboratories and Bhabha Atomic Research Centre) were described and the probability of failure for each containment was calculated. The experiments were compared
102 Jonathan Cyrus Sadeghi
Variable Mean Value, [µ,¯µ] Coefficient of Variation
Concrete tensile strength, Fc [4.3, 4.5] 0.2
Liner yield, Fl [370, 390] 0.2 Rebar yield, Fs [450, 370] 0.2 Tendon yield, Ft [1700, 1800] 0.2 Design Pressure, Pd 0.39 0.2 Radius, R 5537.5 0.2 Concrete area, Ac 312.85 0.2 Liner area, Al 1.6 0.2 Rebar area , As 6.85 0.2 Tendon area, At 3.7 0.2
Table 6.2: Input parameters for Sandia National Laboratories containment test case with additive component strengths.
to a cylindrical concrete containment model, where the area and strength of the concrete, rebar, tendons and liner are modelled as normally distributed random variables. In this example, the Sandia National Laboratories Containment will be analysed with intervalised epistemic uncertainty describing the mean value of the random variables representing yield values of structural materials. This could indicate lack of knowledge about the materials used, i.e. insufficient experiments. The modified properties of the Sandia National Laboratories containment are summarised in Table 6.2.
The performance function of the containment is obtained as a load-strength relationship, i.e.
g = (AsFs+ AtFt+ AlFl+ AcFc)− P R. (6.31) We set the applied pressure to be equal to the design pressure, scaled by a constant.
Using the strength to design load ratio method from Eqn. 6.28 and Eqn. 6.29 with ¯ µS µL = ¯ µAsµ¯Fs+ ¯µAtµ¯Ft+ ¯µAcµ¯Fc+ ¯µAlµ¯Fl µP dµR (6.32) and µS ¯ µL = µAsµFs+ µAtµFt+ µAcµFc+ µAlµFl ¯ µPdµ¯R (6.33) we find that Pf = 0.5 when P ∈ [5.2Pd, 5.24Pd]. In order words, because of our epistemic uncertainty in the structural properties of the system we are unsure which pressure causes
Chapter 6. Analytic Imprecise Probabilistic Safety Analysis 103
Pf = 0.5. Clearly the epistemic uncertainty we have considered does not significantly change the pressure at which Pf = 0.5.
For a more complete understanding of the system (i.e. understanding which pressures cause large and small failure probabilities), advanced simulation methods would be necessary. This is because the strength to design-load ratio method only considers the mean values of the random variable in order to find the pressure at which the structure has Pf = 0.5, and does not consider the variability of the structural components. For example, one could resort to the method proposed by de Angelis et al. [51], where line sampling is applied to structures with epistemic uncertainties.
6.5
Chapter summary
In this chapter, we have demonstrated methods to analytically propagate probability boxes in commonly used Probabilistic Safety Analysis equations. These equations include series and parallel systems with unknown dependencies, lognormal fragility distributions and equations where lognormally distributed factors are multiplied. In addition, Power Law Load load distributions are considered. Crucially, we use intervals to model epistemic uncertainty in the parameters of these distributions. This enables the robust quantification of epistemic uncertainty when performing Probabilistic Safety Analysis, particularly in an industrial context. These distributions are sufficient for the analysis of many industrial problems, but in general the imprecise probability methods proposed could be generalised to other distributions as well.
These expressions are imprecise probabilistic analogues to many of the probabilistic formulae proposed in Kennedy et al. [98], which have become standard expressions used in Probabilistic Safety Analysis. We also demonstrated how similar techniques can be applied to simplified calculations involving more complex models.
Our proposed expressions enable engineers to complete essential design calculations whilst considering epistemic uncertainty, and avoid the impracticalities of double loop Monte Carlo simulation which we believe is a significant barrier to the modelling of epistemic uncertainty in many industrial probabilistic safety assessment workflows. However, the proposed methodology in this chapter cannot be applied for black box models, which do not have an analytic performance function. Therefore, the following chapters propose alternative methodologies which still reduce computational demands, whilst being compatible with more general simulations and models.
Chapter 7
Interval Predictor Models for
Reliability Analysis
7.1
Introduction
As discussed in Chapter 4, metamodels can be used to reduce the computational expense of a Monte Carlo simulation to calculate the probability of failure of a system. However, since an approximate model is used to predict the model response, the surrogate approxi- mation introduces a prediction uncertainty in the model response [178]. Consequently, this prediction uncertainty propagates to uncertainty concerning the computed probability of failure, that has to be effectively estimated and accounted for in such approximate analyses. Interval predictor models (validated by the scenario optimisation theory), as discussed in Chapter 3, are a type of metamodel which provide a robust quantification of their predic- tive uncertainty. This chapter therefore presents a systematic approach to consider such prediction uncertainty in the estimation of small failure probabilities in nonlinear models. Section 7.3.2 describes how IPMs used in the literature can be modified to create more accurate metamodels for performance functions. An analytic case study is performed in Section 7.4 to illustrate the proposed approach, where the performance of interval predictor models is compared to that of Kriging models (i.e. Gaussian Processes). Advanced Monte Carlo methods are used to present a benchmark for the proposed approaches.
The use of interval predictor models for reliability analysis is a novel contribution of the author in Patelli et al. [134]. Subsequently, they have also been applied in Crespo et al. [47].
Chapter 7. Interval Predictor Models for Reliability Analysis 105