this uncertainty analysis, but as previously mentioned, more complex applications would most likely necessitate the use of surrogate models. For the purpose of illustration, it is assumed that the model for calculating the maximum displacement is not perfect. We assume a hypothetical validation study has quantified the error associated with prediction of maximum displacement to be ±0 .3 mm. This is an epistemic uncertainty (i.e. the uncertainty could be reduced by making improvements to the model or collecting more experimental data for validation), and it will be quantified here using an additive error term having a normal distribution with zero mean and a standard deviation of 0.10 mm (that is, the ±0 .3 mm error bound is treated as a ±3σrange). The objective is to analyze the reliability of the beam, which is defined as the probability that the maximum deflection is less than 0.012 m. The four variables, L, w, b, and E, are all subject to aleatory uncertainty, which could be representative of in-service environmental variations for the load and manufacturing variations for the beam. For the purpose of the analysis, we assume that the probability distributions for the four variables must be inferred based on observed sample data. For this example problem, the sample data are simulated from prescribed distributions, which are presumed unknown during the analysis. The moment of inertia of the beam is assumed to be known and is given by I =28.7×10−6 m4. The details of the probability distributions for the variables and the amount of available data are given in Table 1. Note that the probability distribution model forms (e.g. normal or beta) are assumed to be known during the analysis. In actuality, this is an assumption that introduces some additional uncertainty, but for simplicity it is not considered here (see McFarland and Bichon, 2009, for a Bayesian uncertainty analysis that explicitly addresses probability distribution model form uncertainty). Note that the random variables L and b are modeled using beta distributions. This is done because the beta distribution is bounded, and so it prevents extreme values of L and b from causing the solution for maximum displacement to break down. For the purposes of the uncertainty analysis, it is assumed that the bounds for both of these distributions are known, but that the shape and scale parameters (or equivalently, the mean and standard deviation) are uncertain and must be estimated based on the sample data. Table 1 Description of random variables for beam example Variable Units Distributiona Mean COVb Nc L m Beta(l =3, u=5) 3.6 0.056 15 w N/m Lognormal 25×10 3 0.1 25 b m Beta(l =1.5, u=3) 2.2 0.091 25 E Pa Weibull 200×10 9 0.1 200 a For the beta distribution, l and u are the lower and upper bounds b Coefficient of Variation: ratio of standard deviation to mean c Number of observations The objective is to perform a probabilistic analysis and compute a model output statistic (in this case, a probability of failure), but two sources of epistemic uncerJohn McFarland and David Riha 424
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