able approaches for variance decomposition with respect to such parameters are limited. Fortunately, the structured Monte Carlo approach outlined by Saltelli et al (2004) can still be used, provided each θi is treated as a single factor. The drawback is that this will typically require on the order of thousands of evaluations of the model output statistic, for different combinations of the variables θ. In most cases of practical interest, approximation of the deterministic performance function by an efficient surrogate model will be necessary (some relevant examples are given by Bichon et al, 2008a; Kaymaz, 2005; McFarland et al, 2008; Kennedy et al, 2006; McFarland et al, 2009; McFarland, 2008). Other types of epistemic variables can also be treated using the structured Monte Carlo approach. For example, deterministic model uncertainty may be characterized using an error termεhaving some epistemic uncertainty distribution. The structured Monte Carlo approach would then enumerate various combinations of values of ε (along with the other epistemic parameters). Each evaluation of the model output statistic, γ, would then proceed using a particular fixed realization of deterministic model error, ε. 4 Numerical Example In order to illustrate these concepts, the reliability analysis of a statically indeterminate beam in considered. The problem is illustrated in Figure 1, and consists of a simple beam with two supports and a distributed load over part of its length. The random variables are the length of the beam, L, the magnitude of the distributed load, w, the range over which the load is applied, b, and the modulus of elasticity of the beam, E. Fig. 1 Schematic of statically indeterminate beam example The response quantity of interest will be the maximum deflection of the beam. Solution of the maximum displacement can be obtained using the moment-area method (Beer and Johnston, 1992). This requires first finding the location of the maximum displacement along the length of the beam, which involves the solution of a third-order polynomial. For this work, the solution to this polynomial equation is found using the bisection method. Then, magnitude of the maximum displacement is available in closed form, given the location. Further details are given in the Appendix. Calculation of this performance function is fast enough for use directly in Variance Decomposition in the Presence of Epistemic and Aleatory Uncertainty 423
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