Acknowledgements The research reported in this paper was funded as part of an Internal Research and Development project at the Southwest Research Institute. Appendix Here we give further details regarding the solution for maximum displacement of the beam used as the numerical example in Section 4 (see Figure 1). First, the reactions at the supports can be obtained as follows: RA = wb2 2L , RB =wb−RA. (5) Then the following equation can be constructed using the moment-area method: − RAL 2 6EI − wb4 24EIL + RAx 2 m 2EI − w 6EI (xm−L+b) 3 =0 , (6) where xm is the location of the maximum displacement, measured from the left support. For well-conditioned values of Landb, the bisection method can be used to solve for xm. Finally, the magnitude of the maximum displacement, ym, is calculated as ym = RAx 3 m 3EI − w(L−b) 6EI (xm−L+b) 3 − w 8EI (xm−L+b) 4 . (7) References Bae HR, Grandhi RV, Canfield RA (2003) Uncertainty quantification of structural response using evidence theory. AIAA Journal 41(10):2062–2068 Bae HR, Grandhi RV, Canfield RA (2006) Sensitivity analysis of structural response uncertainty propagation using evidence theory. Structural and Multidisciplinary Optimization 31(4):270–291 Beer FP, Johnston ER (1992) Mechanics of Materials, 2nd edn. McGraw-Hill, Inc. Bichon BJ, Eldred MS, Swiler LP, Mahadevan S, McFarland JM (2008a) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA Journal 46(10):2459–2468 Bichon BJ, McFarland JM, Mahadevan S (2008b) Using Bayesian inference and Efficient Global Reliability Analysis to explore distribution uncertainty. In: Proceedings of the 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL Cukier RI, Fortuin CM, Schuler KE, Petschek AG, Schaibly JH (1973) Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. John McFarland and David Riha 428
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