The Neumann-Monte Carlo methodology applied to the quantification of uncertainty in the problem stochastic bending of the Levinson-Bickford beam

This paper proposes to apply the Neumann-Monte Carlo method to obtain the estimates of the statistical moments of the solution for the stochastic bending problem of Levinson-Bickford beams, with uncertainty in beam stiffness. The approximate solutions represent the stochastic displacement processes. For uncertainty modeling, parameterized stochastic processes will be used. The methodology proposed in this work differs from the usual one, as it is developed from theoretical results of existence and uniqueness of the realizations. The consistency of the approximate solutions will be based on studies on the existence and uniqueness of the theoretical solutions to this problem. The uncertainty will be quantified by estimating the statistical moments of the stochastic transverse displacement processes. The Monte Carlo simulation method is used to evaluate the performance of the proposed methodology and, for the numerical examples presented errors were obtained in the approximations of statistical estimators below 1%, and computation times were substantially reduced.