Quantificação da incerteza do problema de flexão estocástica de uma viga de Levinson-Bickford através da metodologia λ-Neumann Monte Carlo

Abstract

In stochastic structural mechanics, the treatment of uncertainty is often the main goal of numerical simulations used to estimate the responses of a system or physical phenomenon. These predictions can form the basis for decision making, and therefore a relevant question to be studied is how reliable they are. In general, uncertainties are evaluated from two aspects: from the available statistical information and by considering the mathematical model that represents the problem in a numerical way. The model identifies a set of relationships usually based on principles, conservation laws, and metrics of physical magnitude. In the case of the stochastic beam bending problem, it is possible to associate randomness to material properties, geometry and loads acting on the structure, and thus the response estimates will be present in the displacement, stress and strain field. In the present work, the variational formulation of the stochastic elliptic boundary value problem with random coefficients is studied in the light of the stochastic version of the Lax-Milgram lemma, and the propagation and quantification of uncertainty are investigated from the recent λ-Neumann Monte Carlo numerical methodology of asymptotic complexity. Numerical simulation results are obtained for the Levinson-Bickford stochastic beam bending problem. This high-order theory has the advantage of meeting the condition of zero shear at the lateral surfaces, and its formulation by the Finite Element method avoids the numerical drawback of shear locking. The solutions are presented over a set of numerical approximations through error estimates of the expected value and variance estimators of the displacement field. Different methods of quantification, uncertainty modeling, boundary conditions, coefficient of variation, and material and geometric properties of the beam are compared.