Uncertainty quantification via λ-Neumann methodology of the stochastic bending problem of the Levinson–Bickford beam

In stochastic structural mechanics, the treatment of uncertainty is often the main objective of numerical simulations to estimate the responses of a system or physical phenomenon. These predictions can form the basis for decision making, and therefore, a relevant issue to be studied is how reliable they are. Uncertainties, in general, are assessed under two aspects: from the available statistical information and considering the mathematical model that represents the problem numerically. The model identifies a set of relationships based on principles, conservation laws, and metrics of physical magnitude. For the stochastic bending problem of elastic and stationary beams, it is possible to associate randomness to the properties of the material, geometry, and loads on the structure. Thus, the estimates of the responses will be present in the field of stresses and deformations. In the present work, the variational formulation of the stochastic problem of the linear elliptical contour value with random coefficients is studied in light of the stochastic version of the Lax–Milgram lemma. The propagation and uncertainty quantification are investigated based on the recent numerical methodology of asymptotic complexity λ-Neumann–Monte Carlo. The results of the numerical simulation are obtained for the stochastic bending beam problem based on the Levinson–Bickford theory. The theory of high-order elastic bending has the advantage of meeting the condition of zero shears on the lateral surfaces of the beam. Numerical results are presented related to the accuracy, convergence, estimators of statistical moments, error estimate, and the processing time of the approximate solution for the high-order beam.