Analysis of Vibration and Buckling Problems in Non-Local Beam Theories Accounting for Parametric Uncertainties

Abstract

The main concern of this work is the study of the uncertainty quantification and its propagation in the static/dynamic response of nano-beams modeled according to first-order shear theories and whose constitutive equations follow the concepts of non-local elasticity. The non-local constitutive relationships in terms of strains and stresses have parameters that could be uncertain depending on the type of material and the constructive procedure of the micro-beam or nano-beam. The structure is conceived as a thin-walled beam with motion in three axes, i.e. bending in two directions and the axial motion and twisting. A deterministic model is developed for isotropic graded beams, and it is employed to calculate the mean measure of buckling loads and vibration patterns in the frequency domain. In order to calculate the propagation of the uncertainty in the response of the beam, the geometrical parameters, such as internal and external characteristic lengths, and/or the material parameter, such as Young modulus, among other, are considered uncertain and random variables are associated to them. The probability density functions of the random variables are obtained by means of the Maximum Entropy Principle according to the available (or assumed) information. The Monte Carlo method is employed to simulate a couple of buckling and vibration problems, whose equations are proposed to be solved in the context of the finite element method. A number of examples are presented in order to show the propagation of the uncertainty associated to the non-local parameters in the buckling response and the vibratory patters of the nano/microbeams.