Analysis of Bernoulli beams with 3D stochastic heterogeneity

The behavior of stochastically heterogeneous beams, composed of isotropic sub-elements of randomly distributed stiffness is studied. Cross sectional as well as longitudinal heterogeneity are included. Average displacements, reaction forces and their statistical variance are found analytically by a functional perturbation method. Ratio of sub-element to beam characteristic size is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. The major aim is to study the relation between various microstructure properties (grain size, shape, modulus, statistical correlation lengths etc.) and the overall behavior of linear elastic Bernoulli beams. For the statically determinate case, only cross sectional 2D microstructure statistics is found to affect the elastic response, so that an equal average displacement can be achieved by an equivalent, non-isotropic homogeneous beam. For the indeterminate case, the average values of macro properties are affected by the 3D morphological features. Therefore, the proper equivalent homogeneous beam has to include non-local elastic properties. A simple reciprocal relation, connecting two separate loading systems is found, relating their external forces and displacement statistical variances. Morphological parameters, like two point probability moments, used in the final results are derived analytically, and their physical interpretations are discussed.