On the Response Variability of Beam Structures with Stochastic Variations of Parameters

Structural mechanics research typically focuses on the improvement of structural and constitutive models of physical behavior, and on the development of computational tools to solve the relevant boundary value problems. These advances, however, do not address the problem of quantifying the uncertainties associated with the system parameters (geometry, loading, material properties) and their effect on the system response (displacements, strains, stresses). Stochastic engineering mechanics addresses these issues using the theory of random fields. In the past, the response variability due to stochastic variations in material and geometric parameters of structures has been analyzed using different stochastic finite element methodologies (SFEM) based on various types of series expansions. These SFEM are problematic if we consider large variations of parameters. As an alternative, a flexibility-based Variability Response Function (VRF) approach has been proposed for the case of statically determinate beams. This alternative approach does not use approximations or series expansions---it is thus exact and not limited by any constraint on the relative magnitude of the variations of the parameters. The extension of the flexibility-based VRF approach to the case of statically indeterminate beams is the main objective of this thesis. In this research work, we introduce a novel methodology that generalizes the flexibility-based VRF concept and is applicable to both statically determinate and indeterminate beams with possibly large stochastic variations of parameters. The result of this methodology is the calculation of Generalized VRFs (GVRFs), which can be utilized in the estimation of the response variability, sensitivity studies, and estimation of upper bounds. Two possible ways to compute the GVRFs are investigated and contrasted: (a) a Monte Carlo-based methodology; and (b) a Fast Monte Carlo-based methodology that is, essentially, a special case of (a) but requires significantly less computational effort. Because of the intensive use of Monte Carlo simulation in this thesis and in SFEM in general, an in-depth examination of numerical simulation methodologies for homogeneous Gaussian and non-Gaussian random fields, with a particular emphasis given to computational performance aspects, is included. Finally, this thesis investigates the following additional topics: the study of monochromatic and related non-Gaussian random fields; the study of the Fast Monte Carlo simulation methodology from a numerical point of view, including the determination of its rate of convergence vis-a-vis brute force Monte Carlo simulation; and the mathematical analysis of a very simple stochastic boundary value problem to determine the conditions under which the problem is mathematically well-posed.