This manuscript contains three main parts which address three different problems in the field of stochastic computational mechanics. Stochastic Galerkin projection, except in the third part where only the primary and necessary ingredient of this approach i.e. the representation of uncertainties in input parameters using (space/time dependent) Hermite Chaos expansions is employed, plays the central role in the propagation of uncertainties in inputs to the response of systems under consideration, In the first part that deals with geometrically non-linear behavior of structural systems with random material property, an asymptotic spectral stochastic paradigm is presented for computing the statistics of equilibrium path in the post-bifurcation regime. The approach combines numerical implementation of Koiter's asymptotic theory with Stochastic Galerkin projection and collocation in stochastic space to quantify uncertainties in the parametric representation of load-displacement relationship in the form of uncertain post-buckling slope and curvature, and a family of stochastic displacements fields. The second part concerns obtaining a probabilistic description for the effective elastic properties of multi-phase periodic composites. A spectral stochastic computational scheme is proposed that links the global elastic properties of the composite to the geometry and randomness in its constituents. The scheme benefits from a combination of homogenization theory built into a Finite Element framework and the Stochastic Galerkin projection where a probabilistic characterization of the solutions to a set of local problems defined on the period cell is first sought. A full stochastic description of the global properties is then obtained by averaging the strains that are associated to these solutions over the unit cell. The last part of this manuscript addresses response of linear dynamic systems to random excitations. In this part a stochastic version of direct integration schemes is constructed based on a general recursive state space formulation. The technique is applicable to evaluating the second order response statistics of systems subjected to non-Gaussian non-stationary random excitations (provided the mean and (cross) covariance functions for the excitation processes are available), and is potentially able to handle non-proportional damping where traditionally used methods such as those based on ordinary modal decomposition fail.
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