Stochastic Dynamic Finite Elements for Applications in Earthquake Engineerin

Presence of inevitable uncertainties and the need to account for them explicitly in our prediction have long been recognized by the earthquake engineering community. Pioneering works by the Pacific Earthquake Engineering Research (PEER) center during the early 2000s saw the development of performance-based design framework to account for those uncertainties. However, numerical simulations of the behavior of solids and structures---which are increasingly being used to feed the performance-based design framework---still remain largely deterministic, amid the presence of huge uncertainties in the system. This is mainly due to the issue of computational tractability of the Monte Carlo approach in solving the governing partial differential equation of solid mechanics with uncertain coefficients and uncertain forcing function.;This dissertation develops an efficient numerical methodology to simulate behavior of uncertain, nonlinear solids and structures subjected to uncertain seismic forcing. The methodology is based on the spectral approach of the stochastic finite element method. It considers the input seismic wave and the input material parameters to be a non-stationary random process and heterogeneous random fields, respectively, and efficiently represents them in terms of multidimensional Hermite polynomial chaos---orthogonal and uncorrelated polynomials of zero-mean, unit variance Gaussian random variables---by taking the advantage of the optimality of the Kosambi-Karhunen-Loeve theorem. The methodology allows for any non-Gaussian marginal distributions and any arbitrary correlation structures for the input processes and fields. The solution random processes (displacement, velocity, and acceleration) are also represented in terms of multidimensional Hermite polynomial chaos expansion whose coefficients, at each time step, are then computed using a stochastic Galerkin approach. A Fokker-Planck-Kolmogorov equation approach and a nonlinear least squares procedure are used to compute the evolution of the stress and tangent stiffness, following any elastic-plastic constitutive model, after each time step, while the time integration is performed via the Newmark's method.;Three sets of examples related to geotechnical earthquake engineering are presented to verify and illustrate the salient features of the methodology. The first set of examples involve a 1-D geotechnical site response analysis with an elastic shear modulus random field and a non-stationary bedrock random motion. The simulation results are in good agreement with the conventional Monte Carlo results, and the computation time shows a speedup of more than 100 over the conventional Monte Carlo approach. The second set of examples involve a large scale 3-D soil-structure interaction (SSI) analysis in exploring the parallel computing efficiency of the formulation. Special block pattern of the global stochastic stiffness matrix is exploited to significantly reduce memory requirement and distribute block matrices among multiple processors for parallel computing. With 766.1 million stochastic DOFs of the uncertain SSI system, it is observed to have 100% efficiency with up to 78 processors and about 50% efficiency with 341 processors. The third set of examples involve a 1-D nonlinear geotechnical site response analysis with uncertain soil shear modulus and uncertain soil shear strength. A von Mises elastic perfectly plastic soil constitutive model has been used to compute the stresses and update the tangent stiffness as the soil has plastified.;A number of parametric studies are conducted by varying the marginal standard deviation, marginal distribution, correlation structure and correlation length of the input material parameters and non-stationary characteristics of the input seismic wave. The parametric studies have yielded several findings that can have significant risk implication towards seismic design of structures. It is observed that the deterministic values of the commonly used ground motion intensity measures and/or structural demand parameters, obtained from the conventional deterministic analyses, can be smaller or larger than the mean values of the same obtained from stochastic analyses of the same systems depending upon which parameters of the system are considered uncertain and their uncertainty characteristics, namely, the marginal coefficient of variation, marginal distribution and the correlation structure. Moreover, the choice of the input uncertainty characteristics is also observed to significantly influence the marginal standard deviations and marginal distributions of the ground motion intensity measures and/or structural demand parameters.