Monte Carlo Simulations with Variance Reduction for Structural Reliability Modeling, Updating and Testing

Abstract

Monte Carlo simulation techniques have emerged as widely accepted computing tools in tackling many problems in modern structural mechanics. Apart from developments in computational hardware, which have undoubtedly made simulation strategies practically feasible, the success of Monte Carlo simulations has also resulted equally significantly from the methodological developments aimed at controlling sampling variance of the Monte Carlo estimates. The study reported in the present thesis is aimed at developing and validating Monte Carlo simulation based approaches with inbuilt variance reduction capabilities to deal with problems of time variant reliability modeling, random vibration testing, and updating reliability models for statically/dynamically loaded instrumented structures. The relevant literature has been reviewed in Chapter 1. Time variant reliability analysis of randomly parametered and randomly driven non-linear vibrating systems has been tackled by combining two Monte Carlo variance reduction strategies into a single framework (Chapter 2). The first of these strategies is based on the application of the Girsanov transformation to account for the randomness in dynamic excitations and, the second approach is fashioned after the subset simulation method to deal with randomness in system parameters. A novel experimental test procedure to estimate the reliability of structural dynamical systems under excitations specified via random process models has been proposed (Chapter 3). The samples of random excitations to be used in the test are modified by the addition of an artificial control force. An unbiased estimator for the reliability is derived based on measured ensemble of responses under these modified inputs based on the tenets of Girsanov’s transformation. The study observes that an acceptable choice for the control force (that can reduce the sampling variance of the estimator) can be made solely based on experimental techniques. This permits the proposed procedure to be applied in the experimental study of time variant reliability of complex structural systems which are difficult to model mathematically. Illustrative example consists of a multi-axes shake table study on bending-torsion coupled, geometrically non-linear, five-storey frame under uni/bi-axial, non-stationary, random base excitation. The first order reliability method (FORM) and inverse FORM have been extended to handle the problem of updating reliability models for existing, statically loaded structures based on measured responses (Chapter 4). The proposed procedures are implemented by combining Matlab based reliability modules with finite element models residing on the Abaqus software. Numerical illustrations on linear and non-linear frames are presented. A solution strategy within the framework of Monte Carlo simulation based dynamic state estimation method and Girsanov’s transformation for variance reduction has been developed to tackle the problem of updating the reliability of instrumented structures based on measured response under random dynamic loading (Chapter 5). For linear Gaussian state space models, the solution is developed based on continuous version of the Kalman filter, while, for non-linear and (or) non-Gaussian state space models, bootstrap particle filters are adopted. Results from laboratory testing of an archetypal five storey bending-torsion coupled frame under seismic base motions form the basis of one of the illustrative examples. A set of three annexures contain details of numerical methods for discretizing Ito’s differential equations (Annexure 1), working of the Girsanov transformation through Kolmogorov’s equations (Annexure 2) and tools for interfacing Matlab and Abaqus codes (Annexure 3).