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Uncertainty Quantification in Vibration-based Structural Health Monitoring using Bayesian Statistics


Although great advancements have been made in structural health monitoring (SHM) for civil structures since 1990s, a lack of accurate and reliable techniques to interpret measured data still challenges the whole community. A common consensus is that raw data cannot directly tell the damage, only features-extracted data can. One major difficulty is that there is too much noise in the SHM data such that the “signal,” even if it conveys the damage information, is buried in the noise. Under the particular circumstance of low signal-to-noise ratio, uncertainty quantification is an invaluable step to determine the influence of uncertainties on predicted values. Armed with Bayesian statistics, this dissertation is devoted to the uncertainty quantification in vibration-based SHM.

A robust optimal sensor placement (OSP) for operational modal analysis is proposed based on the maximum expected utility theory. First, a probabilistic model for OSP considering model uncertainty, load uncertainty and measurement error is introduced, which turns out to be a linear Bayesian normal model. The maximum expected utility theory is then applied with this model by considering utility functions based on three principles: Shannon information, quadratic loss, and K-L divergence. The covariance of modal responses is theoretically derived, and its nearest Kronecker product approximation is developed for fast evaluation of the utility functions. As demonstration and validation examples, sensor placements in a 16-degrees-of- freedom shear-type building, and in Guangzhou (China) TV Tower excited by ground motion and wind load are considered. The results show that, when accounting for prior information, the optimal placement configuration of displacement meter, velocimeter and accelerometer do not have to be congruent, and mixed sensor placement becomes possible. Prior information has less influence on accelerometer placement than on the other sensors, justifying the commonly used mode-shape-based accelerometer placement. The magnitude of input to noise ratio has a great influence on the optimal configuration of sensors, and it connects the kinetic energy-based and Fisher information-based sensor placement approaches.

The uncertainty quantification in the operational modal analysis (OMA) is investigated, where the structural excitations are not directly measured but modeled by band-limited white noise processes. We start with the state-space representation of the dynamical system. By assigning probability distributions to the error terms and specifying prior distributions for the unknown parameters, a probabilistic model, belonging to the conjugate-exponential model, is formally constructed for OMA. The expectation-maximization and the variational Bayes algorithms and the Gibbs sampler are employed to infer the modal parameters from the measured structural responses. For the purpose of restraining the accumulated numerical error in the forward-backward inference, a robust implementation strategy is developed based on square-root filtering and Cholesky decomposition. The proposed approaches are illustrated by application to an example mass-spring system, a laboratory shear-type building model, and the One Rincon Hill Tower in San Francisco. It is observed that the modal frequencies and mode shapes can be identified with small uncertainties comparing to those of identified damping ratios. In addition, the coefficient of variation of the estimated frequency is approximately equal to the standard deviation of the estimated damping ratio in the same mode.

The last problem we consider is the uncertainty quantification in finite element model updating (FEMU) using the measured incomplete and noisy modal data. Based on the generalized eigenvalue decomposition of the stiffness and mass matrices and the assumptions on the error models, a Bayesian probabilistic model for FEMU is formulated, which can incorporate the time-variability, measurement error and model parameter uncertainty. In order to obtain the posterior distributions of the stiffness parameters, a Metropolis-within-Gibbs (MwG) sampler is introduced and a robust implementation strategy is provided as well. The performance of the proposed Bayesian method is illustrated through two examples: a numerical 8-DoF mass-spring system and an experimental 6-story shear-type building. The examples show that the designed MwG sampler accurately recovers the posterior distributions of the stiffness parameters. The posterior variance highly depends on the number of data sets, and correlations between the stiffness parameters represent their physical dependence. It is recommended to use a sufficiently complex model so as to fully explain the measured modal data and include as many modes as possible in estimation to get a more representative model.

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