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The problem of updating a structural model and its associated uncertainties by utilizing dynamic response data is addressed using a Bayesian statistical framework that can handle the inherent ill-conditioning and possible nonuniqueness in model updating applications. The objective is not only to give more accurate response predictions for prescribed dynamic loadings but also to provide a quantitative assessment of this accuracy. In the methodology presented, the updated (optimal) models within a chosen class of structural models are the most probable based on the structural data if all the models are equally plausible a priori. The prediction accuracy of the optimal structural models is given by also updating probability models for the prediction error. The precision of the parameter estimates of the optimal structural models, as well as the precision of the optimal prediction-error parameters, can be examined. A large-sample asymptotic expression is given for the updated predictive probability distribution of the uncertain structural response, which is a weighted average of the predictive probability distributions for each optimal model. This predictive distribution can be used to make model predictions despite possible nonuniqueness in the optimal models.

The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian probabilistic approach is followed to obtain not only the most probable (optimal) values but also the probability distribution of the updated modal parameters. This is very important when one plans to use these estimates for further processing, such as for updating the theoretical finite-element model of the structure, because it provides a rational basis for weighting differently the errors of the various modal parameters, the errors being the differences between the theoretical and identified values of these parameters. The approach is introduced here for a SDOF system. Under the assumption of white noise excitation the statistical properties of an estimator of the spectral density are presented. Based on these statistical results expressions for the updated probability density function (PDF) of the modal parameters are derived. The optimal values of the parameters are the ones at which the updated PDF is maximized. Numerical examples using simulated data are presented to demonstrate the approach.

The problem of model identifiability is addressed. The goal is to find all the structural models within a specified class that produce the same output at a set of observed degrees of freedom as a given model in the class when they are all subjected to the same input. A new algorithm is presented to solve this problem for a class of multi-degree-of-freedom linear structural models, and the approach is illustrated with examples.

The problem of updating a structural model and its associated uncertainties by utilizing measured dynamic response data is addressed. A Bayesian probabilistic formulation is followed to obtain the posterior probability density function (PDF) of the uncertain model parameters for given measured data. The present paper discusses the issue of identifiability of the model parameters and reviews existing asymptotic approximations for identifiable cases. The focus of the paper is on the treatment of the general unidentifiable case where the earlier approximations are not applicable. In this case the posterior PDF of the parameters is found to be concentrated in the neighborhood of an extended and extremely complex manifold in the parameter space. The computational difficulties associated with calculating the posterior PDF in such cases are discussed and an algorithm for an efficient approximate representation of the above manifold and the posterior PDF is presented. Numerical examples involving noisy data are presented to demonstrate the concepts and the proposed method. (C) 1998 Elsevier Science Ltd. All rights reserved.

The problem of updating a structural model and its associated uncertamt1es by utilizing structural response data is addressed using a Bayesian statistical framework which can handle the inherent ill-conditioning and possible nonuniqueness in model updating applications. The model updating is done by using Bayes theorem to update the probability model for the parameters of the structural model set and the parameters of the prediction-error probability model set. From this perspective, model updating is viewed as part of robust analysis where modeling uncertainties are explicitly addressed in the analysis of a system. The exact expressions for updated model predictions are given by multi-dimensional integrals whose direct evaluation is usually computationally prohibitive. Asymptotic approximations are presented for both identifiable and unidentifiable model sets. An illustrative example is given using robust and updated reliabilities to achieve a more effective optimal design of a tuned-mass damper for a simple bridge system.

A Bayesian probabilistic methodology is presented for updating the assessment of the lifetime reliability of a structure based on dynamic test data. Uncertainties from modeling errors are treated in addition to the uncertain excitation the structure will experience during the remainder of its lifetime. Both identifiable and unidentifiable models can be handled.

A new simplified condition is developed for determining the exponential mean-square stability margins of linear stochastic dynamical systems. It is well-known that under parametric wide-band noise disturbances, the governing equations of motion of such a system can be approximated by linear Ito stochastic differential equations (SDE). A necessary and sufficient condition for exponential mean square stability of the resulting Ito SDE is that the real parts of all the eigenvalues of the matrix describing the system of second-order moments are negative. Equivalently, the Routh-Hurwitz procedure provides conditions for stability in the form of several inequalities. In this study, it is shown that a necessary condition for the system configuration to correspond to a point on the stability boundary is that the determinant of the matrix describing the system of second-order moments be zero. This condition is a single algebraic expression allowing for the straight forward calculation of all candidate stability boundaries. In addition, the topological properties of the stability domain are presented and shown to be useful in identifying stability boundaries and stability domains from the developed single stability boundary condition. This simplified condition provides significant advantages in the analytical and numerical estimation of the stability border and stability region of dynamical systems. The usefulness and superiority of the new condition is demonstrated by applications to example dynamical systems, including a long-span bridge model subjected to turbulent wind. (C) 1997 Elsevier Science Ltd.

An asymptotic approximation is developed for evaluating the probability integrals that arise in the determination of the reliability and response moments of uncertain dynamic systems subject to stochastic excitation. The method is applicable when the probabilities of failure or response moments conditional on the system parameters are available, and the effect of the uncertainty in the system parameters is to be investigated. In particular, a simple analytical formula for the probability of failure of the system is derived and compared to some existing approximations, including an asymptotic approximation based on second-order reliability methods. Simple analytical formulas are also derived for the sensitivity of the failure probability and response moments to variations in parameters of interest. Conditions for which the proposed asymptotic expansion is expected to be accurate are presented. Since numerical integration is only computationally feasible for investigating the accuracy of the proposed method for a small number of uncertain system parameters, simulation techniques are also used. A simple importance sampling method is shown to converge much more rapidly than straightforward Monte Carlo simulation. Simple structures subjected to white noise stochastic excitation are used to illustrate the accuracy of the proposed analytical approximation. Results from the computationally efficient perturbation method are also included for comparison. The results show that the asymptotic method gives acceptable approximations, even for systems with relatively large uncertainty, and in most cases, it outperforms the perturbation method.

A modal-based analysis of the dynamic response variability of multiple degree-of-freedom linear structures with uncertain parameters subjected to either deterministic or stochastic excitations is considered. A probabilistic methodology is presented in which random variables with specified probability distributions are used to quantify the parameter uncertainties. The uncertainty in the response due to uncertainties in the structural modelling and loading is quantified by various probabilistic measures such as mean, variance and coefficient of excess. The computation of these probabilistic measures is addressed. A series expansion involving orthogonal polynomials in terms of the system parameters is first used to model the response variability of each contributing mode. Linear equations for the coefficients of each series expansion are derived using the weighted residual method. Mode superposition is then used to derive analytical expressions for the variability and statistics of the uncertain response in terms of the coefficients of the series expansions for all contributing modes. A primary-secondary system and a ten-story building subjected to deterministic and stochastic loads are used to demonstrate the methodology, as well as evaluate its performance by comparing it to existing methods, including the computationally cost-efficient perturbation method.

An asymptotic approximation is developed for calculating a class of probability integrals which arise in the determination of the failure probability of uncertain dynamic systems subject to stochastic excitations. The method is applicable when the failure probability conditional on the system parameters is available. Simple analytical formulas are also derived for the sensitivity of the failure probability to variations in parameters of interest. A structure subjected to a stochastic excitation is used to illustrate the accuracy of the proposed analytical approximation by comparing the results to those obtained by a perturbation method, and an importance sampling technique developed for these types of integrals.

A probabilistic methodology is presented for obtaining the variability and statistics of the dynamic response of multi-degree-of-freedom linear structures with uncertain properties. Complex mode analysis is employed and the variability of each contributing mode is analyzed separately. Low-order polynomial approximations are first used to express modal frequencies, damping ratios and participation factors with respect to the uncertain structural parameters. Each modal response is then expanded in a series of orthogonal polynomials in these parameters. Using the weighted residual method, a system of linear ordinary differential equations for the coefficients of each series expansion is derived. A procedure is then presented to calculate the variability and statistics of the uncertain response. The technique is extended to the stochastic excitation case for obtaining the variability of the response moments due to the variability of the system parameters. The methodology can treat a variety of probability distributions assumed for the structural parameters. Compared to existing analytical techniques, the proposed method drastically reduces the computational effort and computer storage required to solve for the response variability and statistics. The performance and accuracy of the method are illustrated by examples.

The problem of calculating the uncertainty in the dynamic response of a structure due to uncertainties related to the modeling of its dynamic behavior, is addressed. Based on a Bayesian probabilistic approach, a new approximate numerical method is proposed to investigate the resulting uncertainties in the structural response. The proposed method provides a very efficient and accurate approach to the solution of stochastic finite-element models It can be used to quantify the uncertainties in the predicted response of a structure during its design, where engineering judgement is used to quantify the uncertainties in the modeling process.

An asymptotic approximation is developed for evaluating the probability integrals which arise in the determination of the reliability and response moments of uncertain dynamic systems subject to stochastic excitation. The method is applicable when the probabilities of failure or response moments conditional on the system parameters are available, and the effect of the uncertainty in the system parameters is to be investigated. In particular, a simple analytical formula for the probability of failure of the system is derived and compared to some existing approximations, including an asymptotic approximation based on SORM methods. Simple analytical formulas are also derived for the sensitivity of the failure probability and response moments to variations in parameters of interest. Conditions for which the proposed asymptotic expansion is expected to be accurate are presented. Since numerical integration is only computationally feasible for investigating the accuracy of the proposed method for a small number of uncertain system parameters, simulation techniques are also used. A simple importance sampling method is shown to converge much more rapidly than straightforward Monte-Carlo simluation. Simple structures subjected to white noise stochastic excitation axe used to illustrate the accuracy of the proposed analytical approximation. Results from the computationally efficient perturbation method are also included for comparison. The results show that the asymptotic method gives acceptable approximations, even for systems with relatively large uncertainty, and in most cases, it outperforms the perturbation method.

A methodology is presented to investigate global identifiability within a class of structural models to provide the set of models within the given class which are equivalent as far as their input-output dynamic behavior is concerned. As an application, all the equivalent most probable models for given input and output data can be found, and they can then be used to give improved response predictions along with the associated uncertainty.

This is an author's accepted manuscript of: Jin, N., Yang, Y. B., Dimitrakopoulos, E. G., Paraskeva, T. S., & Katafygiotis, L. S. (2021). Application of short-time stochastic subspace identification to estimate bridge frequencies from a traversing vehicle. Engineering Structures, 230, article 111688. https://doi.org/10.1016/j.engstruct.2020.111688