Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation

In a full Bayesian probabilistic framework for “robust” system identification, structural response predictions and performance reliability are updated using structural test data D by considering the predictions of a whole set of possible structural models that are weighted by their updated probability. This involves integrating h(θ)p(θ∣D) over the whole parameter space, where θ is a parameter vector defining each model within the set of possible models of the structure, h(θ) is a model prediction of a response quantity of interest, and p(θ∣D) is the updated probability density for θ, which provides a measure of how plausible each model is given the data D. The evaluation of this integral is difficult because the dimension of the parameter space is usually too large for direct numerical integration and p(θ∣D) is concentrated in a small region in the parameter space and only known up to a scaling constant. An adaptive Markov chain Monte Carlo simulation approach is proposed to evaluate the desired integral that is based on the Metropolis-Hastings algorithm and a concept similar to simulated annealing. By carrying out a series of Markov chain simulations with limiting stationary distributions equal to a sequence of intermediate probability densities that converge on p(θ∣D), the region of concentration of p(θ∣D) is gradually portrayed. The Markov chain samples are used to estimate the desired integral by statistical averaging. The method is illustrated using simulated dynamic test data to update the robust response variance and reliability of a moment-resisting frame for two cases: one where the model is only locally identifiable based on the data and the other where it is unidentifiable.     

Model Selection Using Response Measurements: Bayesian Probabilistic Approach

A Bayesian probabilistic approach is presented for selecting the most plausible class of models for a structural or mechanical system within some specified set of model classes, based on system response data. The crux of the approach is to rank the classes of models based on their probabilities conditional on the response data which can be calculated based on Bayes’ theorem and an asymptotic expansion for the evidence for each model class. The approach provides a quantitative expression of a principle of model parsimony or of Ockham’s razor which in this context can be stated as “simpler models are to be preferred over unnecessarily complicated ones.” Examples are presented to illustrate the method using a single-degree-of-freedom bilinear hysteretic system, a linear two-story frame, and a ten-story shear building, all of which are subjected to seismic excitation.     

Closed-Form Fragility Estimates, Parameter Sensitivity, and Bayesian Updating for RC Columns

Reinforced concrete (RC) columns are the most critical components in bridges under seismic excitation. In this paper, a simple closed-form formulation to estimate the fragility of RC columns is developed. The formulation is used to estimate the conditional probability of failure of an example column for given shear and deformation demands. The estimated fragilities are as accurate as more sophisticated estimates (i.e., predictive fragilities) and do not require any reliability software. A sensitivity analysis is carried out to identify to which parameter(s) the reliability of the example column is most sensitive. The closed-form formulation uses probabilistic capacity models. A Bayesian procedure is presented to update existing probabilistic models with new data. The model updating process can incorporate different types of information, including laboratory test data, field observations, and subjective engineering judgment, as they become available.     

Structural Identification and Damage Detection from Noisy Modal Data

In this paper we present a simple, yet powerful, method for the identification of stiffness matrices of structural and mechanical systems from information about some of their measured natural frequencies and corresponding mode shapes of vibration. The method is computationally efficient and is shown to perform remarkably well in the presence of measurement errors in the mode shapes of vibration. It is applied to the identification of the stiffness distribution along the height of a simple vibrating structure. An example illustrating the method’s ability to detect structural damage that could be highly localized in a building structure is also given. The efficiency and accuracy with which the method yields estimates of the system’s stiffness from noisy modal measurement data makes it useful for rapid, on-line damage detection of structures.     

Bayesian Network Enhanced with Structural Reliability Methods: Application

The enhanced Bayesian network (eBN) methodology described in the companion paper facilitates the assessment of reliability and risk of engineering systems when information about the system evolves in time. We present the application of the eBN: (1) to the assessment of the life-cycle reliability of a structural system; (2) to the optimization of a decision on performing measurements in that structural system; and (3) to the risk assessment of an infrastructure system subject to natural hazards and deterioration of constituent structures. In all applications, observations of system performances or the hazards are made at various points in time and the eBN efficiently includes these observations in the analysis to provide an updated probabilistic model of the system at all times.     

Updating Properties of Nonlinear Dynamical Systems with Uncertain Input

A spectral density approach for the identification of linear systems is extended to nonlinear dynamical systems using only incomplete noisy response measurements. A stochastic model is used for the uncertain input and a Bayesian probabilistic approach is used to quantify the uncertainties in the model parameters. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform and their robustness with respect to the probability distribution of the response signal in order to calculate the updated probability density function for the parameters of a nonlinear model conditional on the measured response. This probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data. The formulation is first presented for single-degree-of-freedom systems and then for multiple-degree-of freedom systems. Examples using simulated data for a Duffing oscillator, an elastoplastic system and a four-story inelastic structure are presented to illustrate the proposed approach.     

Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters

In recent years, Bayesian model updating techniques based on measured data have been applied to system identification of structures and to structural health monitoring. A fully probabilistic Bayesian model updating approach provides a robust and rigorous framework for these applications due to its ability to characterize modeling uncertainties associated with the underlying structural system and to its exclusive foundation on the probability axioms. The plausibility of each structural model within a set of possible models, given the measured data, is quantified by the joint posterior probability density function of the model parameters. This Bayesian approach requires the evaluation of multidimensional integrals, and this usually cannot be done analytically. Recently, some Markov chain Monte Carlo simulation methods have been developed to solve the Bayesian model updating problem. However, in general, the efficiency of these proposed approaches is adversely affected by the dimension of the model parameter space. In this paper, the Hybrid Monte Carlo method is investigated (also known as Hamiltonian Markov chain method), and we show how it can be used to solve higher-dimensional Bayesian model updating problems. Practical issues for the feasibility of the Hybrid Monte Carlo method to such problems are addressed, and improvements are proposed to make it more effective and efficient for solving such model updating problems. New formulae for Markov chain convergence assessment are derived. The effectiveness of the proposed approach for Bayesian model updating of structural dynamic models with many uncertain parameters is illustrated with a simulated data example involving a ten-story building that has 31 model parameters to be updated.     

Application of a Statistical Model Updating Approach on Phase I of the IASC-ASCE Structural Health Monitoring Benchmark Study

This paper addresses the problem of structural health monitoring (SHM) and damage detection based on a statistical model updating methodology which utilizes the measured vibration responses of the structure without any knowledge of the input excitation. The emphasis in this paper is on the application of the proposed methodology in Phase I of the benchmark study set up by the IASC–ASCE Task Group on structural health monitoring. Details of this SHM benchmark study are available on the Task Group web site at 〈http://wusceel.cive.wustl.edu/asce.shm〉. The benchmark study focuses on important issues, such as: (1) measurement noise; (2) modeling error; (3) lack of input measurements; and (4) limited number of sensors. A statistical methodology for model updating is adopted in this paper to establish stiffness reductions due to damage. This methodology allows for an explicit treatment of the measurement noise, modeling error, and possible nonuniqueness issues characterizing this inverse problem. The paper briefly describes the methodology and reports on the results obtained in detecting damage in all six cases of Phase I of the benchmark study assuming unknown (ambient) data. The performance, limitations, and difficulties encountered by the proposed statistical methodology are discussed.     

Fast Bayesian FFT Method for Ambient Modal Identification with Separated Modes

Previously a Bayesian theory for modal identification using the fast Fourier transform (FFT) of ambient data was formulated. That method provides a rigorous way for obtaining modal properties as well as their uncertainties by operating in the frequency domain. This allows a natural partition of information according to frequencies so that well-separated modes can be identified independently. Determining the posterior most probable modal parameters and their covariance matrix, however, requires solving a numerical optimization problem. The dimension of this problem grows with the number of measured channels; and its objective function involves the inverse of an ill-conditioned matrix, which makes the approach impractical for realistic applications. This paper analyzes the mathematical structure of the problem and develops efficient methods for computations, focusing on well-separated modes. A method is developed that allows fast computation of the posterior most probable values and covariance matrix. The analysis reveals a scientific definition of signal-to-noise ratio that governs the behavior of the solution in a characteristic manner. Asymptotic behavior of the modal identification problem is investigated for high signal-to-noise ratios. The proposed method is applied to modal identification of two field buildings. Using the proposed algorithm, Bayesian modal identification can now be performed in a few seconds even for a moderate to large number of measurement channels.     

Bayesian Network Enhanced with Structural Reliability Methods: Methodology

We combine Bayesian networks (BNs) and structural reliability methods (SRMs) to create a new computational framework, termed enhanced BN (eBN), for reliability and risk analysis of engineering structures and infrastructure. BNs are efficient in representing and evaluating complex probabilistic dependence structures, as present in infrastructure and structural systems, and they facilitate Bayesian updating of the model when new information becomes available. On the other hand, SRMs enable accurate assessment of probabilities of rare events represented by computationally demanding physically based models. By combining the two methods, the eBN framework provides a unified and powerful tool for efficiently computing probabilities of rare events in complex structural and infrastructure systems in which information evolves in time. Strategies for modeling and efficiently analyzing the eBN are described by way of several conceptual examples. The companion paper applies the eBN methodology to example structural and infrastructure systems.     

Bayesian Updating Application into Simulation in the North Edmonton Sanitary Trunk Tunnel Project

Simulation models are built on assumptions, approximations, and estimates. Repetitive long-term projects such as tunnel construction provide opportunities to finetune approximations based on input from actual project progress. Bayesian updating techniques are an effective approach for improving the quality of simulation input and output based on what has already been observed. This paper presents a case study in which Bayesian techniques were applied to a simulation model of an actual tunnel project, the North Edmonton Sanitary Trunk. The study shows that using Bayesian techniques greatly improves the quality of projections. The novelty of this work includes the enhancement of the application of Bayesian updating techniques, the demonstration of simulation applications with a fully monitored tunneling project, and the demonstration of the extent of improvement to planning predictions from the use of actual data and the Bayesian updating techniques.     

Software Framework for Parameter Updating and Finite-Element Response Sensitivity Analysis

The finite-element software framework OpenSees is extended with parameter updating and response sensitivity capabilities to support client applications such as reliability, optimization, and system identification. Using software design patterns, member properties, applied loadings, and nodal coordinates can be identified and repeatedly updated in order to create customized finite-element model updating applications. Parameters are identified using a Chain of Responsibility software pattern, where objects in the finite-element model forward a parameterization request to component objects until the request is handled. All messages to identify and update parameters are passed through a Facade that decouples client applications from the finite-element domain of OpenSees. To support response sensitivity analysis, the Strategy design pattern facilitates multiple approaches to evaluate gradients of the structural response, whereas the Visitor pattern ensures that objects in the finite-element domain make the proper contributions to the equations that govern the response sensitivity. Examples demonstrate the software design and the steps taken by representative finite-element model updating and response sensitivity applications.     

Developing Complete Conditional Probability Tables from Fractional Data for Bayesian Belief Networks

A Bayesian belief network (BBN) can be a powerful tool in decision making processes. Development of a BBN requires data or expert knowledge to assist in determining the structure and probabilistic parameters in the model. As data are seldom available in the engineering decision making domain, a major barrier in using domain experts is that they are often required to supply a huge and intractable number of probabilities. Techniques for using fractional data to develop complete conditional probability tables were examined. The results showed good predictability of the missing data in a linear domain by the piecewise representation method. By using piecewise representation, the number of probabilities to be elicited for a binary child node with k binary parent nodes is now 2k rather than 2k.     

Structural Health Monitoring by Recursive Bayesian Filtering

A new vision of structural health monitoring (SHM) is presented, in which the ultimate goal of SHM is not limited to damage identification, but to describe the structure by a probabilistic model, whose parameters and uncertainty are periodically updated using measured data in a recursive Bayesian filtering (RBF) approach. Such a model of a structure is essential in evaluating its current condition and predicting its future performance in a probabilistic context. RBF is conventionally implemented by the extended Kalman filter, which suffers from its intrinsic drawbacks. Recent progress on high-fidelity propagation of a probability distribution through nonlinear functions has revived RBF as a promising tool for SHM. The central difference filter, as an example of the new versions of RBF, is implemented in this study, with the adaptation of a convergence and consistency improvement technique. Two numerical examples are presented to demonstrate the superior capacity of RBF for a SHM purpose. The proposed method is also validated by large-scale shake table tests on a reinforced concrete two-span three-bent bridge specimen.     

Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging

This paper presents a newly developed simulation-based approach for Bayesian model updating, model class selection, and model averaging called the transitional Markov chain Monte Carlo (TMCMC) approach. The idea behind TMCMC is to avoid the problem of sampling from difficult target probability density functions (PDFs) but sampling from a series of intermediate PDFs that converge to the target PDF and are easier to sample. The TMCMC approach is motivated by the adaptive Metropolis–Hastings method developed by Beck and Au in 2002 and is based on Markov chain Monte Carlo. It is shown that TMCMC is able to draw samples from some difficult PDFs (e.g., multimodal PDFs, very peaked PDFs, and PDFs with flat manifold). The TMCMC approach can also estimate evidence of the chosen probabilistic model class conditioning on the measured data, a key component for Bayesian model class selection and model averaging. Three examples are used to demonstrate the effectiveness of the TMCMC approach in Bayesian model updating, model class selection, and model averaging.     

Sensitivity of Shallow Foundation Response to Model Input Parameters

From a predictive point of view, it is desirable to characterize the effect of varying model input parameters on the seismic response of soil-foundation systems. In this paper, this issue is studied for shallow foundation systems in dry dense sand with varying vertical factors of safety, embedment depths, demand levels, and moment to shear ratios. Response parameters considered are the moment, shear, sliding, settlement, and rotation demands of the foundation. First-order sensitivity analyses indicate that among the soil input parameters, the friction angle has the most significant effect on capturing the foundation force and displacement demands. Furthermore, the uncertainty in friction angle contributes 80% of the variance of the settlement demand and 40% of the variance of the moment demand. It is also found that the uncertainty in Poisson’s ratio has a marginal effect in predicting the studied foundation response. Although the findings of this study are limited to the parameter space considered herein and care should be taken for broader applicability, it does shed light on which parameters uncertainty should be minimized.     

Bayesian Updating of Parameters for a Sediment Entrainment Model via Markov Chain Monte Carlo

A Bayesian framework incorporating Markov chain Monte Carlo (MCMC) for updating the parameters of a sediment entrainment model is presented. Three subjects were pursued in this study. First, sensitivity analyses were performed via univariate MCMC. The results reveal that the posteriors resulting from two- and three-chain MCMC were not significantly different; two-chain MCMC converged faster than three chains. The proposal scale factor significantly affects the rate of convergence, but not the posteriors. The sampler outputs resulting from informed priors converged faster than those resulting from uninformed priors. The correlation coefficient of the Gram–Charlier (GC) probability density function (PDF) is a physical constraint imposed on MCMC in which a higher correlation would slow the rate of convergence. The results also indicate that the parameter uncertainty is reduced with increasing number of input data. Second, multivariate MCMC were carried out to simultaneously update the velocity coefficient C and the statistical moments of the GC PDF. For fully rough flows, the distribution of C was significantly modified via multivariate MCMC. However, for transitional regimes the posterior values of C resulting from univariate and multivariate MCMC were not significantly different. For both rough and transitional regimes, the differences between the prior and posterior distributions of the statistical moments were limited. Third, the practical effect of updated parameters on the prediction of entrainment probabilities was demonstrated. With all the parameters updated, the sediment entrainment model was able to compute more accurately and realistically the entrainment probabilities. The present work offers an alternative approach to estimating the hydraulic parameters not easily observed.     

Bayesian State Estimation Method for Nonlinear Systems and Its Application to Recorded Seismic Response

The focus of this paper is to demonstrate the application of a recently developed Bayesian state estimation method to the recorded seismic response of a building and to discuss the issue of model selection. The method, known as the particle filter, is based on stochastic simulation. Unlike the well-known extended Kalman filter, it is applicable to highly nonlinear systems with non-Gaussian uncertainties. The particle filter is applied to strong motion data recorded in the 1994 Northridge earthquake in a seven-story hotel whose structural system consists of nonductile reinforced-concrete moment frames, two of which were severely damaged during the earthquake. We address the issue of model selection. Two identification models are proposed: a time-varying linear model and a simplified time-varying nonlinear degradation model. The latter is derived from a nonlinear finite-element model of the building previously developed at Caltech. For the former model, the resulting performance is poor since the parameters need to vary significantly with time in order to capture the structural degradation of the building during the earthquake. The latter model performs better because it is able to characterize this degradation to a certain extent even with its parameters fixed. For this case study, the particle filter provides consistent state and parameter estimates, in contrast to the extended Kalman filter, which provides inconsistent estimates. It is concluded that for a state estimation procedure to be successful, at least two factors are essential: an appropriate estimation algorithm and a suitable identification model.     

Bayesian Analysis of the Phase II IASC–ASCE Structural Health Monitoring Experimental Benchmark Data

A two-step probabilistic structural health monitoring approach is used to analyze the Phase II experimental benchmark studies sponsored by the IASC–ASCE Task Group on Structural Health Monitoring. This study involves damage detection and assessment of the test structure using experimental data generated by hammer impact and ambient vibrations. The two-step approach involves modal identification followed by damage assessment using the pre- and postdamage modal parameters based on the Bayesian updating methodology. An Expectation–Maximization algorithm is proposed to find the most probable values of the parameters. It is shown that the brace damage can be successfully detected and assessed from either the hammer or ambient vibration data. The connection damage is much more difficult to reliably detect and assess because the identified modal parameters are less sensitive to connection damage, allowing the modeling errors to have more influence on the results.     

Bayesian Probabilistic Inference for Nonparametric Damage Detection of Structures

This paper presents a Bayesian hypothesis testing-based probabilistic assessment method for nonparametric damage detection of building structures, considering the uncertainties in both experimental results and model prediction. A dynamic fuzzy wavelet neural network method is employed as a nonparametric system identification model to predict the structural responses for damage evaluation. A Bayes factor evaluation metric is derived based on Bayes’ theorem and Gaussian distribution assumption of the difference between the experimental data and model prediction. The metric provides quantitative measure for assessing the accuracy of system identification and the state of global health of structures. The probability density function of the Bayes factor is constructed using the statistics of the difference of response quantities and Monte Carlo simulation technique to address the uncertainties in both experimental data and model prediction. The methodology is investigated with five damage scenarios of a four-story benchmark building. Numerical results demonstrate that the proposed methodology provides an effective approach for quantifying the damage confidence in the structural condition assessment.