Assessment of stochastically updated finite element models using reliability indicator

https://doi.org/10.1016/j.ymssp.2016.05.020Get rights and content

Highlights

  • The issue of reliability of stochastically updated finite element models is discussed.

  • An improved perturbation is used for stochastic finite element model updating.

  • The quality of initial model and the updated models are evaluated from reliability indicators.

  • The present study highlights the importance of uncertainty reduction in measurement data for reliable model updating.

Abstract

Finite element (FE) model updating techniques have been a viable approach to correcting an initial mathematical model based on test data. Validation of the updated FE models is usually conducted by comparing model predictions with independent test data that have not been used for model updating. This approach of model validation cannot be readily applied in the case of a stochastically updated FE model. In recognizing that structural reliability is a major decision factor throughout the lifecycle of a structure, this study investigates the use of structural reliability as a measure for assessing the quality of stochastically updated FE models. A recently developed perturbation method for stochastic FE model updating is first applied to attain the stochastically updated models by using the measured modal parameters with uncertainty. The reliability index and failure probability for predefined limit states are computed for the initial and the stochastically updated models, respectively, and are compared with those obtained from the ‘true’ model to assess the quality of the two models. Numerical simulation of a truss bridge is provided as an example. The simulated modal parameters involving different uncertainty magnitudes are used to update an initial model of the bridge. It is shown that the reliability index obtained from the updated model is much closer to true reliability index than that obtained from the initial model in the case of small uncertainty magnitude; in the case of large uncertainty magnitude, the reliability index computed from the initial model rather than from the updated model is closer to the true value. The present study confirms the usefulness of measurement-calibrated FE models and at the same time also highlights the importance of the uncertainty reduction in test data for reliable model updating and reliability evaluation.

Introduction

Developing appropriate and credible finite element (FE) models is indispensable for design and evaluation of complex engineering structures. These models are approximate imitations of real-world systems and they never exactly imitate the real-world systems. Despite the high degree of sophistication in modeling techniques, practical applications often indicate great discrepancies between experimental modal data and the corresponding model predictions due to simplification of complex physical process, inappropriate assignment of parameter values, measurement errors, and incompleteness of available test data, among others. Model updating from test data has become a widely accepted procedure for enhancement of the test/analysis correlation and improvement in the quality of analytical model for a structure [1], [2], [3], [4].

In reality, the modal parameters measured from an operational civil structure show significant variations which are caused by several factors such as natural variations of physical properties in a structure, measurement noise in data, and errors involved in modal identification process. Thus, they are best represented by random variables with uncertainty. Accordingly, stochastic model updating procedures taking into account modal parameter uncertainty have been developed by treating the model parameters as random variables and constructing their probability density functions (PDFs) from the measured uncertainty of modal parameters.

Depending on how the definition of probability is interpreted, two main alternatives of probability description, namely the Bayesian and frequentist description are available for selection. Beck and his colleagues developed a Bayesian-based probabilistic framework for stochastic model updating, where the prior distribution of the model parameters in an initial model and the measured modal data with uncertainty are used to quantify their posterior uncertainties [5], [6], [7]. And the results are often expressed in terms of the most probable value (MPV) of model parameters and their posterior variance/covariance. In the context of frequentist description of probability, the stochastic model updating aims at determining the ensemble statistics of the model parameters based on the modal data with uncertainty; only the mean and covariance of model parameters are of concern when the Gaussian distributions of random variables are followed [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Recently, Au [19] have explored the connections between the Bayesian and frequentist quantification of uncertainty in model updating, and the author demonstrated that the updating results obtained with the two methods are quite close each other when there is no or little model error.

Due to the insensitivity of model parameters to modal data, sparseness of sensor locations and measurement noise, model updating is an ill-posed inverse problem such that small variation in input (modal data) can produce unnecessarily large variation in output (model parameters), which may deteriorate model's predictive performance. The updated finite element model usually produces a more accurate prediction of the low-order modal parameters used for model updating than the initial model, however it is not guaranteed that the updated model give an improved prediction of other independent data set, such high-order modal parameters, specific static deformation and local stress. Thus assessing the quality of the stochastically updated FE models is of significant value for assuring a reliable and useful model updating.

As pointed out in a recent verification and validation standard [20], calibrating parameters by minimizing the residuals between predictions and experiments only indicates model's fitting ability; there is no guarantee of future prediction capability. The assessment of quality of updated models is usually carried out by comparing model predictions with additional independent data set that have not been involved in model updating process [2], [21]. This approach serves well for aerospace and automotive engineering as global dynamic responses are of major concern. However, it may become inadequate for civil engineering applications where structural safety issues such as load-carrying capacity and stresses are of equal importance. For example, as indicated in a recent paper by Aktan and Brownjohn [22], the use of calibrated models with modal data for predictive simulations of local structural response may be problematic in civil engineering disciplines and this is analogous to danger of extrapolating from data that are only robust to interpolation. The problem is further complicated when quality of the stochastic FE models are concerned. Firstly, both the mean and variance/covariance of updating structural parameters contribute to model's predictive performance and model quality. Secondly, direct comparison between prediction and test data cannot be readily applied as the model predictions from the updated model are uncertain or random. Wong and Yao [23] pointed out that structural reliability is a major decision factor for decision making throughout the entire life of a structure, and they emphasized that health monitoring, model updating and damage detection, and reliability evaluation are sequential components for implementation of monitoring-based risk management. Beck and Au [24] performed Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation based on simulation data. Following the Wong and Yao's conceptual framework, Hua et al. [25] have explored the use of the structural reliability as a measure for assessing the quality of a stochastically updated model.

This study examines the use of reliability index calculated from various finite element models for assessing the quality of the updated models in the presence of uncertain modal data. As the exact reliability index of an existing structure is never known, the motivation of this work is mainly to confirm the usefulness of model updating for improved reliability assessment and at the meanwhile to highlight the limitations in model updating in the case of large uncertainty of modal data. A recently developed method that combines the improved perturbation technique and the sensitivity-based model updating algorithm is first adopted for stochastic model updating by using the measured uncertain modal data. The updated results may be further revised according to Bayesian updating when the prior distribution about the model parameters is available. Then structural reliability analysis is conducted on the initial and the updated models to compute the reliability index and failure probability for predefined limit state functions. The reliability indices computed are next compared with those obtained from the ‘true’ model to assess the quality of various FE models. The proposed method is demonstrated by numerical simulations of a truss bridge. Simulated modal data with different uncertainty magnitudes are used to conduct stochastic model updating. The effect of uncertainty magnitudes in the simulated modal data on the quality of updated models is also addressed.

Section snippets

Stochastic finite element updating

In the conventional sensitivity-based FE model updating, inferring model parameter from test data is achieved by solving an optimization problem where the model parameters in an initial model are sought so that the updated model reproduces as closely as possible the test data. When modal data are used, the objective function in the optimization problem is expressed as the residuals between the test modal data and the corresponding model output, as followsMinJθ=ϵTWϵ=W1/2(z˜z(θ))22where ε˜ is

Structural reliability analysis

The stochastic model updating have been performed to reconcile the measured modal data and the corresponding model output, therefore the quality of the updated models have a large reliance on that of the measured modal data. Furthermore, large variability in the updating parameters can be induced due to modal insensitivity to updating parameters, sparseness of measurement locations and high level noise in measurement data. In this study, structural reliability computed from the stochastically

Numerical example

A truss bridge is taken as example to illustrate the proposed methodology for assessing the quality of updated models with uncertainty by using reliability index. Three sets of analytical models are available to represent the bridge, namely the initial model, the updated models, and the true model. The initial model is established with structural design drawings and may not be able to reflect satisfactorily the condition of the bridge due to modeling error, incorrect model parameter values, and

Concluding remarks

This paper has investigated the use of structural reliability as a measure for assessing the quality of the stochastically updated models which are obtained from uncertain modal data. The numerical studies of stochastic model updating of a truss bridge involving different uncertainty magnitudes of modal data are used to update an initial model of the bridge. The results show that the reliability index obtained from the updated model is much closer to true reliability index than that obtained

Acknowledgements

The authors would like to acknowledge the financial support from the National Science Foundation of China for Excellent Young Scholars (No. 51422806).

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