Probabilistic Damage Identification Research Articles

Lamb Wave Probabilistic Damage Identification Based on the Exchanging-Element Time-Reversal Method.

The commonly used baseline-free Lamb wave damage identification methods often require a large amount of sensor data to eliminate the dependence on baseline signals. To improve the efficiency of damage localization, this paper proposes a new Lamb wave damage location method, namely the probabilistic exchanging-element time-reversal method (PEX-TRM), which is based on the exchanging-element time-reversal method (EX-TRM) and the probabilistic damage identification method. In this method, the influence of the damage wave packet migration on the correlation coefficient between the reconstructed signals of each sensing path and the initial excitation signal is analyzed, and the structure is divided into multiple regional units corresponding to the damage to locate damage. In addition, the influence of the number of sensing paths on the location accuracy is also analyzed. A method of damage probability imaging based on structural symmetry is proposed to enhance location accuracy in the case of sparse sensing paths. The experimental and simulation results verify that the method can achieve damage location with fewer excitation times. Moreover, this method can avoid the problem that the damage wave packet is difficult to extract, improve the efficiency of damage location, and promote the engineering application of the Lamb wave damage location method.

Quantitative method for the probability of structural damage based on moment theory

In the context of uncertainty in the inverse problem of damage identification, this study proposes a moment theory based probabilistic damage identification (MbPDI) method. Noise errors in the identification process are treated as uncertain parameters, while structural stiffness parameters are treated as functions of noise errors. A limit state function for damage identification is defined, and its probability is computed. Utilizing the concept of moment theory, the expansion series is performed for the limit state function around the most probable point of failure, with gradient information derived using sensitivity matrices. Based on the gradient information and uncertainty inputs, damage probabilities for various elements can be obtained. Additionally, this paper addresses the calculation of structural damage probabilities under the non-normal distribution of random variables by transforming non-normal variables into equivalent normal variables. The proposed probability index and expected index of structural damage in the probabilistic damage identification method has a clear physical meaning, and the calculation has high accuracy. The proposed method is validated through a mathematical example and a numerical simulation. Meanwhile, the influence of damage extent, noise level, and modal order on identification results are analyzed. Experimental validation of the method is also presented.

Probabilistic damage identification for bridges using multiple damage-sensitive features and FE model update

A specific damage-sensitive feature (DSF) is usually sensitive to damage in a certain area, and the specific location of damage in a specific area can be detected by selecting appropriate DSFs. However, damaged components are seldom known before inspections. This makes it difficult to find an appropriate DSF, and damage identification is sometimes challenging. This paper aims to propose a natural frequency and displacement ratio-based probabilistic damage identification method for bridges using the finite element (FE) model update to solve this issue. The displacement ratio measured at two positions on the bridge is proposed as a new DSF. It is a useful DSF because it is independent of external loads. By integrating displacement ratios and natural frequencies, that is, multiple DSFs, as a decision-level data fusion approach, this paper identifies damaged components without choosing an appropriate DSF beforehand. In addition, probability density functions (PDFs) of structural parameters are estimated from PDFs of derived DSFs through the FE model update to consider errors and uncertainties in measurements. An in-house model bridge experiment is carried out to investigate the feasibility. The results demonstrated that the two kinds of damages in a bearing and girder reproduced in the experiment were successfully identified without false positives even when these damages simultaneously occurred.

A Probabilistic Structural Damage Identification Method with a Generic Non-Convex Penalty

Due to the advantage that the non-convex penalty accurately characterizes the sparsity of structural damage, various models based on non-convex penalties have been effectively utilized to the field of structural damage identification. However, these models generally ignore the influence of the uncertainty on the damage identification, which inevitably reduces the accuracy of damage identification. To improve the damage identification accuracy, a probabilistic structural damage identification method with a generic non-convex penalty is proposed, where the uncertainty corresponding to each mode is quantified using the separate Gaussian distribution. The proposed model is estimated via the iteratively reweighted least squares optimization algorithm according to the maximum likelihood principle. The numerical and experimental results illustrate that the proposed method improves the damage identification accuracy by 3.98% and 7.25% compared to the original model, respectively.

Uncertain damage identification methods based on residual force vector under the influence of measurement noise

This paper proposes probabilistic and interval methods, grounded in the residual force vector technique, to quantify measurement uncertainties and effectively integrate them into the identification process. In the probabilistic method, the measurement noise is assumed to be a zero-mean normal distribution, while the stiffness parameters of structural elements are assumed to follow a normal distribution, represented by mean values and standard deviations. In the interval method, the measurement noise is modelled as interval numbers, and the interval perturbation technique is employed to derive interval bounds for elemental stiffness parameters. A two-step model updating strategy is implemented, separately addressing pristine and damaged structures, to mitigate modeling errors. By using the results of probabilistic damage identification, a novel damage index termed damage expectation capable of capturing both the extent and probability of damage is introduced. Furthermore, an interval damage index is introduced to quantify the damage based on interval damage identification results. The impact of noise levels on the statistical properties of uncertain damage indices is also investigated. Numerical examples reveal that deterministic indexes exhibit a monotonic increase in average values with increasing noise levels. Particularly, as noise levels rise, the standard deviation of the combination index initially surpasses the deterministic damage index but eventually diminishes. Through the proposed methodology, structural damage can be identified even in the presence of uncertainties. The efficacy of the proposed damage identification technique is also validated by experimental data.

Probabilistic damage identification of FGM structures using model updating procedure based on expansion of incomplete FRF data

The spatial incompleteness and uncertainties of measured data are unavoidable factors that could significantly affect the reliability of damage detection results. It is therefore essential to utilize a method that can effectively deal with the spatial incomplete and contaminated data, and presents probabilistic damage identification results instead of deterministic results. In this regard, a probabilistic damage identification approach for functionally graded materials (FGM) structures using model updating procedure based on expansion of incomplete frequency response function (FRF) data with measurement noise is presented. The model updating procedure is formulated as an optimization scheme which is accomplished by minimizing a cost function based on the changes in expanded FRF data. To expand the incompletely measured FRFs, an iterative order reduction method is performed, which makes the identification resistant to the adverse effects of measurement noise. For the minimization process, we adopt a novel meta-heuristic algorithm called bald eagle search algorithm (BES), which has not yet been tested in the field of model updating. Based on the statistical distributions of the identified stiffness parameters in the damaged and undamaged states, the probability of damage existence (PDE) is established to describe the damage probability for each element. The performance of the proposed model updating procedure is verified using three FGM structures: a simple beam, a two-span beam and a cantilever plate. The statistical results indicate that under a high level of noise (15%), the proposed procedure can provide the prediction of damage localization with a high level of confidence and yield damage estimation results with acceptable errors.

Probabilistic damage detection and identification of coupled structural parameters using Bayesian model updating with added mass

Damage detection inevitably involves uncertainties originated from measurement noise and modeling error. It may cause incorrect damage detection results if not appropriately treating uncertainties. To this end, vibration-based Bayesian model updating (VBMU) is developed to utilize vibration responses or modal parameters to estimate structural parameters and the associated uncertainties of those estimates. However, traditional VBMU often assumes that mass is well known and invariant because simultaneous identification of mass and stiffness may yield an unidentifiable problem due to the coupling effect of the mass and stiffness. In addition, the posterior PDF in VBMU is usually approximated by single Markov Chain Monte Carlo (MCMC), leading to a low acceptance rate and limited capability for complex structures. This paper proposed a novel VBMU to address the coupling effect and identify mass and stiffness by adding known mass. Two vibration data sets are acquired from original and modified systems with added mass, giving the new characteristic equations. Then, the posterior PDF is reformulated by measured data and predicted counterparts from new characteristic equations. For efficiently approximating the posterior PDF, Differential Evolutionary Adaptive Metropolis (DREAM) algorithm is adopted to draw samples by running multiple Markov chains parallelly to enhance acceptance rate and sufficiently explore possible solutions. Finally, a numerical example with a ten-story shear building and a laboratory-scale three-story frame structure are utilized to demonstrate the efficacy of the proposed VBMU framework. The results show that the proposed method can successfully identify both mass and stiffness, and their uncertainties. Reliable probabilistic damage detection can also be achieved.

Real-time Bayesian damage identification enabled by sparse PCE-Kriging meta-modelling for continuous SHM of large-scale civil engineering structures

This work presents a surrogate model-based Bayesian model updating (BMU) approach for automated damage identification of large-scale structures, which outperforms methods currently available in the literature by effectively solving the real-time damage identification challenge. The computational difficulties involved in Bayesian inference using intensive numerical models are circumvented by implementing a high-fidelity surrogate model and an adaptive Markov Chain Monte Carlo (MCMC) algorithm. The developed surrogate model combines adaptive sparse polynomial chaos expansion (PCE) and Kriging meta-modelling. The optimal order of the polynomials in the PCE is automatically identified by a model selection technique for sparse linear models, the least-angle regression (LAR) algorithm. Then, the optimal PCE is inserted into a Kriging predictor as the trend term, while the stochastic term is fitted through a global optimization algorithm. Afterwards, the surrogate model bypassing the original numerical model is used for BMU exploiting monitoring data extracted from continuous ambient vibration measurements. The computational demands of the MCMC algorithm are kept minimal by implementing an adaptive Metropolis sampling with delayed rejection (DRAM). The effectiveness of the proposed methodology is demonstrated through three case studies: an analytical benchmark; a planar truss structure; and a real case study of an instrumented historical tower, the Sciri Tower in Italy. The presented results demonstrate that the proposed BMU approach is compatible with real-time Structural Health Monitoring (SHM), providing promising evidence for the development of digital twins with superior probabilistic damage identification capabilities.

A transfer Bayesian learning methodology for structural health monitoring of monumental structures

A critical aspect in structural health monitoring (SHM) applied to civil engineering structures is the lack of diagnostic labels able to assign a damage class to the measured data. In this context, a semi-supervised learning methodology, designated as transfer Bayesian learning (TBL), is proposed with the main objective of labeling post-processed data in a probabilistic way by selecting a limited number of informative elements. The proposed method allows to define multi-class labels by making use of a surrogate model (SM) of the structure considering specific damage-sensitive mechanical parameters. The methodology is applied in a monumental building, the Consoli Palace, located in Gubbio, central Italy. The structure is instrumented with several sensors in order to measure vibrations, temperature and possible variation of existing cracks’ amplitudes. Several nonlinear pushover analyses are carried out on a calibrated finite element (FE) model to use them in conjunction with Engineering judgment for the definition of the damage-sensitive regions. The SM, consisting of a simplified model that continuously exchanges information with the physical reality observed through the measurement system, is then used as a class classifier by means of a sensitivity damage chart (SDC). Finally, a Bayesian model updating of the damage-dependent parameters allows the probabilistic damage identification.

Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage

In this paper, we give an overview of the issues one must consider when designing methods for vibration based health monitoring systems for perforated thin shells especially in relation to frequency response analysis. In particular, we allow either the material parameters or the structure or both to be random. The numerical experiments are computed using the standard high order finite element method with stochastic collocation for the cases with random material and Monte Carlo for those with damaged or random structures. The results display a wide range of responses over the experimental configurations. In perforated shell structures, the internal boundary layers can play an important role especially when damage is allowed within the penetration patterns. The computational methodology advocated here can be used to build statistical databases that are necessary for development of probabilistic damage identification methods.

Probabilistic damage identification incorporating approximate Bayesian computation with stochastic response surface

Abstract Compared with classic probability or statistical theories, Bayesian inference could be a better option for real-world damage identification problems since it can simultaneously utilize both prior knowledge and current measurements of a structure. However for a complex problem, a Bayesian identification procedure should face the challenge of unknown likelihood functions and unacceptable computational expenses. Due to it, approximate Bayesian computation (ABC) is adopted and incorporated with the Metropolis Hastings sampling (MHS) algorithm and stochastic response surface (SRS). Likelihood functions are no longer required during the inference. A stable Markov chain is obtained containing qualified parameter samples, whose random responses are then fast computed by SRSs. Additionally, a nested Bayesian updating procedure with a dynamic tolerance is proposed for fast estimation convergence of posterior probability distributions. After that a probabilistic damage index is defined based on the statistical features of the estimated posterior probability distributions. Lastly, the feasibility of the proposed method has been validated using both numerical and experimental beams. Their damage was indicated by the magnitude variations of the damage index.

Damage Identification of Periodically-Supported Structures Following the Bayesian Probabilistic Approach

This paper presents a probabilistic damage identification methodology tailor-made for periodically-supported structures with finite-length. The free wave motion of a general periodically-supported structure with a single disorder is analyzed through the characteristic receptance approach, and the corresponding frequency characteristic equation is developed. In addition, a concept of nondimensional frequency is introduced, and the sensitivity matrix of the nondimensional frequencies with respect to changes in stiffness of periodic cells is obtained by solving the frequency characteristic equation and utilizing the sensitivity analysis technique. Following the sensitivity-based identification equation with nondimensional frequency information, the probabilistic methodology for identifying the damage occurring in the periodically-supported structures is developed by implementing the Bayesian approach and the Markov chain Monte Carlo (MCMC) simulation with the Metropolis–Hasting sampling algorithm. The validity of the proposed methodology is demonstrated by both numerical simulations for a periodically-supported flanged pipeline example and experimental case studies conducted for a multi-span aluminum beam model endowed with bolted connections in the laboratory.

A Probabilistic Damage Identification Method for Shear Structure Components Based on Cross-Entropy Optimizations

A probabilistic damage identification method for shear structure components is presented. The method uses the extracted modal frequencies from the measured dynamical responses in conjunction with a representative finite element model. The damage of each component is modeled using a stiffness multiplier in the finite element model. By coupling the extracted features and the probabilistic structural model, the damage identification problem is recast to an equivalent optimization problem, which is iteratively solved using the cross-entropy optimization technique. An application example is used to demonstrate the proposed method and validate its effectiveness. Influencing factors such as the location of damaged components, measurement location, measurement noise level, and damage severity are studied. The detection reliability under different measurement noise levels is also discussed in detail.

Probabilistic damage identification of a designed 9-story building using modal data in the presence of modeling errors

Validity and accuracy of model based identification techniques such as linear finite element (FE) model updating are sensitive to modeling errors. Models used for the design and performance assessment of civil structures often contain large modeling errors for certain frequency ranges of response. In other words, modeling errors have unequal effects on different vibration modes of structures. Therefore, the performance of FE model updating for damage identification is sensitive to the type and the subset of data used and to the residual weight factors. This study proposes a process to mitigate the effects of modeling errors by selecting the optimal subset of modes and the optimal modal residual weights. Multiple model updating classes are defined based on different subsets of modes and different weight factors. Structural damage is then identified using Bayesian model class selection and model averaging techniques over the results of all the considered model updating classes. In addition, a new likelihood function is defined to allow damage identification without the need for calibrating a reference FE model. Performance of the proposed damage identification process and the new likelihood function is evaluated numerically at multiple levels of modeling errors and structural damage on the SAC 9-story steel moment frame. It is shown that the structural damages can be identified with negligible bias when the proposed likelihood and updating process is implemented.

Hierarchical Bayesian model updating for structural identification

A new probabilistic finite element (FE) model updating technique based on Hierarchical Bayesian modeling is proposed for identification of civil structural systems under changing ambient/environmental conditions. The performance of the proposed technique is investigated for (1) uncertainty quantification of model updating parameters, and (2) probabilistic damage identification of the structural systems. Accurate estimation of the uncertainty in modeling parameters such as mass or stiffness is a challenging task. Several Bayesian model updating frameworks have been proposed in the literature that can successfully provide the “parameter estimation uncertainty” of model parameters with the assumption that there is no underlying inherent variability in the updating parameters. However, this assumption may not be valid for civil structures where structural mass and stiffness have inherent variability due to different sources of uncertainty such as changing ambient temperature, temperature gradient, wind speed, and traffic loads. Hierarchical Bayesian model updating is capable of predicting the overall uncertainty/variability of updating parameters by assuming time-variability of the underlying linear system. A general solution based on Gibbs Sampler is proposed to estimate the joint probability distributions of the updating parameters. The performance of the proposed Hierarchical approach is evaluated numerically for uncertainty quantification and damage identification of a 3-story shear building model. Effects of modeling errors and incomplete modal data are considered in the numerical study.

Probabilistic identification of simulated damage on the Dowling Hall footbridge through Bayesian finite element model updating

Summary This paper presents a probabilistic damage identification study on a full-scale structure, the Dowling Hall footbridge, through a Bayesian finite element (FE) model updating. The footbridge is located at Tufts University and is equipped with a continuous monitoring system that measures its ambient acceleration response. A set of data is recorded once every hour or when triggered by large vibrations. The modal parameters of the footbridge are extracted from each set of data and are used for FE model updating. In this study, effects of physical damage are simulated by loading a small segment of the footbridge deck with concrete blocks. The footbridge deck is divided into five segments in an FE model of the test structure, and the added mass on each segment is considered as an updating parameter. Overall, 72 sets of data are collected during the loading period, and different subsets of these data are used to find the location and extent of the damage (added mass). The adaptive Metropolis–Hastings algorithm with adaption on the proposal probability density function is successfully used to generate Markov Chains for sampling the posterior probability distributions of the five updating parameters. Effects of the number of data sets used in the identification process are investigated on the posterior probability distributions of the updating parameters. The probabilistic model updating framework accurately predicts the simulated damage and the level of confidence on the obtained results. The maximum a-posteriori estimates of damage in the probabilistic approach are found to be in good agreement with their corresponding deterministic counterparts. Copyright © 2014 John Wiley & Sons, Ltd.

Probabilistic damage identification of structures with uncertainty based on dynamic responses

The probabilistic damage identification problem with uncertainty in the FE model parameters, external-excitations and measured acceleration responses is studied. The uncertainty in the system is concerned with normally distributed random variables with zero mean value and given covariance. Based on the theoretical model and the measured acceleration responses, the probabilistic structural models in undamaged and damaged states are obtained by two-stage model updating, and then the Probabilities of Damage Existence (PDE) of each element are calculated as the damage criterion. The influences of the location of sensors on the damage identification results are also discussed, where one of the optimal sensor placement techniques, the effective independence method, is used to choose the nodes for measurement. The damage identification results by different numbers of measured nodes and different damage criterions are compared in the numerical example.

A Probabilistic Damage Identification Approach for Structures under Unknown Excitation and with Measurement Uncertainties

Recently, an innovative algorithm has been proposed by the authors for the identification of structural damage under unknown external excitations. However, identification accuracy of this proposed deterministic algorithm decreases under high level of measurement noise. A probabilistic approach is therefore proposed in this paper for damage identification considering measurement noise uncertainties. Based on the former deterministic algorithm, the statistical values of the identified structural parameters are estimated using the statistical theory and a damage index is defined. The probability of identified structural damage is further derived based on the reliability theory. The unknown external excitations to the structure are also identified by statistical evaluation. A numerical example of the identification of structural damage of a multistory shear-type building and its unknown excitation shows that the proposed probabilistic approach can accurately identify structural damage and the unknown excitations using only partial measurements of structural acceleration responses contaminated by intensive measurement noises.

Structural Health Monitoring: From Sensing Technology Stepping to Health Diagnosis

Structural health monitoring (SHM) takes a breakthrough of civil engineering by integrating electrical, magnetic, photic, acoustic, thermal and other physical variables, chemical variables, information technology, computer science and technology as well as communication technology into a civil structure to make the structure have self-sensing and self-diagnostic abilities. The SHM is also a basis of smart earth, which may take a profound impact on the behavior and lifestyle of society. The SHM includes sensing technology, data acquisition, transmission and management, and health diagnosis. In this paper, the sensing technologies are developed for monitoring of fatigue, corrosion, score and seismic damages by using integrating piezo-electric ceramic array with optical fiber Bragg Grating sensor array and ultrasonic monitoring technology. The approaches of health diagnosis for civil structures have been proposed by authors and their group. The damage detection approaches considering the uncertainties of civil structures and environmental factors are proposed, such as probabilistic damage identification approach based on dynamic sensitivity analysis and damage detection approach by using information fusion techniques. A multi-scale finite element model (FEM) updating approach is presented for conducting safety evaluation. The modeling approaches for various loads and responses using SHM data are proposed and the framework of SHM-based structure safety evaluation is established. Finally, the role of SHM in developing smart earth in the future is also put forwarded.

A probabilistic damage identification approach for structures with uncertainties under unknown input

To avoid the false positives of damages in the deterministic identification method induced by uncertainties in measurement noise, a probabilistic method is proposed to identify damages of the structures with uncertainties under unknown input. The proposed probabilistic method is developed from a deterministic simultaneous identification method of structural physical parameters and input based on dynamic response sensitivity. The deterministic simultaneous identification method is first derived. The effect of uncertainties caused by measurement noise on the identified parameters is then investigated. The statistical parameters of the identified structural parameters are calculated. The damage index is derived from the statistical parameters of the physical parameters of intact and damaged structure. The probability of identified damage, defined as the probability of identified structural stiffness smaller than that of intact structure, is further derived using the probability method. A twelve-story building and a nine-bay three-dimensional frame structure are, respectively, analyzed numerically and experimentally using the proposed method. The research results indicate that the probabilistic simultaneous identification method for damage and input can decrease the false positives of damages in contrast with the deterministic method under intensive measurement noise, and it can also achieve an accurate identification for structural unknown input.