Structural damage identification of the highway bridge Z24 by FE model updating

The development of a methodology for accurate and reliable condition assessment of civil structures has become very important. The finite element (FE) model updating method provides an efficient, non-destructive, global damage identification technique, which is based on the fact that the modal parameters (eigenfrequencies and mode shapes) of the structure are affected by structural damage. In the FE model the damage is represented by a reduction of the stiffness properties of the elements and can be identified by tuning the FE model to the measured modal parameters. This paper describes an iterative sensitivity based FE model updating method in which the discrepancies in both the eigenfrequencies and unscaled mode shape data obtained from ambient tests are minimized. Furthermore, the paper proposes the use of damage functions to approximate the stiffness distribution, as an efficient approach to reduce the number of unknowns. Additionally the optimization process is made more robust by using the trust region strategy in the implementation of the Gauss-Newton method, which is another original contribution of this work. The combination of the damage function approach with the trust region strategy is a practical alternative to the pure mathematical regularization techniques such as Tikhonov approach. Afterwards the updating procedure is validated with a real application to a prestressed concrete bridge. The damage in the highway bridge is identified by updating the Young's and the shear modulus, whose distribution over the FE model are approximated by piecewise linear functions.     

Statistical damage identification of structures with frequency changes

Model updating methods based on structural vibration data have being rapidly developed and applied to detect structural damage in civil engineering. But uncertainties existing in the structural model and measured vibration data might lead to unreliable damage detection. In this paper a statistical damage identification algorithm based on frequency changes is developed to account for the effects of random noise in both the vibration data and finite element model. The structural stiffness parameters in the intact state and damaged state are, respectively, derived with a two-stage model updating process. The statistics of the parameters are estimated by the perturbation method and verified by Monte Carlo technique. The probability of damage existence is then estimated based on the probability density functions of the parameters in the two states. A higher probability statistically implies a more likelihood of damage occurrence. The presented technique is applied to detect damages in a numerical cantilever beam and a laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and damage level on damage identification results are also discussed.     

Mode shape expansion using perturbed force approach

A new approach for expanding incomplete experimental mode shapes is presented which considers the modelling errors in the analytical model and the uncertainties in the vibration modal data measurements. The proposed approach adopts the perturbed force vector that includes the effect of the discrepancy in mass and stiffness between the finite element model and the actual tested dynamic system. From the developed formulations, the perturbed force vector can be obtained from measured modal data and is then used for predicting the unmeasured components of the expanded experimental mode shapes. A special case that does not require the experimental natural frequency in the mode shape expansion process is also discussed. A regularization algorithm based on the Tikhonov solution incorporating the generalized cross-validation method is employed to filter out the influence of noise in measured modal data on the predictions of unmeasured mode components. The accuracy and robustness of the proposed approach is verified with respect to the size of measured data set, sensor location, model deficiency and measurement uncertainty. The results from two numerical examples, a plane frame structure and a thin plate structure, show that the proposed approach has the best performance compared with the commonly used existing expansion methods, and can reliably produce the predictions of mode shape expansion, even in the cases with limited modal data measurements, large modelling errors and severe measurement noise.     

Nonlinear finite element model updating of an infilled frame based on identified time-varying modal parameters during an earthquake

A model updating methodology is proposed for calibration of nonlinear finite element (FE) models simulating the behavior of real-world complex civil structures subjected to seismic excitations. In the proposed methodology, parameters of hysteretic material models assigned to elements (or substructures) of a nonlinear FE model are updated by minimizing an objective function. The objective function used in this study is the misfit between the experimentally identified time-varying modal parameters of the structure and those of the FE model at selected time instances along the response time history. The time-varying modal parameters are estimated using the deterministic–stochastic subspace identification method which is an input–output system identification approach. The performance of the proposed updating method is evaluated through numerical and experimental applications on a large-scale three-story reinforced concrete frame with masonry infills. The test structure was subjected to seismic base excitations of increasing amplitude at a large outdoor shake-table. A nonlinear FE model of the test structure has been calibrated to match the time-varying modal parameters of the test structure identified from measured data during a seismic base excitation. The accuracy of the proposed nonlinear FE model updating procedure is quantified in numerical and experimental applications using different error metrics. The calibrated models predict the exact simulated response very accurately in the numerical application, while the updated models match the measured response reasonably well in the experimental application.     

STRUCTURAL DAMAGE DETECTION BASED ON A MICRO-GENETIC ALGORITHM USING INCOMPLETE AND NOISY MODAL TEST DATA

This paper describes a procedure for detecting structural damage based on a micro-genetic algorithm using incomplete and noisy modal test data. As the number of sensors used to measure modal data is normally small when compared with the degrees of freedom of the finite element model of the structure, the incomplete mode shape data are first expanded to match with all degrees of freedom of the finite element model under consideration. The elemental energy quotient difference is then employed to locate the damage domain approximately. Finally, a micro-genetic algorithm is used to quantify the damage extent by minimizing the errors between the measured data and numerical results. The process may be either of single-level or implemented through two-level search strategies. The study has covered the use of frequencies only and the combined use of both frequencies and mode shapes. The proposed method is applied to a single-span simply supported beam and a three-span continuous beam with multiple damage locations. In the study, the modal test data are simulated numerically using the finite element method. The measurement errors of modal data are simulated by superimposing random noise with appropriate magnitudes. The effectiveness of using frequencies and both frequencies and mode shapes as the data for quantification of damage extent are examined. The effects of incomplete and noisy modal test data on the accuracy of damage detection are also discussed.     

Numerical and experimental analysis of uncertainty on modal parameters estimated with the stochastic subspace method

Modal parameters of structures are often used as inputs for finite element model updating, vibration control, structural design or structural health monitoring (SHM). In order to test the robustness of these methods, it is a common practice to introduce uncertainty on the eigenfrequencies and modal damping coefficients under the form of a Gaussian perturbation, while the uncertainty on the mode shapes is modeled in the form of independent Gaussian noise at each measured location. A more rigorous approach consists however in adding uncorrelated noise on the time domain responses at each sensor before proceeding to an operational modal analysis. In this paper, we study in detail the resulting uncertainty when modal analysis is performed using the stochastic subspace identification method. A Monte-Carlo simulation is performed on a simply supported beam, and the uncertainty on a set of 5000 modal parameters identified with the stochastic subspace identification method is discussed. Next, 4000 experimental modal identifications of a small clamped–free steel plate equipped with 8 piezoelectric patches are performed in order to confirm the conclusions drawn in the numerical case study. In particular, the results point out that the uncertainty on eigenfrequencies and modal damping coefficients may exhibit a non-normal distribution, and that there is a non-negligible spatial correlation between the uncertainty on mode shapes at sensors of different locations.     

PREDICTION OF DYNAMIC CHARACTERISTICS USING UPDATED FINITE ELEMENT MODELS

Model updating techniques are used to update a finite element model of a structure so that an updated model predicts the dynamics of a structure more accurately. The application of such an updated model in dynamic design demands that it also predict the effects of structural modifications with a reasonable accuracy. This paper deals with updating of a finite element model of a structure and its subsequent use for predicting the effects of structural modifications. Updated models have been obtained by a direct model updating method and by an iterative method of model updating based on the frequency response function (FRF) data. The suitability of updated models for predicting the effect of structural modifications is evaluated by some computer and laboratory experiments. First a study is performed using a simulated fixed-fixed beam. Cases of complete, incomplete and noisy data are considered. Updated models are obtained by the direct and the FRF-based method in each of these cases. These models are then used for predicting the changes in the dynamic characteristics brought about due to a mass and a beam modification. The simulated study is followed by a study involving actual measured data for the case of an F-shape test structure. The updated finite element models for this structure are obtained again by the direct and the FRF-based method. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design. It is found that the predictions based on the iterative method based updated model are reasonably accurate and, therefore, this updated model can be used with reasonable accuracy to perform dynamic design. The predictions on the basis of the direct method based updated model are found to be reasonably accurate in the lower portion of the updating frequency range but the predictions are in a significant error in the remaining portion of the updating frequency range. It is concluded that the updated models that are closer to the structure physically are likely to perform better in predicting the effects of structural modification.     

Structural damage evolution assessment using the regularised time step integration method

This paper presents an approach to identify both the location and severity evolution of damage in engineering structures directly from measured dynamic response data. A relationship between the change in structural parameters such as stiffness caused by structural damage development and the measured dynamic response data such as accelerations is proposed, on the basis of the governing equations of motion for the original and damaged structural systems. Structural damage parameters associated with time are properly chosen to reflect both the location and severity development over time of damage in a structure. Basic equations are provided to solve the chosen time-dependent damage parameters, which are constructed by using the Newmark time step integration method without requiring a modal analysis procedure. The Tikhonov regularisation method incorporating the L-curve criterion for determining the regularisation parameter is then employed to reduce the influence of measurement errors in dynamic response data and then to produce stable solutions for structural damage parameters. Results for two numerical examples with various simulated damage scenarios show that the proposed method can accurately identify the locations of structural damage and correctly assess the evolution of damage severity from information on vibration measurements with uncertainties.     

Finite element model updating of a rod-type ultrasonicmotor based on response surface method

The conventional finite element model(FEM) of a rod-type ultrasonic motor is usually simplified by means of continuous composite structure. Because the actual contact characteristics between the parts of the ultrasonic motor is ignored, there is bigger error between the calculated values and experimental results. Aiming at solving problem, a new modeling method of a rod-type ultrasonic motor is presented to obtain a high-accuracy FEM. The bolt pretension and the normal contact stiffness and friction coefficient of the contact surface of ultrasonic motor are all considered in this method, and the significant parameters of working mode of the motor are selected by the response surface method, and the goal of calculating the structural response rapidly is realized by building the response surface model to replace the FEM. The result of finite element model updating shows that the average error of modal frequencies of updated model drops to 0.21% from 1.20%. The accuracy of FEM is obviously improved, which indicates that the FEM updating based on response surface method is of great application value on the design for a rod-type ultrasonic motor.     

Substructure methods for structural condition assessment

It is commonly known that an accurate analysis of a large structure requires an accurate analytical model. This is also true for the inverse analysis of a structural system where measured structural responses are used as input to assess the structural conditions. However, an accurate model of the structure is always not available in practice. Two substructural identification methods are presented in this paper with the structure divided into substructures and with one substructure assessed at one time. In the first method, an accurate finite element model of the whole structure is assumed known. A state space method is applied to identify the external forces acting on the structure, and a damage identification method is then applied to identify the local damages using time domain information. Iterative model updating method based on the measured acceleration in the selected substructure is employed for the assessment. The second identification method requires only the finite element model of the substructure. The interface forces and the external forces acting on the target substructure are all taken as excitations and they are identified in state space. The substructure is then assessed similar to the first method. Since the target substructure for updating consists of a much reduced number of components and the identification problem is more efficient. The validation of the proposed methods is demonstrated by a truss structure with polluted measured accelerations with promising results.     

DYNAMIC ANALYSIS OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT FLEXURAL RIGIDITY

For the purpose of developing a vibration-based tension force evaluation procedure for bridge cables using measured multimode frequencies, an investigation on accurate finite element modelling of large-diameter sagged cables taking into account flexural rigidity and sag extensibility is carried out in this paper. A three-node curved isoparametric finite element is formulated for dynamic analysis of bridge stay cables by regarding the cable as a combination of an “ideal cable element” and a fictitious curved beam element in the variational sense. With the developed finite element formulation, parametric studies are conducted to evaluate the relationship between the modal properties and cable parameters lying in a wide range covering most of the cables in existing cable-supported bridges, and the effect of cable bending stiffness and sag on the natural frequencies. A case study is eventually provided to compare the measured natural frequencies of main cables of the Tsing Ma Bridge and the computed frequencies with and without considering cable bending stiffness. The results show that ignoring bending stiffness gives rise to unacceptable errors in predicting higher order natural frequencies of the cables, and the proposed finite element formulation provides an accurate baseline model for cable tension identification from measured multimode frequencies.     

A computational approach to calculating frequency-dependent structural operators using the spectral finite element method and a wavenumber-based gridding technique

Spectral finite element methods are used to compute exact vibration solutions of structural models at specific frequencies. The applicability of these methods to certain areas of structural dynamics is limited by two major factors: the lack of separate structural operators (mass, damping, and stiffness matrices), and the subsequent difficulty in computing mode shapes via eigenvalue decomposition. In the work presented in this article, a method is investigated to accurately calculate spectral finite elements while overcoming these limitations. The approach incorporates a two-dimensional, discrete solution utilizing a wavenumber-based gridding technique to compute frequency-dependent local mass, damping, and stiffness matrices which can be assembled into the global structural operators. Computed models are able to be used for precise vibration analysis as well as modal analysis via eigenvalue decomposition of the structural operators.     

A new iterative model updating method using incomplete frequency response function data

A new iterative model updating method is proposed for reduced model using incomplete frequency response function (FRF) data. It uses a modified difference vector between the analytical and experimental FRF data to construct a linear sensitivity updating equation system. To improve the convergence performance of the proposed algorithm, a concept of pseudo master degree-of-freedom (DOF) is put forward and the finite element (FE) model is reduced to the measured and user selected pseudo DOFs. The FRFs at pseudo master DOFs are estimated using the impedance matrix of iteratively modified analytical model and the measured FRFs at master DOFs. They are only used to improve the sensitivity matrix and difference calculation between the analytical and experimental FRF data without introducing additional difference equation. At the end, a 25 truss structure is used to evaluate the performance of the proposed method.     

Structural model updating using incomplete transfer function of strain data

In this paper a model updating algorithm is presented to estimate structural parameters at the element level utilizing frequency domain representation of the strain data. Sensitivity equations for mass and stiffness parameters estimation are derived using decomposed form of the strain-based transfer functions. The rate of changes of eigenvectors and a subset of measured natural frequencies are used to assemble the sensitivity equation of the strain-based transfer function. Solution of the derived sensitivity equations through the least square method resulted in a robust parameters estimation method. Numerical examples using simulated noise polluted data of 2D truss and frame models confirm that the proposed method is able to successfully update structural models even in the presence of mass modeling errors.     

基于薄层单元与弹簧单元的滚动直线导轨副动力学建模

针对某大型光机装置中使用的LM型直线导轨-滑块的结合面,采用薄层单元与弹簧单元模拟其接触面,建立了大行程传输结构模拟件的有限元模型。通过模型修正方法,根据模态试验结果对薄层单元弹性模量以及弹簧刚度进行了识别。模型修正后,结构前三阶模态频率计算结果与试验结果最大差别为2.23%,地脉动载荷下各测点位移响应计算结果与试验结果最大差别为7.61%。计算结果与试验结果具有较好的一致性,验证了模型的有效性。     

Added mass matrix estimation of beams partially immersed in water using measured dynamic responses

An added mass matrix estimation method for beams partially immersed in water is proposed that employs dynamic responses, which are measured when the structure is in water and in air. Discrepancies such as mass and stiffness matrices between the finite element model (FEM) and real structure could be separated from the added mass of water by a series of correction factors, which means that the mass and stiffness of the FEM and the added mass of water could be estimated simultaneously. Compared with traditional methods, the estimated added mass correction factors of our approach will not be limited to be constant when FEM or the environment of the structure changed, meaning that the proposed method could reflect the influence of changes such as water depth, current, and so on. The greatest improvement is that the proposed method could estimate added mass of water without involving any water-related assumptions because all water influences are reflected in measured dynamic responses of the structure in water. A five degrees-of-freedom (dofs) mass-spring system is used to study the performance of the proposed scheme. The numerical results indicate that mass, stiffness, and added mass correction factors could be estimated accurately when noise-free measurements are used. Even when the first two modes are measured under the 5 percent corruption level, the added mass could be estimated properly. A steel cantilever beam with a rectangular section in a water tank at Ocean University of China was also employed to study the added mass influence on modal parameter identification and to investigate the performance of the proposed method. The experimental results demonstrated that the first two modal frequencies and mode shapes of the updated model match well with the measured values by combining the estimated added mass in the initial FEM.     

Inverse substructure method for model updating of structures

Traditional model updating of large-scale structures is usually time-consuming because the global structural model needs to be repeatedly re-analyzed as a whole to match global measurements. This paper proposes a new substructural model updating method. The modal data measured on the global structure are disassembled to obtain the independent substructural dynamic flexibility matrices under force and displacement compatibility conditions. The method is extended to the case when the measurement is carried out at partial degrees-of-freedom of the structure. The extracted substructural flexibility matrices are then used as references for updating the corresponding substructural models. An orthogonal projector is employed on both the extracted substructural measurements and the substructural models to remove the rigid body modes of the free–free substructures. Compared with the traditional model updating at the global structure level, only the sub-models at the substructural level are re-analyzed in the proposed substructure-based model updating process, resulting in a rapid convergence of optimization. Moreover, only measurement on the local area corresponding to the concerned substructures is required, and those on other components can be avoided. The effectiveness and efficiency of the proposed substructuring method are verified through applications to a laboratory-tested frame structure and a large-scale 600 m tall Guangzhou New TV Tower. The present technique is referred to as the inverse substructuring model updating method as the measured global modal data are disassembled into the substructure level and then the updating is conducted on the substructures only. This differs from the substructuring model updating method previously proposed by the authors, in which the model updating is still conducted in the global level and the numerical global modal data are assembled from those of substructures. That can be referred to as the forward substructuring model updating method.     

Identification of embedded horizontal cracks in beams using measured mode shapes

Mode shapes (MSs) have been extensively used to detect structural damage. This paper presents two new non-model-based methods that use measured MSs to identify embedded horizontal cracks in beams. The proposed methods do not require any a priori information of associated undamaged beams, if the beams are geometrically smooth and made of materials that have no stiffness discontinuities. Curvatures and continuous wavelet transforms (CWTs) of differences between a measured MS of a damaged beam and that from a polynomial that fits the MS of the damaged beam are processed to yield a curvature damage index (CDI) and a CWT damage index (CWTDI), respectively, at each measurement point. It is shown that the MS from the polynomial fit can well approximate the measured MS and associated curvature MS of the undamaged beam, provided that the measured MS of the damaged beam is extended beyond boundaries of the beam and the order of the polynomial is properly chosen. The proposed CDIs of a measured MS are presented with multiple resolutions to alleviate adverse effects caused by measurement noise, and cracks can be identified by locating their tips near regions with high values of the CDIs. It is shown that the CWT of a measured MS with the n-th-order Gaussian wavelet function has a shape resembling that of the n-th-order derivative of the MS. The crack tips can also be located using the CWTs of the aforementioned MS differences with second- and third-order Gaussian wavelet functions near peaks and valleys of the resulting CWTDIs, respectively, which are presented with multiple scales. A uniform acrylonitrile butadiene styrene (ABS) cantilever beam with an embedded horizontal crack was constructed to experimentally validate the proposed methods. The elastic modulus of the ABS was determined using experimental modal analysis and model updating. Non-contact operational modal analysis using acoustic excitations and measurements by two laser vibrometers was performed to measure the natural frequencies and MSs of the ABS cantilever beam, and the results compare well with those from the finite element method. Numerical and experimental crack identification can successfully identify the crack by locating its tips.