Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique

The numerical results from a finite element (FE) model often differ from the experimental results of real structures. FE model updating is often required to identify and correct the uncertain parameters of FE model and is usually posed as an optimisation problem. Setting up of an objective function, selecting updating parameters and using robust optimisation algorithm are three crucial steps in FE model updating. In this paper, a multiobjective optimisation technique is used to extremise two objective functions simultaneously which overcomes the difficulty of weighing the individual objective function of more objectives in conventional FE model updating procedure. Eigenfrequency residual and modal strain energy residual are used as two objective functions of the multiobjective optimisation. Only few updating parameters are selected on the basis of the prior knowledge of the dynamic behaviours of the structure and eigenfrequency sensitivity study. The proposed FE model updating procedure is first applied to the simulated simply supported beam. This case study shows that the methodology is robust with an effective detection of assumed damaged elements. The procedure is then successfully applied to the updating of a precast continuous box girder bridge that was tested on field under operational conditions.     

Statistical moments of autoregressive model residuals for damage localisation

Monitoring structural health is a problem with significant importance in the world today. Aging civil infrastructure and aircraft fleets have made non-destructive evaluation an important research topic. Non-destructive techniques based on dynamic signatures have struggled to gain widespread acceptance due to the perceived difficulty in applying these methods, as well as the mixed results they can produce. A simple and reliable method that is useful without in-depth knowledge of the structure is necessary to transition dynamic response-based health monitoring into the industrial mainstream.Modal parameters, including shifting frequencies, damping ratios, and mode shapes have received considerable attention as damage indicators. The results have been mixed and require an expert to carry out the testing and interpretation. Detailed knowledge of the structure before it becomes damaged is required, either in the form of experimental data or an analytical model.A method based on vector autoregressive (ARV) models is proposed. These models accurately capture the predictable dynamics present in the response. They leave the unpredictable portion, including the component resulting from unmeasured input shocks, in the residual.An estimate of the autoregressive model residual series standard deviation provides an accurate diagnosis of damage conditions. Additionally, a repeatable threshold level that separates damaged from undamaged is identified, indicating the possibility of damage identification and localisation without explicit knowledge of the undamaged structure. Similar statistical analysis applied to the raw data necessitates the use of higher-order moments that are more sensitive to disguised outliers, but are also prone to false indications resulting from overemphasising rarely occurring extreme values.Results are included from data collected using an eight-degree of freedom damage simulation test-bed, built and tested at Los Alamos National Laboratory (LANL). Confidence bounds on each moment are computed for the available data sets and are included to illustrate “significant” differences.     

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

Damage identification by multi-model updating in the modal and in the time domain

Computational model updating techniques are used to adjust selected parameters of finite element models in order to make the models compatible with experimental data. This is done by minimizing the differences (residuals) of analytical and experimental data, for example, natural frequencies and mode shapes by numerical optimization procedures. For a long-time updating techniques have also been investigated with regard to their ability to localize and quantify structural damage. The success of such an approach is mainly governed by the quality of the damage model and its ability to describe the structural property changes due to damage in a physical meaningful way. Our experience has shown that due to unavoidable modelling simplifications and measurement errors the changes of the corresponding damage parameters do not always indicate structural modifications introduced by damage alone but indicate also the existence of other modelling uncertainties which may be distributed all over the structure. This means that there are two types of parameters which have to be distinguished: the damage parameters and the other parameters accounting for general modelling and test data uncertainties. Although these general parameters may be physically meaningless they are necessary to achieve a good fit of the test data and it might happen that they cannot be distinguished from the damage parameters. For complex industrial structures it is seldom possible to generate unique structural models covering all possible damage scenarios so that one has to expect, that the parameters introduced for describing the damage will not be fully consistent with the physical reality. Even then the change of such parameters identified from test data taken continuously or temporarily over the time may serve as a feature for structural health monitoring. It is well known that low-frequency modal test data or static response data are not very well suited for detecting and quantifying localized small size damage. Time domain response data from impact tests carry high-frequency information which usually is lost when experimental modal data are utilized for damage identification. Even so only little literature was found addressing the utilization of experimental time histories for model updating in conjunction with damage identification.In the present paper we summarize the methodology of computational model updating and report about our experience with damage identification using two different model updating techniques. The first is based on classical modal residuals (natural frequencies and mode shapes) which is extended to allow for simultaneous updating of two models, one for the initial undamaged structure and the second for the damaged structure using the test data of both states (multi-model updating). The second technique uses residuals composed of measured and analytical time histories. Time histories have the advantage of carrying high-frequency information which is beneficial for the detection of local damage and which usually is lost when modal residuals are used. Both techniques have been applied to the same beam structure consisting of two thin face sheets which were bonded together by an adhesive layer. It was the aim of this application to study the performance of the two techniques to localize and quantify the damage which was introduced locally in the adhesive layer.     

Finite element model updating and structural damage identification using OMAX data

The main limitations in the finite element (FE) model updating technique lie in the ability of the FE model to represent the true behavior of the structure (modelling problem), and in the ability to identify enough modal parameters with sufficient accuracy, especially for large structures that are tested in operational conditions (identification problem). In this paper, the identification problem is solved with an OMAX approach, where an artificial force is used in operational conditions and a structural model is identified that takes both the forced and the ambient excitation into account. From an extensive case study on a real three-span bridge, it is observed that, while updating the FE model using the experimental output-only data yields a good fit, discrepancies show up when the more extensive set of OMAX data is used for validation, or even for updating. It can be concluded that an OMAX approach not only increases the well-posedness of the updating problem, it also allows to detect potential inaccuracies in the FE model.     

Structural crack detection without updated baseline model by single and multiobjective optimization

Identification, localization and quantification of structural damage can be performed through a model-updating procedure. Model-updating methods require a baseline finite element (FE) model of the undamaged structure, which imposes a restriction on their applicability and can become very problematic especially for large and complex civil structures. Modeling errors in the baseline model whose effects exceed the modal sensitivity to damage are critical and make an accurate estimation of damage impossible. This paper presents an identification algorithm using modal data for assessing structural damage that is based on FE-updating procedures and takes modeling error into account. To overcome its influence, differences of mode shapes and frequencies before and after damage for both numerical model and experimental measurements are used instead of the mode shapes and frequencies themselves. To formulate the objective function, two different approaches have been considered taking into account how these differences are grouped: a single-objective approach and a multiobjective approach. The effectiveness of both approaches is verified against numerical and experimental results.     

Damage detection of damaged beam by constrained displacement curvature

Structural damage detection technique involves the problem of how to locate and detect damage that occurs in a structure by using the observed changes of its dynamic and static characteristics. The objective of this study is to present an analytical method for damage detection by utilizing displacement curvature and all static deflection data to be expanded from the measured deflection data. Utilizing the measured displacements as displacement constraints to describe a damaged beam, minimizing the change of the displacement vector between undamaged and damaged states, and neglecting the variation between the test data and the analytical results, the deflection shape and displacement curvature of the damaged beam can be estimated, and the damage can be detected by the curvature. The validity and effectiveness of the proposed method are illustrated through comparison with the experimental results of a simple cantilever beam test.     

运用波传播和子结构技术检测结构损伤

提出了一种基于波传播和子结构技术来检测大型周期结构损伤的方法,分析了存在单一扰乱单元的有限周期结构的自由波传播,讨论了单元柔度变化对结构自振频率变化的敏感性。通过求解敏感性方程,用测得的自振频率的变化检测了大型周期结构的损伤位置和程度,并通过与子结构方法的结合,进一步提高了大型周期结构损伤检测的准确性和计算效率。对周期弹簧－质量结构损伤识别的数值结果说明,该方法不仅对结构损伤的位置和大小能够作出良好地预测,而且不需要知识未损伤结构的原始信息,只需要测得结构损伤前后的少数几个自振频率的变化,这对于实际的工程运用具有一定的吸引力。     

Damage detection in frame structures using damage locating vectors

Among the numerous vibration based damage detection methods, some are using dynamically measured flexibility matrix to detect and locate damage in the structure. One of the methods in this category is damage locating vector (DLV). It is based on extracting a set of load vectors from flexibility shifts that cause the same deformation in damaged and undamaged structures. As a result, the damaged member can be located by studying the characterizing stress of all members. In this study, the application of DLV method on damage detection of 3D frame structure was conducted. Some factors like the number of measured DOFs and the accuracy of modal data were assumed to be ideal so that other parameters could be examined. The characterizing stress for a 3D structure and the accuracy of damage index in different damage scenarios are discussed.     

Damage classification and estimation in experimental structures using time series analysis and pattern recognition

Developed for studying long sequences of regularly sampled data, time series analysis methods are being increasingly investigated for the use of Structural Health Monitoring (SHM). In this research, Autoregressive (AR) models were used to fit the acceleration time histories obtained from two experimental structures: a 3-storey bookshelf structure and the ASCE Phase II Experimental SHM Benchmark Structure, in undamaged and limited number of damaged states. The coefficients of the AR models were considered to be damage-sensitive features and used as input into an Artificial Neural Network (ANN). The ANN was trained to classify damage cases or estimate remaining structural stiffness. The results showed that the combination of AR models and ANNs are efficient tools for damage classification and estimation, and perform well using small number of damage-sensitive features and limited sensors.     

Characteristics of transmission error and vibration of broken tooth contact

One of the most important research topics relating to the development of a vibration-based condition monitoring system for a gearbox is that of quantitative analysis and testing of the effect of gear tooth damage on gearbox vibration. This paper presents a finite element model of the Transmission error (TE) and its associated characteristics, together with experimental results of vibration of broken tooth contact in a spur gear. The two-dimensional finite element model is developed to analyze the TE of the spur gear, and the TE for both the normal contact and broken tooth contact state of the spur gear pair are analyzed. As a result, the developed FE model well represents the difference in the static TE for a single-tooth and double-tooth contact of the undamaged and damaged gear pair. The Peak-to-peak magnitude of the static TE of the full-tooth fractured gear pair is significantly higher than that of undamaged gear pair. While the Peak-to-peak TE (PPTE) of the damaged gear tooth increases with increasing load, the ratio of PPTE of the undamaged gear pair and the full-tooth fractured gear pair decreases exponentially with increasing load. This shows that, to determine the fault level of gearbox, it is necessary to carefully apply the condition indicator when analyzing the impulse character of a vibration signal.

     

Damage identification in composite materials using ultrasonic based Lamb wave method

The paper presents an ultrasonic based Lamb waves propagation method for identifying and measuring the damage location in a material for SHM. The present work determines the experimental and analytical effects of various parameters on the sensitivity of damage detection and a methodology is proposed for estimating and measuring the location of damage in the test specimens. An experimental setup is used for generating A_o Lamb waves by calibrating ultrasonic pulse generation for optimal value of the parameters. The experiment is performed on two carbon fiber reinforced plastic bars in both undamaged and damaged state, where the two damaged states are (1) having a cut partway through the bar, perpendicular to the long axis of the bar and (2) having a circular hole. The Lamb wave propagation parameters are calibrated using the ultrasonic pulse generator test setup and the method was compared with direct measured values of ultrasonic instrument.     

Dynamic characteristics and model updating of damaged slab from ambient vibration measurements

This paper reports a field investigation using ambient vibration testing on a damaged floor slab of a reinforced concrete frame building. Due to unexpected heavy rainfall, the hill slope at the rear of building failed triggering a major landslide and causing major damage to the perimeter beams and parts of the slab on the first floor. The modal parameters namely natural frequencies and mode shapes were acquired using output only identification technique and the results obtained from the undamaged and damaged floor slabs were compared. It was observed that there was a 25–53% drop in natural frequencies of the damaged slab compared to the undamaged slab, with a much bigger drop for the lower modes. The irregularities in mode shapes identified correlates with the location of the cracks as revealed from visual examination on the damaged slab. Two finite element models of the slab were created using a finite element software package. The damaged slab was updated manually so as to match the modal parameters obtained experimentally. The results from this study further highlight the possibility and feasibility of using non-destructive vibration testing for condition monitoring of structures over more conventional testing techniques.     

基于模型修正的大跨斜拉桥损伤识别方法

首先,根据斜拉桥的结构特点,将主梁等效为弹性地基梁,由此可推知结构损伤而导致的内力重分布只影响损伤部位附近单元,以此作为结构损伤位置的初步判断的依据;然后,由于内力重分布程度与结构损伤前的初始位移成线性关系,可知汽车荷载作用下的索力变化对损伤更为敏感.根据基准有限元模型,采用优化算法,以损伤单元的损伤程度为设计变量,以实测索力变化和理论索力变化之差为目标函数,对有限元模型进行修正,进而实现结构损伤程度的识别;最后,以苏通长江大桥为例,对4种损伤工况进行了分析识别.计算结果表明,根据同一汽车荷载作用下的索力变化,可以有效地识别主梁单元的损伤位置和程度.     

运用改进残余力向量法的结构损伤识别研究

针对结构损伤识别时测试结构模态振型不完备的情况,采用了自由度凝聚方法进行模型凝聚,并给出了缩聚模型的残余力向量计算公式,证明了残余力向量对结构损伤单元的敏感性.为减小自由度凝聚及测量等因素带来的误差影响,提出了一种改进的残余力向量法.该方法首先通过测试损伤结构多阶模态得到残余力向量,并计算相对于损伤前结构的残余力向量得到残余力向量差矩阵,选取向量差矩阵各行元素中最大绝对值组成改进的残余力向量.损伤识别时,先运用改进的残余力向量确定可能损伤单元,再运用筛选法计算单元损伤程度.数值仿真算例说明,采用该方法只需结构的前几阶低阶频率和振型,就可以快速准确地识别出结构的损伤,显示了该方法具有较好的抗噪性.     

A study on finite element of pre damaged stress concentration factor for a composite laminate member with central circular

The study investigates the pre damaged stress concentration factor (SCF) for a composite laminate member (CLM) with central circular hole subjected to tensile loading. The presence of holes yields high stress concentrations in the structural members termed as stress raiser and often regarded as an important design deriver. The pre damaged SCF refers to the stress concentrations within elastic (undamaged) range of the CLM prior to the failure initiation. The traditional experimental and analytical methods for the estimation of SCF offer macro level behaviour which is not considered appropriate for CLM where lamina by lamina behaviour is more significant. A meso level finite element (FE) model is presented to capture the lamina by lamina influence on the overall SCF for a CLM using commercial software Abaqus. The paper deals with the fundamental influencing factors such as laminae orientation effect, stress distribution effect around periphery of the hole, decaying effect and ligament effect on SCF for CLM. Analytical models have been formulated to validate the FE models. The results of FE models have been found in close agreement with the analytical results.     

DAMAGE ASSESSMENT USING MODE SHAPE SENSITIVITIES

In this paper, a method for the localisation and assessment of damage is presented. The method is based on the use of mode shape sensitivities to changes in mass or stiffness in the test structure. With these sensitivities, differences in the dynamical behaviour of the structure in its undamaged and damaged conditions can be translated into damage information (location and amount of changes in mass or stiffness). Since the sensitivities are calculated on the basis of the experimentally determined mode shapes, there is no need for a prior finite element model of the test structure. The applicability of this technique is discussed on the basis of damage detection experiments performed on a beam-like structure (laboratory conditions) as well as data from the experiments on the I-40 highway bridge in Albuquerque, New Mexico. A comparison is made with a variety of existing damage detection techniques which do not require a finite element model.     

Experimental characterization and finite element modeling of critical thrust force in cfrp drilling

Composite laminates are used in many applications in ae-rospace/defense industries due to their high strength-to-weight ratio and corrosion resistance properties. In general, composite materials are hard-to-machine materials which exhibit low drilling efficiency and drilling-induced delamination damage at exit. Hence, it is important to understand the drilling processes for composite materials. This article presents a comprehensive study involving experimental characterization of drilling process to understand the cutting mechanism and relative effect of cutting parameters on delamination during drilling of carbon fiber reinforced plastic (CFRP). Thrust force and torque data are acquired for analyzing the cutting mechanism, initiation and propagation of delamination, and identification of critical thrust force below which no damage occurs. An FE model for prediction of critical thrust force has been developed and validated with experimental results. A [0/90] composite laminate is modeled simulating the last two plies in exit condition and a thin interface layer is inserted in between the plies to capture delamination extent. The tool geometry is modeled as “rigid body” with geometric features of twist drill used in experiments. The tool is indented on the workpiece to simulated tool feeding action into the workpiece. The FE model predicts the critical thrust force within 5% of the experimentally determined mean value.     

An experimental study on damage detection of structures using a timber beam

Using vibration methods for the damage detection and structural health monitoring in bridge structures is rapidly developing. However, very little work has so far been reported on timber bridges. This paper intends to address such shortcomings by experimental investigation on a timber beam using a vibration based method to detect damage. A promising damage detection algorithm based on modal strain energy was adopted and modified to locate/evaluate damage. A laboratory investigation was conducted on a timber beam inflicted with various damage scenarios using modal tests. The modal parameters obtained from the undamaged and damaged state of the test beam were used in the computation of damage index, were then applied using a damage detection algorithm utilising modal strain energy and a statistical approach to detect location of damage. A mode shape reconstruction technique was used to enhance the capability of the damage detection algorithm with limited number of sensors. The test results and analysis show that location of damage can be accurately identified with limited sensors. The modified method is less dependent on the number of modes selected and can detect damage with a higher degree of confidence.     

A continuum mechanics for damaged anisotropic solids

This paper develops a theory of continuum damage mechanics for anisotropic solids on the basis of both the strain energy equivalence principle and the equivalent (fictitious) line crack damage modeling. The strain energy equivalence principle is used to develop the effective continuum elastic properties of a damaged solid in terms of the undamaged anisotropic elastic properties and a scalar damage variable. The equivalent line crack representation of local damage provides, a means by which the effective direction of damage propagation can be identified from the local stresses and strains that are available in the course of continuum damage analysis. A scalar damage variable is defined as the effective volume fraction of a damaged zone associated with an equivalent line crack. Finally, an iterative numerical approach to continuum damage analysis is introduced.