Uncertainty and Sensitivity Analysis of Damage Identification Results Obtained Using Finite Element Model Updating

Abstract: A full‐scale seven‐story reinforced concrete shear wall building structure was tested on the UCSD‐NEES shake table in the period October 2005–January 2006. The shake table tests were designed so as to damage the building progressively through several historical seismic motions reproduced on the shake table. A sensitivity‐based finite element (FE) model updating method was used to identify damage in the building. The estimation uncertainty in the damage identification results was observed to be significant, which motivated the authors to perform, through numerical simulation, an uncertainty analysis on a set of damage identification results. This study investigates systematically the performance of FE model updating for damage identification. The damaged structure is simulated numerically through a change in stiffness in selected regions of a FE model of the shear wall test structure. The uncertainty of the identified damage (location and extent) due to variability of five input factors is quantified through analysis‐of‐variance (ANOVA) and meta‐modeling. These five input factors are: (1–3) level of uncertainty in the (identified) modal parameters of each of the first three longitudinal modes, (4) spatial density of measurements (number of sensors), and (5) mesh size in the FE model used in the FE model updating procedure (a type of modeling error). A full factorial design of experiments is considered for these five input factors. In addition to ANOVA and meta‐modeling, this study investigates the one‐at‐a‐time sensitivity analysis of the identified damage to the level of uncertainty in the identified modal parameters of the first three longitudinal modes. The results of this investigation demonstrate that the level of confidence in the damage identification results obtained through FE model updating, is a function of not only the level of uncertainty in the identified modal parameters, but also choices made in the design of experiments (e.g., spatial density of measurements) and modeling errors (e.g., mesh size). Therefore, the experiments can be designed so that the more influential input factors (to the total uncertainty/variability of the damage identification results) are set at optimum levels so as to yield more accurate damage identification results.     

Calculation of Posterior Probabilities for Bayesian Model Class Assessment and Averaging from Posterior Samples Based on Dynamic System Data

Abstract: In recent years, Bayesian model updating techniques based on dynamic data have been applied in system identification and structural health monitoring. Because of modeling uncertainty, a set of competing candidate model classes may be available to represent a system and it is then desirable to assess the plausibility of each model class based on system data. Bayesian model class assessment may then be used, which is based on the posterior probability of the different candidates for representing the system. If more than one model class has significant posterior probability, then Bayesian model class averaging provides a coherent mechanism to incorporate all of these model classes in making probabilistic predictions for the system response. This Bayesian model assessment and averaging requires calculation of the evidence for each model class based on the system data, which requires the evaluation of a multi‐dimensional integral involving the product of the likelihood and prior defined by the model class. In this article, a general method for calculating the evidence is proposed based on using posterior samples from any Markov Chain Monte Carlo algorithm. The effectiveness of the proposed method is illustrated by Bayesian model updating and assessment using simulated earthquake data from a ten‐story nonclassically damped building responding linearly and a four‐story building responding inelastically.     

Calibrating a high-fidelity finite element model of a highway bridge using a multi-variable sensitivity-based optimisation approach

This article presents the implementation of a calibration procedure for a finite element (FE) model of a state highway bridge using sensory data measured on the bridge. The objective is to modify the high-fidelity FE model of the bridge so that its dynamic behaviour matches, as closely as possible, that of the bridge under analysis. The bridge under investigation is a steel–concrete composite bridge that is instrumented with a wireless monitoring system to collect its vibration response under ambient vibrations. A detailed three-dimensional FE model of the bridge was developed to represent the bridge as realistically as possible. The detailed modelling can minimise the amount of uncertainty in the model and the number of parameters that require updating. A multi-variable sensitivity-based objective function is used to minimise the error between the experimentally measured and the FE-computed modal characteristics. An iterative optimisation approach has been undertaken to find the optimum structural parameters of the FE model that minimise the selected objective function. It is shown that developing a high-fidelity FE model may help simplify the model calibration process for a large structure, which is contrary to conventional thinking on the subject.     

Probabilistic analysis of tunneling-induced building safety assessment using a hybrid FE-copula model

For investigating the uncertainty and reliability of building disturbance induced by tunnel excavating, the finite element (FE) method and copula technique are nicely combined in this paper. Six existing buildings located just beside the tunnel axis in Wuhan Yangtze River Metro Tunnel were taken as case study. In this research, the tunnel–building interaction was first precisely described by deterministic and stochastic FE method. The probability and uncertainty of surrounding circumstance were considered as input parameters and imported into FE model, while 180 simulated data were generated by the FE method. Then, in order to precisely appraise the potential risk of pre-existing building, the multi distributions of S_max and I_max were constructed by different copula functions, and the best-fit copula function was identified by AIC and BIC criterions. Finally, the safety risk levels of the six concerned buildings were evaluated by considering both the importance of the examined buildings and the limited value of building reliability index β. The conclusions of this research can help appraising the potential risk of these buildings during tunnel excavating and improving the security of the metro construction.     

悬索桥结构基于敏感性分析的动力有限元模型修正

提出一种悬索桥结构基于特征值敏感性分析的有限元模型修正方法.由结构特征值关于模型参数的灵敏度矩阵建立理论/试验频率差与模型参数摄动之间的关系式.根据待修正参数的不确定性,将参数摄动限制于预设的范围,并将参数修正转化为不等式约束优化问题迭代求解.通过一1/150比例的悬索桥模型的试验分析,验证了这一模型修正方法的可行性.     

桥梁损伤识别的实用模型修正方法研究

对面向损伤识别的桥梁结构模型修正实用方法进行了研究。提出了基于振动特性测量的三步模型修正策略和综合利用通用有限元程序ANSYS的优化功能进行模型修正的方法。为了缩减待修正的参数,根据计算目标函数对每个单元参数的敏感性,进行子结构划分,通过对子结构参数的修正进行结构损伤的大致定位,然后对确定为损伤的子结构的每个单元进行参数修正,进行结构的损伤定量识别和状态评估。修正算法的优化方法采用ANSYS一阶优化方法和随机搜索方法,敏感性分析和模型修正完全基于ANSYS软件进行,较适合于实际工程的应用。为了说明方法的可行性,以某一实际三跨预应力钢筋混凝土连续箱梁桥为仿真算例,以结构模型的单元刚度衰减来模拟损伤,进行损伤识别,达到了较好的效果。     

Effects of model updating on the estimation of stochastic seismic response of a concrete-filled steel tubular arch bridge

The objectives of this paper are to integrate the structural health monitoring (SHM) technique with the structural seismic analysis, and to make the SHM technique serve, benefit and promote the structural seismic analysis integrally. Therefore, considering a concrete-filled steel tubular arch bridge structure, the SHM technique is used to calibrate the finite element (FE) model through the model-updating scheme to minimise the structural response differences caused by FE model errors. Effects of model updating on structural seismic responses are investigated using the stochastic vibration analysis approach. It is observed that effects of model updating are significant on structural seismic responses, and these effects may become more evident in structural nonlinear dynamic analysis. Hence, it is of prime importance to calibrate the FE models through the SHM technique for seismic evaluations of some operational critical structures.     

Finite Element Model Updating Using Evolutionary Strategy for Damage Detection

Abstract: Structural health monitoring through the use of finite element model updating techniques for dispersed civil infrastructures usually deals with minimizing a complex, nonlinear, nonconvex, high‐dimensional cost function with several local minima. Hence, stochastic optimization algorithms with promising performance in solving global optimization problems have received considerable attention for finite element model updating purposes in recent years. In this study, the performance of an evolutionary strategy in the finite element model updating approach was investigated for damage detection in a quarter‐scale two‐span reinforced concrete bridge system which was tested experimentally at the University of Nevada, Reno. The damage sequence in the structure was induced by a range of progressively increasing excitations in the transverse direction of the specimen. Intermediate nondestructive white noise excitations and response measurements were used for system identification and damage detection purposes. It is shown that, when evaluated together with the strain gauge measurements and visual inspection results, the applied finite element model updating algorithm of this article could accurately detect, localize, and quantify the damage in the tested bridge columns throughout the different phases of the experiment.     

Dynamic investigation of a concrete footbridge using finite element modelling and modal analysis

In this article, the application of a manual updating method for finite element (FE) model updating of a concrete footbridge using modal analysis approach is described in detail. An FE model was developed using DIANA (FEM software package) to estimate the response of structure under free-vibration analysis. Afterwards, ambient vibration test (AVT) was conducted to extract the dynamic properties. The fundamental mode shapes of the structure were successfully identified applying ARTeMIS (modal analysis computer program). The mode shape pairs of initial FE model and a complete set of test results were employed for manual updating. A parametric study was carried out to specify the most sensitive parameters of the model. For this purpose, boundary conditions, mass density and Young's modulus of elasticity were examined as uncertain parameters. Attempts to calibrate the primary FE model revealed that the spring constants of supports were the most effective parameters for updating process. The FE model was calibrated considering three main criteria consisting of combination of natural frequencies/mode shapes and modal assurance criteria (MAC)/mode shapes. The calibration strategy performed in the present study, including parametric study on uncertain parameters of initial FE model, parameter and target response selection and MAC calculation based on modified formulation, has been discussed. The updated FE model and the measured mode shape counterparts exhibited very good correlation.     

基于健康监测的连续刚构桥有限元模型确认(Ⅰ)——基于响应面法的有限元模型修正

有限元模型修正不能考虑不确定性和进行响应预测,而模型确认是模型修正的发展,该系列论文建立了桥梁有限元模型确认的基本方法。主要讨论模型确认的关键步骤之一——基于响应面的桥梁有限元模型修正方法,包括试验设计、参数筛选、响应面模型选择以及模型检验等;基于下白石大桥的实时健康监测,利用响应面方法成功地对下白石大桥的有限元模型进行修正。结果表明:有限元模型修正后的计算结果与实测结果比较吻合,最大相对误差不超过6%,修正后的有限元模型可以进一步应用于模型确认,对桥梁健康监测和安全评估有益。     

Kernel Smoothing Method Applicable to the Dynamic Calibration of Traffic Flow Models

Abstract: Real time traffic flow simulation models are used to provide traffic information for dynamic traffic management systems. Those simulation models are supplied by traffic data in order to estimate and predict traffic conditions in unobserved sections of a traffic network. In general, most of recent real time traffic simulators are based on the macroscopic model because the macroscopic model replicates the average traffic behavior in terms of observable variables such as (time–space) flow and speed at a relatively fast computational time. Like other simulation models, an important aspect of the real time macroscopic simulator is to calibrate the model parameters online. The most conventional way of the online calibration is to add a random walk to the parameters to constitute an augmentation of the traffic variables and the model parameters to be estimated. Actually, this method allows the parameters to vary at every time step and, therefore, describes the adaptation of the model to the prevailing traffic conditions. However, it has been reported that the use of the random walk results in a loss of information and an increase of the covariance of parameters, which consequently leads to posteriors that are far more diffuse than the theoretical posteriors for the true parameters. To this end, this article puts forward a Kernel density estimation technique in the calibration process to handle the covariance issue and to avoid the information loss. The Kernel density estimation technique is embedded in the particle filter algorithm, which is extended to the calibration problems. The proposed framework is investigated using real‐life data collected in a freeway in England.     

焊接结构响应的多目标模拟及模型修正敏感参数分析

本文主要研究复杂结构模拟和模型修正过程中的两个关键问题,即多级目标一致逼近模型的建立和模型修正的敏感参数分析.研究过程针对一个焊接结构钢桁架的缩尺试样进行,利用该钢桁架结构试验提供的多目标一致逼近建模和模型修正所需的结构细节和试验测试结果,研究这类焊接结构的多目标模拟过程,使建立的模型能够同时逼近三个测试目标,即钢桁架结构的整体动力特性、构件名义应力和焊接细节处的局部热点应力;并通过影响因素计算分析探讨该模型修正过程中需要考虑的主要敏感参数.研究结果表明,钢桁架结构的约束参数、焊接结构的焊缝厚度和焊缝弹性模量是模型修正中应该考虑的主要参数.以上分析结果为这类结构的多目标模型修正奠定了基础.     

Finite element model updating on small-scale bridge model using the hybrid genetic algorithm

Finite element (FE) model-based dynamic analysis has been widely used to predict the dynamic characteristics of civil structures. FE model updating method based on the hybrid genetic algorithm, by combining genetic algorithm and the modified Nelder–Mead's simplex method, is presented to improve bridge structures' FE model. An objective function is formulated as a linear combination of fitness functions on natural frequencies, mode shapes and static deflections using measurements and analytical results to update both stiffness and mass simultaneously. A commercial FE analysis tool, which can utilise previously developed element library and solution algorithms, is adopted for applications on diversified and complex structures. The validity of the proposed method is verified by using a simply supported bridge model with three I-shaped girders. FE models such as grid, beam-shell and shell model are considered to modify initial FE models on the experimental structure. Experimental results suggest that the proposed method can be applied efficiently to various FE models and is feasible and effective when this method is applied to identify FE modelling errors.     

消能减震建筑结构的附加有效阻尼比估计

为估计一栋实际消能减震建筑结构在地震作用下的附加有效阻尼比,提出了一种基于有限元模型修正技术的阻尼比估计和验证方法。考虑到结构初始有限元模型存在较大模型误差,采用基于结构振动模态参数的直接模型更新方法修正初始有限元模型,其中,对于实测振型不完整问题,利用振型扩阶方法补充完整振型。基于模态应变能概念,利用整体结构模态参数识别值,推导了油阻尼器支撑系统附加给结构的有效阻尼比和有效频率的计算公式,并估计了主体结构的频率和阻尼比。以一组实际地震动监测数据为例,采用建议方法估计有效阻尼比和有效频率,通过对比修正模型的预测响应与实际监测数据,验证了建议方法的有效性。     

Structural parameter identification and damage detection for a steel structure using a two-stage finite element model updating method

A two-stage eigensensitivity-based finite element (FE) model updating procedure is developed for structural parameter identification and damage detection for the IASC-ASCE structural health monitoring benchmark steel structure on the basis of ambient vibration measurements. In the first stage, both the weighted least squares and Bayesian estimation methods are adopted for the identification of the connection stiffness of beam-column joints and Young’s modulus of the structure; then the damage detection is conducted via the FE model updating procedure for detecting damaged braces with different damage patterns of the structure. Comparisons between the FE model updated results and the experimental data show that the eigensensitivity-based FE model updating procedure is an effective tool for structural parameter identification and damage detection for steel frame structures.     

Bayesian methodology for updating geomechanical parameters and uncertainty quantification

A generic framework for the updating geomechanical parameters is presented. It is based on Bayesian probabilities, and considers several levels of uncertainty. It allows one to properly update the probability distribution function of a given parameter when new data are available. This framework is applied to the case of deformability modulus updating in a large underground structure. Two different approaches were tested in terms of initial knowledge about the parameter, namely no knowledge, and a prior distribution of the parameter based on geological/geotechnical data and application of analytical solutions based on the empirical classification systems. The updating was carried out using the framework together with the results of high quality in situ tests. The Bayesian framework was shown to be a mathematically valid way to deal with the problem of the geomechanical parameter updating and of uncertainty reduction related to the parameter's real value.     

Local finite element model updating with forced vibration measurements

Finite element (FE) model updating based on the method of selective sensitivity analysis is applied to a girder bridge. The response patterns are chosen in accordance with the selective sensitivity criteria and result in a set of four independent local girder bridge response patterns. The excitation of these patterns is conducted with a four-shaker system. This effectively reduces the degrees of freedom of the corresponding FE model from 486 to 7. Excitations and responses are measured at three nodes of each selective sensitive pattern. Local finite element model updating is conducted by minimising the influence of the rest of the structure and by maximising the relative sensitivities with the inclusion of the fourth shaker. This excitation is shown to be model independent; therefore, updating is separated from the measurement process. An independent verification shows the ability of the proposed method to identify submodel parameters. The sensitivities of the accelerometers used were very small, which impacted the data quality. Updating results are discussed considering these limits.     

在役T构桥梁的有限元模型修正

有限元模型修正是建立精确的基准有限元模型的基础。以在役T构桥梁——324国道乌龙江大桥为工程背景,利用ANSYS软件建立了全桥结构的三维有限元模型,进行了结构静、动力数值模拟分析,并与实测结果进行了比较;结果表明,未修正的有限元模型计算结果与实测结果存在较大误差。通过参数灵敏度分析,确定了对桥梁结构静、动力特性影响均较大的参数;采用零阶和一阶算法,基于自振频率与静力挠度组合的目标函数,对乌龙江大桥有限元模型进行了修正。修正后的有限元模型能较真实地反映结构的实际状态,可作为该桥梁长期健康监测与状态评估的基准有限元模型。     

基于虚拟变形方法的结构有限元模型修正

虚拟变形方法是一种快速精确的结构重分析方法。基于静力虚拟变形理论的结构有限元模型修正方法,推导了梁单元影响矩阵和基于虚拟变形的结构位移灵敏度。利用初始结构的静力响应和影响矩阵,就可根据需要对感兴趣的结点响应进行重分析,实现有限元模型的修正,无需修改结构的刚度矩阵;计算位移灵敏度时也与传统算法不同,不需要对整体刚度矩阵求逆,根据虚拟变形方法就可求出相应结点的静力位移灵敏度,提高了模型修正过程中的计算效率。最后分别利用数值仿真和荷载试验数据对一平面框架结构进行了模型修正。     

Probabilistic updating of fishbone model for assessing seismic damage to beam–column connections in steel moment‐resisting frames

This article presents a probabilistic method of updating fishbone models for assessing seismic damage on beam–column connections in steel moment‐resisting frames. Fishbone models enable explicitly identifying the rotational stiffness of beams, which is not possible with a shear building model commonly used in Bayesian model updating approaches. Necessary formulations to utilize fishbone models for model updating with measured floor accelerations under small‐amplitude loadings including ambient excitations were first formulated. To accommodate the incompleteness of modal data to update unknown parameters of fishbone models, that is, the stiffness of rotational springs, a hierarchical Bayesian model updating algorithm is implemented. Seismic damage of beam–column connections are estimated by making a comparison of the identified rotational stiffness of springs in the fishbone models before and after earthquakes. The effectiveness of the proposed method is first examined with numerical simulations using a 10‐story building model. Then, the applicability to realistic beam damage, that is, not artificially introduced, is evaluated through a full‐scale steel frame test at the E‐Defense shaking table facility. The article also discusses coefficients of variation of the identified stiffness and influence of modeling error on estimation of realistic damage to check the credibility of the method for real‐life applications.