Finite Element Model Updating of Civil Engineering Structures Under Operational Conditions

FINITE ELEMENT MODEL UPDATING OF CIVIL ENGINEERING STRUCTURES UNDER OPERATIONAL CONDITIONS Supervisor: Prof. Wei-Xin Ren

By Bijaya Jaishi

A dissertation submitted to the College of Civil Engineering and Architecture FUZHOU UNIVERSITY in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in STRUCTURAL ENGINEERING

May, 2005

基于环境振动的土木工程结构有限元模型修正 博士生：Bijaya Jaishi 导

师：任伟新 教授

中文详细摘要 有限元（FE）理论自出现以来，人们在新型的有限单元、有效的数值求解方法、 模型网格划分以及前后处理等方面做了大量的研究工作。然而，建立结构有限元分析 模型时势必要对结构几何、材料和边界条件等进行一定的假定和近似处理。直接建立 的结构有限元模型分析预测的结果通常和实际结构或试验结果存在差别，有时这种误 差会很大。有限元模型修正就是一个试图通过识别或修正有限元分析模型中的参数, 使有限元计算结果与实际结构尽可能接近的过程，通常认为属于优化问题范畴。 有限元模型修正理论最初用于力学系统动力学模型的修改与精化，大多从某种试 验/理论残差（目标函数）的最小化过程出发。在结构工程领域，一般采用试验模态 分析结果（如频率、振型等），修正有限元理论模型的质量、刚度等参数，使得修正 后有限元模型的振动特性参数趋于试验值。有限元模型修正过程不仅需要满足分析结 果和试验结果的对应关系，而且修正后的参数还要有实际的物理意义。确定目标函数， 选取修正参数和应用有效的优化算法是结构有限元模型修正中的三个关键步骤。 对土木工程结构进行有限元模型修正，必须考虑土木工程结构的特点。大型土木 工程结构的动力特性一般由现场的振动试验确定，对于桥梁一类的土木工程结构，在 正常工作条件（operational condition）下，风、车辆、行人等是一种自然的环境激励 (ambient excitation)方式。直接利用环境激励时桥梁的振动响应数据进行模态参数识 别，具有明显的优点：不需额外的人工激励，不必中断交通，更符合结构实际的边界 条件与工作状态，可以实现实时的监测等。因此基于环境振动的土木工程结构有限元 模型修正方法更具有实际意义。 有限元模型修正对象、优化目标以及约束条件的不同，决定了特定的有限元模型 修正方法只适应于特定的问题。基于环境振动的土木工程结构有限元模型修正方法研 究，尽管已有许多的处理方法和研究成果，但还有许多关键问题或难点没有很好地解 决，比如： z

有限元模型修正概念上简洁，但在实践中并非易事，一个主要的困难在于大型土 木工程结构可观测的动力参数（频率、振型）对局部刚度的变化或结构参数微小 i

的改变不敏感。如何选择模型修正过程中优化目标函数，使之既能够反映土木工 程结构的特点，又能够反映结构参数微小的改变，尚需进一步的研究和探讨。 z

大型土木工程结构有限元分析模型单元众多，对每个单元中的每一个参数都进行 修正，实际应用时由于修正的参数太多而变得不现实。在修正算法上如何选择修 正参数，而且修正后的参数还要符合实际的物理意义，研究实用的大型土木工程 结构有限元模型修正算法，尚有大量工作要做。

z

环境振动结构模态参数识别得到的是工作模态振型（operational mode shape）,它 既不以质量矩阵归一化，也不以刚度矩阵归一化，仅仅是一个相对量，很难比较。

z

结构有限元分析模型的自由度要远大于模态试验所实测的自由度，因此，进行有 限元模型修正时，需要对有限元模型进行模型简缩(model reduction)或对试验模态 进行模态扩展 (modal expansion)。

z

有限元模型可以描述结构的详细信息，为桥梁结构提供完整的理论模态参数集， 而环境振动试验模态参数识别所提供的信息可能是不完备的。任何有限元模型修 正方法均需处理这两种不同层次信息的差异。 本论文就是基于这样的背景，旨在对处于工作环境下的土木工程结构有限元模型

进行修正，使其与现场环境振动的试验结果尽可能地接近，最终建立适用于土木工程 结构的有限元模型修正实用方法。重点研究目标函数的确定，修正参数的选取方法和 实用的优化算法。研究结果可广泛应用于土木工程结构的损伤识别、既有结构的承载 力评定和结构的长期健康监测，具有较大的理论意义和工程实用价值。 全文共分 8 章，各章的主要内容包括： 第一章是引言，详细介绍了问题的提出、工程背景、研究目的、论文的主要内容 和贡献。对结构有限元模型修正研究的进展进行了文献综述，重点讨论了确定目标函 数，选取修正参数和优化算法这些结构有限元模型修正中的关键问题。 第二章介绍了一个自主开发的基于 Matlab 的有限元理论模态分析工具箱－ MBMAT，旨在为实现所提出的结构有限元模型修正算法提供一个平台。应用此工具 箱，有限元模型的所有信息，例如单元或整体的质量和刚度矩阵、以及边界条件等， 都可以方便地提取并加以修正，所有优化算法均可方便地在 Matlab 框架内实现。 MBMAT 工具箱的原理、功能和编程实现均在本章中进行了介绍，最后用两个结构动 力分析算例，对 MBMAT 工具箱进行了验证。 第三章详细讨论了基于环境振动的结构有限元模型修正所涉及到的主要方面和 技术，如结构有限元建模、环境振动试验和工作模态参数识别等。重点介绍了各种评 判理论计算和试验结果相关程度的方法、结构有限元模型的简缩方法和试验模态扩展 的方法。在本章中还建议了两个新的试验模态扩展方法：基于模态柔度的扩展方法和 基于正则化模态差（Normalized Modal Difference－NMD）的扩展方法，并对其有效 ii

性用模拟的简支梁结果进行了验证。 第四章详细讨论了实现结构有限元模型修正算法的主要步骤。除了常用的频率残 差和振型残差，建议和讨论了两个新的残差：模态柔度残差和模态应变能残差及其相 应的优化目标函数。重点讨论了与结构有限元模型修正有关的问题，如目标函数、目 标函数的梯度（灵敏度）、各种残差及其权重、各种修正参数和参数选择等。对有限 元模型修正中使目标函数达到极小的优化算法进行了讨论，最后还详细介绍了第六章 中求解多目标优化问题所采用的连续二次规划（Sequential Quadratic Programming – SQP）方法的原理。 第五章建立了基于单目标优化（Single-Objective Optimization）函数的土木工程 结构有限元模型修正方法，此时，对基于不同残差的各种目标函数合成为一个单一的 目标函数进行有限元模型修正。重点研究了三种不同的目标函数：频率目标函数、模 态振型目标函数和模态柔度目标函数，其中模态柔度目标函数是本文提出的新的目标 函数。由于环境振动试验无法直接得到质量归一化模态振型，因此采用有限元模型质 量矩阵，应用 Guyan 简缩方法得到质量归一化的模态振型来计算模态柔度。单目标 优化问题的求解采用罚函数法，即子问题逼近方法（Subproblem Approximation Method）和一阶优化方法（First-Order Optimization Method），前者是直接（全局）的 优化算法，后者需要涉及梯度计算，较费时。基于单目标优化的结构有限元模型修正 方法，首先用一数值模拟的简支梁进行了验证计算，各种目标函数的不同组合的结果 表明：由特征频率、模态振型和模态柔度三个残差的组合是最佳的目标函数，特别是 模态柔度的应用，可提高有限元模型修正的精度，说明模态柔度对结构局部微小的变 化较敏感。随后，由模态频率，模态振型和模态柔度残差组合成的单目标函数，成功 应用到一座钢管混凝土拱桥的有限元模型修正上，该桥的动力特性由现场环境振动试 验得到。通过这一实例，建立了基于特征频率的敏感度分析和工程经验选取修正参数 的过程，并对有限元模型修正迭代过程的收敛性进行了讨论。在单目标优化过程中， 不同的残差按照不同的权重组合成一个单目标函数，但是权重的选择并没有明确的规 则，因此在计算过程中必须经过反复改变权重的数值，直到找到合适的解为止。 第六章建立了基于多目标优化（Multi-Objective Optimization）函数的土木工程结 构有限元模型修正方法。此时基于不同残差的各种目标函数作为独立的目标函数，不 需合成为一个单一的目标函数进行有限元模型修正，因此不需考虑不同目标函数的权 重。本章采用模态频率和模态应变能作为两个独立的目标函数进行结构有限元模型修 正，其中模态应变能目标函数是本文提出的新的目标函数。多目标优化算法采用第四 章中介绍的连续二次规划算法。在多目标优化方法中，最优化的概念并不明显，因为 通常能将所有目标个体最小化的解向量并不存在。因此用 Pareto 最优的概念表现目标 的特征，优化完成以后，对每个修正参数组单独进行一维优化，直到修正参数满足 Pareto 解的特征。本章首先用简支梁模拟算例，验证了多目标优化有限元修正过程， iii

然后成功地对一座预制连续混凝土箱梁桥的初始有限元分析模型进行了修正，主要的 修正参数为桥面板的弹性模量和氯丁橡胶支座的弹簧刚度，该桥的动力特性是由现场 环境振动试验得到的。结果表明，本文建立的基于多目标优化的有限元模型修正方法 是有效的，可应用于处于工作状态的土木工程结构的有限元模型修正中。 第七章作为结构有限元模型修正在土木工程结构中的一个应用，建立了一个基于 模态柔度的有限元模型修正损伤识别方法。用有限元分析模态柔度和试验模态柔度的 残差作为目标函数，采用 Fox 和 Kapoor 的方法，推导出了模态柔度敏感度（梯度） 矩阵的解析表达式。优化算法采用标准置信区间牛顿方法（standard trust region Newton method），从而使优化算法更加有效，减少病态问题。首先也是通过简支梁的 模拟算例，验证了所建立的结构损伤识别过程，同时还研究了噪声对有限元模型修正 损伤识别算法的影响，结果表明：本章所提出的基于模态柔度的有限元模型修正损伤 识别方法，在有噪声的情况下仍能识别损伤，结果可以接受。随后对实验室进行的钢 筋混凝土梁损伤试验，采用该算法成功地进行了损伤识别。在事先对梁的损伤模式不 作任何假定，即将有限元模型中的每一个单元的参数均选为修正参数时，损伤识别结 果与试验结果基本一致。因此本文所建立的基于模态柔度的有限元模型修正损伤识别 方法，可用于实际结构，同时也再一次证明了模态柔度对结构损伤较敏感。 第八章是结论和今后进一步研究工作的展望。 本文的主要贡献和创新之处： 1. 建议了两个新的试验模态扩展的方法：基于模态柔度的扩展方法和基于正则化模 态差（Normalized Modal Difference－NMD）的扩展方法。 2. 提出了两个新的用于结构有限元模型修正的残差量及其相应的优化目标函数：模 态柔度和模态应变能；分别推导出了模态柔度和模态应变能目标函数的敏感度（梯 度）矩阵的解析表达式。 3. 建立了基于单目标优化（Single-Objective Optimization）函数的土木工程结构有限 元模型修正方法。重点研究了三种不同的目标函数：特征频率目标函数、模态振 型目标函数和模态柔度目标函数，其中模态柔度目标函数是本文提出的新的目标 函数。简支梁数值模拟结果表明，由特征频率、模态振型和模态柔度三者的组合 是最佳的单目标函数。特别是模态柔度的应用，可显著提高有限元模型修正的精 度，说明模态柔度对结构局部微小的变化较敏感。 4. 由频率，模态振型和模态柔度残差组合的单目标函数，采用本文所建立的基于单 目标优化有限元模型修正方法，成功地应用于一座钢管混凝土拱桥实桥的有限元 模型修正，该桥的动力特性由现场环境振动试验得到，并对修正迭代过程的收敛 性进行了讨论。该实例的成功实现，为大型土木工程结构有限元模型修正的实用 算法提供了很好的借鉴作用。 iv

5. 在单目标优化过程中，不同的残差按照不同的权重组合成一个单目标函数，但是 权重的选择并没有明确的规则，因此在计算过程中必须经过反复改变权重的数值， 直到找到合适的解。为克服这个问题，本文建立了基于多目标优化（Multi-Objective Optimization）函数的土木工程结构有限元模型修正方法。此时，基于不同残差的 各种目标函数作为独立的目标函数进行优化计算，不需考虑不同目标函数的权重。 6. 采用特征频率和模态应变能两个独立的目标函数，进行了多目标优化有限元模型 修正计算，其中模态应变能目标函数是本文提出的新的目标函数。多目标优化结 构有限元修正过程，成功地对一座既有的连续混凝土箱梁桥的初始有限元分析模 型进行了修正，该桥的动力特性是由现场环境振动试验得到的。结果表明，本文 建立的基于多目标优化的有限元模型修正方法是有效的，可应用于处于工作状态 的土木工程结构的有限元模型修正中。 7. 有限元模型修正的成功与否，修正参数的选取至关重要。有限元解析结果和试验 结果之间差异的目标函数应该是这些参数的敏感函数。否则，修正的参数有时需 要严重偏离它们的初始值，才能达到可接受的对应关系，且参数会失去其物理意 义。同时为了避免数值计算出现病态，应尽可能少修正参数的数量。因而，选取 参数时需要透彻了解待修正结构的特性。通过实例研究，本文建立了基于特征频 率的敏感度分析和工程经验相结合的选取修正参数的过程。 8. 作为结构有限元模型修正在土木工程结构中的一个应用，本文建立了一个基于模 态柔度的有限元模型修正损伤识别方法。利用模拟简支梁算例研究了噪声对识别 算法的影响，随后应用于在实验室进行的钢筋混凝土梁损伤试验。在事先对梁的 损伤模式不作任何假定，即将有限元模型中的每一个单元的参数均选为修正参数 时，损伤识别结果与试验结果基本一致。因此本文所建立的基于模态柔度的有限 元模型修正损伤识别方法，可用于实际结构，同时也再一次证明了模态柔度对结 构损伤较敏感这一特性。

关键词：有限元，模型修正，目标函数，优化，结构动力学

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ABSTRACT The analytical predictions from a finite element model often differ from the experimental results of a real civil engineering structures. Finite element model updating is an inverse problem to identify and correct uncertain parameters of finite element model that leads to the better predictions of the dynamic behavior of actual structure. And it is usually posed as an optimization problem. In model updating process, one requires not only satisfactory correlations between analytical and experimental results, but also the updated parameters should have a physical significance in practice. Setting-up of an objective function, selecting updating parameters and using robust optimization algorithm are three crucial steps in model updating. To implement the proposed algorithms for structural finite element model updating, a finite element toolbox is developed in Matlab environment which is used to carry out the analytical modal analysis of engineering structures. From this toolbox, the information about the finite element model like the global mass and stiffness matrices can be extracted and whole updating and damage detection work can be realized in Matlab framework. Finite element model updating procedure using single-objective optimization is first investigated in this thesis. The use of dynamically measured flexibility matrices using ambient vibration method is proposed and investigated for model updating. The issue related to the mass normalization of mode shapes obtained from ambient vibration test is investigated and applied to use the modal flexibility for finite element model updating. The algorithms of penalty function method, namely subproblem approximation method and first-order optimization method are explored, which are then used for finite element model updating. The model updating is carried out using different combinations of possible residuals in the objective functions and the best combination is recognized with the help of simulated case study. It is demonstrated that the combination that consists of three residuals, namely eigenvalue, mode shape related function and modal flexibility with weighing factors assigned to each of them is recognized as the best objective function. In single-objective optimization, different residuals are combined into a single objective function using weighting factor for each residual. A necessary approach is required to solve the problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained since there is no rigid rule for selecting the weighting factors. The single-objective optimization with eigenfrequecy residual, mode shape related function and modal flexibility residual is applied successfully for the finite element model updating of a vi

real concrete filled steel tubular arch bridge in which eigensensitivity method with engineering judgment is used for updating parameter selection. Finite element model updating procedure using multi-objective optimization technique is then proposed. The weighting factor for each objective function is not necessary in this method. The implementation of dynamically measured modal strain energy using ambient vibration method is investigated and proposed for model updating. The eigenfrequencies and modal strain energies are used as the two objective functions of multi-objective optimization technique. The multi-objective optimization method, called goal attainment method is used to solve the optimization problem. The Sequential Quadratic Programming algorithm is used in the goal attainment method. In multi-objective optimization technique, the notion of optimality is not obvious since in general, a solution vector that minimizes all individual objectives simultaneously does not exist. Hence, the concept of Pareto optimality must be used to characterize the objectives. Hence, in goal attainment problem, one-dimensional optimization on each of the components of the updated parameters obtained after optimization are carried out to see if one can do better by changing that one component, using the definition of a Pareto point. The procedure is repeated with different values of weights and goals until the updated parameter obtained from goal attainment method satisfies the characteristics of the Pareto solution. The finite element model updating procedure using the multi-objective optimization method is illustrated with the examples of both simulated simply supported beam and a real case study. The latter is used to estimate the elastic modulus of the deck of the bridge and spring stiffness of neoprene support of precast continuous box girder bridge. The success of finite element model updating depends heavily on the selection of updating parameters. The updating parameter selection should be made with the aim of correcting uncertainties in the model. Moreover, the objective function which represents the differences between analytical and experimental results needs to be sensitive to such selected parameters. Otherwise, the parameters deviate far from their initial values and lose their physical foundation in order to give acceptable correlations. To avoid the ill-conditioned numerical problem, the number of parameters should be kept as low as possible. Thus, the parameter selection requires considerable physical insight into the target structure, and trial-and-error approaches are used with different set of selected parameters. In this study, the eigenvalue sensitivity of the different possible parameters is calculated and then the most sensitive parameters with some engineering intuition are elaborately selected as the candidate parameters for updating. The convergence process during iteration for finite element model updating is also discussed in the thesis. vii

As an application of finite element model updating in structural dynamics, a damage detection algorithm is developed from finite element model updating using modal flexibility. The Guyan reduced mass matrix of analytical model is used for mass normalization of ambient vibration mode shapes to calculate the modal flexibility. The objective function is formulated in terms of difference between analytical and experimental modal flexibility. Analytical expressions are developed for the flexibility matrix error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The optimization algorithm used to minimize the objective function is the standard trust region Newton method which makes the algorithm more robust to reduce the ill-conditioning problem. The procedure of damage detection is demonstrated with the help of the simulated example of simply supported beam. The effect of noise on the updating algorithm is studied using the simulated case study. The procedure is thereafter successfully applied for the damage detection of laboratory tested reinforced concrete beam with known damage pattern.

Keywords: Finite element, Model updating, Objective function, Optimization, Structural dynamics

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CONTENTS Abstract in Chinese

i

Abstract

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Contents

ix

List of Symbols

xiii

List of Figures

xx

List of Tables

xxii

1.

Introduction

1

1.1 1.2 1.3 1.4

Background……………………………………………………………………... Objectives and Scopes of the Research………………………………………… Contributions of the Research………………………………………………….. Literature Survey……………………………………………………………….. 1.4.1 Direct Methods…………………………………………………………. 1.4.2 Sensitivity Based Methods……………………………………………... 1.4.2.1 Parameter Identification of Structures………………………... 1.4.2.2 Damage Detection of Structures……………………………… 1.5 Organization of Dissertation…………………………………………………….

1 3 3 5 5 6 7 8 11

2

14

Finite Element Modal Analysis Toolbox Development

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Introduction………….……………………………………………………..…… Shape Functions……….………………………………………………………... Finite Element Mass and Stiffness Matrices……………………………………. Governing Equation and Solution……………………….……………………... Element Types…………………………..……………….……………………… Program Realization in Matlab…………..……………….…………………….. Program Verification…..………………………………….…………………….. 2.7.1 Time Period of Simply Supported Beam………….……………………. 2.7.2 Plane Frame – Bathe and Wilson Eigenvalue Problem…………….…... 2.8 Chapter Conclusions…………………….……………….……………………...

14 15 16 20 22 24 25 25 27 29

3

30

Finite Element Model Updating in Structural Dynamics

3.1 Finite Element Modeling, Modal Testing, and System Identification for Model Updating………………………………………………………………………... 3.1.1 Finite Element Modeling…….…………………………………………. 3.1.2 Modal Testing and System Identification….…………………………… ix

30 30 31

3.2 Techniques for Comparison and Correlation for Model updating.……………... 3.2.1 Direct Natural Frequency Correlation….………………………………. 3.2.2 Visual Comparison of Mode Shapes…………………………………… 3.2.3 Direct Mode Shape Correlation………………………………………… 3.2.4 Modal Assurance Criterion……………………………………………... 3.2.5 Normalized Modal Difference………………………………………….. 3.2.6 Coordinate Modal Assurance Criterion………………………………… 3.2.7 Orthogonality Methods…………………………………………………. 3.2.8 Energy Comparison and Force Balance………………………………... 3.3 Incompatibility in Measured and Finite Element Data…………………………. 3.3.1 Model Reduction.………………………………………………………. 3.3.2 Mode Shape Expansion………………………………………………… 3.3.2.1 Kidder Dynamic Expansion………………………………….. 3.3.2.2 Modal Expansion Method……………………………………. 3.4 Two Proposed Methods for Mode Shape Expansion…………………………... 3.4.1 Modal Flexibility Method……………………………………………… 3.4.2 Normalized Modal Difference Method………………………………… 3.4.3 Performance Metrics…………………………………………………… 3.4.4 Simulated Case Study………………………………………………….. 3.5 Three Key Issues of Finite Element Model Updating…………………………. 3.6 Chapter Conclusions…………………….……………….……………………..

34 34 34 35 35 36 37 37 38 38 38 40 41 42 43 43 44 44 44 47 49

4

50

Finite Element Model Updating Procedure

4.1 Theoretical Procedure.………………………………………………………….. 4.1.1 Objective Function……………………………………………………... 4.1.1.1 Eigenfrequencies……………………………………………... 4.1.1.2 Mode Shapes…………………………………………………. 4.1.1.3 Modal Flexibility…………………………………………….. 4.1.1.4 Modal Strain Energy…………………………………………. 4.1.2 Weighting………………………………………………………………. 4.1.3 Gradient of Objective Function………………………………………… 4.2 Finite Element Model Updating Parameters……………………………………. 4.2.1 Physical Parameters……………………………………………………. 4.2.2 Substructure Parameters……………………………………………….. 4.3 Selection of Updating Parameters……………………………………………… 4.3.1 Empirically Based Selection of Updating Parameters…………………. 4.3.2 Sensitivity Based Selection of Updating Parameters………………….. 4.4 Optimization algorithm………………………………………………………… 4.4.1 Search Direction……………………………………………………….. 4.4.2 Line Search and Trust Region Strategies………………………………. 4.4.3 Sequential Quadratic Programming.………..………………………….. 4.4.3.1 Updating the Hessian Matrix of the Lagrange Function…...… 4.4.3.2 Solution of Quadratic Programming Problem………………... 4.4.3.3 Line Search and Merit Function……………………………… 4.5 Chapter Conclusions…. ………………………………………………………... x

50 52 53 54 55 58 60 61 63 63 64 64 65 66 67 69 71 72 72 73 75 76

5

Finite Element Model Updating Using Single-Objective Optimization

77

5.1 Mass Normalization of Operational Mode Shapes. ……………………………. 5.1.1 Sensitivity Based Method………………………………………………. 5.1.2 Using Orthogonality of Modes with Mass Matrix……………………… 5.1.3 Finite Element Model Approach………………………………………... 5.2 Objective Functions and Constraints. ………………………………………….. 5.3 Optimization Techniques……………………………………………………….. 5.3.1 Subproblem Approximation Method…………………………………… 5.3.2 First-Order Optimization Method………………………………….…… 5.4 Simulated Simply Supported Beam. …………………………………………… 5.5 Concrete Filled Steel Tubular Arch Bridge…………………………………….. 5.5.1 Bridge Description and Finite Element Modeling……………………... 5.5.2 Ambient Vibration Testing, Modal Parameter Identification and Model Correlation.. ……………………………………………………………. 5.5.3 Parameters Selection for Finite Element model Updating……………... 5.5.4 Finite Element Model Updating………………………………………... 5.5.5 Physical Meaning of Updated Parameters……………………………… 5.5.6 Conclusions from the Updating of Full Scale Arch Bridge……………. 5.6 Chapter Conclusions…………………………………………………………….

77 78 78 79 80 81 82 85 88 93 93 96 104 105 109 110 111

6

Finite Element Model Updating Using Multi-Objective Optimization

112

6.1 Introduction……………………………………………………………………... 6.2 Multi-objective Optimization…………………………………………………... 6.3 Theoretical Procedure for Multi-objective Optimization………………………. 6.3.1 Formulation of Objective Functions and Constraints…...……………… 6.3.2 Objective Function Gradient…………………………………………… 6.4 Simulated Simply Supported Beam…. ………………………………………… 6.5 Precast Continuous Box Girder Bridge. ……………………………………….. 6.5.1 Bridge Description and Finite Element Modeling……………………… 6.5.2 Ambient Vibration Measurements, Modal Parameter Identification and Model Correlation……………………………………………………… 6.5.3 Parameters Selection for Finite Element Model Updating……... ……... 6.5.4 Finite Element Model Updating... ……………………………………... 6.5.5 Conclusions from Updating of a Continuous Girder Bridge.. …………. 6.6 Chapter Conclusions…………………………………………………………….

112 113 117 117 119 120 121 121

7

126 131 132 135 135

Damage Detection by Finite Element Model Updating Using Modal Flexibility……………………………………………………………………….

137

7.1 Theoretical Background………………………………………………………… 7.1.1 Objective Function and Minimization Problem………………………... 7.1.2 Objective Function Gradient…………………………………………… 7.1.3 Optimization algorithm………………………………………………… 7.2 Simulated Simply Supported Beam……………………………………………..

137 137 139 140 143

xi

7.3 Experimental Beam…………………………………………………………….. 7.3.1 Description of Experimental Beam and Modal Parameter Identification 7.3.2 Model Updating and Damage Detection……………………………….. 7.3.3 Conclusions from Experimental Beam…………………………………. 7.4 Chapter Conclusions…………………………………………………………….

146 146 149 153 153

8

154

Conclusions and Future Work

8.1 Conclusions……………………………………………………………………... 8.2 Significance of the Study……..………………………………………………… 8.3 Future Research………………………………………………………...….……

154 157 159

References

161

Appendices

172

Acknowledgements

178

Curriculum Vitae

179

xii

LIST OF SYMBOLS

Acronyms COMAC DOFs FE FRF IRF MAC MOP MSE MSF NMD SQP

Coordinate modal assurance criterion Degrees of freedom Finite element Frequency response function Impulse response function Modal assurance criterion Multi-objective optimization Modal strain energy Modal scale factor Normalized modal difference Sequential quadratic programming

General conventions ∂f ( • ) ∂ai

Partial derivative of • wrt to parameters

∇ (•)

Gradient vector with first-order partial derivatives of •

∇2 (•)

Hessian matrix with second-order partial derivatives of •

diag ( •....)

Diagonal matrix with ( •....) as diagonal elements

det ( • )

Determinant of matrix •

I

Identity matrix

(•) −1 (•)

Transpose of •

•

Absolute value of •

T

(•) •

+1

2

•

F

Inverse of matrix •

Pseudo inverse of matrix • l2 norm of vector • Frobenius norm of matrix •

•k

Quantity • evaluated at iteration k

∑

Sum operator

∫

Integration operator xiii

General symbols a

Vector of normalized updating parameters

ai , ai

Upper and lower limit for the design variable a

a0 , ai , bij

Coefficients used in optimization

Ai , B j ,Ck Ak

State variables related parameters during optimization Active constraints at the solution point

Ai

i -th row of the m-by-n matrix A

A

Cross sectional area of an element

An

Area of the neoprene bearing

ai , bi

Coefficients of updating parameters

bn

Width of neoprene bearing

bi

Vector that shows right side of linear system of equations

[ B]

Derivative of the shape function matrix

c

Constant vector

cn

Length of neoprene bearing

C

Constants that are internally chosen between 0 and 1

c1,.., c4

Constants associated with objective function

[ D]

Matrix of material constants

dk dˆ

Search direction at k -th iteration

d1,.., d 4

Constants associated with penalty function W

d ( j)

Parameter showing the direction to the line search

En2

Weighted least square error norm for the objective function

E

Modulus of elasticity of material of an element

Ecomp

Modulus of elasticity of composite material

ei

Vector with i -th element equal to 1, and zero elsewhere

frj

Frequency corresponding to j -th mode

frej , fraj

Experimental and analytical frequencies of j -th mode

k

Search direction for internal loop at k -th iteration

f (a) , f fˆ

Function approximation

f0

Reference objective function value

F ( x, Pk )

Unconstrained objective function

f1 ( a0 )

Initial values of objective function f1 when a =0

f (x)

Vector of objective functions

Objective function

xiv

Vector of design goals i.e., f * = { f1* , f 2* ,...., f m*}

f* F (t )

Dynamic force in a system

{F } , {F } ej

aj

A static force balance for j -th measured and analytical mode shapes

{Fa } {F ( t ) a }

Dynamic forces corresponding to master DOFs.

Gs

Shear modulus

Vector of applied static loads

[G ] [Gn ] [Gr ] [Gmm ] = ⎡⎣Gexp ⎤⎦

Modal flexibility matrix Modal flexibility, formed from the measured modes Residual flexibility Measured flexibility matrix

Gana

Analytical flexibility matrix corresponding to the measured DOFs

⎡⎣G f ⎤⎦

Flexibility matrix due to flexible modes

G ( a0 )

Flexibility estimate at initial parameters values

gi ( x )

Inequality constraints

gi , h j , wk

State variables(equality and inequality constraints) Upper and lower bounds of state variables

gi , h j

G, H ,W ∧

∧

∧

Penalty functions for state variable constraints

g, h, w

State variables approximations

H k = ∇ fk

Hessian matrix at k -th iteration

I zz , I yy

Z and Y moment of inertia

I cg

Moment of inertia of the individual bearings

i, j , k

Indices

2

[J ] [K ] [ K e ] , ⎡⎣ Ke' ⎤⎦ [ Ka ] , [ K N ]

Jacobian matrix between the global and a local coordinate Global stiffness matrix Elemental stiffness matrix in original and transformed system Reduced and full stiffness matrix

⎡ K• ⎤ ⎣ ⎦ Kv

Vertical stiffness of neoprene support

K rot

Rotational stiffness of bearing

K1 , K 2

Spring stiffness of two springs

L1

Lower limit to constrain the MAC

le

Length of element

l

Number of analytical modes considered in modal expansion

lc

Number of active constraints

Stiffness corresponding to • DOFs when unit displacement at

xv

DOFs.

Ld

Distance between two springs in bearing

Mk

Quadratic model at k -th iteration

mo

Number of objective functions

M

Global mass matrix of system

M1

Bending moment

me

Number of equality constraints

m

Total number of constraints

ms

Number of mode shapes considered

mf

Number of eigenfrequency residual

mk

Component of diagonal mass matrix

m

Mass per unit length

m1 + m2 + m3

Number of state variables

MSE j

Modal strain energy corresponding to j -th mode

MSFj

Modal scale factor corresponding to j -th mode

MACi

MAC number for the i th mode pair

[M a ] [M ] , [M N ] [ M e ] , [ M e' ]

Analytical global mass matrix reduced to measurement DOfs Global mass matrix

[M • ]

Mass corresponding to • DOFs when unit displacement is at

N elem

Number of finite elements in system

Nm

Number of elements in substructures

N

Total number of finite element DOFs

nr

Number of residuals

n

Number of updating parameters

Ni

Shape function corresponding to the i -th node

nd

Number of the measurement points

ns

Number of subproblem iterations

Ns

Maximum number of iterations

N si

Maximum number of sequential infeasible design sets

nc

Current number of design sets

p

Number of experimental modes considered in modal expansion

pv

Some vector

Px , Pg , Ph , Pw

Penalties applied to the constrained design and state variables Unitary matrix so that Ak = Q * R

Q

Elemental mass matrix in original and transformed system

xvi

DOFs

Q ( x, q )

Unconstrained objective function with response surface parameter ( q )

Q f , Qp

Function related to objective function and penalty constraint respectively ∇f ( xk +1 ) − ∇f ( xk )

qk q (d )

Objective function which is function of search direction Upper triangular matrix of the same dimension as Ak

R

r

Radius of gyration rz = sqrt ( I zz A ) , ry = sqrt ( I yy A )

rn

Number of nodes

rf

Eigenfrequency residual

rs

Mode shape residual

ri

Penalty parameter

rj

Residual corresponding to j -th mode

S ji =

∂rj ∂ai

Sensitivity matrix

sk

xk +1 − xk

Sj

Line search parameter

s*j

Largest possible step size for the line search of the current iteration

Smax

Maximum (percent) line search step size

Sn

Shape factor of neoprene bearing

t

Time period

tn

Thickness of bearing

T

Transformation matrix

Tpp

Transformation matrix for modal expansion

u ,u

Displacement and acceleration of system

u , v, w

Displacements of a point

u1 u2

un

Degrees of freedom

ui

Deflection coefficient at point i

UL

Upper limit whose value can be set as absolute error of i -th eigenvalue

uaj

Analytical uniform load surface

uej

Experimental uniform load surface

{u}

Vector of resulting static responses

v

Any vector

νk

Constant at k -th iteration

wj

Square root of weighting factor of residual rj

wc

Constant scalar

w

Vector of weighting functions, i.e., w = {w1 , w2 ,...., wm } xvii

x, y , z

Coordinates of a point

xa , xo

Master

x

Updating parameters or design variables

xi , xi

Upper and lower bounds on the design variables

x*

Optimal design set, noninferior solution

X

Penalty function used to enforce design variable constraints

xj

Design variable at j -th iteration

yk

Mass normalized trial vector at k -th iteration

zj

Aanalytical modal parameters at j -th iteration

z Zk

Experimental modal parameters Matrix formed from the QR decomposition of the matrix Ak

αk

Step length during k -th iteration

αj

Weighting factor for j -th eigenvalue

αs

Line search parameter

βj

Weighting factor for j -th modeshape

χ

Tolerances for state variable gi

δi

Mass normalization constant

∈p

Error in potential energy

∈k

Error in kinetic energy

∈φ

An overall mode shape error indicator

∈ortho

Error from orthogonality check

∈f j

Error between j -th analytical and experimental mode

∈

Very small positive number

(a)

and slave ( o ) degrees of freedom

( φe ) jr

* indicates the complex conjugate of element in mode shape

φaj , φej

j -th analytical and experimental mode shape vectors

*

Φe

Experimental mode shape matrix

Φ

Analytical mode shape matrix corresponding to the experimental DOFs

φaj

j -th measured mode shape vector

Φ nm =Φ , Φ rm

Measured and unmeasured mode shapes at the measured DOFs

φi

Mode shape vector obtained from ambient vibration test

φj Φ ij

j -th mode shape vector

Φ ik

Normalized i -th component of the j -th modal vector i -th coefficient of unit mass normalized modal vector for mode k

Φf

Flexible mode shapes

{ϕi },{ϕ j }

Any two modal vectors xviii

γj

Modal scale factor (MSF) of j -th modal vector

γ

Dummy argument to minimize the vector of objectives

κ

Large integer

λj

j -th eigenvalue

λaj , λej

j -th finite element and experimental eigenvalue

ρ

Mass density

ρi

Design variable tolerance

ρk

Ratio of the actual reduction in the objective function over the predicted reduction at k -th iteration

τ

Objective function tolerance

ω

Angular frequency

ωj

j -th measured natural frequency

ξ1 , ξ 2 , ξ3

Local coordinates

ψ

Weight associated with design set j

j

Ψ ( x ) Ψ ( x, γ )

Merit functions

∆a

Step length

∆K

Updated stiffness matrix

∆M

Updated mass matrix

∆D

Forward difference step size ( %)

∆x

Increment in x

∆ = ∆k

Radius of the region in which the quadratic model is trusted

∆1 , ∆ 2

Displacements of springs

Λ =ω

2

Matrix of eigenvalue

Λ n = Λ, Λ r

Eigenvalues matrix of measured and unmeasured modes.

Λf

Eigenvalues matrix corresponding to the flexible modes

Π (γ )

Feasible region for the objective function space

ϒi

Lagrange’s multiplier

Ω

Feasible region in parameter space x ∈ℜn

ℜn

Parameter space which represents real n-vectors ( n × 1 matrices)

xix

LIST OF FIGURES 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6

4.1 4.2 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15

Hermite shape functions of plane beam element…………………………….. Local and global coordinate system…………………………………………. Degrees of freedom…………………………………………………………… Simply supported beam to verify MBMAT……………………………………. First bending mode of simply supported beam as seen in the output window of MBMAT…………………………………………………………………… Bending modes of simply supported beam obtained from MBMAT………….. Nine storey, ten bay plane frame to verify MBMAT…………………………. First bending mode of frame as seen in the output window of MBMAT……... Bending modes of plane frame obtained from MBMAT……………………… Relationship between FE modelling, testing and system identification for FE model updating……………………………………………………………. Standard simulated simply supported beam before and after introducing damage……………………………………………………… MAC values between the actual and expanded mode shapes…………………. Norm errors for different expansion methods…………………………….…… Norm of eigenvector differences……………………………………………… Schematic diagram to show the main issues of model updating (a) poor selection of updating parameters (b) poor setting up of objective function………………………………………. The general procedure of the FE model updating method…………………….. Graphical interpretation of quasi-Newton method…………………………….. A simulated simply supported beam…………………………………………... Correlation of simulated beam after updating with frequency residual, MAC function and flexibility residual in objective function………………….. Photo of Beichuan river concrete-filled steel tubular arch bridge…………….. Elevation and plan of arch bridge……………………………………………... Cross section of arch rib and deck beam connection of arch bridge…………... Three dimensional FE model of arch bridge………………………………….. Details of measurement points of arch bridge…………………………………. Data acquisition system and arrangement of accelerometers in vertical direction of arch bridge………………………………………………………... Raw measurement data of Point 9 for vertical direction of arch bridge……….. Re-sampled data and modified power spectral density of point 9 for vertical direction of arch bridge……………………………………………….. Average normalized power spectral densities for full data in vertical direction of arch bridge………………………………………………………... Typical stabilization diagram for vertical data of arch bridge…………………. First six mode shapes obtained from FE analysis and test of arch bridge…..…. Frequency and MAC correlation of arch bridge before updating……………... Eigenvalues sensitivity of arch bridge to potential parameters………………... xx

16 18 19 25 26 27 28 28 29 33 45 46 46 47

48 51 70 89 90 93 94 95 96 97 98 99 99 100 100 101 103 104

5.16 5.17 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Convergence of six FE eigenvalues during updating of arch bridge………….. Frequency and MAC correlation of arch bridge after updating……………….. Mapping from parameter space into objective function space………………… Set of inferior solutions……………………………………………………….. Geometrical interpretation of goal attainment problem for 2D problems……... Plot of different solution points……………………………………………….. Location and severity of damage in simulated beam after FE model updating... Photo of Hongtang bridge…………………………………………………….. Details of Hongtang bridge and bearings with measurement points for ambient vibration test…………………………………………………………. Diagram to calculate the equivalent rotational stiffness of support…………… Finite element model of the Hongtang bridge…………………………………. The arrangement of accelerometers in vertical direction of Hongtang bridge… Raw measurement data of Point 29 for vertical direction of Hongtang bridge... Re-sampled data and modified power spectral density of point 29 for vertical direction of Hongtang bridge………………………………………… Average normalized power spectral densities for full data in vertical direction. Typical stabilization diagram for vertical data of Hongtang bridge…………… First five vertical mode shapes obtained from FE analysis and Test of Hongtang bridge………………………………………………………. Frequency correlation of Hongtang bridge before updating…………………... Frequency correlation of Hongtang bridge after updating…………………….. Location and severity of damage in simulated beam after FE model updating for different cases…………………………………………………… Static test arrangement and cross section of experimental beam……………… Observed cracks of the experimental beam in each load step…………………. Dynamic test setup of experimental beam…………………………………….. Identified mode shapes of experimental beam………………………………… Descritization of experimental beam………………………………………….. Location and severity of damage after FE model updating (reference state) …. Location and severity of damage after FE model updating (damaged state) …..

xxi

107 108 114 114 115 117 120 122 123 124 125 126 127 127 128 128 130 130 133 144 146 147 148 149 150 151 151

LIST OF TABLES 2.1 2.2 3.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 7.6

Comparison of time period (sec) with Clough and Penzien and Mario Paz…… Comparison of eigenvalues with Bathe and Wilson 1972 and Peterson 1981… Comparison of experimental (assumed damage) and initial analytical modal properties of simulated simply supported beam………………………… Comparison of experimental (assumed damage) and analytical modal properties of simulated beam before updating………………………………….. Results of simulated beam after updating with different residuals in objective function……………………………………………………………………………….….. Error detection after updating when frequency, MAC related function and flexibility residual are used in objective function………………………….. Test setup in vertical direction of arch bridge……………………………………… Comparison of experimental and analytical modal properties of arch bridge before updating…………………………………………………………. Comparison of experimental and analytical modal properties of arch bridge after updating……………………………………………………………. Value of updating parameters of arch bridge before and after updating………... Comparison of experimental (assumed damage) and analytical modal properties of simulated beam after updating…………………………….. Comparison of experimental and analytical modal properties of Hongtang bridge before updating…………………………………………… Comparison of experimental and analytical modal properties of Hongtang bridge after updating……………………………………………… Value of updating parameters of Hongtang bridge before and after updating….. Comparison of experimental (assumed damage) and analytical modal properties of simulated beam after updating…………………………….. Comparison of experimental (assumed damage) and analytical modal properties of simulated beam with 3% noise after updating…………….. Static load steps for experimental beam………………………………………... Bending frequencies of beam (Hz) …………………………………………….. Comparison of experimental and analytical modal properties of experimental beam before updating……………………………………………. Comparison of experimental and analytical modal properties of experimental beam after updating………………………………………………

xxii

27 29 45 89 91 92 97 102 108 109 121 129 133 134 144 145 147 148 150 152

Finite element model updating of civil engineering structures under operational conditions

CHAPTER 1 INTRODUCTION

CHAPTER SUMMARY

This chapter introduces the problems, objectives and scopes of the thesis with own contributions. The up to date literature review is carried out to show the state-of-the-art of the finite element model updating in civil engineering application. At the end of the chapter, the organization of the dissertation is presented.

1.1 Background The finite element (FE) method is widely used in the design and the analysis of civil engineering structures. The FE model of a structure is constructed on the basis of highly idealized engineering blue prints and designs that may not truly represent all the physical aspects of an actual structure. When field dynamic tests are performed to validate the analytical model, inevitably their results, commonly natural frequencies and mode shapes, do not coincide well with the expected results from the analytical model. These discrepancies originate from the uncertainties in simplifying assumptions of structural geometry, materials as well as inaccurate boundary conditions and experimental errors. The problem of how to modify the analytical model from the dynamic measurements is known as the model updating in structural dynamics. In other words, structural model updating is the process of using test measurements to refine a mathematical model of a physical structure. Basically, FE model updating is an inverse problem to identify and correct uncertain parameters of FE model and it is usually posed as an optimization problem. This is typically done to improve the ability of the model to predict the response of the structure under various conditions. The updated models are used in many applications of civil engineering structures like damage detection, health monitoring, structural control, structural evaluation and assessment. In a model updating process, not only the satisfactory correlation is required between analytical and experimental results, but also the updated parameters should preserve the physical significance. Closely associated with the model updating problem is the problem of parameter identification and damage detection. This is because many algorithms for parameter -1-

Ph.D. dissertation of Fuzhou University

identification and damage detection rely on differences between models of the structure correlated before and after the damage occurs. Discrepancies between the two models are used to localize and determine the extent of the damage. Damage identification by means of FE model updating has the advantage that it is a general approach. Unlike many other damage identification methods which are often developed for specific modal quantities, the algorithm can in principle be applied to any modal feature which is sensitive to damage, such as eigenfrequencies, mode shape displacements, modal curvatures, modal strain energy, etc. and their combinations. In this dissertation, the focus is on the model updating problem, i.e., improving the predictive performance of structural models. However, since one can view damage detection as a special case of the model updating problem, many of the results are applicable to damage detection as well. A FE model updating approach is used to propose a damage detection algorithm by using modal flexibility in this work. There are many criteria which tell the differences between analytical and experimental results. In general, they are combined into a single objective function using weighting factors. There are no general straightforward rules for selecting the weighting factors since the relative importance among the criteria is not obvious and specific for each problem. Thus, a necessary approach is to solve the same problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained. The other alternative is to use the multi-objective or multi-criteria optimization technique to evaluate the error criteria without combining the multi-objective functions in a single one. All real structures have an infinite number of degrees of freedom (DOFs), and natural frequencies and modes. But the data obtained from modal test are incomplete. The number of measured DOFs is limited. It is not possible to measure as many natural frequencies and mode shapes as required, because the available transducers and data acquisition hardware limit the frequency range that can be measured. On the other hand, FE models consist of many finite elements, extending in many cases to several thousands. Thus, due to the inherent limitations of experimental data, the number of parameters which can be used to modify the FE model far exceeds that of the measured data. Hence, there can be numerous modified or updated FE models that agree with the incomplete test data [1]. If the aim of model updating is not simply to mimic the incomplete test results, there must be some restrictions on the selection of updating parameters and their allowable changes so that the updated model retains its physical foundations. Updating parameters should be selected with the aim of correcting modeling errors. And the objective functions or criteria to be minimized for model improvements should be sensitive to such selected parameters. Otherwise, to give acceptable results, the updating parameters may deviate far from their initial values and lose their physical meaning. -2-

Finite element model updating of civil engineering structures under operational conditions

1.2 Objectives and Scopes of the Research The purpose of the current research is to present and develop a robust FE model updating technique that can be applied to the full-scale civil engineering structures by using ambient vibration based experimental modal data. To accomplish this, the research is focused on several basic objectives. They are itemized as follows. •

To propose the theory and procedure necessary for robust FE model updating of civil engineering structures. The capability of proposed algorithms are implemented and demonstrated via numerical simulations and full-scale bridges.

•

To investigate the existing optimization techniques and use the most effective and robust algorithms and strategies available for efficient FE model updating.

•

To investigate and propose the use of dynamically measured flexibility matrices and modal strain energy index obtained from ambient vibration method for model updating, which leads the development of the new algorithm in this field.

•

To investigate the use of different possible error residuals alone or in combined form in the objective functions, which is cast in a single-objective optimization framework using weighting factor for each residual and to propose the best combination for FE model updating.

•

To investigate and propose the use of multi-objective optimization technique for FE model updating that does not need weighing factor for each error residual like in single-objective optimization.

•

To propose an algorithm for damage detection from FE model updating technique using modal flexibility. Despite the presence of experimental errors in vibration test data, it is generally

assumed that the experimental data are more accurate than the analytical predictions. Thus, in this research, it is assumed that the experimental data is accurate and the FE model is modified or updated to better represent the experimental results. And it is further assumed that such errors in the model are mainly due to the inaccuracy in modeling parameters. Other causes of the errors are not further dealt with in this research.

1.3 Contributions of the Research The primary contribution of this dissertation is the development of a robust FE model updating technique that can be applied to identify unknown properties in civil engineering structures using ambient vibration based experimental modal data. The core of the thesis forms the FE model updating method and damage detection application. The basic -3-

Ph.D. dissertation of Fuzhou University

procedure of FE model updating is improved using the findings known from the mathematical optimization study. Apart from the eigenfrequency residuals and mode shape residuals, two new residuals, namely modal flexibility and modal strain energy are investigated and proposed to use for FE model updating. The FE model updating procedure is investigated in single-objective and multi-objective optimization framework. A fair amount of applications to full-size civil engineering structures is carried out. Real case studies represent a surplus value to simulated examples, since many identification methods fail when applied to real test data. More specifically, the original contributions of the thesis are itemized below. •

FE model updating procedure in single-objective optimization framework is cast and studied. The model updating is carried out using different combinations of possible residuals in objective function and best combination is recognized. From the study, the combination that consists of three residuals, namely eignevalue, mode shape related function and modal flexibility with weighing factors assigned to each of them is recognized and proposed as the best objective function for model updating. The proposed single-objective optimization framework is successfully applied for the FE model updating of a real concrete filled steel tubular arch bridge measured by ambient vibration tests.

•

FE model updating procedure using multi-objective optimization technique is investigated and proposed. The multi-objective optimization method, called the goal attainment method, is used to solve the multi-objective optimization problem. The use of dynamically measured modal strain energy is proposed for model updating. The analytical gradient of modal strain energy is derived. The eigenfrequencies and modal strain energies are used as the two independent objective functions to be minimized by the multi-objective optimization technique. The developed multi-objective FE model updating technique is successfully used to identify the elastic modulus of deck and the stiffness of the neoprene supports of a precast concrete highway bridge that was dynamically tested in the field.

•

A damage detection algorithm is developed from FE model updating using modal flexibility. The gradient of the analytical modal flexibility is derived in order to implement the sensitivity based updating techniques. The optimization algorithm to minimize the objective function is realized by using trust region strategy that makes the algorithm more robust to reduce ill-conditioning problem. The procedure is successfully applied for the simulated case studies with and without noise as well as laboratory tested reinforced concrete beam with known damage pattern. -4-

Finite element model updating of civil engineering structures under operational conditions

•

In the structural FE model updating, the updating parameter selection requires a considerable physical insight into the target structure, and trial-and-error approaches are recommended with different set of selected parameters. The procedure that consists of eigenvalue sensitivity study of the different parameters and selects the most sensitive parameters with some engineering intuition is proposed and applied for updating parameters selection in the model updating of practical civil engineering structures.

•

To implement the proposed algorithms for structural FE model updating, a FE toolbox is developed in Matlab environment for the analytical modal analysis of engineering structures.

1.4 Literature Survey There has been a significant amount of work on FE model updating over the past few years and several hundred papers have been published. Among them, Imregun and Visser [2], Mottershead and Friswell [3] and Friswell and Mottershead [4] give extensive reviews of the various model updating methods that have been developed. The standard references are the books of Friswell and Mottershead [4] and Maia and Silva [5]. Most often modal data, such as eigenfrequencies and mode shapes, are used for model updating of civil structures since they can be identified from output-only data obtained from ambient vibrations and do not require the structure to be excited by artificial forces, e.g. by a shaker. An alternative approach in mechanical engineering is to use frequency response functions (FRF), as in the work of Fritzen et al. [6]. The state-of-the-art in model updating technology has long been based on modal-based model updating procedures as these are, in general, numerically more robust and better suited to cope with larger applications. Model updating methods can be classified into two broad groups, namely direct methods and sensitivity based methods as explained below.

1.4.1 Direct Methods The updated model is expected to match some reference data, usually consisting of an incomplete set of eigenvalues and eigenvectors derived from measurements. Such approaches are called direct or representation models as presented in the work of Zhang et al. [7]. The direct methods compute a closed-form direct solution for the global stiffness and/or mass matrices using the structural equations of motion and the orthogonality equations. Baruch [8] described these methods as reference basis methods, since one of the three quantities (the measured modal data, the mass or stiffness matrices) is assumed to be exact, i.e., the reference and the other two are updated. Caesar [9] extended this approach -5-

Ph.D. dissertation of Fuzhou University

and produced a range of methods based on optimizing a number of objective functions. Wei [10] updated the mass and stiffness matrices simultaneously, instead of one quantity at a time. Friswell et al. [11] extended the direct methods to update both stiffness and viscous damping matrices based on measured complex modal data. The main advantages of these direct methods are: •

The convergence is assured since the methods do not need any iterations.

•

The central processing unit (CPU) time is usually less than that required by the iterative methods.

•

The methods try to produce the reference data set exactly. On the other hand, the main drawbacks of the direct methods are obviously:

•

High quality measurements and accurate modal analysis are needed.

•

The mode shapes must be expanded to the size of FE model.

•

The methods are usually unable to keep the connectivity of the structure and the updated matrices are usually fully populated.

•

The resulting updated FE model may not provide any physical meaning since all the elements in the system matrices are changed separately. There is no guarantee for the positive definiteness of the updated mass and stiffness matrices. Friswell and Mottershead [12] stated furthermore that forcing the model updating

procedure to reproduce the measured modal data exactly, causes the measurement errors to be propagated to the parameters. For these drawbacks, this method is seldom used in structural dynamics [4].

1.4.2 Sensitivity Based Methods Nowadays the sensitivity based methods are the most popular since they overcome the limitations of the direct methods. In these methods, the model updating problems are posed as optimization problems. They set the errors between analytical and experimental data as an objective function, and try to minimize the objective function by making changes to the pre-selected set of physical parameters of FE model. The optimum solution is obtained using sensitivity-based optimization methods. Because of the nonlinear relation between the vibration data and the physical parameters, an iterative optimization process is performed. This approach is able to update the relevant physical parameters and to locate erroneous regions of the model. Link [13] gives a clear overview of the sensitivity-based updating methods. These methods can be further classified according to the data used in optimization process as modal domain methods [7, 14-18] and frequency domain methods [19,20]. -6-

Finite element model updating of civil engineering structures under operational conditions

In frequency domain methods, one can use the input error, the output error or the error in frequency response data [13, 21]. Since the number of measured DOFs is generally much smaller than the number of analytical DOFs, it is necessary for most residual types to expand the measured vector to full model size or to condense the model order down to the number of measured DOFs. Furthermore, an implicit weighting is performed which depends on the proximity of the chosen frequency points to resonance. Therefore, the weighting is less controllable. In civil engineering, however, the approach using modal data is most often applied since the frequency response functions of heavy civil structures are not available over a wide frequency domain. The FE model updating can also be performed with a neural networks algorithm, as reported in Atalla and Inman [22]. A major problem in model updating is the relatively low information content of the measured data. Rade and Lallement [23] and Nalitolela et al. [24] increase the information content of the data, by testing the structure in different configurations so that the areas of model uncertainty are stressed in different ways. The alternative is to reduce the number of updating parameters, which is done in Teuguels et al. [25] through the use of damage functions. Fritzen and Bohle [26] proposed a parameter reduction technique for damage identification problems based on the correlation between the change in the dynamic stiffness matrix and the residual vectors. Parameterization is the key issue in FE model updating. It is important that the chosen parameters should be able to clarify the ambiguity of the model, and in that case it is necessary for the model output to be sensitive to the parameters. Element stiffness parameters, such as the element’s Young’s modulus, are most often used as updating parameter as in Friswell and Mottershead [4] and Link [13]. Mottershead et al. [27] used the geometric parameters, such as offsets in beam elements, for the updating of mechanical joints and boundary conditions. Gladwell and Ahmadian [28] and Ahmadian et al. [29] demonstrated how an element stiffness matrix can be adjusted by modifications to its eigenvalues and eigenvectors, and Mottershead et al. [30] used both, the geometric as well as the element modal parameters, in a generic element method to update mechanical joints. FE model updating is used for the parameter estimation and damage assessments of structures. The works that have been reported in the literature in this aspect are summarized below. 1.4.2.1 Parameter Identification of Structures

The sensitivity-based FE model updating technique can be used as a parameter identification technique and belongs to the class of inverse problems. Inverse problems -7-

Ph.D. dissertation of Fuzhou University

typically involve the estimation of some quantities based on indirect measurements of these quantities [31]. In parameter identification process, the inverse operation is performed in which the model parameters are determined by fitting the model to the measured output values. Zhang et al. [32] identified various structural parameters, including connections and boundary conditions, of a cable-stayed bridge in Hong Kong by minimizing the discrepancies in eigenfrequencies and in literature [33], they reported similar work on a scaled suspension bridge model. Brownjohn et al. [34] quantified the effectiveness of upgrading works on a short-span highway bridge in Singapore through subsequent model updating, i.e., before and after the refurbishing and strengthening of the bridge. Ventura et al. [35] updated the Heritage Court building structure in Vancouver, Canada, by adjusting the stiffness and mass properties. Brownjohn and Xia [36] investigated the application of the model updating technology to the dynamic assessment of a cable-stayed bridge in Singapore, by adjusting the Young’s modulus of the concrete and the structural geometry. Gentile and Cabrera [37] performed a similar study on a curved cable-stayed bridge at Malpensa airport in Milan. 1.4.2.2 Damage Detection of Structures

As FE model updating procedures are used to identify unknown physical properties and to build a representative FE model applicable to structural dynamics, they can also be used to detect and identify damage on structures. In 1996, Doebling et al. [38] made a detailed review of the vibration based damage identification literature. It gave a brief overview of global nondestructive methods based on the fact that structural damage usually causes a decrease in the structural stiffness, which produces changes in the vibration data of the structure. Fritzen et al. [6] examined the problem of detecting the location and extent of structural damage from measured vibration test data using FE model updating. It is noted that the mathematical model used in the model updating is usually ill posed and the special attention is required for an accurate solution. Wang et al. [39] implemented FE model updating to establish the baseline modal values (modal frequencies and mode shapes) for a long-span bridge. They suggested that model updating might be used in automated on-line monitoring on bridges. FE model updating method was successfully applied to the damage assessment of structures using frequency and mode shape residual with the introduction of damage functions [25, 40]. Quantitative and objective condition assessment for infrastructure protection has been a subject of strong research within the engineering community. To achieve this aim, methodologies of the routine inspections with fixed intervals or the continuous monitoring, -8-

Finite element model updating of civil engineering structures under operational conditions

which provide constant information on safety, reliability or remaining lifetime of the structure, have been under development in recent years. Inspection of structural components for damage is vital to take decisions about their repair or retirement. Visual inspection is tedious and often does not yield a quantifiable result [41]. For some components a visual inspection is virtually impossible. Methods which are based on pure signal processing have only a limited capability for the early detection of damage and often do not allow unique conclusions to be drawn on the sources of the damage [6]. The importance and difficulty of the damage detection problem has caused a great deal of research on the quantitative methods of damage detection based upon physical testing. Among those physical tests, the use of the modal tests has emerged as an effective tool to use in damage detection. The possibility of using measured vibration data to detect changes in structural systems due to damage has gained increasing attention [42, 43]. The methods are predominantly based on the change in eigenfrequencies, as in the paper of Hanselka et al. [44], Williams and Messina [45]. In an earlier work by Cawley and Adams [46], it was shown that the ratio of frequency changes in different modes was only a function of damage location and not the magnitude of damage. Salawu [47] reviewed the different methods of structural damage detection through changes in natural frequencies. He emphasized the simplicity and low cost of this approach, but at the same time pointed out the factors that could limit successful application of vibration monitoring to damage detection and structural assessment since the changes in natural frequencies cannot provide the spatial information about structural damage. In order to localize the damage, mode shapes are used which provide spatial information about structural damage. Analysis of changes in mode shapes due to damage represents another subgroup of modal-based methods. Natke and Cempel [48] used changes in eigenfrequencies and mode shapes to detect damage in a cable-stayed steel bridge. Based on changes in frequencies and mode shapes of vibration, Ren and Roeck [49,50] proposed a damage identification technique for predicting damage location and severity. However, a large number of measurement locations are required to accurately characterize the mode shape vectors and to provide a sufficient resolution to find the damage location. As an alternative for obtaining spatial information, Pandey et al. [51] introduced the use of mode shape curvatures and Maeck and De Roeck [52] extended this approach by using mode shape curvatures in a direct stiffness calculation technique which they applied to the damage identification in a prestressed concrete bridge. Ho and Ewins [53] states that the derivatives of mode shapes are more sensitive to damage, but the differentiation process enhances the experimental errors inherent in mode shapes, yielding a large statistical -9-

Ph.D. dissertation of Fuzhou University

uncertainty. Changes in strain energy were used as an indicator to represent damage in many works. Modal strain energy has been studied previously by Lim and Kashangaki [54] and Doebling et al. [55] in the identification of structural behavior and location of structural damage. Kim et al. [56] evaluated damage detection and localization algorithms based on changes in eigenfrequencies, mode shapes and modal strain energy. Stubbs and Kim [57] directly used the modal strain energy as a damage indicator. Shi and Law [58] and Ren and Roeck [59] studied the change of the elemental modal strain energy before and after the occurrence of damage in the structure, and they verified that this parameter would be a very efficient indicator in structural damage localization. Another class of damage identification methods uses the dynamically measured modal flexibility matrix. Catbas and Aktan [60] and Bernal [61] proposed the use of the flexibility matrix as damage indicator. Aktan et al. [62] proposed the use of the measured flexibility as a condition index to indicate the relative integrity of a bridge. Two bridges were tested and the measured flexibility was compared to the static deflections induced by a set of truck-load tests. Pandey and Biswas [63] presented a damage detection and location method based on changes in the measured modal flexibility of the structure. This method is applied to several numerical examples and to an actual spliced beam where the damage is linear in nature. Results of the numerical and experimental examples showed that estimates of the damage condition and the location of the damage could be obtained from just the first two measured modes of the structure. It is demonstrated that the modal flexibility is more sensitive to damage than the natural frequency or mode shape. Similarly, in the study of Zhao and DeWolf [64], the sensitivity study is carried out to compare the use of natural frequencies, mode shapes and modal flexibilities for damage detection and concluded that modal flexibilities are more likely to indicate damage than either natural frequencies or mode shapes. Reisch and Park [65] proposed a method of structural health monitoring based on relative changes in localized flexibility properties and applied for the damage detection of elevated highway bridge column. Topole [66] developed an algorithm to calculate the contribution of the flexibility of the structural members to the sensitivity of the modal parameters to change on the flexibilities of the members and applied to detect the damage of simulated structure with truss member. The literatures for other specific issues of model updating are discussed in most relevant places throughout the thesis.

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Finite element model updating of civil engineering structures under operational conditions

1.5 Organization of Dissertation The dissertation is organized in 8 chapters. Each chapter begins with a summary of the contents and ends with chapter conclusions. Chapter 1 introduces the background, objectives and scopes of the thesis with own contributions. The up to date literature review is carried out to show the state of art of the FE model updating in structural dynamics. Chapter 2 discusses the modal analysis toolbox developed in Matlab environment which is named as MBMAT to implement the proposed algorithms. The general expressions of mass and stiffness matrices in terms of shape function are first explained and these matrices for different elements are presented in matrix form as used in program coding. The theory of eigensolution technique adopted in the program is presented. The program realization process is explained and the types of finite elements included are shown. At last, two well known examples are solved using the program MBMAT to demonstrate the accuracy of the program. Chapter 3 deals with different aspects and techniques needed to carry out FE model updating in structural dynamics. The role of modeling, testing, and system identification for model updating are discussed and their relationship is illustrated. Various available methods for correlating analytical and experimental data, reducing analytical mode shapes and expanding experimental mode shapes for successful FE model updating are investigated. Two new methods for modal expansion are proposed and their effectiveness is demonstrated with the help of simulated case study. At last, three important issues of model updating are explained. Chapter 4 deals with the FE model updating procedure carried out in this thesis. The theoretical exposition on FE model updating is presented. Two new residuals, namely modal flexibility and modal strain energy are proposed and formulated to use in FE model updating. Many related issues including the objective functions, the gradients of the objective function, different residuals and their weighting and possible parameters for model updating are investigated. The issues of updating parameters selection procedures adopted in this work are discussed. The ideas of optimization to be used in model updating application are explained. The algorithm of Sequential Quadratic Programming is explored which will be used to solve the multi-objective optimization problem of Chapter 6. Chapter 5 deals about the FE model updating using single-objective optimization. The use of dynamically measured flexibility matrices is proposed for model updating. The issue related to the mass normalization of mode shapes obtained from ambient vibration test is investigated and applied to use the modal flexibility for FE model updating. The - 11 -

Ph.D. dissertation of Fuzhou University

algorithms of penalty function methods, namely subproblem approximation method and first-order optimization method are explored, which are then used for FE model updating. The model updating is carried out using different combinations of possible residuals in the objective functions and the best combination is recognized with the help of simulated case study. It is demonstrated that the combination that consists of three residuals, namely eignevalue, mode shape related function and modal flexibility with weighing factors assigned to each of them is recognized as the best objective function. In single-objective optimization, different residuals are combined into a single objective function using weighting factor for each residual. A necessary approach is required to solve the problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained since there is no rigid rule for selecting the weighting factors. The single-objective optimization with eigenfrequecy residual, mode shape related function and modal flexibility residual is applied successfully for the FE model updating of a full-scale concrete filled steel tubular arch bridge that was tested by means of ambient vibration. The eigensensitivity method with engineering judgment is used for updating parameter selection. Chapter 6 deals with the FE model updating procedure using multi-objective optimization technique (MOP). The weighting factor for each objective function is not necessary in this method. In MOP, the notion of optimality is not obvious since in general, a solution vector that minimizes all individual objectives simultaneously does not exist. Hence, the concept of Pareto optimality is used to characterize the objectives. The multi-objective optimization method, called the goal attainment method is used to solve the optimization problem. The Sequential Quadratic Programming algorithm is used in the goal attainment method. The implementation of the dynamically measured modal strain energy identified from ambient vibration measurements is investigated and proposed for model updating. The eigenfrequencies and modal strain energies are used as the two independent objective functions of multi-objective optimization technique. The FE model updating procedure is illustrated with the examples of both simulated simply supported beam and a practical precast continuous box girder bridge that was dynamically measured under operational conditions. Chapter 7 deals with the damage detection application of FE model updating procedure using modal flexibility. The Guyan reduced mass matrix of analytical model is used for mass normalization of ambient vibration mode shape, to calculate the modal flexibility. The objective function is formulated in terms of difference between analytical and experimental modal flexibility. Analytical expressions are developed for the flexibility - 12 -

Finite element model updating of civil engineering structures under operational conditions

matrix error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The optimization algorithm to minimize the objective function is realized by using trust region strategy that makes the algorithm more robust to reduce ill-conditioning problem. The procedure of damage detection is demonstrated with the help of the simulated example of simply supported beam. The effect of noise on the updating algorithm is studied using the simulated case study. The procedure is thereafter successfully applied for the damage detection of a laboratory tested reinforced concrete beam with known damage pattern. Chapter 8 summarizes the conclusions and significances of the research work and suggests some topics for future research.

- 13 -

Ph.D. dissertation of Fuzhou University

CHAPTER 2 FINITE ELEMENT MODAL ANALYSIS TOOLBOX DEVELOPMENT

CHAPTER SUMMARY

To implement the proposed algorithms for structural FE model updating, a FE toolbox is developed in Matlab environment, which is used to carry out the analytical modal analysis of engineering structures. This chapter discusses the developed toolbox, which is named as MBMAT. The general expressions of mass and stiffness matrices in terms of shape function are first explained and these matrices for different types of finite elements are presented in matrix form, as used in the program coding. The theory of eigensolution technique adopted in the program is discussed. The program realization process is explained and the types of elements included are shown. At last, two well known examples are solved using the program MBMAT to demonstrate the accuracy of the program. From this toolbox, the information about the FE model like the global mass and stiffness matrices can be easily extracted and whole updating and damage detection work can be realized in Matlab framework.

2.1 Introduction The proposed FE model updating method using modal strain energy and damage detection algorithm using modal flexibility in this thesis, requires global mass and stiffness matrices of an analytical model of a structure to calculate the objective function gradient matrix, and eigenproblem must be solved in every iteration. It is very difficult and sometime impossible to extract these information from commercial FE software in which the program code is not exposed for ordinary users. It is the main motivation behind the development of this FE toolbox in Matlab [67] environment. The FE method has been established as the universally accepted analysis method for dynamic analysis and structural design. The method leads to the construction of a discrete system of matrix equations to represent the mass and stiffness of a continuous structure. The matrices are usually banded and symmetric. No restriction is placed upon the geometrical complexity of the structure because the mass and stiffness matrices are - 14 -

Finite element model updating of civil engineering structures under operational conditions

assembled from the contributions of the individual finite elements with simple shapes. Thus, each finite element possesses a mathematical formula which is associated with a simple geometrical description, irrespective of the overall geometry of the structure. Accordingly, the structure is divided into discrete areas or volumes known as elements. Element boundaries are defined when nodal points are connected by a unique polynomial curve. In most elements, the same polynomial description is used to relate the internal element displacements to the displacements of the nodes. This process is generally known as shape function interpolation.

2.2 Shape Functions In most FE formulations, the shape functions are used to express both the coordinates and the displacements of an internal point in terms of values at the nodes. Thus, if coordinates of a point are denoted by ( x, y, z ) and the displacements by ( u , v, w ) , then: x=

rn

∑N

xi

(2.1)

∑Nu

(2.2)

i =1

u =

i

rn

i =1

i

i

where xi is the x coordinate of the i -th node and ui is the displacement of this node. Similar

expressions

can

be

written

for

the

coordinates y and z and

the

displacements v and w . The summation in Equations (2.1) and (2.2) is taken over rn nodes and N i is the shape function corresponding to the i -th node. The shape functions N i are functions of position and for reasons of generality are given in terms of the local coordinates (ξ1 , ξ 2 , ξ3 ) such that the boundaries of the element describe a cube ( 2 × 2 × 2 ) in the ξ1 , ξ 2 , ξ3 frame. Thus, at each of the surface of the cube, a single local coordinate will take a constant value of ± 1 . As an illustration, the shapes functions for a plane beam element are shown in Figure 2.1 and in Equation (2.3). These shape functions are used for the derivation of masses and stiffness matrices of the finite elements. The details of the numerical procedures and derivations in this work are taken from standard literatures [68,69].

- 15 -

Ph.D. dissertation of Fuzhou University

Figure 2.1: Hermite shape functions of plane beam element

1 2 (1 − ξ ) ( 2 + ξ ) 4 1 2 N 2 = (1 − ξ ) (ξ + 1) 4 1 2 N 3 = (1 + ξ ) ( 2 − ξ ) 4 1 2 N 4 = (1 + ξ ) (ξ − 1) 4 N1 =

(2.3)

2.3 Finite Element Mass and Stiffness Matrices The general formulation of the structural mass and stiffness matrices, when the shape functions are defined in the local coordinate system, is given as:

[ M e ] = ∫− 1 ∫− 1 ∫− 1 [ N ] 1

1

1

T

ρ [ N ] d et ( J ) d ξ 1 d ξ 2 d ξ 3

[ K e ] = ∫−1 ∫−1 ∫−1[ B ] [ D ][ B ] det ( J ) dξ1dξ 2 dξ3 1

1

1

T

(2.4) (2.5)

where ρ is the mass density, [ N ] is the shape function matrix, [ B ] is the derivative of the shape function matrix, [ D ] is the matrix of material constants and J is the Jacobian matrix between the global and a local element coordinate system. In the case of one dimensional Euler beam, Equations (2.4) and (2.5) can be simplified to obtain:

- 16 -

Finite element model updating of civil engineering structures under operational conditions

⎛ dx ⎞ ⎟d ξ1 ⎝ dξ1 ⎠

[ M e ] = ρ A∫−1 N T N ⎜ 1

⎛ dx ⎞ [ K e ] = EI ∫−1 B B ⎜ ⎟ dξ1 ⎝ dξ1 ⎠ 1

(2.6)

T

where B contains terms which are the second derivatives of the shape function with respect to x , EI denotes bending rigidity and A denotes cross sectional area. The mass matrix formulation using Equation (2.4) gives a so-called, consistent mass matrix and predictions using this formulation are usually more accurate than predictions using a lumped mass matrix formulation. The lumped mass matrix formulation is a simple concentration of the mass at the translational degrees of freedom that leads to a diagonal form of mass matrix. The elemental mass and stiffness matrices derived using the shape functions for different types of finite elements are summarized in matrix form in pages 18-19 in local coordinate system corresponding to DOFs as shown in Figures 2.2 and 2.3 respectively. The mass and stiffness matrices in local coordinate systems obtained as explained above are transformed to the global direction before they are assembled using the relationship: ⎡⎣ M e' ⎤⎦ = ⎡⎣ T T ⎤⎦ ⎡⎣ M e ⎤⎦ [T ] ⎡⎣ K e' ⎤⎦ = ⎡⎣ T T ⎤⎦ ⎡⎣ K e ⎤⎦ [T ]

(2.7)

where ⎡⎣ M e' ⎤⎦ and ⎡⎣ K e' ⎤⎦ are mass and stiffness matrices in global coordinate system and [T ] is the transformation matrix. The overall mass and stiffness matrices are obtained by assembling mass and stiffness matrices of the individual elements at the common nodes and are defined by the expressions as shown in Equations (2.8) and (2.9). N elem

[M ] = ∑

i =1

[K ] =

N elem

∑

i =1

⎡ M e' ⎤ ⎣ ⎦

(2.8)

⎡ K e' ⎤ ⎣ ⎦

(2.9)

At nodes where a number of individual elements meet, the motion experienced at each of the element nodal degree of freedom in turn must be identical if separation does not take place. This is the constraint, which ties elements together, and results in individual element mass and stiffness terms being added to mass and stiffness terms of - 17 -

Ph.D. dissertation of Fuzhou University

other elements at nodes, which are shared between those elements at each degree of freedom in turn. (a) Truss3d element ⎡1 ⎢ ⎢0 E A ⎢0 Ke = e e ⎢ le ⎢ −1 ⎢0 ⎢ ⎢⎣ 0 ⎡2 ⎢ ⎢0 ⎢ ρ A l ee 0 me = ⎢ 6 ⎢1 ⎢0 ⎢ ⎢⎣ 0

0 0 −1 0 0 ⎤ ⎥ 0 0 0⎥ 0 0 0⎥ ⎥ 0 0 1 0 0⎥ 0 0 0 0 0⎥ ⎥ 0 0 0 0 0 ⎥⎦ 0 0 0 0

0 0 1 0 0⎤ ⎥ 2 0 0 1 0⎥ 0 2 0 0 1⎥ ⎥ 0 0 2 0 0⎥ 1 0 0 2 0⎥ ⎥ 0 1 0 0 2 ⎥⎦

Figure 2.2: Local and global coordinate system (b) Beam3d ⎡ EA ⎢ l ⎢ e ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ e K =⎢ ⎢ − EA ⎢ le ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢⎣ ⎢

0

0

0

0

0

12 EI z

0

0

0

6 EI z

le3 0

12 EI y

0 0

0 −

6 EI z

GI x le

6 EI y

0

le2

−

6 EI y le2 0

4 EI y le

EA le 0

0

0

0

EA le

12 EI z

0

0

0

0 0 6 EI z le2

0 −

6 EI y le2 0

−

GI x le 0 0

6 EI y le2 0 2 EI y le 0

0

0

0

0

0

0

0

le3

0

le3

0

0

12 EI y

12 EI z

0

0

−

0

0

4 EI z le

0

0

0

0

−

0

0

0

le3

−

0

0

0

le2

−

0

le3

le2

−

−

6 EI z

−

12 EI y

0

le3 0 6 EI y le2

−

GI x le 0

−

6 EI y le2 0

2 EI y le

0

0

0

0

0

0

0

0

12 EI z

0

0

0

0

0

0

0

0

0

0

0

0

0

6 EI z

6 EI z le2

2 EI z le

- 18 -

le2

le3

−

le2

12 EI y le3 0 6 EI y le2 0

0 GI x le 0 0

6 EI y le2 0 4 EI y le 0

⎤ ⎥ ⎥ 6 EI z ⎥ ⎥ le2 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥ ⎥ 2 EI z ⎥ ⎥ le ⎥ ⎥ 0 ⎥ ⎥ ⎥ 6 EI z ⎥ − 2 ⎥ le ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ 4 EI z ⎥ ⎥ le ⎥⎦ ⎥ 0

Finite element model updating of civil engineering structures under operational conditions

Figure 2.3: Degrees of freedom ⎡1 ⎢3 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢ ⎢0 ⎢ e m = ρ Ae le ⎢ 1 ⎢ ⎢6 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢⎢ ⎣⎢ ⎢

0

0

0

0

13 6 rz2 + 65 le2

0

0

0

0

13 6 rz2 + 65 le2

0

0

0 ⎛ 11 r2 − le ⎜ + z ⎜ 210 10 l 2 e ⎝

0 ⎛ 11 r2 le ⎜ + z ⎜ 210 10 l 2 e ⎝

⎞ ⎟ ⎟ ⎠

I yy + I zz 3A ⎞ ⎟ ⎟ ⎠

0

⎛ 11 r2 − le ⎜ + z ⎜ 210 10 l 2 e ⎝

0

0

0

0

0

0

0

9 6r 2 − z 70 5le2

0

0

0

0

6r 2 9 − z 70 5le2

0

0

0 ⎛ 13 r2 le ⎜ − z ⎜ 420 10 l 2 e ⎝

0 ⎛ 13 r2 − le ⎜ − z ⎜ 420 10 l 2 e ⎝

⎞ ⎟ ⎟ ⎠

0

I yy + I zz 3A ⎞ ⎟ ⎟ ⎠

0

0

⎛ 13 ry2 − le ⎜ − ⎜ 420 10 le2 ⎝

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

0

0

0

0

0

9 6r 2 − z 70 5le2

0

0

0

0

0

0

9 6r 2 − z 70 5 le2

0

0

0

0

0

0

0

0

⎞ ⎟ ⎟ ⎠

⎛ 1 2 ry2 l2 ⎜ + ⎜ 105 15l 2 ⎝

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

⎛ 13 ry2 le ⎜ − ⎜ 420 10 l 2 e ⎝

⎞ ⎟ ⎟ ⎠

3A ⎞ ⎟ ⎟ ⎠

0

⎛ 1 ry2 − le2 ⎜ + ⎜ 140 30 le3 ⎝

0

0

0

0

0

0

0

13 6 rz2 + 65 5le2

0

0

0

0

0

0

13 6 rz2 + 65 5 le2

0

0

0

0

0

0

0

0

⎞ ⎟ ⎟ ⎠

⎛ 1 ry2 − le2 ⎜ + ⎜ 140 30 le3 ⎝

⎞ ⎟ ⎟ ⎠

- 19 -

0

⎛ 11 r2 − le ⎜ + z ⎜ 210 1 0 l 2 e ⎝

⎛ 11 r2 le ⎜ + z ⎜ 210 10 l 2 e ⎝ ⎞ ⎟ ⎟ ⎠

0

I yy + I zz 3A ⎞ ⎟ ⎟ ⎠

0

0

⎛ 11 r2 le ⎜ + z ⎜ 210 10 l 2 e ⎝

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

0 ⎛ 1 2 ry2 le2 ⎜ + ⎜ 105 15le2 ⎝ 0

⎤ ⎥ ⎥ 2 ⎞⎥ ⎛ 13 rz ⎟⎥ − le ⎜ − ⎜ 420 10 l 2 ⎟ ⎥ e ⎠ ⎝ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 2 ⎞⎥ ⎛ 1 r y ⎟⎥ + − le2 ⎜ ⎜ 140 30 l 3 ⎟ ⎥ e ⎠⎥ ⎝ ⎥ ⎥ 0 ⎥ ⎥ 2 ⎛ 11 rz ⎞ ⎥ ⎜ ⎟ − le + ⎥ ⎜ 210 10 l 2 ⎟ ⎥ e ⎠ ⎝ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 2 ⎞ ⎥ ⎛ 1 2 rz ⎟ ⎥ + le2 ⎜ ⎜ 105 15l 2 ⎟ ⎥ e ⎠ ⎥⎦ ⎝ ⎥ ⎥ 0

0

0

⎛ 13 ry2 l⎜ − ⎜ 420 10 l 2 ⎝ ⎞ ⎟ ⎟ ⎠

0

⎛ 13 ry2 − le ⎜ − ⎜ 420 10 le2 ⎝

I yy + I zz

⎛ 13 r2 − z le ⎜ ⎜ 420 10 l 2 e ⎝

1 3

0

0 ⎛ 1 ry2 − le2 ⎜ + ⎜ 140 30 le3 ⎝

0

⎛ 11 r2 + z l⎜ ⎜ 210 10 l 2 ⎝

0 ⎛ 1 2 ry2 + le2 ⎜ ⎜ 105 15le2 ⎝

1 6

0

⎞ ⎟ ⎟ ⎠

Ph.D. dissertation of Fuzhou University

The overall mass and stiffness matrices are generally sparsely populated and the degree to which the matrices are banded can often be significantly affected by the arrangement and ordering of the degree of freedom in the assembled system of equations. Matlab includes a number of very efficient sparse matrix routines that can be quite effective in dealing with the kinds of global stiffness matrices generated by FE problems. In this work, Matlab sparse matrices are used to store global mass and stiffness matrices to improve efficiency and capacity of FE program.

2.4 Governing Equation and Solution The equation of motion for a structural system which ha N DOFs, without considering damping is given as:

[ M ]{u} + [ K ]{u} = {F ( t )}

(2.10)

where M and K are global mass and stiffness matrices respectively and F ( t ) is the forcing system. The generalized form of the eigenproblem can be written in the form:

([ K ] − ω [ M ]){φ} = {0} 2

(2.11)

The mass normalized mode shapes satisfy the orthogonality conditions with respect to mass and stiffness matrices as defined in the following expressions:

[Φ ] [ M ][Φ ] = [ I ] T [Φ ] [ K ][Φ ] = [ Λ ] T

(2.12a) (2.12b)

where Λ = ω N2 is the matrix of eigenvalues. The solution of Equation (2.11) is the most important part of the modal analysis toolbox development. The generally applied eigenvalue solution techniques are either iterative or based upon the repeated application of similarity transformations. The former can be considered as techniques for locating the roots of the characteristic polynomial, and include power methods, inverse iteration and shifting techniques. The latter involves the use of Jacobi, Householder and QR approached to obtain tridiagonal eigenvalues. The method of Lanczos [70] and the subspace iteration technique [71] are well suited to the solution of large scale FE eigenproblem. In this work, the method of Lanczos is implemented for the eigensolution. In addition, the special function to solve the eignevlaue problem provided in Matlab can also be used as the - 20 -

Finite element model updating of civil engineering structures under operational conditions

alternative option. The Lanczos method uses an orthogonal triangular decomposition (QR) type approach to generate a sequence of tridiagonal matrices with the property that the extremal eigenvalues provide a progressively better estimate of the extremal eigenvalues of the original problem. A key feature of the method is that the banded form of the equation is preserved. Furthermore, the triple diagonal matrix need not be formed completely, and the eigenvalues of this converge to the extremal eigenvalues as more of the triple diagonal is formed. The equations which are applied sequentially to produce the tridiagonal matrices have been given in the following form by Bathe [69]. Beginning with a mass normalized trial vector y1 , a sequence of vectors yk , k = 2,3... is calculated according to the steps shown in Equation (2.13). At the k-th step, the tridiagonal matrix is as shown in Equation (2.14) and the matrix Y = [ y1 , y2 ,...., yk ] satisfies the relationship as shown in Equation (2.15). K yk = M yk −1

α k −1 = yk M yk −1 . yk = yk − α k −1 yk −1 − β k −1 yk − 2 , β1 = 0

(2.13)

β k = ykT M yk yk =

yk

βk

⎡ α1 β 2 ⎢β α β 2 3 ⎢ 2 ⎢ β3 α 3 Tr = ⎢ ⎢ ⎢ ⎢ ⎢⎣

α k −1 βk

Y T ( MK −1M ) Y = Tr

φ j = Yφ j

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ βk ⎥ ⎥ α k ⎥⎦

(2.14)

(2.15) (2.16)

Thus, the eigenvectors φ j of Tr are related to those of the structural eigenproblem Equation (2.11) by the linear transformation as shown in Equation (2.16) and by combining Equations (2.11),(2.15) and (2.16) it can be shown that the eigenvalues of Tr ( k = n ) are the reciprocals of λ j , j = 1,...., n ,where λ j is the eigenvalue. The analysis above relates strictly to the case when the stiffness matrix is positive definite. - 21 -

Ph.D. dissertation of Fuzhou University

2.5 Element Types The program is arranged in such a way that new elements can be added in the future. All the elements use the consistent mass matrix. The following elements are included in current MBMAT to generate the FE model of the structure. (a) Spring A spring is a simple one-dimensional element, only capable of extensional modes. Spring elements use only one DOF at each node. •

A spring element requires one constant: c1 = spring stiffness

•

A spring element requires two nodes at the ends of the element.

(b) Truss2d Truss2d elements are two dimensional elements which should lie on an X-Y plane. Truss2d elements have two translational DOFs at each node. Hence, a truss2d element has four DOFs. •

A truss2d element requires three constants: c1 = Cross-sectional area c 2 = Young's modulus of elasticity c3 = Mass density

•

A truss2d element requires two nodes at the ends of the element.

(c) Truss3d Truss3d elements are three dimensional truss elements. Truss3d elements have three translational DOFs at each node. Hence, truss3d elements have six DOFs. The mass and stiffness matrices for the truss3d element in local coordinate system are shown in page 18. •

A truss3d element requires three constants: c1 = Cross-sectional area c 2 = Young's modulus of elasticity c3 = Mass density

•

A truss3d element requires two nodes at the ends of the element.

(d) Beam2d Beam2d elements are two dimensional elements which should lie on an X-Y plane. Beam2d elements are capable of bending and extensional modes. The DOFs used by beam2d elements are X and Y translation and Z-rotation at each node. Hence a beam2d - 22 -

Finite element model updating of civil engineering structures under operational conditions

element has six DOFs. •

A beam2d element requires four constants: c1 = Cross-sectional area c 2 = Young's modulus of elasticity c3 = Moment of Inertia c 4 = Mass density

•

A beam2d element requires two nodes located at the ends of the beam.

(e) Beam3d Beam3d elements are three dimensional beam elements and are capable of bending in two axes, extensional modes and torsional (twisting) modes. Beam3d elements use all six DOFs at each node, and hence have 12 DOFs overall. The stiffness and mass matrix of the beam3d element in local coordinate system is shown in page 18 and 19 respectively. •

A beam3d element requires four constants: c1 = Cross-sectional area c 2 = Young's modulus of elasticity c3 = Moment of Inertia ( I ZZ ) c 4 = Mass density

•

The following four constants are optional c5 = axial angle, which defaults to zero c6 = shear modulus, which is calculated assuming Poisson's ratio 0.3. c7 = moment of inertia ( I YY ), which is assumed equal to ( I ZZ ). c8 = moment of inertia ( I xx ) which is assumed equal to ( I YY + I ZZ ).

•

A beam3d element requires two nodes at the ends of the beam.

(f) Mass2d It is a simple two-dimensional point mass and has up to three DOFs, namely X and Y translation and Z-rotation. •

A mass2d element may have between one and three constants which define the mass in the X and Y directions and the rotational mass as follows: Constants c1 , c 2 , c3

•

X-mass = c1 , Y-mass = c3 , Z-mass = c 2

A mass2d element requires only one node.

(g) Mass3d It is a simple three-dimensional point mass and has up to six DOFs, namely X,Y,Z translations and rotations. - 23 -

Ph.D. dissertation of Fuzhou University

•

A mass3d element may have between one and six constants which define the mass as follows: Constants c1 - c6

•

X= c1 , Y= c 2 , Z= c3 , Xrot= c 4 , Yrot= c5 , Zrot= c6

A mass3d element requires only one node.

2.6 Program Realization in Matlab The Matlab programming language allows one to code numerical methods faster and has a vast predefined mathematical library. The matrix, vector and many linear algebra tools are already defined and the developer can focus entirely on the implementation of the algorithm not defining these data structures. The extensive mathematics and graphics functions further free the developer from the drudgery of developing these functions themselves or finding equivalent pre-existing libraries. Most of these Matlab functions are state-of-the-art and highly efficient. A simple two dimensional FE program in Matlab need only be a few hundred lines of code whereas in FORTRAN or C++ one might need a few thousand. Hence, Matlab environment is chosen for this work. MBMAT is the short form of Matlab Based FE Modal Analysis Toolbox. The natural frequencies and mode shapes of the model are calculated and the mode shapes can be animated. Like any general FEM program, MBMAT consists of three main phases. (a) Preprocessing The input file containing the initial text description of the FE model in specified format is read by the function, which translated it into matrix of number and string. These matrices are the reflection of the all inputs by the user to generate and model the FE model. Hence, these matrices consist of nodal information, element type information, connectivity, material properties, constraints, DOFs information etc. Then, the other function checks the possible duplicates of DOFs and other relevant information. Similarly, the node matrix and element matrix is put in order. At last, the element mass and stiffness matrices are derived from the information of nodes, elements, material constants and element types. The size of these matrices are 24 by (24 × number of elements). Each elemental matrix is padded with zeros (if necessary) to make it into a 24 by 24 matrix. The mass and stiffness matrices for every element are joined together to form elemental mass and stiffness matrix. There is other connectivity matrix of size number of elements × 24 which gives details of how to construct the global system matrices from the elemental matrices.

- 24 -

Finite element model updating of civil engineering structures under operational conditions

(b) Modal analysis In this phase, the function generates the full system matrices [ K ] and [ M ] from the information of elemental matrices and connectivity matrices using the sparse function of Matlab. Similarly, other matrix is also formed that describes which DOF corresponds to each row of the global system matrices. Then, the constraints are applied for the system matrices [ K ] and [ M ] from the information of constraint list matrix. The eigenvalues and eigenvectors are found by performing an eigensolution on the [ K ] and [ M ] matrices. There are two options for the eigensolution, namely the Lanczos method and the build in command of the Matlab. For comparatively large scale model, the first option is suggested. Then, the modes are sorted and normalized with respect to the mass matrix to get the mass normalized mode shape. (c) Post processing In this phase, the frequencies and mode shape are given in ascending order. Then the mode shapes can be plotted and animated.

2.7 Program Verification 2.7.1 Time Period of Simply Supported Beam The MBMAT eigenvalue computations are verified using vibrations of a simply supported beam. This example uses 6 m long simply supported concrete beam as shown in Figure 2.4. The material and sectional properties are shown along with the figure in consistent unit. The first five bending eigenmodes for the model are compared with the independent solution provided in Clough and Penzien [72] and Paz [73].

L=6m

h=0.2m b=0.25m

Material properties

Typical sectional properties

E=3.2 E + 10 N / m 2

A = 0.05m 2

Density = 2500 kg / m3

I = 1.6666 E − 04m 4

Figure 2.4: Simply supported beam to verify MBMAT

- 25 -

Ph.D. dissertation of Fuzhou University

The simulated simply supported beam is equally divided into 15 two dimensional beam elements yielding 48 DOFs of which three are grounded. The input file is prepared and FE model is developed. The modal analysis is then performed. The time period obtained corresponding to first five bending modes are shown in Table 2.1. The first five mode shapes obtained from MBMAT are shown in the Figures 2.5 and 2.6 in which the former shows the mode shapes as seen in the output window of MBMAT.

Figure 2.5: First bending mode of simply supported beam as seen in the output window of MBMAT

The independent results are calculated based on formulas presented in page 380 of Clough and Penzien [72] and page 422 of Paz [73] for a simply supported beam with uniformly distributed mass and constant bending stiffness EI . The expression for the angular frequency of the beam is given by the Equation (2.17).

ωn =

n2 × π 2 L2

EI m

(2.17)

where n =1,2,3… for first mode, second mode etc and m is the mass per unit length. The time period corresponding to first five modes are calculated using the Equation (2.17) and are presented in Table 2.1. - 26 -

Finite element model updating of civil engineering structures under operational conditions

Second mode

Third mode

Fourth mode

Fifth mode

Figure 2.6: Bending modes of simply supported beam obtained from MBMAT

It is clearly seen from the Table 2.1 that the MBMAT results show an acceptable match with the independent solution with the error less than 1.04% for all the five modes considered. Table 2.1: Comparison of time period (sec) with Clough and Penzien and Mario Paz Mode 1 2 3 4 5

MBMAT 0.111225 0.027843 0.012402 0.006996 0.004493

Independent 0.111175 0.027793 0.012352 0.006948 0.004447

Difference (%) 0.0449 0.1799 0.4047 0.6908 1.0344

2.7.2 Plane Frame – Bathe and Wilson Eigenvalue Problem A ten-bay, nine-story, two-dimensional, fixed base frame structure solved in Bathe and Wilson [74] is analyzed for the first three eigenvalues. The MBMAT results are compared with independent results presented in Bathe and Wilson [74] as well as independent results presented in Peterson [75]. The material and section properties and the mass per unit length used for all members, shown in Figure 2.7 are consistent with those used in the two above mentioned references. For the considered frame, the input file is prepared and FE model is developed. The modal analysis is then performed. The eigenvalues to first three bending modes are presented in Table 2.2.

- 27 -

9 @ 10' = 90'

Ph.D. dissertation of Fuzhou University

10 @ 20' = 200' Material properties

Typical section properties

E=432000 k ft 2

A = 3 ft 2

Mass per unit length = 3k − sec ft 2

I = 1 ft 4

Figure 2.7: Nine storey, ten bay plane frame to verify MBMAT

Figure 2.8: First bending mode of frame as seen in the output window of MBMAT

- 28 -

Finite element model updating of civil engineering structures under operational conditions

Second mode shape

Third mode shape

Figure 2.9: Bending modes of plane frame obtained from MBMAT

The first three mode shapes obtained from MBMAT are shown in the Figures 2.8 and 2.9 in which the former shows the mode shapes as seen in the output window of MBMAT. The independent results are taken from the work of Bathe and Wilson 1972 [74] and Peterson 1981 [75]. The comparison is shown in Table 2.2. It is clearly seen from the table that the MBMAT results show an acceptable match with the independent solution with the error less than 1.14% for the eigenvalus of all the three modes considered. Table 2.2: Comparison of eigenvalues with Bathe and Wilson 1972 and Peterson 1981 Mode

MBMAT

1 2 3

0.589814 5.550460 16.775700

Bathe and Wilson 1972 Independent Difference (%) 0.589541 0.0463 5.526950 0.4253 16.587800 1.1327

Peterson 1981 Independent Difference (%) 0.589541 0.0463 5.526960 0.4251 16.587900 1.1321

2.8 Chapter Conclusions A simple toolbox is developed in Matlab environment for analytical modal analysis of engineering structures. The sparse function of Matlab is used to deal with the kinds of global stiffness and mass matrices generated by finite element problems to improve efficiency and capacity of FE program. Two options, namely Lanczos method and function provided by Matlab for the eigensolution are provided. The input file is first created in some specified format, and the program will read and carry out modal analysis with frequency and mode shape as output. The program realization process is explained and the types of finite elements included are discussed. At last, two well known examples are solved using the program MBMAT to demonstrate the accuracy of the program. It is observed that MBMAT results show an acceptable match with the independent solution reported in the literatures. - 29 -

Ph.D. dissertation of Fuzhou University

CHAPTER 3 FINITE ELEMENT MODEL UPDATING IN STRUCTURAL DYNAMICS

CHAPTER SUMMARY

This chapter deals with different aspects and techniques needed to carry out FE model updating in structural dynamics. The role of modeling, testing and system identification is first explored. Various available techniques for correlating analytical and experimental data and expanding experimental mode shapes for successful FE model updating are investigated. Two new methods for modal expansion are proposed and their effectiveness is demonstrated with the help of simulated case study. At last, three important issues of model updating are explained.

3.1 Finite Element Modeling, Modal Testing and System Identification for Model Updating 3.1.1 Finite Element Modeling Models are mathematical representations which provide a means for predicting the response characteristics of a structure without actually building it and subjecting the structure to the maximum loads or disturbances it is being designed to withstand. In most cases of practical interest, the model takes the form of a FE model. In a FE model, the physical continuous domain of a complex structure is discretized into small components called finite elements, a term first used by Clough [76] in 1960. The FE method is extensively used in research and industrial applications as it can produce a good representation of a true structure. However, the prediction from FE method is not always accurate. Inaccuracies and errors in an FE model may arise due to: •

Inaccurate estimation of the physical properties of the structure.

•

Discretisation errors of distributed parameters due to faulty assumptions in individual element shape functions and/or a poor quality mesh.

•

Poor approximation of boundary conditions.

- 30 -

Finite element model updating of civil engineering structures under operational conditions

•

Approximation or omission of damping representation, or assumption of proportional damping.

•

Inadequate modeling of joints.

•

Introduction of additional inaccuracies during the solution phase such as the reduction of large models to a smaller size. In reality, structures always differ in some way from the idealizations assumed when

modeling them. The material and geometric properties may vary or be uncertain and there may be nonlinearities, damping mechanisms, and coupling effects that are not taken into account in the model. In most cases, little confidence can be placed in the model until it can be validated from some form of testing of the structure.

3.1.2 Modal Testing and System Identification Testing is performed to increase the knowledge and understanding of the behavior of a structure. This is accomplished by observing the response of a structure to a set of known conditions. Currently, the most popular dynamic testing technique is modal testing or experimental modal analysis [77, 78]. Experimental modal analysis is used to obtain an experimental model of a structure which describes its dynamic behavior through a set of natural frequencies, modes shapes, and damping ratios. This information is obtained from a modal test of the structure during which the structure is excited and the responses of the structure are captured by a set of sensors. For the experimental modal analysis of structures, there are three main types of dynamic tests: (1) forced vibration tests (2) free vibration tests, and (3) ambient vibration tests. In the first method, the structure is excited by artificial means and correlated input-output measurements are performed. Impulse hammers, drop weights and electro-dynamic shakers are the main excitation equipments. The successes of forced vibration tests are limited for relatively small structures. In case of large and flexible bridges like cable-stayed or suspension bridges, it often requires heavy equipments and involves important resources to provide a controlled excitation at enough high levels [79], which becomes difficult and costly. Free vibration tests can be done by a sudden release of a heavy load or mass appropriately connected to the bridge deck [80]. Both forced and free vibration tests, however, need the artificial means to excite the bridges and the traffic has to be shut down. This could be a serious problem for intensively used bridges. During the past few years, operational modal testing proved to be a valuable alternative for the use of classic forced vibration testing. Instead of using one or more artificial excitation devices, in-operation modal testing makes use of the freely available ambient - 31 -

Ph.D. dissertation of Fuzhou University

excitation caused by natural excitation sources on or near the test structure. Especially in the case of civil engineering structures, the latter can be considered as an important advantage, since the use of artificial excitation devices (large shakers, drop weights) can be considered expensive and impractical. Another advantage is that the test structure remains in its operating condition during the test, which can differ significantly from laboratory conditions. Compared with traditional forced vibration testing, the ambient vibration testing using natural or environmental vibrations induced by traffic, winds and pedestrians is more challenging to the dynamic testing of bridges. It corresponds to the real operating condition of the bridge. However, relatively long records of response measurements are required and the signal levels are considerably low in ambient vibration testing. The experimental modal analysis by ambient vibrations was successfully applied to many structures, like the Golden Gate Bridge [81], the Roebling Suspension Bridge [82], Tennessee River Arch Bridge [83], CFT Arch Bridge [84], temple structures in Nepal [85] and Qingzhou cable-stayed bridge [86]. Varieties of methods exist to obtain modal parameters from these measurements and are generally classified as either time domain or frequency domain methods [77,78,87-89]. There are several ambient vibration system identification techniques developed by different investigators for different uses like single-degree-of-freedom identification method [90], peak-picking from the power spectral densities [91], auto regressive-moving average (ARMA) model based on discrete-time data [92], natural excitation technique (NExT) [93], and stochastic subspace identification [94,95]. The stochastic subspace identification (SSI) method is probably the most advanced operational modal parameter identification technique up to date. SSI is a time domain method that directly works with time data, without the need to convert them to correlations or spectra. Vibration measurements are taken directly from a physical structure, without any assumptions about the structure, and as such they are considered to be more reliable than their FE counterparts. However, limitations and errors in the experimental approach can occur due to: •

The maximum number of measurement locations is limited and the size of the experimental model is always less than that of the analytical model.

•

In general, it is not possible to measure some degrees of freedom, such as rotational and internal ones.

•

The number of identified modes is limited by the frequency range.

•

Measured data are contaminated by a certain level of noise.

- 32 -

Finite element model updating of civil engineering structures under operational conditions

•

Some modes of the structure may not be excited during the test or, even if excited, some modes may not be identified. • Estimate geometric & material properties • Simplifying assumptions • Boundary conditions • Judgment of modeler

• • • • •

Transducer selection Selection of measurement points Environmental effects Available resources Judgment of test engineer

Testing Analytical modeling process

• Parameter error • Model form error • Discretization error

Raw data FE model

[ K ][ M ]

System identification

Experimental model

[ Λ ] [Φ ]

• Random errors • Systematic errors • Modal and spatial incompleteness

Model updating process

• Model reduction/eigenvector expansion • Choose error residual • Choose Updating parameters • Choose optimization algorithm Validation/Correlation

Refined model and updated parameters ⎡ K new ⎤ ⎡ M new ⎤ ⎣ ⎦⎣ ⎦

Figure 3.1: Relationship between FE modeling, testing and system identification for FE model updating

Taking the modeling and testing uncertainties into account and developing a refined model that offers good predictions under conditions of interest, is the primary challenge associated with the modeling, testing and system identification process. In this research, it is assumed that the experimental data is accurate and the FE model is modified or updated - 33 -

Ph.D. dissertation of Fuzhou University

to better represent the experimental results. The traditional relationship between modeling, testing, system identification and model updating is illustrated in Figure 3.1. The process of using information from an experimental model to refine an analytical model is commonly referred to as the model updating or test/analysis correlation problem. This part of the process is the subject of this dissertation.

3.2 Techniques for Comparison and Correlation for Model Updating Correlation can be defined as the initial step to assess the quality of the analytical model. Test data are considered to be more accurate and thus used as reference to assess the quality of the available FE model. Before updating an analytical model, it is a common practice to compare the experimental and analytical data sets to obtain some insight as to whether both sets are in reasonable agreement so that updating is at all possible. The correlation methods form a set of techniques to compare the analytical modal data with the experimental modal data. This section gives an overview of the most often used correlation techniques [5, 77, 96-100].

3.2.1 Direct Natural Frequency Correlation The most common and simplest approach to correlate two modal models is the direct comparison of the natural frequencies. If a plot of the experimental values against analytical ones lies on a straight line of slope 1, the data are perfectly correlated. A percentage difference can be defined as shown in Equation (3.1) and an overall frequency scatter indicator may be used as presented in Equation (3.2). ∈f j =

frej − fraj frej

× 100

⎡ mf ⎤ ⎢ ∑ ( frej − fraj ) ⎥ j =1 ⎥ × 100 ∈f = ⎢ mf ⎢ ⎥ frej2 ⎢ ⎥ ∑ ⎢⎣ ⎥⎦ j =1

where frej and fraj are the experimental and analytical frequencies of

(3.1)

(3.2)

j -th mode

respectively and m f is the number of measured frequencies.

3.2.2 Visual Comparison of Mode Shapes Visual comparison between two sets of modal data includes a process, which involves - 34 -

Finite element model updating of civil engineering structures under operational conditions

the analyst’s non-quantitative visual assessment of any kind of graphically presented data. It usually consists of simultaneous animation of one mode shape from each of the two sets and direct comparison of their natural frequencies. This method basically consists of a visual comparison of the patterns of two different mode shapes and a non-quantitative analyst’s assessment of differences or similarities between two mode shape patterns. A problem arises when one experimental mode appears to match two or more theoretical modes. Although this can happen for several reasons, a more detailed inspection is necessary in order to identify the correlated mode pairs. Mostly, visual comparisons of mode shapes are followed by numerical comparison techniques which are easy to implement in automatic correlations.

3.2.3 Direct Mode Shape Correlation Mode shapes can also be compared by plotting the analytical ones against experimental ones. As before, for a perfect correlation, the resulting curve should lie on a straight line of slope one. The slope of the best straight line through the data points of two correlated mode can be defined as the modal scale factor (MSF) proposed by Allemang & Brown [96]:

φajT φej MSFj ( γ j ) = T φej φej

(3.3)

in which, φaj and φej are the analytical and experimental mode shapes respectively. MSF also provides a means of normalizing all estimates of the same modal vector. Since the mass distribution of the FE model and that of the actual structure may be different, the experimental and analytical mode shapes should be scaled correctly. When two modal vectors are scaled similarly, elements of each vector can be averaged, differenced, or sorted to provide a better estimate of the modal vector or to provide an indication of the type of error vector superimposed on the modal vector.

3.2.4 Modal Assurance Criterion Mode pairing is one of the most critical tasks, when the updating is based on modal data. The matching of modes can be a very difficult task especially for structures with high modal densities. The modal assurance criterion (MAC), defined by Allemang & Brown [96] is often used in automatic pairing and comparing analytical and experimental mode shapes. It is easy to apply and does not require mass and stiffness matrices. MAC is defined by:

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Ph.D. dissertation of Fuzhou University

MAC j =

(φ

T

aj

(φ

aj

T

φej )

2

(3.4)

φaj ) (φej T φej )

where φaj is the analytical eigenmode that has been paired with the j -th experimental mode φej . The value of the MAC is bound between 0 and 1. A value of 1 means a perfect correlation. A MAC value equal to 0 indicates that the two modes do not show any correlations. The experimental and analytical mode shapes must contain the same number of elements, although their scaling does not have to be the same. If the transducers are placed at the nodes of the FE model, then the application of the MAC merely requires choosing the elements in the full analytical mode shapes that correspond to the measurement locations. Usually all the analytical modes are correlated with all the measured modes and the results are placed in a matrix. If the mode pairs are in numerical order then the diagonal should show high MAC values (> 0.9) for a good correlation and value less than 0.05 for uncorrelated modes [77]. ⎡ 1 ∈φ = ⎢1 − ⎢ ms ⎣

ms

∑ ( MAC ) j =1

⎤ ⎥ × 100 ⎥ j ⎦

2

(3.5)

An overall mode shape error indicator may be calculated from Equation (3.5), in which ms is the number of measured mode shapes in the frequency range of interest.

3.2.5 Normalized Modal Difference Normalized Modal Difference (NMD) [101] is the more discriminating comparison technique between the mode shapes obtained from experimental and analytical modal analysis. NMD is proposed in quantifying the accuracy of modal data without the use of

FE system matrices. The NMD between experimental {φej } and analytical {φaj } mode shape is defined as:

(

)

NMD j {φaj } , {φej } =

{φ } − γ {φ } γ {φ } aj

j

j

ej

ej

2

(3.6)

2

where Modal Scale Factor ( γ j ) is given in Equation (3.3) and

2

is the l2 norm of a

vector defined in appendix A. The NMD is closely related to the MAC by the following formula:

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Finite element model updating of civil engineering structures under operational conditions

1 − MAC j

NMD j =

MAC j

(3.7)

In practice, the NMD is a close estimate of the average difference between the components of both vectors φaj and φej

and is much more sensitive to mode shape

differences than the MAC [77]. Hence, NMD can also be used as an alternative correlation criterion, but since the NMD is not bounded by unity, the comparison becomes more difficult for weakly correlated modes.

3.2.6 Coordinate Modal Assurance Criterion The coordinate modal assurance criterion (COMAC) was developed by Lieven and Ewins [100] from the original MAC concept, in such a way that the correlation is now related to the degrees of freedom of the structure rather than to mode numbers. Having first constructed the set of ms mode pairs via MAC, COMAC calculates the amount of correlation at each coordinate over all correlated mode pairs as: ms

COMAC j =

∑ (φ ) (φ )

2

*

a

r =1 ms

e

jr

jr

ms

∑ (φ ) ∑ (φ ) r =1

2

a

jr

r =1

e

(3.8) 2 jr

where * indicates the complex conjugate of element. With values ranging from 0 to 1, low values of COMAC indicate very little correlation between the modes and high values indicate very good correlation.

3.2.7 Orthogonality Methods The self compatibility of a set of measured vibration modes is usually checked by the mass orthogonality defined by Targoff [102] as shown in Equation (3.9).

[∈ortho ] = ⎡⎣ΦTe ⎤⎦ [ M a ][Φ e ]

(3.9)

in which φe is the experimental mode shape and M a is the analytical mass matrix. A commonly-accepted goal is to keep the off-diagonal terms of [∈] to 0.1 or less and to have diagonal elements greater than 0.9 as reported in the paper of Chu and DeBroy [103]. Since the order of the mass matrix is generally greater than the number of test coordinates, the mass matrix is usually reduced before the mass orthogonality check.

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Ph.D. dissertation of Fuzhou University

3.2.8 Energy Comparison and Force Balance The kinetic and potential energies stored in each mode for both experimental and FE model can be computed using the following expressions [104].

1 T {φ} j [ M ]{φ} j 2 1 T Potential energy = {φ} j [ K ]{φ} j 2 Kinetic energy =

(3.10) (3.11)

The kinetic and potential energies can be compared as: T T 1 1 φej } [ M a ]{φej } − {φaj } [ M a ]{φaj } { 2 2 T T 1 1 ∈p = {φej } [ K a ]{φej } − {φaj } [ K a ]{φaj } 2 2

∈k =

(3.12)

where the index j denotes the mode number. A static force balance for j -th measured and analytical mode shapes is proposed by Wada [105] as:

{F } = [ K ]{φ } {F } = [ K ]{φ } ej

a

ej

aj

a

aj

(3.13)

where high unbalance forces indicate coordinates that need updating. The energy comparison and force balance techniques are not widely used as MAC and COMAC.

3.3 Incompatibility in Measured and Finite Element Data As explained earlier in the text, in most practical cases, the number of coordinates defining the FE model exceeds by far the number of measured coordinates. The lack of measured degrees of freedom can be solved in two ways, either by reducing the FE model to the size of experimental DOFs by choosing the measured degrees of freedom as masters, or by expanding the experimental data to include the unmeasured degrees of freedom in the FE model. The model reduction and expansion methods are briefly described below.

3.3.1 Model Reduction Due to the large size mismatch between the analytical and experimental DOFs, substantial effort has been devoted to the investigation of the effects of model reduction. The most popular technique is the static condensation of Guyan [106]. Other main techniques are dynamic reduction method, improved reduction system (IRS) method, and - 38 -

Finite element model updating of civil engineering structures under operational conditions

system equivalent reduction expansion process (SEREP). In all reduction techniques, there exists a relation between the measured or master ( a ) degrees of freedom and the unmeasured or slave ( o ) degrees of freedom: ⎧{ x }⎫ { xN } = ⎪⎨ a ⎪⎬ = [T ]{ xa } ⎩⎪{ xo }⎭⎪

(3.14)

where { x} = physical displacement, [T ] = transformation matrix and N = total number of FE DOFs. The reduced mass and stiffness matrices are then given by:

[ M a ] = [T ] [ M N ][T ] T [ K a ] = [T ] [ K N ][T ] T

(3.15) (3.16)

Different methods of reduction differ in the way of defining [T ] matrix. The Guyan reduction technique partitioned [ M ] and [ K ] matrices in time domain equation of motion into measured and slave DOFs and neglecting inertia terms as shown in Equation (3.17). Using the lower set of equation in Equation (3.17), one easily gets the Equation (3.18). ⎡[ K aa ] ⎢ ⎣[ K oa ]

[ K ao ]⎤ ⎧⎪{ xa }⎫⎪ = ⎪⎧{F ( t )a }⎪⎫ [ Koo ]⎥⎦ ⎩⎨⎪{ xo }⎭⎪⎬ ⎨⎪{F ( t )o }⎬⎪ ⎩

⎭

{ xs } = − [ Koo ] [ K oa ]{ xa } + [ Koo ] −1

(3.17)

−1

{ F ( t )o }

(3.18)

Assuming that there are no external forces at the slave DOFs, the Guyan reduction transformation matrix can be obtained as: ⎤ [I ] ⎥ −1 ⎢⎣ − [ K oo ] [ K oa ]⎥⎦ ⎡

[T ] = ⎢

(3.19)

Since the inertia terms are neglected, this technique is also called static reduction. The choice of master coordinates is of paramount importance to the success of the reduction and one should refrain from choosing coordinates as masters because they happen to coincide with the measurement coordinates. The reduction techniques have some significant disadvantages, which are given below. •

The measurement points often are not the best points to choose as masters as they are always on the surface of the structure while for dynamic condensation, it is vital to select masters corresponding to large inertia properties.

•

All reduction techniques yield system matrices where the connectivity of the original model is lost and thus the physical representation of the original model disappears. - 39 -

Ph.D. dissertation of Fuzhou University

•

There may not be enough measurement coordinates to be used as masters.

•

The reduction introduces extra inaccuracies since it is only an approximation of the full model. Hence, it should be borne in mind that reduction techniques such as Guyan’s were

formulated in order to be able to obtain the eigensolution of large matrix and not for model updating purposes. Hence it is not surprising to discover that the problem of model updating is further compounded by several additional problems due to model reduction. Instead of reduction technique, modal expansion is the better alternative to use in model updating application.

3.3.2 Mode Shape Expansion In most of the cases, it is necessary to know the measurement at all DOFs of the structure under consideration. The potential cases may include (i) for correlation of test and analysis results (ii) for FE model updating and damage detections using system matrices (iii) to visualize the mode shapes obtained from experimental modal analysis effectively, and (iv) to predict the response at unmeasured DOFs for structural integrity and reliability assessment to dynamic loads. Existing mode shape expansion methods can be divided into four broad categories. The first approach involves the interpolation or extrapolation of the measured DOFs to those of the full model [107]. These methods use the FE model geometry to infer the mode shape at unmeasured locations and are very sensitive to spatial discontinuities and are mainly used for plate-like structures such as aircraft wings [108]. The second approach uses the FE model properties, such as mass and stiffness, to obtain a closed-form solution of the mode shapes at unmeasured DOFs. These methods include the Guyan static expansion [106], which assumes that the inertial forces at the unmeasured DOFs are negligible, and the Kidder dynamic expansion [109] which uses the full dynamic equations to infer the mode shapes at the unmeasured DOFs. The third approach is presented in some literatures [110, 111] and based on the assumption that the measured mode shapes can be expressed as a linear combination of the analytical ones. Another expansion method using the analytical mode shapes and the MAC matrix has been suggested by Lieven and Ewins [112]. The validity and performance of these expansion techniques are highlighted in some literatures [112, 113]. A systematic study of the second and third approach explained above is carried out by Imregun & Ewins [108] to define the validity boundaries of the methods. It is concluded that the quality of the expanded mode shapes are case-dependent. To account for uncertainties in the - 40 -

Finite element model updating of civil engineering structures under operational conditions

measurements and in the prediction, new expansion techniques based on least squares minimization techniques with quadratic inequality constraints (LSQI) are proposed by west et al. [114]. The most commonly used two methods are explained below. 3.3.2.1 Kidder Dynamic Expansion

This method is based on the eigenvalue equation. Partitioning the mass and stiffness matrix from the FE model into measured a and unmeasured o coordinates and substituting the measured natural frequency and mode shape, ⎛ ⎡ Ka a ⎜⎜ ⎢ ⎝ ⎣ Ko a

Kao ⎤ ⎡M aa − ω 2j ⎢ ⎥ Ko o ⎦ ⎣M oa

M a o ⎤ ⎞ ⎧⎪φ a j ⎫⎪ ⎧0 ⎫ ⎟⎨ ⎬ = ⎨ ⎬ M oo ⎥⎦ ⎟⎠ ⎩⎪φo j ⎭⎪ ⎩0 ⎭

(3.20)

where ω j and φ a j represents j -th measured natural frequency and corresponding mode shape at the measured coordinates and φo j represents the estimated mode shape at unmeasured DOFs. The estimates of the unmeasured DOFs may be obtained using the lower or upper part of matrix Equation (3.20) or combination of the two which leads the three different methods. Equation (3.20) can be rewritten as two sets of simultaneous equations as shown in Equations (3.21) and (3.22). In the first method, Equation (3.22) is used which leads Equation (3.23) and in the second method, Equation (3.21) is used which gives Equation (3.24).

( ⎡⎣ K ( ⎡⎣ K

){ } ( ) ⎤⎦ − ω ⎡⎣ M ⎤⎦ ) {φ } + ( ⎡⎣ K ⎤⎦ − ω ⎡⎣ M ⎤⎦ ) {φ } = {0} {φ } = − ( ⎡⎣ K ⎤⎦ − ω ⎡⎣ M ⎤⎦ ) ( ⎡⎣ K ⎤⎦ − ω ⎡⎣ M ⎤⎦ ){φ } {φ } = − ( ⎡⎣ K ⎤⎦ − ω ⎡⎣ M ⎤⎦ ) ( ⎡⎣ K ⎤⎦ − ω ⎡⎣ M ⎤⎦ ){φ } aa

⎤⎦ − ω 2j ⎡⎣ M a a ⎤⎦ φ a j + ⎡⎣ K a o ⎤⎦ − ω 2j ⎡⎣ M a o ⎤⎦ {φo j } = {0}

oa

2 j

aj

oa

2 j

oo

−1

oj

oo

2 j

oo

oj

ao

2 j

ao

oo

oj

(3.21) (3.22)

oa

2 j

oa

aj

(3.23)

aa

2 j

aa

aj

(3.24)

+

where + denotes the pseudo inverse. To use third method, from Equation (3.21) and (3.22) one can define two matrices, A1 and A2 as shown in Equation (3.25). Hence, Equation (3.21) can be written as shown in Equations (3.26) and (3.27). ⎡ ⎡ K a a ⎤ − ω 2j ⎡ M a a ⎤ ⎤ ⎣ ⎦ ⎣ ⎦⎥ A1 = ⎢ ⎢ ⎡ K o a ⎤ − ω 2j ⎡ M o a ⎤ ⎥ ⎦ ⎣ ⎦⎦ ⎣⎣

⎡ ⎡ K a o ⎤ − ω 2j ⎡ M a o ⎤ ⎤ ⎣ ⎦ ⎣ ⎦⎥ A2= ⎢ ⎢ ⎡ K o o ⎤ − ω 2j ⎡ M o o ⎤ ⎥ ⎦ ⎣ ⎦⎦ ⎣⎣

[ A1 ]{φ a j } + [ A2 ]{φo j } = {0} {φo j } = [ A2 ]+ [ A1 ]{φ a j }

(3.25) (3.26) (3.27)

The second method that uses Equation (3.24) involves a pseudo inverse and that may - 41 -

Ph.D. dissertation of Fuzhou University

successfully reproduce the mode shape properties at the unmeasured DOFs when the number of unmeasured DOFs is not greater than the number of measured DOFs. In contrast to the second method, the generalized inverse of rectangular matrix [ A2 ] used in Equation (3.27) of the third method in general, satisfies the relationship [ A2 ] [ A2 ] = [ I ] . +

Hence, the vector {φo j } can be uniquely determined by using Equation (3.27).

3.3.2.2 Modal Expansion Method

In this method, the measured modes are assumed to be a linear combination of the analytical modes and the transformation matrix Tpp is defined by: ⎡ Φ a p ⎤ = ⎡Φ a p ⎤ ⎡Tp p ⎤ ⎦⎣ ⎦ ⎣ ⎦ ⎣

(3.28)

where p is the number of modes considered, Φ a p is the measured mode shape and Φ a p is the analytical mode shape corresponding to measurement DOFs. Applying pseudo inverse to Equation (3.28): +

⎡⎣Tp p ⎤⎦ = ⎡⎣Φ a p ⎤⎦ ⎡Φ a p ⎤ ⎣ ⎦

(3.29)

This transformation matrix is then used to expand the measured mode shape to unmeasured DOFs according to Equation (3.30). ⎡ Φ o p ⎤ = ⎡ Φ o p ⎤ ⎡Tp p ⎤ ⎦⎣ ⎦ ⎣ ⎦ ⎣

(3.30)

Similar method is proposed by Lipkins and Vandeurzen [111]. In this method, the measured modes are assumed to be a linear combination of the analytical modes and transformation coefficient is given by the following relationship: ⎡ ⎡Φ1e ⎤ ⎤ ⎡ ⎡Φ1a ⎤ ⎤ ⎢ ⎣ ⎦ nd × p ⎥ = ⎢ ⎣ ⎦ nd ×l ⎥ T ⎢ ⎡Φ e ⎤ ⎥ ⎢ ⎡Φ a ⎤ ⎥ l× p 2 2 ⎣ ⎦ ⎣ ⎦ ⎥ ⎣⎢ ⎥ ( N − nd ) × p ⎦ ( N − nd ) ×l ⎦ ⎣⎢

(3.31)

in which e and a represent the experimental and analytical quantities respectively, N is the number of DOFs, nd is the number of measured DOFs, p is the number of mode shapes identified, l is the number of mode shapes that are used for expansion, and T is the transformation matrix. As long as n ≥ l , the coefficient T can be obtained in a least square sense as shown in Equation (3.32).

(

T

T = ⎡⎣Φ1a ⎤⎦ ⎡⎣Φ1a ⎤⎦

)

−1

- 42 -

T

⎡⎣Φ1a ⎤⎦ ⎡⎣ Φ1E ⎤⎦

(3.32)

Finite element model updating of civil engineering structures under operational conditions

The mode shapes at the unmeasured DOFs, i.e., ⎡⎣Φ e2 ⎤⎦ , then can be easily computed from Equation (3.31). In general, the number of analytical mode shapes is set equal to the identified mode shapes, i.e., l = p .

3.4 Two Proposed Methods for Mode Shape Expansion In this chapter, two possible ways for mode shape expansion are proposed. The first method minimizes the modal flexibility1 error between the experimental and analytical mode shapes corresponding to the measured DOFs to find the transformation matrix, which can be treated as the least-squares minimization problem. In the second method, normalized modal difference (NMD) is used to calculate transformation matrix using the analytical DOFs corresponding to measured DOFs. This matrix is then used to expand the measured mode shape to unmeasured DOFs. A simulated simply supported beam is used to demonstrate the performance of the methods. These methods are then compared with two most promising existing methods, namely Kidder dynamic expansion and Modal expansion methods. The details of the methods are presented below.

3.4.1 Modal Flexibility Method The modal flexibility is the accumulation of the contribution from all available mode shapes and corresponding natural frequencies. The modal flexibility matrix [G ]n×n is defined as [115]:

[G ] = [ Φ ] ⎡⎣ Λ −1 ⎤⎦ [ Φ ]

T

(3.33)

in which, [ Φ ] is the mass normalized mode shape matrix and Λ is the matrix of eigenvalue. The purpose of the method is to identify the transformation matrix ⎡⎣Tpp ⎤⎦ to minimize the Frobenius Norm2 of difference between experimental and analytical modal flexibility at coordinates corresponding to measured DOFs in least square sense, such that the experimental eigenvalue equals the analytical eigenvalue at each mode considered. Mathematically, the problem can be cast as: min Φ a p Λ −p1p Φ Ta p − (Φ a p Ap p ) Λ −p1p (Φ a p Ap p )T Tpp

F

Such that

Λ

pp

= Λpp

(3.34)

When substituting so called constraint into the objective function, the problem is cast

1

2

The detail discussion of modal flexibility is presented in chapter 4. Frobenius Norm is defined in appendix A. - 43 -

Ph.D. dissertation of Fuzhou University

into unconstrained form which can be easily solved to get the transformation matrix ⎡⎣Tp p ⎤⎦ . This transformation is then used to expand the measured mode shape to unmeasured DOFs according to Equation (3.30).

3.4.2 Normalized Modal Difference Method Physically, the NMD represents the error fraction on average by which each DOF differs between the two modes. So, this error fraction obtained from the measurement and corresponding analytical DOFs can be used to estimate the mode shape at unmeasured DOFs with the help of corresponding analytical mode shape. The NMD between experimental ⎡Φ⎤ ⎣ ⎦ and analytical [ Φ ] mode shapes can be calculated in matrix form similar to those shown in Equation (3.6). Then, Equation (3.35) is simply used to expand the measured mode shape at unmeasured DOFs, in which, C1 = diag (1 − diag ( NMD ) ) . ⎡ Φ o p ⎤ = ⎡Φ o p ⎤ [C1 ] ⎦ ⎣ ⎦ ⎣

(3.35)

3.4.3 Performance Metrics Three performance metrics are defined in this work to see the accuracy of different methods for mode shape expansion. The orthogonality properties of eigenvectors, as inferred in MAC can be used as a performance metric. The MAC between known full eigenvector and expanded counterpart can be calculated using Equation (3.4). The second and third performance metrics to compare the expanded and known exact measured mode shape is given by Equations (3.36) and (3.37):

Error1 (%) =

[Φ ]exact − [Φ ]exp anded [ Φ ]exact

Error2 (%) =

[ Φ ]exact − [Φ ]exp anded [ Φ ]exact

*100

*100

(3.36)

(3.37)

Error1 gives a global appreciation of the error between the two mode shapes, while Error2 is more sensitive with the localized error.

3.4.4 Simulated Case Study This simulated beam is used to demonstrate the performance of the proposed methods. A simulated example has the advantage that the expected answer is known. A standard

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Finite element model updating of civil engineering structures under operational conditions

simulated simply supported beam is shown in Figure 3.2 with its geometrical and material properties. The simulated beam of 6 m length is equally divided into 15 two dimensional beam elements. The density and elastic modulus of the material of the beam are 2500 kg / m3 and 3.2 E + 10 N / m 2 respectively. Similarly area of cross section and moment of inertia of simulated beam are 0.05 m 2 and 1.66 E − 04 m 4 respectively. 1

2

3

4

5

6

7

8

9

10

DAM

DAM

11

12

13

14

15

DAM

L=6m

h=0.2m b=0.25m

Before damage Material properties

Typical section properties

E=3.2 E + 10 N / m 2

A = 0.05 m 2

Density = 2500 kg / m3

I = 1.6666 E − 04 m 4

Damage applied Reduce E of element 3

= 20 %

Reduce E of element 8 = 50 % Reduce E of element 10 = 30 %

Figure 3.2: Standard simulated simply supported beam before and after introducing damage Table 3.1: Comparison of experimental (assumed damage) and initial analytical modal properties of simulated simply supported beam Mode 1 2 3 4 5 6 7 8 9 10

Damaged beam 8.245 34.920 75.080 137.508 209.028 313.581 405.839 547.260 671.483 836.938

Natural frequency (Hz) Undamaged beam 8.990 35.914 80.632 142.930 222.532 319.160 432.532 562.405 708.677 871.146

Error (%) 9.035 2.846 7.394 3.943 6.460 1.779 6.577 2.767 5.539 4.087

MAC % 99.918 99.869 99.216 99.588 97.497 99.528 97.444 99.107 98.424 98.068

Modal analysis is carried out using MBMAT [116] to get the FE frequencies and mode shapes, which are shown in Table 3.1. All mode shapes have been normalized with respect to the analytical mass matrix. To get assumed experimental modal parameters, several damages are introduced as shown in Figure 3.2. The modal analysis is again carried out in this damaged beam to get the assumed experimental modal parameters and is presented in Table 3.1. It is observed that, the maximum error that appeared in frequency is 9.04% and minimum MAC is 97.44%. - 45 -

Ph.D. dissertation of Fuzhou University 1

0.95

MAC

0.9

0.85

0.8

Kidder Dynamic Expansion Modal Expansion Expansion using modal Flexibility Expansion using NMD

0.75

1

2

3

4

5 6 Modes

7

8

9

10

Figure 3.3: MAC values between the actual and expanded mode shapes 15

Error1(%)

10

5

0

Mode 1 to 10 respectively

1

1-Kidder Method

2

3

2-Modal Expansion

3-Using Modal Flexibility

4

4-Using NMD

Figure 3.4: Norm errors for different expansion methods

The vertical DOFs are assumed as measured ones, and hence mode shape vectors of damaged case corresponding to vertical DOFs are used for modal expansion and remaining DOFs are used to check the result of different expansion methods. At first, the expansion is carried out by the proposed modal flexibility method. The fminsearch function of the optimization toolbox of Matlab [67] is used for minimization, that predicted the value of matrix ⎡⎣Tpp ⎤⎦ which is used to obtain expanded mode shapes from remaining DOFs of the analytical model. This function uses the Nelder-Mead simplex algorithm [117, 118] which is one of the most widely used methods for non-linear unconstrained optimization. Similarly, for the NMD method, the modal scale factor (MSF) is calculated and NMD value is predicted between the measured DOFs and corresponding analytical counterparts.

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Finite element model updating of civil engineering structures under operational conditions

Then, mode shape expansion is carried out. To compare the results, the modal expansion is also carried out using Kidder dynamic modal expansion method. For Kidder dynamic method, Equation (3.23) is used, which is the most standard one. The MAC values between the actual and expanded mode shapes for all four methods are plotted in Figure 3.3. It is observed that the performance of Kidder’s method is best except for 5th and 7th mode. The result can also be correlated with the initial MAC value of simulated beam presented in Table 3.1, which is the value obtained between actual and analytical mode shape that is used for modal expansion. For the first four modes, the initial correlation before expansion is good. It is seen that the expanded result of these four modes from all methods are good. The similar trend can be observed for higher modes. 25

Error2(%)

20

15

Mode 1 to 10 respectively

10

5

0

1 2 3 4 1- Kidder Method 2- Modal Expansion 3- Using Modal Flexibility 4- Using NMD

Figure 3.5: Norm of eigenvector differences

The other performance metrics Error1 and Error2 defined in Equations (3.36) and (3.37) are plotted in Figures 3.4 and 3.5 respectively. One must take care with the definition of Error2, which highlights error in small modal displacements. Figures 3.4 and 3.5 clearly show that the performance of modal flexibility method is comparable with those of the existing methods. NMD method also has the potential to expand the mode shapes, although it is seen more sensitive to the distribution of error between FE model and actual test data. It is because, for the well correlated modes with higher value of initial MAC, the error in expanded mode shape is less which can be observed from above figures.

3.5 Three Key Issues of Finite Element Model Updating The FE model updating method considered in this dissertation is the iterative method. One common approach of iterative methods is to consider an objective function that

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Ph.D. dissertation of Fuzhou University

quantifies the differences between analytical and experimental results. It is common to adjust the selected parameters to minimize the objective function, thus it becomes a typical optimization problem. The success of the FE model updating method depends on the accuracy of the FE model, the quality of the modal test, the definition of the optimization problem, and the mathematical capabilities of the optimization algorithm.

ℜ1

FEM up

ℜ2 FEM opt

FEM init

ℜ3

(a)

ℜ1

ℜ2 FEMinit

FEM opt

ℜ3

FEM up

(b) Figure 3.6: Schematic diagram to show the main issues of model updating (a) poor selection of updating parameters (b) poor setting up of objective function

In a model updating process, one requires not only satisfactory correlations between analytical and experimental results but also maintaining physical significance of updated parameters. Thus, setting-up of an objective function, selecting updating parameters and using robust optimization algorithm are three crucial steps in structural FE model updating. They require deep physical insight and usually trial-and error approaches are commonly used. Figure 3.6 highlights their relationship and importance as shown in reference [119]. - 48 -

Finite element model updating of civil engineering structures under operational conditions

•

Region ℜ1 contains all the possible FE models of a structure.

•

Region ℜ2 contains all the FE models that correlate well with experimental results. One of these models, FEM opt gives the best possible description of dynamic behavior of the structure.

•

Region ℜ3 is a set of models that can be derived from the initial FE model, FEM init , by varying the selected updating parameters. Both the initial FE model, FEM init , and the updated model, FEM up , are the members of ℜ3 . The dimension of ℜ3 is determined from the initial model, which is deeply related with

the selection of updating parameters. A bad selection of updating parameters will not result a common space between ℜ3 and ℜ2 (Figure 3.6(a)). As a consequence, the updated model having good correlation with experimental results cannot be obtained even if an objective function is properly set up. Conversely, a very good selection of updating parameters will give FEM opt within the common space of ℜ2 and ℜ3 (Figure 3.6(b)). Then, whether FEM up will converge to FEM opt depends mainly on used optimization algorithm. A poor set-up of an objective function will never allow FEM up to move toward FEM opt .Since FE model updating is basically an inverse process, one can hardly distinguished the causes of poor updated results. These may come from a poor selection of updating parameters or an inappropriate objective function or both. Thus, when updated results are not satisfactory, the model updating process should be solved repeatedly with a modified objective function and with a different set of updating parameters until appropriate results are derived.

3.6 Chapter Conclusions This chapter deals with different aspects and techniques needed to carry out FE model updating in structural dynamics. The role of modeling, testing and system identification for model updating is first explored. Various available techniques for correlating analytical and experimental data and expanding experimental mode shapes for successful FE modal updating are investigated. Two new methods for modal expansion are proposed using modal flexibility and NMD. Their effectiveness is demonstrated by comparing the predicted result with the two existing methods using simulated simply supported beam. It is demonstrated that the modal flexibility method gives good results and NMD method also has the potential to expand the mode shapes although it is seen more sensitive to the distribution of error between FEM and actual test data. At the last of chapter, three important issues of model updating, namely objective function, parameter selection and optimization algorithm are explained.

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Ph.D. dissertation of Fuzhou University

CHAPTER 4 FINITE ELEMENT MODEL UPDATING PROCEDURE

CHAPTER SUMMARY

This chapter deals with the FE model updating procedure carried out in this thesis. The theoretical exposition on FE model updating is presented. Two new residuals, namely modal flexibility and modal strain energy are proposed and formulated to use in FE model updating. Many related issues including the objective functions, the gradients of the objective function, different residuals and their weighting and possible parameters for FE model updating are investigated. The issues of updating parameters selection process adopted in this work are discussed. The ideas of optimization to be used in FE model updating application are explained. The algorithm of Sequential Quadratic Programming (SQP) is explored which will be used to solve the multi-objective optimization problem of chapter 6.

4.1 Theoretical Procedure The general outline of the FE model updating procedure carried out in this thesis is shown in Figure 4.1. Initially, the FE model is developed using the initially estimated values for the unknown model parameters. FE Modal analysis is then carried out to obtain the FE modal data. For the ambient vibration testing of the structure, the optimum points for the placement of sensors are chosen and test data are recorded. Experimental modal analysis is then carried out by using stochastic subspace identification (SSI) technique [120] to get the modal parameters. Before the numerical and experimental modal parameters are compared, they must be paired correctly, i.e., the modal parameters must relate to the same modes. Arranging the eigenfrequencies in ascending order is not sufficient, since due to incorrect parameter estimates, the order of the modes in both models may differ. Furthermore, some modes of the structure may be measured inaccurately due to the placement of an accelerometer close to a node of a particular mode shape, or sometime, the mode shape may even not be excited. The most common and easy way to pair mode shapes correctly is the use of modal assurance criterion (MAC) as defined in Equation (3.4). For an experimental mode, the - 50 -

Finite element model updating of civil engineering structures under operational conditions

corresponding analytical mode is defined as the FE mode that shows the highest MAC value with respect to that experimental mode. Define design parameters

Create FE model, apply boundary conditions

Ambient vibration test planning

Ambient vibration test

Select updating parameters, Sensitivity analysis, Initialization Initial values x0 ; j = 0

FE analysis Computation of analytical modal parameters

zi = z ( x j )

System identification Experimental modal parameters: z

Correlation Automatic pairing of mode shapes using MAC criteria

Evaluation of objective function, Sensitivity matrix and Gradient and weighting factors

f = zj − z

2

Optimization step Updated values x j +1

j = j +1

No

Convergence ?

Yes

Result: Identified updated parameters, x = x j +1 , Frequencies, Mode shapes

Figure 4.1: The general procedure of the FE model updating method

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Ph.D. dissertation of Fuzhou University

No expansion of the experimental mode shapes is needed to calculate the MAC. MAC can be utilized to automatically pair the mode shapes at each iteration. Once the correct mode shape pairing is ensured, in an iterative process, the unknown model parameters are adjusted until the discrepancies between the numerical and experimental modal data are minimized. In this way, the FE model is corrected such that, it better represents the real dynamic characteristic of target structure and at the same time the unknown parameters are identified. The issues related to the objective functions, gradients of objective functions and weighting of different residuals are briefly explained below.

4.1.1 Objective Function The authors, Friswell and Mottershead [4] and Maia and Silva [5] propose the FE model updating procedure by solving a least squares problem including other approaches as presented in [4]. The least squares approach is very efficient and has become the common way to solve the updating problem, as shown in the work of Link [13,21,121, 122], Mottershead et al. [30]. An objective function f reflects the deviation between the analytical prediction and the real behavior of a structure. The FE model updating can be posed as a minimization problem to find x* design set such that:

( )

f x* ≤ f ( x ) ,

xi ≤ xi ≤ xi ,

∀x

(4.1)

i = 1, 2,3,......n

where the upper ( xi ) and lower ( xi ) bounds on the design variables are required. The objective function in an ordinary least squares problem is defined as a sum of squared differences: 2

nr

nr

f ( x ) = ∑ ⎡⎣ z j ( x ) − z j ⎤⎦ = ∑ rj ( x ) j =1

2

(4.2)

j =1

where each z j ( x ) represents an analytical modal quantity which is a nonlinear function of the optimization or design variables x ∈ℜ n and z refers to the measured modal parameters. In order to obtain a unique solution, the number of residuals nr should be greater than the number n of unknown parameters x . The updating parameters are the uncertain physical properties of the numerical model. Instead of the absolute value of each uncertain variable x , its relative variation to the initial value x0 is chosen as dimensionless updating parameter a . By using the normalized parameters a , problems of numerical ill-conditioning due to large relative differences in

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Finite element model updating of civil engineering structures under operational conditions

parameter magnitudes can be avoided. ai = −

x i − x0i x0i

xi = x0i (1 − a i )

(4.3a) (4.3b)

The objective of FE model updating problem is to find the value of vector a i of Equation (4.3) which minimizes the error between the measured and analytical modal parameters. Hence, Equation (4.2) becomes: nr

f ( a ) = ∑ rj ( a )

2

(4.4)

j =1

Equation (4.4) represents the basic least squares function, but several forms exist and are worked out in literatures. For example, in a weighted least squares problem, the residual vector is multiplied with a weighting matrix in order to take into account the relative importance of the different types of residuals and their accuracy. The least squares criterion is solidly grounded in statistics. In Nocedal [123], it is shown that under certain statistical assumptions, the use of the objective function defined as a sum of squared differences minimizes the statistical errors in the identified parameters, evolving from the measurement errors. In general, the residual vector r contains the differences in the identified modal data and some derived quantities, such as the eigenfrequencies, the mode shapes etc. Mode shape expansion and reduction operation is not necessary for calculating the residuals or modal sensitivities which may cause additional inaccuracies. The relative weighting between the different residual types can be controlled by the definition of the residual functions and by an additional weighting matrix in order to account for the measurement and identification errors. Two new residuals, namely modal flexibility and modal strain energy are proposed and formulated below in addition to frequency and mode shapes residuals to use in objective function for FE model updating. 4.1.1.1 Eigenfrequencies

The most important residual vector of the FE model updating in structural dynamics is the differences between the numerical and experimental undamped eigenfrequencies. The eigenfrequency residual r f is formulated as:

rf ( a ) =

λaj − λej λej

j ∈ {1,........., m f }

- 53 -

(4.5)

Ph.D. dissertation of Fuzhou University

with eigenvalue λ j = ( 2 ∗ π ∗ frj ) where frj is the eigenfrequency corresponding to j -th 2

mode. λaj and λej are analytical and corresponding experimental eigenvalue, respectively.

m f refers to the number of identified eigenfrequencies that are used in the updating process. Relative differences are taken in r f in order to obtain a similar weight for each eigenfrequency residual, since higher eigenfrequency gives the higher absolute difference between the analytical and experimental quantity. Eigenfrequencies can be measured and identified more accurately during testing. The eigenfrequencies provide global information of the structure. They are indispensable quantities to be used in the updating process and have a favorable effect on the condition of the optimization problem. But, the higher natural frequencies are not measured as accurately as the lower frequencies. An objective function with only a limited set of eigenfrequencies is a too poor basis for the definition of the dynamic behavior of the structure. Hence, other residuals are also necessary to form the full objective function for FE model updating. 4.1.1.2 Mode Shapes

Mode shapes contain spatial information about the dynamic behavior of the structure. Therefore, the residual vectors with differences in mode shape displacements are other possibility to use in objective function. Since the civil engineering structures are most often measured in operational conditions, the exciting forces come from ambient sources (wind, traffic, etc.) and thus are unknown. As a result, the identified experimental mode shapes cannot be absolutely scaled. Hence scaling and normalization of the mode shapes obtained from ambient vibration testing is an important issue, which is not standard in conventional updating. Friswell and Mottershead [4] propose to scale the measured mode shape to the analytical one by multiplying it with the modal scale factor (MSF) defined in Equation (3.3). This results in the mode shape residual formulation as: rs ( a ) = φaj ( x ) − MSFj × φej ( x )

j ∈ {1,........., ms }

(4.6)

In some work, the mode shape is normalized in one reference node. Since an absolute scaling factor is missing for the experimental mode shapes, the numerical and experimental mode shapes are normalized to 1 in a reference node, which is a node at which the mode shapes have their largest amplitude or at least large amplitude and is chosen for each mode separately [25]. There are many alternatives forms of mode shapes to use in objective function. Gentile et al. [37] uses the normalized modal difference (NMD) to define the mode shape residuals. The mode shape residual formulation applied by Gentile is shown in

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Finite element model updating of civil engineering structures under operational conditions

Equation (4.7). Unlike in the previous approach, the authors do not square the residuals rs when including them in the objective function. rs ( x ) = NMD (φaj ,φej )

j ∈ {1,........., ms }

(4.7)

After trying several expressions, Moller and Friberg [124] proposed the following residual expression related to mode shapes.

rs2

(1 − (a) =

MAC j

)

2

MAC j

j ∈ {1,........., ms }

(4.8)

where MAC is defined in Equation (3.4). In this thesis, this form of mode shape residual is used for FE model updating in single-objective optimization framework presented in chapter 5. The experimental mode shape normalization as explained above has no effect on MAC calculation. So, the MAC value calculated with out normalization can be used in this formulation. As stated above, in addition to the global information from the eigenfrequencies, mode shapes provide spatial information of the structure, which is necessary to uniquely identify local parameters in a structure. But, large numbers of measurement locations are required to accurately characterize the mode shapes. However, due to the spatial information related to them, the mode shapes are desirable and in most cases even indispensable quantities in the updating process, even though they may have an unfavorable effect on the stability of the optimization problem. 4.1.1.3 Modal Flexibility

To introduce the modal flexibility, consider the simple relationship from the structural mechanics:

{u} = [G ]{Fa }

(4.9)

where { Fa } is the vector of applied static loads, and {u} is the vector of resulting static responses. Then matrix [G ] which is the inverse of structural stiffness matrix is called the matrix of static flexibility influence coefficients. By inspection of Equation (4.9), it is seen that the j -th column of [G ] is the displacement pattern observed when a unit load is applied at the j -th structural DOF, i.e., when the j -th entry of { Fa } is 1 and all other entries in { Fa } are zero. Thus, the structure is characterized by N flexibility shapes corresponding to the N columns of

[G ] . It can be shown using Maxwell’s reciprocity - 55 -

Ph.D. dissertation of Fuzhou University

theorem [125] that the flexibility influence coefficient matrix is symmetric for linear systems, such that: Gij = G ji

(4.10)

Thus, the displacement pattern observed at all DOFs due to a unit load at i is the same as the displacements observed only at DOF i as the unit load is applied at each DOF successively. This condition of reciprocity plays an important role in understanding the full meaning of flexibility. A more general motivation for using the flexibility matrix in the structural dynamic applications is that the columns of the flexibility matrix have a very straightforward physical interpretation which is the displacement response due to an applied unit load. The first issue in the computation of the flexibility matrix from identified modal parameters is the estimation of the flexibility matrix from the eigensolution of the system using inverse vibration. Suppose that the undamped free vibration of a structural dynamic system is described by the ( N × N ) second-order differential equation as shown in Equation (2.10). The eigensolution of this system consists of the eigenvalue matrix [ Λ ] , which is a diagonal matrix of the squared natural frequencies diag {ωk2 } and the

eigenvector matrix [ Φ ] , which is mass normalized, i.e., scaled such that Equation (2.12) is

satisfied. Solving the Equation (2.12b), the stiffness matrix can be written in modal form as:

[ K ] = [Φ ] [ Λ ][Φ ] −T

−1

(

= [ Φ ][ Λ ]

−1

[Φ ]

T

)

−1

(4.11)

The flexibility is defined as the inverse of the stiffness matrix as shown in Equation (4.12).

[G ] ≡ [ K ]

−1

(4.12)

Substituting Equation (4.12) into Equation (4.11) yields the inverse vibration representation of the flexibility matrix as:

[G ] ≡ [ Φ ][ Λ ] [ Φ ] −1

T

(4.13)

When the structure has one or more rigid body modes with associated zero frequencies, [G ] is infinite. In this case, the flexible contribution to the flexibility may be defined similarly as:

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Finite element model updating of civil engineering structures under operational conditions −1

⎡⎣G f ⎤⎦ ≡ ⎡⎣Φ f ⎤⎦ ⎡⎣ Λ f ⎤⎦ ⎡⎣Φ f ⎤⎦

T

(4.14)

where ⎡⎣ Λ f ⎤⎦ contains only those eigenvalues corresponding to the flexible modes of the system and ⎡⎣Φ f ⎤⎦ contains the corresponding mass-normalized flexible mode shapes. If all the mode shapes and frequencies are available at all the DOFs, Equation (4.13) gives the modal flexibility matrix. The flexibility matrix can be separated into modal component and residual component. The contribution of the unmeasured vibration modes to flexibility is called the residual flexibility. The eigensolution used to form [G ] in Equation (4.13) is the full eigensolution for the system. In practice, however, only a few lower mode shapes and frequencies are actually measured during vibration testing. Defining the measured modal set as n and unmeasured set as r the eigensolution can be partitioned as:

[G ] = [Gn ] + [Gr ]

(4.15)

where [Gn ] is the modal flexibility, formed from the measured modes and frequencies as:

[Gn ] = [Φ n ][ Λ n ] [Φ n ] −1

T

(4.16)

and [Gr ] is the residual flexibility formed from the residual modes and frequencies as:

[Gr ] = [Φ r ][ Λ r ] [ Φ r ] −1

T

(4.17)

In practice, the measured flexibility matrix is not computed for the full DOF set, because only a limited number of measurements are available. Partitioning the full DOF set into measured m and non measured DOFs and multiplying the partitioned, it is shown in Doebling [126] that:

[Gmm ] = ⎡⎣Φ nm ⎤⎦ [ Λ n ]−1 ⎡⎣Φ nm ⎤⎦

T

−1 + ⎡⎣Φ rm ⎤⎦ [ Λ r ] ⎡⎣Φ rm ⎤⎦

T

(4.18)

where [Gmm ] is called the measured flexibility matrix, Φ nm and Φ rm are respectively the measured and unmeasured mode shapes of the structure at the measured DOFs, and Λ n and Λ r corresponds to the eigenvalues of measured and unmeasured modes. The first and second portion of the Equation (4.18) indicates the modal and residual contribution to measured flexibility respectively. Doebling [126] developed a method to accurately estimate the structural flexibility of the structure considering residual flexibility from forced vibration case, in which the input of the system is known. For the case of known input, there is no problem to compute the - 57 -

Ph.D. dissertation of Fuzhou University

residual flexibility. The issue considered in this work is related to the ambient vibration work, where the input of the system is not measured. For the case of unknown input with measurements carried out only in certain locations, the situation is much more different and the calculation of residual flexibility is not straightforward. In general, the measured modes are typically those that are lower in frequency and therefore contribute the most to the flexibility. Therefore a good estimate of the flexibility matrix may be obtained from only a few low frequency modes corresponding to measured DOFs [63,115,127,128,129]. Hence, Equation (4.18) becomes,

[Gmm ] = ⎡⎣Φ nm ⎤⎦ [ Λ n ]−1 ⎡⎣Φ nm ⎤⎦

T

(4.19)

In other way, Equation (4.19) can also be expressed as: N

[G ] = ∑ φ jφ Tj Λ −j 1

(4.20)

j =1

For the sake of clarity, the measured flexibility matrix [Gmm ] will be referred to simply as ⎡⎣Gexp ⎤⎦ , Φ nm will be referred as Φ , and Λ n will be referred as Λ in the remaining portion of the thesis. Similarly, the notation for matrix [ ] is removed for convenience. Equation (4.19) is used to compute the modal flexibility and used for the model updating procedure in this thesis. The analytical modal flexibility is estimated using the analytical eignevalue and mode shapes which are partitioned corresponding to the measured DOFs. The most important issue to use Equation (4.19) is the mass normalization of mode shapes obtained from ambient vibration test. This issue is elaborated in most relevant chapter 5. Different form of modal flexibility residual can be used for model updating purpose which will be dealt in later chapters. In general, the modal flexibility error residual may be given by the expression: 2 rflex ( a ) = Gexp − Gana

2

(4.21)

where Gexp is the measured modal flexibility matrix obtained at the measurement DOFs,

Gana is the analytical flexibility matrix corresponding to the measured DOFs and a is the updating parameters which is a column matrix. In this thesis, the modal flexibility index is investigated to use in FE model updating and damage detection and whole procedure is developed and presented in chapter 5 and 7 respectively. 4.1.1.4 Modal Strain Energy

The work performed on a structure through deformation is stored as potential energy, - 58 -

Finite element model updating of civil engineering structures under operational conditions

which is called the strain energy. Strain energy is a measure of deformation of the structure by a load. The strain energy is equal to the work done in distorting the system. Thus, strain energy =

1 N 1 Fj u j = F T u ∑ 2 j =1 2

(4.22)

The strain energy stored in any structure may be expressed conveniently in terms of either the flexibility or the stiffness matrix. By substituting Equation (4.9), Equation (4.22) gives the strain energy in terms of flexibility matrix as: strain energy =

1 T F GF 2

(4.23)

Similarly, invoking the relationship between load F and stiffness K of the system: F = Ku

(4.24)

Transposing Equation (4.22) and substituting Equation (4.24) leads to the strain energy expression in terms of stiffness matrix as: 1 strain energy = u T Ku 2

(4.25)

Equations (4.23) and (4.25) give the expression of static strain energy in terms of flexibility and stiffness of the system respectively. Equation (4.25) is convenient and standard expression for strain energy. If the modal displacements are used in Equation (4.25) the corresponding strain energy is called the modal strain energy. Hence, the modal strain energy (MSE) of the j -th mode of the structure can be defined as: 1 MSE j = φ Tj Kφ j 2

(4.26)

where φ is the mode shape vector and K is the global stiffness matrix. Hence, the modal strain energy residual can be cast in the form shown in Equation (4.27). ⎛ φajT Kφaj ⎞ 2 renergy ( x ) = ∑ ⎜ T − 1⎟ ⎜ ⎟ j =1 ⎝ φej K φej ⎠ ms

2

(4.27)

in which φaj and φej are the analytical and experimental mode shapes respectively and the analytical stiffness matrix is used in place of experimental stiffness matrix as an approximation [58,130]. The modal strain energy index is used as new a new residual for FE model updating in - 59 -

Ph.D. dissertation of Fuzhou University

this thesis. The detail methodology is presented in chapter 6. This thesis focuses on the use of conventional eigenfrequency and mode shape residuals as well as new residuals, namely modal flexibility and modal strain energy for FE model updating.

4.1.2 Weighting The least squares problem formulation allows the residuals to be weighted separately according to their importance and accuracy. The weight factors influence the result only in case of an overdetermined set of equations, i.e., when the number of residuals is higher than the number of design variables. Furthermore, only the relative proportion of the weighting factors is important, not their absolute values. The ability to weight the different data sets gives the method its power and versatility, but at the same time requires engineering insight to provide the correct weights. In a weighted least squares problem the following minimization problem is solved: nr

min ∑ ⎡⎣ w j rj ( x ) ⎤⎦

2

(4.28)

j =1

where w j is the square root of weighting factor of residual rj and nr is the number of residuals. As explained earlier, the experimental eigenfrequencies are in general the most accurate experimental data that are available. Experimental mode shapes on the other hand are more noisy. In a typical vibration test, the natural frequencies are obtained to within 1% and the mode shapes to within 10% at best [4]. An appropriate weighting is therefore necessary. The eigenfrequency residuals in Equation (4.5) are already equally weighted by their definition as relative differences. Similar is the case for mode shape residuals as shown in Equation (4.8). It is the general practice to assign more weight for modal parameters corresponding to lower modes due to their more confidence on identification result. It is difficult to state beforehand or in a general way which relative weighting factor should be assigned to the different residuals. Due to the modeling and measurement errors, different results will always be obtained for different weighting factors, hence no unique ideal solution exists. The most likely and realistic result should be selected based on engineering insight. Appropriate weights can be identified in a trail and error basis. If for the obtained result, the eigenfrequencies correspond fully but the mode shapes show a considerable discrepancy, it can be assumed that too much weight is given to the eigenfrequency residuals. On the other hand, if a very non-smooth result is obtained which refers to a too - 60 -

Finite element model updating of civil engineering structures under operational conditions

high influence from the mode shape measurement errors that correspond with eigenfrequencies which deviate much from the experimental eigenfrequencies, the weight for the mode shapes should be decreased. Although it is easy to state what kind of solution is satisfactory considering the correlations of the initial FE model with the experimental data, the importance of individual modal properties, and measurement uncertainties, it is very difficult to identify the weighting factors that would produce satisfactory solutions. There are many residuals which tell the differences between analytical and experimental modal parameters. In general model updating procedure, they are combined into a single objective function using weighting factor for each residual. There are no hard and fast rules for selecting the weighting factors since the relative importance among the criteria is not obvious and specific for each problem. Thus, a necessary approach is to solve the problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained [131]. This kind of FE model updating procedure is investigated in chapter 5. But, due to the uncertainty of weighting coefficients, it usually takes long time to finally obtain satisfactory weights. As an alternative, multi-objective optimization technique is introduced and applied in this thesis using strain energy and eigenfrequencies as two objective of multi-objective optimization technique and is presented in chapter 6.

4.1.3 Gradient of Objective Function The nonlinear optimization problem as shown in Equation (4.4) is solved with a gradient (sensitivity) based iterative optimization method. Therefore, the gradient matrix needs to be calculated in each iteration. The objective function gradient can be calculated with the finite difference approximation. But, some optimization algorithm needs the analytically calculated gradient for robust performance and to treat the ill-conditioning problem. Taking the first derivative of objective function in Equation (4.4) with respect to correction parameter a , nr ∂r ( a ) ∂f ( a ) = 2∑ rj ( a ) j ∂ai ∂ai j =1

(4.29)

In matrix form, this can be expressed as,

∂f ( a ) ∂ai where the matrix

∂rj ( a ) ∂ai

= 2 S ji ( a ) rj ( a ) T

(4.30)

that contains the first-order derivatives of each residual rj ( a ) in

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Ph.D. dissertation of Fuzhou University

the residual vector with respect to each correction parameter ai is called the sensitivity matrix. This matrix can be expressed as: ⎡ ⎢ S ji = ⎢ . ⎢ ⎢ ⎢⎣

. ∂rj ∂ai .

⎤ ⎥ .⎥ ⎥ ⎥ ⎥⎦

⎧⎪ j = 1,......., nr ( no. of residuals considered ) ⎨ ⎪⎩ i = 1,......, n ( no. of updating parameters )

(4.31)

When the residual contains eignvalues and eigenvectors and other modal indices (which are function of eigenvalue and eigenvecotors like modal flexibility, modal strain energy), their first derivatives are needed to calculate the objective function gradient as shown in Equation (4.30). The calculation of eigenvalue and eigenvector derivatives has been extensively studied and reported in many papers. In this study, the expressions derived by Fox and Kapoor [132] are used. Differentiating the generalized eigenvalue problem with respect to the design variables and using the orthogonalization properties of eigenvectors as presented in appendix B, one arrives at: ∂λ j

⎡ ∂K ∂Μ ⎤ = φ Tj ⎢ − λj ⎥φj ∂ai ∂ai ⎦ ⎣ ∂ai

(4.32)

It is seen that Equation (4.32) includes only the eigenvalue and eigenvector under consideration, therefore a complete solution of eigenproblem is not needed to obtain these derivatives. The modal vector derivative may be expressed as a linear combination of all eigenvectors, i.e.,

∂φ j ∂ai

d

= ∑ β jq φq

(4.33)

q =1

where the coefficients β jq are determined using the generalized eigenvalue problem and orthogonalization properties of eigenvectors. Provided that the eigenvectors have been normalized to unit modal masses, as shown in appendix B, one can get

⎧ T ⎡⎛ ∂K ∂Μ ⎞ − λj ⎪φq ⎢⎜ ⎟ ∂a ∂ai ⎠ ⎪ β jq = ⎨ ⎣⎝ i ⎪− 1 φ T ∂Μ φ ⎪ 2 j ∂a j , i ⎩

(λ

j

⎤ − λq ) ⎥ φ j , q ≠ j ⎦

(4.34)

q= j

Because the full eigensystem is not available and far too expensive to solve for, the summation in Equation (4.33) is in practice over number 1) indicate the miss-modeled element. This sort of correction method does not need a close link with the FE model code. They result in, however, the updated matrices that is hard to be interpreted in terms of physically meaningful parameters. It is the main drawback of the method. Hence, in this thesis, the first method using physical parameters and boundary conditions are used as updating parameters for model updating.

4.3 Selection of Updating Parameters In FE model updating, as mentioned above, the unknown physical parameters such as material or geometrical properties or model parameters, e.g. Young’s modulus, moment of inertia, spring stiffness, etc. are adjusted in order to obtain correct system matrices in the updated FE model. The success of FE model updating depends heavily on the selection of updating parameters. The updating parameter selection is basically made with the aim of correcting uncertainties in the model. In the solution of inverse problems, like in FE model updating, the sensitivity matrix is prone to be ill-conditioned. Insensitive parameters should be avoided since they also yield an ill-conditioned matrix. From the viewpoint of parameter identification, it is desirable that small changes in the design variables cause large deviations in the modal data, which means that the residuals are highly sensitive to a change in the design variables. Typically, the number of potential erroneous parameters may be huge. However, in order to ensure a well-conditioned estimation problem, the number of parameters should be relatively small. Only those parts of the model that are actually erroneous should be updated otherwise the updated model will become physically unreasonable. Engineering insight is therefore necessary to determine which parts of the - 64 -

Finite element model updating of civil engineering structures under operational conditions

model and which properties have to be adjusted. The treatment of ill-conditioned, noisy systems of equations is a problem central to FE model updating and is dealt in many papers [4,5,133,134]. These authors mainly focus on the conventional regularization techniques. The conventional regularization techniques initially developed by Tikhonov [135] seek to improve the problem condition in a merely mathematical way and require the determination of a regularization parameter. Consequently these techniques are less practical. In this work, to avoid the ill-conditioned numerical problem, only a few updating parameters are selected on the basis of the prior knowledge about the structural dynamic behavior and eigenfrequency sensitivity study and explicit bound constraints are introduced for updating parameters. Similarly, exact analytical expression for objective function gradient is derived and used which has a favorable effect on the efficiency of optimization problem. Although the final decision should be taken based on a detailed understanding of the dynamics of the structure, some automatic localization methods are developed in literature. For example, Lallement and Piranda [136] and Zhang and Lallement [137] propose to use the error in the characteristic equation of motion, if the eigenvalues and eigenvectors are measured. This is an approach that localizes errors on a degree of freedom basis and thus indicates the areas of the model that should be further investigated. An alternative approach is subset selection, presented by Friswell et al. [138], where the optimum subset of a large number of candidate parameters is chosen, that is best able to fit the measured data. It is usually difficult to rely totally on the automatic methods and hence they can only be used as an additional tool. In many civil engineering applications, however, the erroneous model regions and properties can be assumed based on engineering insight. The fact that the modal data are sensitive to a parameter does not imply that this parameter should be included in the updating process. If the parameter is likely to be estimated accurately in the initial model, there is no reason to update it. Considerable physical insight is required in order to improve the model not only in its ability to mimic the measurement data, but also in its feature to reflect the physical meaning of the parameters. Hence initial selection of updating parameters adopted in this thesis can be divided in to two basic approaches and these are: (a) an empirical approach and (b) a sensitivity-based approach. Both approaches are manual, i.e., selection is carried out manually by the analyst.

4.3.1 Empirically Based Selection of Updating Parameters This type of initial selection of updating parameters is based on knowledge of the FE - 65 -

Ph.D. dissertation of Fuzhou University

model of the structure and approximations built into the initial model. In most practical cases, the analyst will compare technical drawings of a structure with the structure itself, and after thorough inspection of the structure an initial FE model will be generated. Using this process, the analyst can select several regions of the structure that are not approximated as accurately as the reminder of the model. These regions of the structure are then selected in a few updating parameters according to the level of approximation in the initial model. It is important to notice here that no other knowledge than the level of approximation of the initial model is used for this type of selection of updating parameters. The empirical updating parameter selection approach is a fundamentally correct and appropriate method to detect the genuine errors in an initial model. A major advantage of this approach is that it is based on an empirical knowledge of the initial model and the structure itself. The method is expected to select the regions of structure which have the largest errors providing that sufficient knowledge about the initial model and structure is available. Unfortunately, this condition may not be easy to meet in real practical situations, i.e., sometimes it is not possible to inspect a structure, or even if the structure is available it may be impossible to inspect every detail or some regions of structure may be extremely difficult to assess. Also, an initial FE model of a structure may be very complicated, assembled from several sources that were generated by different people or the initial model may be based on technical drawings that may not be exactly identical to built structure. This process of assessment of both structure and initial model is extremely dependent on human factors (the analyst’s experience) and it cannot be easily quantified.

4.3.2 Sensitivity Based Selection of Updating Parameters Sensitivity analysis is carried out to see the most sensitive parameters for FE model updating. A widely used means of identifying potential error locations in the FE model is the use of eigenvalue-sensitivities. These frequently accompany parameter studies of dynamic structures [139] and represent the rate of change in eigenvalue for a unit change of a given design parameter. Normally, the sensitivities of each finite element associated with a selected design parameter are computed and compared. Based on this comparison, the analyst may then select the most sensitive elements as updating parameters. Equation (4.32) can be used to calculate the eigenvalue sensitivity of various potential parameters. The procedure explained above for the analytical calculation of eigenvalue sensitivity may not be easy when study is carried out using the commercial software whose program code is not available and system matrices cannot extracted easily. In that case, finite difference approximation is one of the alternatives for the calculation of eigensensitivity. In - 66 -

Finite element model updating of civil engineering structures under operational conditions

this approach, the eigenvalue sensitivity matrix is approximated using the forward difference of the function with respect to each parameter considered.

∂λ j ∂ai

=

∆ai =

λ ( a + ∆ai e ) − λ ( a )

(4.37)

∆ai ∆D ai − ai 100

(

)

(4.38)

where ∆D is a forward difference step size (in %), taken 0.2 general and a i , a i are the upper and lower limit for the design variable a respectively. Customarily, the sensitivities of a number of modes are analyzed with respect to a selected set of design parameters. Unless only one particular mode is under scrutiny, the process of locating the errors (i.e., identifying highly sensitive regions) consists of as many sensitivity

studies

as

there

are

modes

of

concern.

However,

the

use

of

eigenvalue-sensitivities for localizing miss-modeled elements must be handled with care. The sensitivity term defined in Equation (4.32) ignores the measured information and is a purely analytical expression. This is somewhat contradictory as it is aimed at identifying elements which are potentially able to minimize the discrepancy between the measurements and the predictions. Therefore, highly sensitive design parameters do not necessarily bring about the response changes that actually minimize the errors. Or in other words, Equation (4.32) is insensitive to the direction to which the predicted eigenvalue should change. It is difficult to conclude which method of selection of updating parameters is more suitable in the general case. If, for instance, only the empirical selection approach is used but the dynamic properties under consideration are not sensitive to the selected updating parameters, there is little chance of a successful final result. If, however, only the sensitivity-based selection approach is used, then there is a possibility that accurately modeled regions of a structure are selected as updating parameters and this will reduce the confidence in the final updated model. A proper balance of the two methods is used in this thesis, i.e., both sets are selected independently and overlaid them in order to select updating parameters.

4.4 Optimization Algorithm

{

}

Optimization is used to find a set of design parameters, x = x1 x2 x3 …… xn , that can be defined as optimal. In general, the objective function, f ( x ) , to be minimized are subjected to constraints in the form of equality constraints, gi ( x ) = 0 ( i = 1,......, me ) ,

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Ph.D. dissertation of Fuzhou University

inequality constraints, gi ( x ) ≤ 0 ( i = me + 1,....., m ) and lower and upper parameter bounds

x, x respectively. The general optimization problem is stated as: minimize x ∈ ℜn subject to

f ( x)

gi ( x ) = 0,

i = 1,...., me

gi ( x ) ≤ 0,

i = me + 1,...., m

(4.39)

x≤x≤x

where x is the vector of design parameters, ( x ∈ ℜn ) , f ( x ) is the objective function that

returns a scalar value ( f ( x ) : ℜn → ℜ ) , and the vector function g ( x ) returns the values of

the equality and inequality constraints evaluated at x ( g ( x ) : ℜn → ℜm ) . The special form

of the problem stated below is also considered for the optimization problem in this thesis. Minimize

f = f ( x)

subject to

gi ( x ) ≤ g i

wk ≤ wk ( x ) ≤ wk

( i = 1, 2,3,...m1 ) ( j = 1, 2,3,....m2 ) ( k = 1, 2,3....m3 )

xi ≤ xi ≤ xi

(i = 1, 2,3,...n)

h j ≤ hj ( x)

(4.40)

where x is the vector of design variable with parameter bounds x, x and gi , h j , wk represent the state variables (equality and inequality constraints) containing with under bar and over bar representing lower and upper bounds respectively and m1 + m2 + m3 = number of state variables. Penalty function approach is used to solve the optimization problem in the form of Equation (4.40), which is dealt in chapter 5. The optimization problems can be classified into constrained and unconstrained form depending on whether or not constraints are imposed. In constrained optimization, which is the case in many practical problems, the design variables cannot be chosen arbitrarily, they rather have to satisfy certain explicit requirements. Optimization algorithms seek an approximate solution by proceeding iteratively. They begin with an initial guess of the optimal values of the variables and generate a sequence of improved estimates until they reach the solution. Hence, the name hill- climbing or downhill methods are popular, since the iterations go gradually uphill (for maximization) or downhill (for minimization) on the surface of the objective function. The strategy used to move from one iterate to the next distinguishes one algorithm from another. Although a wide spectrum of methods exists for optimization, methods can be broadly categorized in terms of the derivative information that used or not during optimization. In - 68 -

Finite element model updating of civil engineering structures under operational conditions

general, the methods can be classified into three categories:

•

Using only functional evaluations (direct methods)

•

Using gradient evaluations

•

Using Hessian3 evaluations as well as gradient and function evaluations Let us consider the typical iteration procedure represented as:

xk +1 = xk + α k d k

(4.41)

where d k is a search direction and α k > 0 is chosen so that f k +1 < f k .Search methods that use only function evaluations (e.g., the simplex search of Nelder and Mead [118]) are most suitable for problems that are very nonlinear or have a number of discontinuities. Similarly, subproblem approximation method which is also called advanced zero-order method does not need the derivative information and it will be explained in chapter 5. In gradient based methods, it is assumed that at least the gradient of f is available at a reasonable price. The methods to obtain search direction d k in gradient based methods are explained below.

4.4.1 Search Direction In method of steepest descent, the search direction for Equation (4.41) is given as:

d k = −∇f k

(4.42)

The steepest descent direction appears as an ingredient in typical trust region method. If one needs to use it in stand-alone mode, then it must be incorporated with some line search technique. Similarly, for Newton’s method, let us use Taylor's expansion to model f locally by a quadratic expression as: 1 f ( xk + d ) ≈ f k + d T ∇f k + d T H k d = M k ( d ) 2

(4.43a)

If matrix of second derivative of objective function H k is positive definite4, then the minimum of M k ( d ) is at its critical point, ∇f k + H k d k = 0 . Thus, the Newton's direction is given as:

d k = − H k−1∇f k

(4.43b)

The Newton’s direction of Equation (4.43b) has two drawbacks. One is that the computed direction d k is not necessarily a descent direction unless ∇ 2 f k is positive 3

4

Hessian is defined in appendix A. Positive definite property is defined in appendix A. - 69 -

Ph.D. dissertation of Fuzhou University

definite. Another is that, an explicit use of second derivative information is made, this may be hard to evaluate, expensive to evaluate and expensive to invert all depending on the application. Thus, in quasi-Newton method, one looks for approximate Hessian matrix Hk which is symmetric positive definite, easily computable and invertible, and somehow approximates the action of ∇ 2 f k . Hk is not computed from scratch at each iteration but updated using gradient information from the most recent step.

Hk dk = −∇fk

(4.44)

Given the current iterate xk and the approximate Hessian matrix Hk at xk , the linear system is solved to generate the direction d k . Updates of Hk are calculated using the fact that changes in the gradient provide information about the second-order derivative of f along the search direction. In the quasi-Newton methods, the new Hessian approximation H k +1 satisfies the quasi-Newton condition or secant equation, defined with the help of Figure 4.2 as:

Figure 4.2: Graphical interpretation of quasi-Newton method

H k +1 sk = qk sk = xk +1 − xk

(4.45)

qk = ∇f ( xk +1 ) − ∇f ( xk )

Typically, some additional requirements on H k +1 (or its inverse form) are imposed such as positive definiteness, symmetry and a limited difference between the successive approximations. Generally, the formula of Broyden [140], Fletcher [141], Goldfarb [142], and Shanno [143] (BFGS) is thought to be the most effective for use in a general purpose - 70 -

Finite element model updating of civil engineering structures under operational conditions

method. The formula is given by

H k +1 = H k +

qk qkT H kT skT sk H k − T qkT sk sk H k sk

(4.46)

As a starting point, H 0 can be set to any symmetric positive definite matrix, for example, the identity matrix I . To avoid the inversion of the Hessian H , one can derive an updating method in which the direct inversion of H is avoided by using other formula that makes an approximation of the inverse Hessian H −1 .

4.4.2 Line Search and Trust Region Strategies The term line search refers to a procedure for choosing α k in Equation (4.41). If a Newton or a quasi-Newton method is used, then update Equation (4.47) yields fast convergence which is quadractic 5 or superlinear 6 respectively provided x0 is close enough to the minimum solution x* .

xk +1 = xk + d k

(4.47)

But to obtain global convergence, which means essentially dropping the close enough clause, the basic update must be modified. A sufficient decrease is required in f k +1 as compared to f k . If this is not achieved by xk + d k then in Equation (4.41), α k is found by line searching, or modifying the direction d k altogether using a trust region approach. The trust region algorithm to solve the optimization problem is explained and utilized in chapter 7. In line search methods, the search direction d k is hold fixed and searching for a step length α k to define the next iterate according to Equation (4.41) is carried out. An exact line search is not performed, i.e., the one dimensional minimization problem min f ( xk + α d k ) is not solved due to the solution being expensive. Rather, a weak line α search is performed, accepting as α k the first α , to be found which provides sufficient decrease in the objective function f . The minimum along the line formed from this search direction is generally approximated using a search procedure (e.g., Fibonacci, Golden Section) or by a polynomial method involving interpolation or extrapolation (e.g., quadratic, cubic). These concepts of search direction and line search are used to carry out the constrained optimization of Equation (4.39) as explained below. 5 6

quadractic convergence is defined in appendix A. superlinear convergence is defined in appendix A. - 71 -

Ph.D. dissertation of Fuzhou University

4.4.3 Sequential Quadratic Programming In constrained optimization, the general aim is to transform the problem into an easier subproblem that can be solved and used as the basis of an iterative process. A characteristic of a large class of early methods is the translation of the constrained problem to a basic unconstrained problem by using a penalty function for constraints, which are near or beyond the constraint boundary. In this way, the constrained problem is solved using a sequence of parameterized unconstrained optimizations, which in the limit of the sequence converge to the constrained problem. There are other methods that have focused on the solution of the Kuhn-Tucker (KT) equations. Referring to GP (Equation (4.39)), the KT equations can be stated as:

( ) i=1 ∇gi ( x* ) = 0 m

( )

f x* + ∑ λ*i .∇gi x* = 0

λ*i ≥ 0

i = 1,..., me

(4.48)

i = me + 1,..., m

The solution of KT equations forms the basis to many nonlinear programming algorithms. These algorithms attempt to compute directly the Lagrange multipliers. These methods are commonly referred to as SQP methods since a QP sub-problem is solved at each major iteration. The SQP method, which is a well-known direct method, is explained in this chapter which is used to solve the multi-objective optimization problem of chapter 6. Then, important indirect method, namely the penalty function methods is explained and utilized in chapter 5. SQP methods represent the state-of-the-art in nonlinear programming methods as explained in Schittowski [144]. Based on the work of Biggs [145], Han [146], and Powell [147], the method allows one to closely mimic Newton’s method for constrained optimization just as is done for unconstrained optimization. The SQP implementation used in this thesis consists of three main stages, which are discussed briefly in the following sub-sections:

•

Updating of the Hessian matrix of the Lagrangian function

•

Solution of quadratic programming problem

•

Calculation of line search and merit function

4.4.3.1 Updating the Hessian Matrix of the Lagrange Function

At each major iteration, a positive definite quasi-Newton approximation of the Hessian - 72 -

Finite element model updating of civil engineering structures under operational conditions

of the Lagrangian function, H , is calculated using the BFGS method shown in Equation (4.46) in which, n n ⎛ ⎞ qk = ∇f ( xk +1 ) + ∑ ϒi .∇gi ( xk +1 ) − ⎜ ∇f ( xk ) + ∑ ϒi .∇gi ( xk ) ⎟ ⎜ ⎟ i =1 i =1 ⎝ ⎠

(4.49)

where ( ϒi , i = 1,....., m ) is an estimate of the Lagrange multipliers and n is the number of design parameters. Powell [147] recommends keeping the Hessian positive definite even though it may be positive indefinite at the solution point. A positive definite Hessian is maintained providing qkT sk is positive at each update and that H is initialized with a positive definite matrix, where sk is defined in Equation (4.45). When qkT sk is not positive, qk is modified on an element by element basis so that qkT sk > 0 . The general aim of this modification is to distort the elements of qk , which contribute to a positive definite update, as little as possible. Therefore, in the initial phase of the modification, the most negative element of qk * sk is repeatedly halved. This procedure is continued until qkT sk is greater than or equal to 1e-5 in this application. If after this procedure, qkT sk is still not positive, qk is modified by adding a vector v multiplied by a constant scalar wc , as shown in Equation (4.50) and w is systematically increased until qkT sk becomes positive.

qk = qk + wc v

where, vi = ∇gi ( xk +1 ) .gi ( xk +1 ) − ∇gi ( xk ) .gi ( xk ) if

( q k )i .wc < 0 and ( q k )i . ( sk )i < 0 ( i = 1,...m )

(4.50)

vi = 0 otherwise 4.4.3.2 Solution of Quadratic Programming Problem

At each major iteration of the SQP method, a QP problem is solved by the form shown in Equation (4.51), where Ai refers to the i -th row of the m-by-n matrix A . 1 minimize q ( d ) = d T Hd + cT d 2 d ∈ℜn Ai d = bi i = 1,...me

Ai d ≤ bi

(4.51)

i = me + 1,...m

The method used is the active set strategy similar to that of Gill et al. [148]. It has been modified for both linear programming (LP) and QP problems. The solution procedure involves two phases: the first phase involves the calculation of a feasible point, the second phase involves the generation of an iterative sequence of feasible points that converge to the solution. In this method, an active set is maintained, Ak , which is an estimate of the - 73 -

Ph.D. dissertation of Fuzhou University

active constraints at the solution point. Ak is updated at each iteration, k , and this is used to form a basis for a search direction dˆ . Equality constraints always remain in the active set, A . The notation for the k

k

variable, dˆk ,is used here to distinguish it from d k in the major iterations of the SQP method. The search direction, dˆk , is calculated and minimizes the objective function while remaining on any active constraint boundaries. The feasible subspace for dˆ is k

formed from a basis Z k , whose columns are orthogonal to the estimate of the active set Ak (i.e., Ak Z k = 0 ). Thus, a search direction, which is formed from a linear summation of any combination of the columns of Z k , is guaranteed to remain on the boundaries of the active constraints. The matrix Z k is formed from the last m − lc columns of the QR decomposition of the matrix Ak , where lc is the number of active constraints and lc < m . That is, Z k is given by:

Z k = Q [:, lc + 1: m ] ⎡R⎤ where QT AkT = ⎢ ⎥ ⎣0⎦

(4.52)

where R is an upper triangular matrix of the same dimension as Ak and Q is a unitary matrix so that Ak = Q * R . Having found Z k , a new search direction dˆk is sought that minimizes q ( d ) where dˆ is in the null space of the active constraints, that is, dˆ is a k

k

linear combination of the columns of Z k : dˆk = Z k p for some vector pv .Then, if quadratic is considered as a function of p , by substituting for dˆ ,one gets Equation (4.53). k

Differentiating this with respect to pv yields Equation (4.54).

q ( pv ) =

1 T T pv Z k HZ k pv + cT Z k pv 2

∇q ( pv ) = Z kT HZ k pv + Z kT c

(4.53) (4.54)

where ∇q ( pv ) is referred to as the projected gradient of the quadratic function because it is the gradient projected in the subspace defined by Z k . The term zkT HZ k is called the projected Hessian. Assuming the Hessian matrix H is positive definite, then the minimum of the function q ( pv ) in the subspace defined by Z k occurs when ∇q ( pv ) = 0 , which is the solution of the system of linear equations as shown in Equation (4.55). A step is then taken of the form as shown in Equation (4.56).

Z kT HZ k pv = − Z kT c

- 74 -

(4.55)

Finite element model updating of civil engineering structures under operational conditions

xk +1 = xk + dˆk

where dˆk = Z kT p

(4.56)

At each iteration, because of the quadratic nature of the objective function, there are only two choices of step length α . A step of unity along dˆk is the exact step to the minimum of the function restricted to the null space of Ak . If such a step can be taken, without violation of the constraints, then this is the solution to QP (Equation (4.52)). Otherwise, the step along dˆk to the nearest constraint is less than unity and a new constraint is included in the active set at the next iterate. The distance to the constraint boundaries in any direction dˆ is given by: k

⎧⎪ − ( Ai xk − b ) ⎫⎪ α = min ⎨ ⎬ Ai dˆk ⎪⎩ i ⎭⎪

( i = 1,...., m )

(4.57)

which is defined for constraints not in the active set, and where the direction dˆk is towards the constraint boundary, i.e., A dˆ > 0, i = 1,...., m . When n independent constraints are i k

included in the active set, without location of the minimum, Lagrange multipliers, ϒ k are calculated that satisfy the nonsingular set of linear equations

AkT ϒ k = c

(4.58)

If all elements of ϒ k are positive, xk is the optimal solution of QP (Equation (4.52)). However, if any component of ϒ k is negative, and it does not correspond to an equality constraint, then the corresponding element is deleted from the active set and a new iterate is sought. 4.4.3.3 Line Search and Merit Function

The solution to the QP sub-problem produces a vector d k , which is used to form a new iteration xk +1 = xk + α k d k . The step length parameter is determined in order to produce a sufficient decrease in a merit function. The choice of the distance to move along the search direction d k is not as clear as in the unconstrained case, where simply a step length that sufficiently decreases f along this direction is chosen. For constrained problems, one would like the next iterate not only to decrease f but also to come closer to satisfying the constraints. Often these two aims conflict, so it is necessary to weight their relative importance and define a merit function, which is used as a criterion for determining whether or not one point is better than another. The merit function used by Han and Powell [149] of the form as shown in Equation (4.59) has been used in this

- 75 -

Ph.D. dissertation of Fuzhou University

implementation. They also recommend setting the penalty parameter as shown in Equation (4.60). me

Ψ ( x ) = f ( x ) + ∑ ri .g i ( x ) + i =1

m

∑

i =me +1

(

ri .max {0, gi ( x )}

(4.59)

)

(4.60)

⎧ 1 ⎫ ri = ( rk +1 )i = max ⎨ ϒi , ( rk )i + ϒi ⎬ , i = 1,..., m ⎭ i ⎩ 2

This allows positive contribution form constraints that are inactive in the QP solution but were recently active. In this implementation, initially the penalty parameter r i is set to:

ri =

where .

∇f ( x )

∇g i ( x )

(4.61)

represents the l2 norm 7 . This ensures larger contributions to the penalty

parameter from constraints with smaller gradients, which would be the case for active constraints at the solution point.

4.5 Chapter Conclusions This chapter deals with the FE model updating procedure carried out in this thesis. The theoretical exposition on FE model updating is presented. Many related issues including the objective functions, the gradients of the objective function, different residuals and their weighting and possible parameters for FE model updating are investigated. The eignefrequency residual, mode shape related function, modal flexibility residual and strain energy residuals are formulated which are used in FE model updating in later chapters. Analytical formula of Fox and Kapoor is used to calculate the modal sensitivities. The potential types of parameters and the issues of updating parameters selection process adopted in this work are presented. The physical parameters, geometrical parameters and boundary conditions of FE model are probable updating parameters. Empirically based and sensitivity based updating parameter selection procedure are recognized and formulated to use for FE model updating of real civil engineering structures. Then, the ideas of optimization to be used in FE model updating application are explained. The algorithm of SQP is explored which will be used to solve the multi-objective optimization of chapter 6.

7

l2 norm is defined in appendix A. - 76 -

Finite element model updating of civil engineering structures under operational conditions

CHAPTER 5 FINITE ELEMENT MODEL UPDATING USING SINGLE-OBJECTIVE OPTIMIZATION

CHAPTER SUMMARY

This chapter deals with the FE model updating using single-objective optimization. The procedure of model updating outlined in chapter 4 is utilized. The use of dynamically measured flexibility matrices obtaining from ambient vibration measurements is proposed for FE model updating. The issue related to the mass normalization of mode shapes obtained from ambient vibration test is investigated and applied to use the modal flexibility for FE model updating. The algorithms of penalty function method, namely subproblem approximation method and first-order optimization method are explored, which are then used for FE model updating. Frequency residual only, mode shape related function only, modal flexibility residual only and their combinations are studied independently. The comparative study of the influence of different possible residuals in a single objective function is carried out with the help of simulated case study. It is demonstrated that the combination that consists of three residuals, namely eignevalue, mode shape related function and modal flexibility with weighting factors assigned to each of them is recognized as the best objective function. The single-objective optimization with the best objective function is successfully applied for the FE model updating of a full-size concrete filled steel tubular arch bridge that was tested under operational conditions.

5.1 Mass Normalization of Operational Mode Shapes An important drawback of operational modal analysis is that some modal parameters can no longer be determined. Since the ambient forces that excite the structure are not being measured under operational conditions, the modal participation factors cannot be determined. Consequently, the estimated operational mode shapes are not correctly scaled since their scaling factor will depend on the unknown ambient excitation. This so-called incompleteness of the operational modal model somewhat restricts its use in certain application domains. To use the modal flexibility for the FE model updating and damage detection, it requires the mass normalized mode shapes of the structure. - 77 -

Ph.D. dissertation of Fuzhou University

Until recently, no straightforward technique is available for the mass normalization of operational mode shapes purely on the basis of output-only data. The investigation is carried out by recalling the definition of flexibility matrix as shown in Equation (4.20). Expressing the mass normalized modes in terms of arbitrarily normalized ones, one gets:

φi = φiδ i

(5.1)

where φi is the mode shape obtained from operational measurements and δ i is the mass normalization constants. Since the arbitrarily scaled modes and the eigenvalues are readily obtained from the identification, the problem of assembling the flexibility is simply one of the devising a way to compute the constants in Equation (5.1). All existing ways to find δ i can be divided into three broad categories as explained below.

5.1.1 Sensitivity Based Method A recently proposed approach to extract the constants δ i is based on resting the structure with a perturbed mass matrix (by adding a known mass at a certain location) and exploiting the eingenvalue sensitivity equations. This sensitivity-based method was introduced for the normalization of operational mode shapes on a basis of in-operation modal models only [150]. It was shown that by adding, for instance, one (or more) masses (with well-known weights) to the test structure, the operational mode shapes can be experimentally normalized by means of the measured shift in natural frequencies between the original and mass-loaded condition. The method was tested on mechanical [151,152] and civil [153] engineering structures.

5.1.2 Using Orthogonality of Modes with Mass Matrix This technique is based on the orthogonality of the modes with respect to the mass matrix which computes the constants δ i to within a missing multiplier common to all the modes. The partition of the inverse of the mass associated with the measured coordinates is given by: N

M −1 = ∑ φiφi T δ i2

(5.2)

i =1

The possibility is to use any information that may be available regarding M −1 to set up equations to compute δ i . It may be reasonable to assume that M −1 is diagonal and in this case one can use the off diagonal zeros as a constraints [154,155,156]. Hence, one can - 78 -

Finite element model updating of civil engineering structures under operational conditions

compute δ i in terms of some scalar common to all modes. For damage detection using modal flexibility, these scalar factors for undamaged and damaged states can be expressed as a ratio and can be used efficiently. But for model updating problems, the FE model flexibility does not possess any scalar but the derived mass normalized flexibility has scalar multiplier. This is the main problem to use this method for the FE model updating process.

5.1.3 Finite Element Model Approach In case of FE model updating application, the approach of using the FE mass matrix to normalize the experimental mode shape is straightforward due to the fact that the detail analytical model of the complex structure is readily available. In the paper of Doebling and Farrar [128], a number of FE model based normalization techniques are compared by means of experiments performed on a bridge. All included methods involve the use of a FE model of the structure. One of the effective methods is the Guyan reduced mass normalization (GRM) technique. This method uses a FE model mass matrix, reduced to the measured DOFs, to normalize the mode shapes such that Equation (2.12a) is satisfied. The reduction is done according to Guyan [106], and assumes that the inertial forces at the eliminated DOFs are negligible. This assumption typically makes the GRM method valid for only the lower-frequency modes. It is pointed out in Duan et al. [157] that, the analytical model can be used to compute mass normalized mode shapes from arbitrarily scaled one, but for a real complex structure, it is not easy to set up an analytical model with confidence. But in model updating, the analytical model is readily available within some confidence. Hence in this research work, the mass matrix of analytical model is used for mass normalization of operational mode shapes as reported in [158]. The Guyan-reduced mass normalization technique is used in this work. The expression shown in Equation (5.3a) is an especially convenient normalization for a general system and for a system having a diagonal mass matrix, it may be written as shown in Equation (5.3b) where ϕ is the mode shape obtained from operational vibration test and M is the FE model mass matrix reduced to measured DOFs.

Φ ij = Φ ij =

ϕ ij {ϕ i }T [ M ]{ϕ i }

ϕij

(5.3b)

n

∑m ϕ k =1

(5.3a)

k

- 79 -

2 kj

Ph.D. dissertation of Fuzhou University

5.2 Objective Functions and Constraints The general objective function formulated in terms of the discrepancy between FE and experimental eigenvalues and mode shapes related functions as explained in chapter 4, are respectively shown below: ⎛ λaj − λej f1 ( x ) = ∑ α j ⎜ ⎜ λ j =1 ej ⎝ mf

ms

f2 ( x ) = ∑ β j j =1

(1 −

2

⎞ ⎟⎟ 0 ≤ α j ≤ 1 ⎠

MAC j

)

(5.4)

2

MAC j

0 ≤ βj ≤1

(5.5)

where α j and β j are the weight factors to impose a relative difference between eigenvalue and mode shape deviations respectively, because these entities may have been measured with different accuracy. λaj and λej are the FE and experimental eigenvalue of the j -th mode respectively and MAC j is the modal assurance criteria of the j -th analytical and corresponding experimental modes as defined in Equation (3.4). It has been reported that the modal flexibility is more sensitive to local damage than the mode shapes and natural frequencies [64,159]. The modal flexibility is the accumulation of the contribution from all available mode shapes and corresponding natural frequencies as defined in Equation (4.19). If the deflection vector ui under uniformly distributed unit load, called the uniform load surface (ULS), is defined in Equation (5.6), the objective function f 3 ( x ) considering the modal flexibility residual can be presented as shown in Equation (5.7). nd

ms

ui = ∑ k =1

( Φ ik ) ∑ ( Φ kj ) j =1 2 k

(5.6)

ω

m nd ⎡ u − u ⎤ f3 ( x ) = s * ∑ ⎢ aj ej ⎥ nd j =1 ⎢⎣ uej ⎥⎦

2

(5.7)

where uaj and uej are analytical and experimental uniform load surface respectively. Similarly, nd and ms are number of the measurement DOFs and number of mode shapes considered respectively. It is necessary to have a mass-normalized mode shapes to use the measured flexibility matrix in the FE model updating. To realize mass normalization, the Guyan-reduced mass normalization technique is used in this work as shown in Equations (5.3a) and (5.3b). - 80 -

Finite element model updating of civil engineering structures under operational conditions

Equations (5.4), (5.5) and (5.7) are the objective functions considering frequency residual only, mode shape related function only and modal flexibility residual only. In FE model updating, different objective function as indicated above can be used independently or in combined form with suitable weighting factors. Hence, one possible objective function is their combination as shown in Equation (5.8). The constraints as shown in Equations (5.9) and (5.10) are imposed on objective functions in this study. f ( x) = f1 ( x) + f 2 ( x) + f 3 ( x)

(5.8)

0 ≤ λaj − λej ≤ UL

(5.9)

L1 ≤ MAC ≤ 1

(5.10)

where UL is the upper limit whose value can be set as absolute error of the j -th eigenvalue and L1 represents the lower limit to constrain the MAC value. In single-objective optimization carried out in this work, different residuals are combined into a single objective function using weighting factors for each residual. There is no rigid rule for selecting the weighting factors. Thus, a necessary approach is to solve the problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained. Different form of objective functions with the constraints imposed as explained above will be studied with the help of simulated simply supported beam and best form of objective function will be suggested.

5.3 Optimization Techniques FE model updating is carried out to solve a minimization problem whose aim is the minimization of the objective function f ( x ) , under the constraints defined in Equations (5.9) and (5.10). In this work, the constrained optimization problem of the form as shown in Equation (4.40) is used, which for the sake of convenience is again presented below. Minimize

f = f ( x)

subject to

gi ( x ) ≤ g i

wk ≤ wk ( x ) ≤ wk

( i = 1, 2,3,...m1 ) ( j = 1, 2,3,....m2 ) ( k = 1, 2,3....m3 )

xi ≤ xi ≤ xi

(i = 1, 2,3,...n)

h j ≤ hj ( x)

(5.11)

where x is the vector of design variable with parameter bounds x, x and gi , h j , wk represent the state variables(equality and inequality constraints) containing under and over bar representing lower and upper bounds respectively and m1 + m2 + m3 = number of state - 81 -

Ph.D. dissertation of Fuzhou University

variables. Two methods, namely subproblem approximation method [160] and first-order optimization method [161, 162] are utilized in this thesis to solve the constrained optimization problem of Equation (5.11). In these optimization algorithms, the penalty function concept is used. Penalty function methods generally use a truncated Taylor series expansion of the modal data in terms of unknown parameters. These methods are briefly explained below.

5.3.1 Subproblem Approximation Method This method of optimization can be described as an advanced zero-order method, which requires only the values of the dependent variables and not their derivatives. The dependent variables are first replaced with approximations by means of least squares fitting and the constrained minimization problem is converted to an unconstrained problem using penalty functions. Minimization is then performed in every iteration on the approximated penalized function (called the subproblem) until convergence is achieved or termination is indicated. For this method, each iteration is equivalent to one complete analysis loop. Since this method relies on approximation of the objective function and each state variable, a certain amount of data in the form of design sets is needed. Three main steps of the subproblem optimization method are described below. •

Function Approximations

The dependent variables are first replaced with approximations as shown below using notation ^ by means of least square fitting. fˆ ( x) = f ( x) + error

(5.12a)

∧

g ( x ) = g ( x ) + error ∧

h( x ) = h(x ) + error

(5.12b)

∧

w( x ) = w( x ) + error

The most complex form that the approximations can take on is a fully quadratic representation with cross terms. Using the example of objective function, ∧

n

n

n

i

i

j

f = a0 + ∑ ai xi + ∑∑ bij xi x j

(5.13)

where ai and bij are coefficients, whose value is determined by weighted least squares technique. For example, the weighted least square error norm for the objective function has the form: - 82 -

Finite element model updating of civil engineering structures under operational conditions nc

E = ∑ψ 2 n

j =1

( j)

( f ( ) − fˆ ( ) ) j

j

2

(5.14)

where, ψ j is the weight associated with design set j and nc is the current number of design sets. Similar, En2 norms are formed for each state variable. The coefficients in Equation (5.14) are determined by minimizing En2 with respect to the coefficients. •

Minimizing the Subproblem Approximations

With the function approximation available, the constrained minimization problem can be recast as follows: Minimize fˆ = f ( x)

(5.15a)

subject to

xi ≤ xi ≤ x i

( i = 1, 2,3...., n )

∧

( i = 1, 2,3...., m1 )

g i ( x ) ≤ gi + Ai ∧

h j − Bj ≤ h j ( x)

( j = 1, 2,3...., m2 )

∧

w k − Ck ≤ w k ( x ) ≤ w k + Ck

(5.15b)

(k = 1, 2,3,..., m3 )

where Ai , B j , Ck represent the state variables related parameters after function approximation available during optimization. The constrained minimization problem in Equation (5.15a) is converted to the unconstrained problem using penalty functions leading to the following subproblem statement: Minimize ∧ m m1 m ⎡ n ⎛ ∧ ⎞ 2 ⎛ ∧ ⎞ 3 ⎛ ∧ ⎞⎤ F ( x, Pk ) = f + f 0 Pk ⎢ ∑ X ( xi ) + ∑ G ⎜ gi ⎟ + ∑ H ⎜ h j ⎟ + ∑ W ⎜ wk ⎟ ⎥ ⎝ ⎠ j =1 ⎝ ⎠ k =1 ⎝ ⎠ ⎦ i =1 ⎣ i =1

(5.16)

where F ( x, pk ) represents the unconstrained objective function that varies with the design variables and parameter Pk , X is the penalty function used to enforce design variable constraints, and G , H , W are penalty functions for the state variable constraints. The reference objective function value f 0 is introduced to achieve consistence unit. It is clear that the unconstrained objective function (also termed a response surface) F ( x, pk ) vary with the design variables and the quantity Pk ,which is a response surface parameter. A Sequential Unconstrained Minimization Technique [163] is used to solve Equation (5.16) at each design iteration. All the penalty functions used are of the extended interior type. For example, near the upper limit, the design variable penalty function is formed as: - 83 -

Ph.D. dissertation of Fuzhou University

⎪⎧c + c ( x − xi ) if x i < x − ∈ ( x − x ) ⎪⎫ X ( xi ) = ⎨ 1 2 ⎬ ⎩⎪ c3 + c4 ( xi − x ) if x i ≥ x − ∈ ( x − x ) ⎭⎪

( i = 1, 2,3,..., n )

(5.17)

where, c1, c2 , c3 , c4 are constants that are internally calculated and ∈ is very small positive number. State variable penalties take a similar form. For example, again near the upper limit: ⎪⎧d1 + d 2 ( wi − wi ) if wˆ i < wi − ∈ ( wi − wi ) ⎪⎫ W ( wi ) = ⎨ ⎬ ⎪⎩ d3 + d 4 ( wi − wi ) if wˆ i ≥ wi − ∈ ( wi − wi ) ⎪⎭

( i = 1, 2,3,..., m1 )

(5.18)

where, d1, d 2 , d3 , d 4 are constants that are internally calculated and similar form can be written for G and H . A Sequential Unconstrained Minimization Technique (SUMT) is employed to reach the minimum unconstrained objective function f ( that is, x to f

( j)

→x

( j)

f

( j)

→ f

( j)

where x

( j)

j)

at design iteration j

is the design variable vector corresponding

( j)

. The final step performed at each design iteration is the determination of the design j+1 variable vector to be used in the next iteration j+1 . Vector x ( ) is determined according

(

)

to the following equation. x(

j +1)

(

= x( ) + C x b

( j)

− x(

b)

)

(5.19)

b where x( ) is the best design set constants and C is internally chosen to vary between 0

and 1 based on the number of infeasible solutions. •

Convergence

Minimization is then performed at every iteration on the approximated penalized function until convergence is achieved. Convergence is assumed when either the present design set x ( j ) , or the previous design set x ( j −1) or the best design set x (b ) is feasible and one of the following conditions is satisfied. f ( j ) − f ( j −1) ≤ τ

(5.20a)

f ( j ) − f (b) ≤ τ

(5.20b)

xi ( j ) − xi ( j −1) ≤ ρi

( i = 1, 2,3,..., n )

(5.20c)

xi ( j ) − xi (b ) ≤ ρi

( i = 1, 2,3,..., n )

(5.20d)

Equations (5.20a) and (5.20b) corresponds to difference in objective function values and Equations (5.20c) and (5.20d) to design variable difference. If the satisfaction of Equations (5.20a)-(5.20d) is not realized, then termination occurs if either of the below two

- 84 -

Finite element model updating of civil engineering structures under operational conditions

conditions are reached. ns = N s

(5.21)

nsi = N si

(5.22)

where ns is the number of subproblem iterations, nsi is the number of sequential infeasible design set, N s is the maximum number of iterations and N si is the maximum number of sequential infeasible design sets.

5.3.2 First-Order Optimization Method This method of optimization calculates and makes use of derivative (gradient) information. The constrained problem statement expressed in Equation (5.11) is transformed into an unconstrained problem via penalty functions. Derivatives are formed for the objective function and state variable penalty functions, leading to a search direction in design space. Various steepest descent and conjugate direction searches are performed during each iteration until convergence is reached. Each iteration is composed of sub iterations that include search direction and gradient computations. In other words, one first-order design optimization iteration will perform several analysis loops. Compared to the subproblem approximation method, this method is usually seen to be more computationally demanding and more accurate. With regard to the first-order optimization method, three major steps involved are explained below. •

The Unconstrained Objective Function

The constrained problem statement expressed in Equation (5.11) is transformed into an unconstrained one using penalty functions. An unconstrained form of Equation (5.11) is formulated as follows. m3 m2 n ⎡ m1 ⎤ f Q( x, q ) = + ∑ Px ( xi ) + q ⎢ ∑ Pg ( gi ) + ∑ Ph (h j ) + ∑ Pw ( wk ) ⎥ f 0 i =1 j =1 k =1 ⎣ i =1 ⎦

(5.23)

where Q( x, q ) is the dimensionless unconstrained objective function; Px , Pg , Ph , Pw are the penalties applied to the constrained design and state variables and f 0 refers to the reference objective function value, that is selected from the current group of design sets. Constraint satisfaction is controlled by a response surface parameter q . Exterior penalty functions Px are applied to the design variables. State variables constraints are represented by extended interior penalty functions Pg , Ph , Pw . For example, for state variables constrained by an upper limit, the penalty function is written as:

- 85 -

Ph.D. dissertation of Fuzhou University

⎛ gi ⎞ Pg ( gi ) = ⎜ ⎟ ⎝ gi + χi ⎠

2κ

(5.24)

where κ is the large integer so that the function will be very large when the constraint is violated and very small, when it is not violated and χ is tolerances for state variable gi . The functions used for remaining penalties are of similar form. As search directions are devised, a certain computational advantage can be gained, if the function Q is rewritten as the sum of two functions Q f ( x ) and Q p ( x, q) as defined in Equations (5.25) and (5.26). Then, Equation (5.23) takes the form as shown in Equation (5.27).

Q f ( x) =

f f0

m3 m2 n ⎡ m1 ⎤ Q p ( x, q ) = ∑ Px ( xi ) + q ⎢ ∑ Pg ( gi ) + ∑ Ph (h j ) + ∑ Pw ( wk ) ⎥ i =1 j =1 k =1 ⎣ i =1 ⎦ Q ( x, q ) = Q f ( x ) + Q p ( x, q )

(5.25) (5.26) (5.27)

where functions Q f and Q p relate to the objective function and the penalty constraints respectively.

•

The Search Direction

Derivatives are formed for the objective function and the state variable penalty functions leading to the search direction in design space. For each optimization iteration

( j ) a search

direction vector d ( j ) is devised. The next iteration ( j + 1) is obtained from Equation (5.28). In this equation, measured from x( j ) ,the line search parameter S j corresponds to the minimum value of Q in the direction d ( j ) . x ( j +1) = x ( j ) + S j d ( j )

(5.28)

The solution for S j uses a combination of a golden-section algorithm and local quadratic fitting technique. The range for S j is limited to:

S 0 ≤ s j ≤ max s*j 100

(5.29)

where s*j is the largest possible step size for the line search of the current iteration (internally computed) and Smax is the maximum(percent) line search step size. The key to the solution of the global minimization of the Equation (5.27) relies on the sequential generation of the search directions and the internal adjustments of the response surface parameter ( q ) . For the

- 86 -

Finite element model updating of civil engineering structures under operational conditions

initial iteration ( j = 0 ) , the search direction is assumed to be the negative of the gradient of the unconstrained objective function.

)

(

0 0 0 ( 0) d ( ) = −∇Q x( ) , q = d f + d (p )

(5.30)

in which q = 1 and

( )

( )

0 0 0 ( 0) d f = −∇Q f x( ) and d (p ) = −∇Q p x( )

(5.31)

Clearly for the initial iteration, the search method is that of steepest descent. For subsequent iterations

( j > 0)

,conjugate directions are formed according to the

Polak-Ribiererecursion formula [164].

)

(

j j j −1 d ( ) = −∇Q x( ) , qk + r j −1d ( )

)

(

( (

) )

T

(

⎡∇Q x( j ) , q − ∇Q x( j −1) , q ⎤ ∇Q x( j ) , q ⎢ ⎥⎦ r j −1 = ⎣ 2 j −1 ∇Q x ( ) , q

(5.32)

)

(5.33)

It should be noticed that when all design variable constraints are satisfied, Px ( xi ) = 0 . This means that q can be factored out of Q p , and can be written as:

)

(

( )

j j Q p x( ) , q = q Q p x( ) if xi ≤ xi ≤ xi ( i = 1, 2,3,...n )

(5.34)

If suitable corrections are made, q can be changed from iteration to iteration without destroying the conjugate nature of Equation (5.32). Adjusting q provides internal control of state variable constraints, to push constraints to their limit values as necessary, as convergence is achieved. The justification for this becomes more evident once Equation (5.32) is separated into two direction vectors as shown in Equation (5.35), where each direction has a separate recursion relationship as shown in Equations (5.36) and (5.37). j j ( j) d ( ) = d f + d (p )

( ) d ( ) = − q∇Q ( x( ) ) + r

(5.35)

( j −1) j ( j) d f = −∇Q f x( ) + r j −1d f j

p

j

p

- 87 -

( j −1)

j −1d p

(5.36) (5.37)

Ph.D. dissertation of Fuzhou University

The algorithm is occasionally restarted by setting r j −1 = 0 , forcing steepest descent iteration. Restarting is employed whenever ill-conditioning is detected, convergence is nearly achieved, or constraint satisfaction of critical state variables is too conservative. So far, it has been assumed that the gradient vector is available. The gradient vector is computed using an approximation as follows:

( ) ≈ Q ( x ( ) + ∆x e ) − Q ( x ( ) )

j ∂Q x( )

j

j

i

∂xi

∆xi

(5.38)

where, e is the vector with 1 in its i -th component and 0 for all other component and ∆D is the forward difference (in percent) step size and ∆xi = •

∆D ( xi − xi ) 100

(5.39)

Convergence

Various steepest descent and conjugated direction searches are performed during each iteration, until the convergence is reached. Convergence is assumed when comparing the current iterations design set ( j ) to the previous ( j + 1) set and the best ( b ) set as shown in Equation (5.40), in which τ is the objective function tolerance. j j −1 f ( ) − f ( ) ≤τ

and

j b f ( ) − f ( ) ≤τ

(5.40)

It is also a requirement that the final iteration used a steepest descent search. Otherwise, additional iterations are performed. In other words, steepest descent iteration is forced and convergence is rechecked. The termination occurs when ni = N1

(5.41)

where ni is the number of iterations, N1 is the allowed number of iterations.

5.4 Simulated Simply Supported Beam A simulated simply supported beam is aimed at demonstrating a comparative study of the influence of different possible residuals on objective function for FE model updating and their sensitivity to the detection of damaged elements. The simulated simply supported beam with a length of 6 m is discretized as shown in Figure 5.1.The density and modulus of elasticity of the beam are 2500 kg / m3 and 3.2×104 MPa respectively. Similarly, the area and moment of inertia of the cross section are 0.05 m2 and 1.66×10-4 m4 respectively. - 88 -

Finite element model updating of civil engineering structures under operational conditions 1

2

3

4

5

6

7

8

9

DAM

DAM

10

11

12

13

14

15

DAM

L=6m

h=0.2m b=0.25m

Figure 5.1: A simulated simply supported beam

Analytical modal analysis is first carried out to get the FE frequencies and mode shapes. To get the simulated experimental modal parameters, three damage locations are assumed in the beam as shown in Figure.5.1, in which the elastic modulus and moment of inertia of beam elements 3, 8 and 10 are reduced by 20%, 50% and 30% respectively. The modal analysis is again carried out on this damaged beam to get the assumed experimental modal parameters. The initial values of frequencies and corresponding errors and MAC of first ten modes selected in this study are shown in Table 5.1. The maximum error that appeared in frequency is 23.8% and minimum MAC is 90.3% due to damage. Table 5.1: Comparison of experimental (assumed damage) and analytical modal properties of simulated beam before updating Mode 1 2 3 4 5 6 7 8 9 10

Damaged beam 7.257 33.683 68.706 131.100 141.200 193.560 305.210 382.460 420.960 522.300

Natural frequency (Hz) Initial FE model 8.990 35.914 80.632 142.930 149.140 222.530 319.160 432.530 449.050 562.420

Error (%) 23.80 6.62 17.35 9.02 5.62 14.96 4.57 13.09 6.67 7.68

MAC % 99.5 99.4 96.2 97.7 99.8 90.3 97.7 92.4 99.3 94.3

Updating of the FE model of the undamaged beam is to correlate the modal parameters with the damaged beam and to identify the damage severity and location. Both inertia moment and elastic modulus of individual elements are chosen as updating parameters (UPs). The numbers of UPs are selected with respect to the numbers of modes considered where three cases are considered: •

UPs=6 case where E and I of damaged elements 3, 8 and 10 are selected as UPs and numbers of UPs are less than the mode numbers considered. It is the case that the structural damaged locations are exactly known.

- 89 -

Ph.D. dissertation of Fuzhou University

•

UPs=10 case where E and I of elements 5 and 13 in addition to damaged elements 3, 8 and 10 are selected as UPs and numbers of UPs are equal to the mode numbers considered. It is the case that the structural damaged locations are partly known. UPs=30 case where E and I of all 15 elements are selected as UPs and numbers of UPs are larger than the mode numbers considered. It is the case that the structural damaged

Frequencies obtained from assumed damaged beam(Hz)

locations are not known. 600 UPs=6 UPs=10 UPs=30

500

400

300

200

100

0

0

100

200 300 400 Updated frequncies(Hz)

500

600

(a) Frequencies correlation 1 0.98 0.96 MAC

•

0.94 0.92

UPs=6 UPs=10 UPs=30

0.9 0.88 1

2

3

4 5 6 Mode Number

7

8

9

10

(b) Mode shapes correlation Figure 5.2: Correlation of simulated beam after updating with frequency residual, MAC function and flexibility residual in objective function

Five different cases of objective functions consisting of frequency residual only, MAC

- 90 -

Finite element model updating of civil engineering structures under operational conditions

related function only, flexibility residual only, combination of frequency and MAC related function, and combination of frequency, MAC and flexibility residuals under constraints shown in Equations (5.9) and (5.10) are studied independently, each having three cases depending upon the number of updating parameters. Table 5.2: Results of simulated beam after updating with different residuals in objective function Residuals in objective function Frequency Only MAC related function Only Frequency + MAC related function Flexibility Only Frequency + MAC related function + flexibility

Number of updating parameters 6 10 30 6 10 30 6 10 30 6 10 30 6 10 30

Max. error in frequency tuning (%) 0.10 0.30 3.50 0.18 0.20 0.42 0.40 0.37 3.05 0.29 0.40 2.36 0.26 0.27 1.29

MAC Tuning Min.

Max.

0.99 0.98 0.97 0.99 0.99 0.99 0.99 0.99 0.97 0.99 0.99 0.97 0.99 0.99 0.97

1.00 0.99 0.99 1.00 1.00 0.99 0.99 0.99 0.99 1.00 1.00 0.99 1.00 0.99 0.99

Max. error in damage detection (%) 3.5 18.5 26.7 21.1 22.5 23.1 7.6 23.0 31.4 3.2 14.9 24.5 3.1 12.3 18.3

The first-order method is used to carry out the optimization and results are also checked by using the subproblem method, so that the algorithm does not trap in local minima. The analytical and corresponding experimental mode shapes are automatically paired with the help of MAC criteria. The weighing factors are not used in this simulated case study. The updating parameters are estimated during an iterative process. The tuning process is over when the tolerances were achieved or pre-defined numbers of iterations are reached. The tuning results of frequency and MAC with different five cases are studied. The tuning results of frequency and MAC with the objective function considering the combination of frequency residual, MAC related function and flexibility residuals for the conceived three UPs cases after model updating are shown in Figure 5.2. The maximum errors in frequency tuning and maximum and minimum MAC values within first ten modes after updating for remaining four cases are summarized in Table 5.2. It is shown in Figure 5.2 that there is significant improvement on the tuning of frequencies in each three cases laying all points of frequencies in the diagonal line (Figure 5.2a) compared to the results before updating which is shown in Table 5.1. It is also demonstrated that there is an enough improvement on MAC values with more than 0.96 for each mode (Figure5.2b).

- 91 -

Ph.D. dissertation of Fuzhou University Table 5.3: Error detection after updating when frequency, MAC related function and flexibility residual are used in objective function Elem. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UPs =6 Error (%) E

0.1

I

UPs =10 Error (%) E I

1.5

1.2

12.3

3.2

2.5

0.8

0.7

5.5

5.3

2.8

3.1

0.8

0.8

6.9

5.1

UPs =30 Error (%) E I 1.6 0.2 3.3 0.6 10.0 12.7 0.6 0.7 2.7 2.0 13.3 13.1 3.4 5.1 0.2 4.0 0.0 3.4 13.2 18.3 6.8 3.4 3.7 0.6 12.5 10.5 1.6 0.5 0.2 0.2

The most important aspect in FE model updating is the error appeared in damage detection. To deal with this issue in this simulated case study, a damage detection result for all three cases is shown in Table 5.3, for objective function that consists of frequency, MAC related function and modal flexibility residual. From Table 5.3, maximum error in damage detection are picked up and put in Table 5.2, which is underlined as an example for one typical case. Same procedure is repeated for remaining four cases of objective functions and final result is presented in Table 5.2. It is clearly seen in each case of all five objective functions that as the number of UPs in the FE model goes on increasing, tuning on the frequency and MAC values go on decreasing. Similarly, an accurate identification of damage location and severity becomes difficult if a large number of parameters and elements are selected. It is observed that the tuning on modal parameters alone can be achieved even any one residual explained above is used in objective function. However, in the damage detection part, the introduction of modal flexibility residual in the objective function with other residuals, considerably improves the damage detection, which supports the fact that the modal flexibility term is sensitive to local damage. For example, when the number of updating parameter is 10 and 30, maximum error in damage detection in the last case is 12.3% and 18.3% respectively which is considerably less than the remaining three cases as shown in Table 5.2. Therefore, in view of tuning as well as damage detection, the objective function considering frequency residual, MAC related function and flexibility residual is the best for FE model updating. This full objective function with constraints imposed as shown in Equations (5.9) and (5.10) is utilized for FE model updating of real bridge structure. - 92 -

Finite element model updating of civil engineering structures under operational conditions

5.5 Concrete Filled Steel Tubular Arch Bridge 5.5.1 Bridge Description and Finite Element Modeling The Beichuan River Bridge is a concrete filled steel tubular half-through tied arch bridge. It is located at the center of Xining City, China and the span of the bridge is 90 m. Figure 5.3 shows the photograph of the bridge. This bridge is constructed over the existing old bridge. The superstructure of the bridge consists of the vertical load bearing system, the lateral bracing configuration, and the floor system. The cross-section of two main arch ribs comprises of four concrete-filled tubes, with the dimension of 650 × 10 mm. The depth of main arch rib is 3000 mm. The remaining connecting tubes of superstructure are of hollow steel tubes.

Figure 5.3:

Photo of Beichuan river concrete-filled steel tubular arch bridge

There are 32 main suspenders of steel wire ropes that are vertically attached on main arch rib and floor system is suspended through it. Each of these ropes consists of 127 smaller bars each with a diameter of 5.5 mm. The floor system consists of a 250 mm thick concrete slab supported directly by cross girders at a spacing of 5 m c/c. The typical rectangular cross section of the cross girder is 0.36 × 1.361 m. The length of each cross girder is 21.6 m between the suspenders. The main arch ribs are fixed at two abutments, and connected by 4 pre-stressed strands each side in the longitudinal direction, which acts as tie bars. Each - 93 -

Ph.D. dissertation of Fuzhou University

strands are prestressed by 2200 kN force. Expansion bearings are constructed at the joints of bridge and pre-existing road at the two ends of bridge deck. Elevation and plan of the bridge is shown in Figure 5.4 and the details of cross section of arch rib and deck beam connection is shown in Figure 5.5. 0 0 0 9 0 5 7

0 0 5 5 1

0 5 7

×

n a g n i P 4 6 . 4 4 2 2

Huangyuan

e g d i r b f o e n i l l a r t n e c r a b d e i T

0 2 . 7 3 2 2

old bridge

0 2 . 2 1 2 2

(a) Elevation

s egu s s n r i r e t t v r s o l n p e a p e r u t t s s

0 5

re e d v i i s r

0 5

0 0 5 0 0 5 0 0 5 0 0 5 0 0 5 0 0 5 0 0 5 0 0 5

0 5 8 0 0 5 85 1 0 5 8

0 0 0 9

(b) Plan Figure 5.4: Elevation and plan of arch bridge

- 94 -

Finite element model updating of civil engineering structures under operational conditions

(a) Cross-section of arch rib

(b) Connection of bridge deck and floor beam Figure 5.5: Cross section of arch rib and deck beam connection of arch bridge

Three-dimensional elastic FE model of the bridge was constructed using ANSYS [165]. The arch members, cross girders, and bracing members were modeled by two-node beam elements (BEAM4) having three translational DOFs and three rotational DOFs at each node. All suspenders and prestressed tie bars were modeled by the truss elements (LINK10). The deck slab of the bridge was modeled as shell elements (SHELL63). Solid elements (SOLID45) were used to simulate the abutments of arch. The connection between the cross-girders and bridge deck was established by using spring element (combine14) in the transverse direction. The value of spring stiffness is assumed as 500000 N m based on the previous experience on similar bridges.

- 95 -

Ph.D. dissertation of Fuzhou University

Figure 5.6: Three dimensional FE model of the bridge

The vertical and longitudinal translational freedoms of all nodes were coupled, and the three rotational degrees of freedoms were released. The foundations of the piers were simplified and modeled as fixed end supports. Hence, the full FE model consisted of 1,434 beam elements, 68 link elements, 1,092 shell elements, 564 solid elements and 288 constraint elements. As a result, 3,120 nodes, 3,446 elements and 14,060 active DOFs were recognized on the model. Figure 5.6 shows the full 3-D view of the FE model of the arch bridge.

5.5.2 Ambient Vibration Testing, Modal Parameter Identification and Model Correlation •

Ambient Vibration Testing

Just prior to officially opening, the field dynamic testing on the Beichuan River arch bridge was carried out under operational conditions where the bridge was excited by ambient vibration. The equipment used for the tests included accelerometers, signal cables, and a 32-channel data acquisition system with signal amplifier and conditioner. Accelerometers convert the ambient vibration responses into electrical signals. Cables are used to transmit these signals from sensors to the signal conditioner. Signal conditioner unit is used to improve the quality of the signals by removing undesired frequency contents (filtering) and amplifying the signals. The amplified and filtered analog signals are converted to digital data using an analog to digital (A/D) converter. The signals converted to digital form are stored on the hard disk of the data acquisition computer. Measurement points were chosen to both sides of the bridge at a location near the joint of suspenders and deck. As a result, a total of 32 locations (16 points per side) were selected as shown in - 96 -

Finite element model updating of civil engineering structures under operational conditions

Table 5.4 and Figure 5.7. The force-balance (891-IV type) accelerometers and INV306 data acquisition system as shown in Figure 5.8(a) were used. Table 5.4: Test setup in vertical direction of arch bridge Setup 1 2 3 4

Measurement points 1，2，3，4，5，6，7，8 9，10，11，12，13，14，15，16 17，18，19，20，21，22，23，24 25，26，27，28，29，30，31，32

Reference point (at same location) R1 R2 R3 R4

Reference Point 16(32)

1(17)

2(18) 4(20) 6(22) 8(24) 10(16) 12(28) 14(30) 3(19) 5(21) 7(23) 9(25) 11(17) 13(29) 15(31)

Tied bar Accelerometer

16

32

15 14 13 12 11

31 30 29 28 27

10 Upstream 9 8 7 6 5 4 3 2 1

26 25 24 23 22 21 20 19 18 17

Bridge Surface

t i n o P e c n e r e f e R

Figure 5.7: Details of measurement points of arch bridge

The accelerometers were installed on the surface of the bridge in the vertical and transverse directions. Four test setups for vertical measurements and four test setups for transverse measurements were conceived to cover the planned testing locations of the bridge. One reference location was selected near one side of abutments for each setup. - 97 -

Ph.D. dissertation of Fuzhou University

Each setup consisted of eight moveable accelerometers and one fixed reference accelerometer. The accelerometers placed for vertical acceleration measurements are shown in Figure 5.8(b). The sampling frequencies on site for vertical data and transverse data were 80 Hz and 200 Hz respectively and corresponding recording time was 15 min and 20 min respectively. During all tests, normal traffic was simulated by using a truck to go and back in random manner not in the controlled way.

(a)

(b) Arrangement of accelerometers

INV306 data acquisition system

Figure 5.8: Data acquisition system and arrangement of accelerometers in vertical direction of arch bridge

•

Modal Parameter Identification

Once the measured time domain data are available from testing, the next work is the modal parameter identification from these data. Ambient excitation does not lend itself to frequency response functions (FRFs) or impulse response functions (IRFs) calculations because the input force is not measured in an ambient vibration test. Therefore, a modal identification procedure will need to base itself on output-only data. Two complementary modal analysis methods were implemented in this study. They were peak picking (PP) method in the frequency domain and stochastic subspace identification (SSI) method in the time domain. The data processing and modal parameter identification were carried out by MACEC, a modal analysis for civil engineering construction [166]. The raw measurement data of point 9 visualized in both time and frequency domain for vertical direction is shown in Figure 5.9. The measured data were de-trended which caused the removal of the DC-components that could badly influence the identification results. For most bridges, the frequency range of interest lies between 0 and - 98 -

Finite element model updating of civil engineering structures under operational conditions

10 Hz, containing at least the first ten frequencies within this range. So re-sampling of the raw measurement data is necessary.

Figure 5.9: Raw measurement data of Point 9 for vertical direction of arch bridge

Figure 5.10: Re-sampled data and modified power spectral density of point 9 for vertical direction of arch bridge

- 99 -

Ph.D. dissertation of Fuzhou University

Figure 5.11: Average normalized power spectral densities for full data in vertical direction of arch bridge

Figure 5.12: Typical stabilization diagram for vertical data of arch bridge

During system identification, re-sampling also leads preprocessing steps much faster due to the reduced amount of data. For vertical data, a re-sampling and filtering from 80Hz to 20Hz was carried out which leads (=72,704/4) 18,176 data points with a frequency range from 0 to 10Hz. Similarly, for transverse data, a re-sampling and filtering from 200Hz to 25Hz was carried out which leads (=24,0640/8)30,080 data points with a frequency range from 0 to 12.5Hz. A much smoother spectrum could be obtained by adjusting the power

- 100 -

Finite element model updating of civil engineering structures under operational conditions

spectral density (PSD) parameters. A window length of 1024 data points was then selected. Subsequently, the PSD was taken for all succeeding blocks of 1024 data points and an excellent noise free PSD was obtained. Re-sampled data and modified PSD of point 9 in vertical direction is shown in Figure 5.10. Mode

Mode shapes Obtained from FE analysis

Mode shapes Identified from ambient test

First vertical

Second vertical

Third vertical

Fourth vertical

First torsion (Elevation)

First transverse (Plan)

Figure 5.13: First six mode shapes obtained from FE analysis and test of an arch bridge - 101 -

Ph.D. dissertation of Fuzhou University

The peak picking technique was first applied to the data, thus the average normalized power spectral densities (ANPSDs) for all measurement data were obtained. The ANPSDs for full data in vertical direction is shown in Figure 5.11. The peak points could be clearly seen and then the frequencies were picked up. Though the peak picking method provides a good identified frequency in most of the cases, sometimes they cannot reflect enough good mode shapes. The stochastic subspace identification in time domain was then applied to the re-sampled data. The stabilization diagrams were constructed effectively. The stabilization diagram is the plot of frequency and system order containing stable system poles that can aid to select the true modes. The typical stabilization diagram is shown in Figure 5.12 for the re-sampled data. The frequencies identified from peak picking and SSI methods are very near, but the mode shapes obtained from SSI are better than peak picking. So, in this work, the modal parameters obtained from SSI method are used for FE model updating which are shown in Table 5.5 and Figure 5.13. •

Model Correlation

The modal analysis is carried out on developed FE model to get the analytical eigenfrequencies and mode shapes which are shown in Table 5.5 and Figure 5.13 respectively. The mode shapes obtained from initial FE model of the arch bridge are paired with those identified from field ambient vibration measurements as shown in Figure 5.13. It is clearly seen from the visual inspection of mode shapes that, good mode shapes of the bridge were extracted by the SSI from ambient vibration output-only data and they are paired well in each considered modes. It should be kept in mind that usually only limited number of modes could be excited using ambient vibration in practice and quite likely the spatial resolution of these modes could be poor. Considering this fact, in this work, first six modes of frequencies up to 3.86 HZ are considered for updating purpose. Table 5.5: Comparison of experimental and analytical modal properties of arch bridge before updating Mode First vertical Second vertical First torsion First transverse Third vertical Fourth vertical

Experiment 2.002 2.511 2.827 2.780 3.473 3.864

Natural frequency (Hz) Initial FE model 1.743 2.210 2.391 2.669 2.778 3.541

Error(%) -12.93 -11.98 -15.42 -3.99 -20.01 -8.35

MAC % 93.0 96.0 96.8 62.1 75.1 79.6

To evaluate the correlation of mode shapes, MAC as defined in Equation (3.4) is widely used since it is easy to apply and does not need an estimation of the system matrices. The MAC values of initial FE and experimental mode shapes are shown in Table 5.5. It demonstrates that the frequencies correlation is not so good with the maximum error of 20.01 % in the third bending mode. Figure 5.14(a) clearly shows the pairing of frequencies - 102 -

Finite element model updating of civil engineering structures under operational conditions

between initial FE model and tests emphasizing errors as departure from a diagonal line with unit slope. It can be seen that the point representing the third bending mode with a maximum difference has the largest departure from diagonal line, whereas the first transverse mode with a least error of 3.99% is near diagonal line showing well matching of that frequency. However, Table 5.5 shows that the correlation of mode shapes expressed by MAC values seems good except for the first transverse mode shape which is only 62.1%. Figure 5.14 (b) presents a plot of the MAC matrix that illustrates the orthogonal conditions between all combinations of analytical and experimental mode shapes. For well-paired modes, the MAC values are high and off diagonal values have the magnitudes Frequency obtained from ambient vibration test (Hz)

near zero. 4.5 M ode pair

4 3.5 3 2.5 2 1.5 1.5

2 2.5 3 3.5 4 Frequency obtained from FEM (Hz) (a) Frequencies correlation

4.5

(b) Mode shapes correlation

Figure 5.14: Frequency and MAC correlation of arch bridge before updating - 103 -

Ph.D. dissertation of Fuzhou University

5.5.3 Parameters Selection for Finite Element Model Updating The choice of parameters is an important step in model updating. The crucial step is how many parameters to be selected and which parameters from many possible parameters are used in FE model updating. If too many parameters are included in the FE model updating, the problem may appear ill-conditioned because only few modes are correctly recognized in the ambient vibration testing. Sensitivity analysis is carried out to see the sensitivity of parameters to various modes of interest. 1

0.8

Eigenvalue sensitivity

0.6

E of concrete filled tubular arch rib E of material of deck Thickness of deck Density of concrete filled tubular arch rib Density of deck material I of concrete filled tubular arch rib A of concrete filled tubular arch rib

0.4

0.2

0

-0.2

-0.4 1V

2V

1 Tor

1 Tra

3V

4V

3V

4V

Modes

1 0.8

Eigenvalue sensitivity

0.6 0.4 0.2

E of steel used in hollow tube E of cross girder I of cross girder about major axis Area of suspender Spring stiffness in transverse direction E of wall above deck Density of wall above deck A of prestressed cable

0 -0.2 -0.4 -0.6 -0.8 1V

2V

1 Tor

1 Tra Modes

Figure 5.15: Eigenvalues sensitivity of arch bridge to potential parameters - 104 -

Finite element model updating of civil engineering structures under operational conditions

To perform sensitivity analysis, it is better to start from all possible parameters [36] and then identify the most sensitive and non sensitive parameters to response. The possible parameters for the bridge structure may include Young modulus of elasticity, mass density of reinforced concrete components, cross sectional area and inertia moment of beam elements, thickness of deck elements and boundary conditions. In this case study of one span arch bridge, the boundary conditions are not very complicated. The abutments rest on stone strata through piles. The two ends of the deck are simply rested on the piers of two sides. So it is not taken as an updating parameter. Out of possible parameters, the eigenvalue sensitivity analysis with respect to initial estimation of parameters is performed for 15 influential parameters as shown in the Figure 5.15 and further 10 most sensitive and logical parameters are selected for updating purpose. The sensitivity coefficients are calculated using the Equation (4.37). Only the sensitivity criterion is not enough to select the updating parameters for real structures. Parameters chosen should have physical meaning and they should be able to model the errors in the FE model. If the selection of updating parameters is purely based on the sensitivity analysis, the updated model may have no physical meaning. It can be seen from Figure 5.15 that the mass density of concrete-filled steel tubular arch ribs, deck thickness, deck mass density and other selected parameters are very sensitive to most of the modes considered, whereas the parameters like the inertia moment of arch ribs, the sectional area of cable connecting two abutments are not so sensitive. The parameters which are sensitive to certain modes can be effectively updated with these sensitive modes while neglecting the others which are not affected [167]. Spring stiffness in transverse direction is selected as an updating parameter to update it separately with respect to transverse mode only. The initial values of the parameters are taken from the design blue prints and related codes. The elastic modulus of arch is obtained by considering transformed contribution from the steel tubes. The values of elastic modulus of cross girder and bridge deck are taken from code [168] corresponding to their grade of concrete.

5.5.4 Finite Element Model Updating To carry out updating of parameters, nature and number of mode shapes to be used are first confirmed. Then, an objective function and state variables are defined. In this study of a real bridge, the objective function considering the frequency residual, MAC related function and modal flexibility residuals shown in Equation (5.8) and state variables defined by Equations (5.9) and (5.10) are implemented. The value of weighting coefficients should be chosen in the objective function to reflect the relative accuracy among the measured - 105 -

Ph.D. dissertation of Fuzhou University

modes. Appropriate weights can be identified in an iterative way, for example, if for the obtained results, the eigenfrequencies correspond fully but the mode shapes show a significant discrepancy, it can be assumed that too much weight is given to the eigenfrequency residuals. Typically, the frequencies of the lower few modes are measured more accurately than those of the higher modes. By assigning proper values for α i , the difference between analytical and the measured eigenvalues of the lower modes can be further minimized. In this work, based on the iterative procedures, α i

values

corresponding to first four modes are set to be 5 times larger than the remaining modes and weighing factors for mode shape residuals are not applied. Although it is very hard to estimate the variation bound of the parameter during updating, it is assumed according to some engineering judgement. In the studies of Zhang et al. [32, 33], the maximum variation of ±40% is given for some uncertain parameters. In this work, the variation ± 20% is allowed for the thickness of deck and ±30% for all other remaining parameters. Similarly, suitable tolerances for the objective function, updating parameters as well as state variables are confirmed and at last the number of iterations to complete the optimization is defined. These values depend on the nature of the problems, so there is no fixed and fast rule to set the magnitude of these values. An iterative procedure for model tuning was then carried out. One important issue to be aware is that one has to be able to pair the mode shapes in each iteration. This is done in this paper, with the help of MAC criterion between FE mode shapes and experimental mode shapes. The selected updating parameters were estimated during an iterative process. The tuning process is over when the tolerances were achieved or pre-defined number of iterations was reached. For subproblem approximation, the optimizer initially generates random designs to establish the state variable and objective function approximations. The convergence may be slow due to random designs. It is necessary sometimes to speed up convergence by providing more than one feasible starting design. It can be simply achieved by running a number of random design tools and discarding all infeasible designs. Compared to the subproblem approximation method, the first-order method is seen to be more computationally demanding and more accurate. However, the high accuracy does not always guarantee the best solution. Here are some situations to be watched: •

It is possible for the first-order method to converge with an infeasible design. In this case, it has probably found a local minimum, or there is no feasible design space. If this occurs, it may be useful to run a subproblem approximation analysis, which is a better measure of full design spaces. Also, one may try to generate the random designs to locate the feasible design space (if any exists), and then reruns the first-order method

- 106 -

Finite element model updating of civil engineering structures under operational conditions

using a feasible design set as a starting point. So two optimization algorithms can be used complementarily. •

The first-order method is more likely to hit a local minimum since it starts from one existing point in the design space and works its way to the minimum. If the starting point is too near to a local minimum, it may find that point instead of the global minimum. If it is suspected that a local minimum has been found, one may try using the subproblem approximation method or random design generation, as described above.

•

An objective function tolerance that is too tight may cause a high number of iterations to be performed. Because the method solves the actual FE representation, it will strive

Ratio of FE to EX eigenvalue in each iteration

to find an exact solution based on the given tolerance. 1.3 1.2 1.1 1 0.9 First bending Second bending First torsion First transverse Third bending Fourth bending

0.8 0.7 0.6 70

80

90

100

110

120

130

No of iterations Figure 5.16: Convergence of six FE eigenvalues during updating of arch bridge

In this study, to carry out the optimization more than one feasible starting design was performed to run a number of random designs tools and discard all infeasible designs. The first-order optimization was first used until the convergence was achieved. The optimization is then carried out using subproblem method to see the minimization process which gives some guide line to see whether the first-order method traps in local minima or not. The changes of eigenvalues only after 70 iterations are shown in Figure 5.16 to show it more clearly. The first 30 iterations correspond to the generation of random sets, which is actually not optimization. Then, from 31 iterations, the first-order optimization is carried out. Because further iterations after 129 did not yield any progress, it was decided to terminate the optimization after 129 iterations.

- 107 -

Ph.D. dissertation of Fuzhou University Table 5.6: Comparison of experimental and analytical modal properties of arch bridge after updating Experiment 2.002 2.511 2.827 2.780 3.473 3.864 Frequency obtained from ambient vibration test (Hz)

Mode First vertical Second vertical First torsion First transverse Third vertical Fourth vertical

Natural frequency (Hz) After updating 1.962 2.493 2.815 2.770 3.256 4.027

Error(%) -1.99 -0.69 -0.42 -0.35 -6.23 4.18

MAC % 93.7 96.5 97.3 76.9 90.9 80.9

4.5 Mode pair 4 3.5 3 2.5 2 1.5 1.5 2 2.5 3 3.5 4 4.5 Frequency obtained from updated FEM (Hz)

(a) Frequencies correlation

(b) Mode shapes correlation Figure 5.17: Frequency and MAC correlation of arch bridge after updating

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Finite element model updating of civil engineering structures under operational conditions

The final correlation of frequencies and mode shapes after FE model updating is shown in Table 5.6, which shows that the difference between FE and experimental frequencies are reduced below 7% . The errors on the first four frequencies fall below 2%, which is a significant improvement comparing to the initial FE model result (Table 5.5). The correlation of mode shapes is also improved as all MAC are over 80% except for first transverse mode which also has improvement on the MAC of 76.6% from initial value of 62.1%. The well pairing of frequencies and MAC indictors are plotted in Figure 5.17. It is clearly seen that all pair points are close to the diagonal. Careful inspection of MAC matrix of Figure 5.17 (b) shows that there is an improvement on the MAC values since every mode considered has the magnitude more than the initial value shown in Figure 5.14(b).

5.5.5 Physical Meaning of Updated Parameters The changes in selected updating parameters are shown in Table 5.7. One of the most important issues in FE model updating is to check the physical meaning of updated parameters against normal practice. In this case study, it is clearly seen from Table 5.5 that all the test frequencies are more than the values from initial FE model. In most of the cases, it is observed in the Table 5.7 that there is increase in value of stiffness related parameters and decrease in value of mass related parameters after updating which is as expected. Especially, the updated values of Young’s modulus of concrete components are all increased, which are identical to the fact that the dynamic Young’s modulus of concrete is larger than the static one. The increase in the value of the inertia moment of cross girders shows that there is good interaction between cross girders and slab although they are cast separately during construction. Table 5.7: Value of updating parameters of arch bridge before and after updating Parameters selected to update

Initial values

Elastic modulus of arch (Pa)

4.56×1010

5.30×1010

16.3

3.45×10

10

10

23.5

3.00×10 0.0756 0.25 2871.0 2500.0 0.4311 0.0025 500000

10

10

30.0

Elastic modulus of cross girders (Pa) Elastic modulus of deck (Pa) 4

moment of Inertia of cross girder (m ) Thickness of bridge deck (m) Mass density of arch (kg/m3) Mass density of deck (kg/m3) Sectional area of arch (m2) Sectional area of suspender (m2) Spring stiffness in lateral direction(N/m)

- 109 -

Updated values 4.26×10

Change (%)

3.90×10 0.0972 0.246 2010.0 2144.0 0.3384 0.0021 516550

28.5 -1.6 -30.0 -14.2 -21.5 -16.0 3.3

Ph.D. dissertation of Fuzhou University

The significant reduction in the value of mass density of the composite arch is found. The fact is that the density of concrete is not always constant. The water cement ratio of concrete mix and many other uncertainties related with concrete may cause the variation of concrete density. It may also be due to the over-estimation of the actual mass density of the particular composite arch while using in the FE model. The initial value of density of the composite arch is taken according to code [168]. Similarly, there is some variation on the values of remaining parameters like sectional area of arch, area of suspender, value of spring stiffness in lateral direction, although the deviations are not much significant.

5.5.6 Conclusions from the Updating of Full Scale Arch Bridge The following conclusions are drawn from the FE model updating of a real bridge that was tested under operational conditions: •

This work presented a sensitivity based FE model updating method for real bridge structures using the test results obtained by ambient vibration technique. The objective function consisting of combination of eigenvalue residual, mode shape considered function and modal flexibility residual is used for FE model updating which is the main contribution of this work.

•

An eigenvalue sensitivity study is feasible to see the effect of various parameters to the concerned modes, according to which the most sensitive parameters can be selected for updating. Only the sensitivity criterion is not enough to select the updating parameters for real structures. Parameters chosen should also have physical meaning.

•

Since the presented FE model updating method is a sensitivity-based technique, it can be trapped in local minimum. Appropriate initial values for the updating parameters are required. This can be done by creating a random generation of a set of parameters before carrying out real optimization. Two optimization algorithms, subproblem approximation

method

and

first-order

optimization

method,

can

be

used

complementarily. •

The updated FE model of true concrete filled steel tubular arch bridge is able to produce natural frequencies in close agreement with the experiment results with enough improvement on the frequencies and MAC values of the concerned modes and still preserve the physical meaning of updating parameters.

•

Successful updating of the real bridge presented in this work demonstrates that, even for the big model, the cost of calculation is not too high and this method is practical for daily use of engineers. - 110 -

Finite element model updating of civil engineering structures under operational conditions

5.6 Chapter Conclusions This chapter deals with the FE model updating using single-objective optimization. The procedure of model updating outlined in chapter 4 is utilized. The use of dynamically measured flexibility matrices using ambient vibration method is proposed for FE model updating. The issue related to the mass normalization of mode shapes obtained from ambient vibration test is investigated and applied to use the modal flexibility for FE model updating. The algorithms of penalty function methods, namely subproblem approximation method and first-order optimization method are explored, which are then used for FE model updating. The model updating is carried out using different combinations of possible residuals in the objective functions and the best combination is recognized with the help of simulated case study. It is demonstrated that the combination that consists of three residuals, namely eignevalue, mode shape related function and modal flexibility with weighing factors assigned to each of them is recognized as the best objective function. In single-objective optimization, different residuals are combined into a single objective function using weighting factor for each residual. A necessary approach is required to solve the problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained since there is no rigid rule for selecting the weighting factors. Appropriate weights can be identified in an iterative way, for example, if for the obtained results, the eigenfrequencies correspond fully but the mode shapes show a significant discrepancy, it can be assumed that too much weight is given to the eigenfrequency residuals. For the issue of parameter selection, an eigenvalue sensitivity study is feasible to see the effect of various parameters to the concerned modes, according to which the most sensitive parameters can be selected for updating. Only the sensitivity criterion is not enough to select the updating parameters for real structures. Parameters chosen should have physical meaning. The procedure developed is applied for the FE model updating of real concrete filled steel tubular bridge. The updated FE model of true concrete filled steel tubular bridge is able to produce natural frequencies in close agreement with the experiment results with enough improvement on the frequencies and MAC values of the concerned modes and still preserve the physical meaning of updating parameters. Successful updating of the real bridge presented in this chapter demonstrates that, even for the big model, the cost of calculation is not too high and this method is practical for daily use of engineers. - 111 -

Ph.D. dissertation of Fuzhou University

CHAPTER 6 FINITE ELEMENT MODEL UPDATING USING MULTI-OBJECTIVE OPTIMIZATION

CHAPTER SUMMARY

In this chapter, FE model updating procedure using multi-objective optimization technique (MOP) is proposed. The implementation of the dynamically measured modal strain energy is investigated and proposed for model updating. The eigenfrequencies and modal strain energies are used as the two independent objective functions of multi-objective optimization technique. The weighting factor for each objective function is not necessary in this method. In MOP, the notion of optimality is not obvious since in general, a solution vector that minimizes all individual objectives simultaneously does not exist. Hence, the concept of Pareto optimality is used to characterize the objectives. The multi-objective optimization method, called the goal attainment method is used to solve the optimization problem. The Sequential Quadratic Programming algorithm is used in the goal attainment method. The FE model updating procedure is illustrated with the examples of both simulated simply supported beam and a real precast continuous box girder bridge that was dynamically measured under operational conditions.

6.1 Introduction FE model updating problems are often formulated as an optimization problem in which the objective functions are built up into a single objective function using weighting factors. Standard optimization techniques are then used to find the optimal values of the parameters that minimize that single objective function. The results of the optimization are influenced by the weighting factors assumed. The choice of the weighting factors depends on the model adequacy and the uncertainty in the available measured data, which are not known a priori. Hence, the selection of the weighting factors is a subjective task, since the relative importance among the data is not obvious but specific for each problem. The relative importance among objectives is not generally known until the system’s best capabilities are determined and trade-offs between the objectives are fully understood. As the number of objectives increases, the trade-offs are likely to become complex and less easily - 112 -

Finite element model updating of civil engineering structures under operational conditions

quantified. In this thesis, FE model updating problem is formulated in a multi-objective context using the goal attainment method of Gembicki [169], which allows the simultaneous minimization of the multiple objectives, eliminating the need for using trial and error method to find out weighting coefficients.

6.2 Multi-Objective Optimization Multi-objective optimization is concerned with the minimization of a vector of objective functions f ( x ) subject to a number of constraints or bounds. A multi-objective optimization problem is formulated as follows:

{

}

f ( x ) = f1 ( x ) , f1 ( x ) ,....., f mo ( x )

minimize x ∈ ℜn

subject to gi ( x ) = 0 i = 1,...., me

(6.1)

gi ( x ) ≤ 0 i = me + 1,...m x≤x≤x

where x is the vector of design parameters, i.e., x = { x1, x2 ....xn } , correlations

or

objective

functions

which

are

to

be

{f

}

are the

and

vector

f ......, f mo

1, 2,

minimized

function g ( x ) returns the values of the equality and inequality constraints, which are also called hard constraints and x and x are lower and upper parameter bounds respectively. Because f ( x ) is a vector, if any of the components of f ( x ) are competing, there is no unique solution to this problem. In single-objective optimization, the notion of optimality scarcely needs any explanation. The lowest value of an objective will be the target. In MOP, however, the notion of optimality is not at all obvious since in general, a solution x that minimizes all individual objectives f i ( x ) simultaneously does not exist. Instead, the concept of noninferiority [170] (also called Pareto optimality [171]) must be used to characterize the objectives.

Ω = { x ∈ℜn } subject to gi ( x ) = 0 i = 1,...., me gi ( x ) ≤ 0 i = me + 1,...m Π = { y ∈ℜ

n

}

(6.2)

x≤x≤x where

y = f ( x)

subject to x ∈ Ω

(6.3)

A noninferior solution is one, in which an improvement in one objective requires a degradation of another. To define this concept more precisely, a feasible region, Ω ,is considered in the parameter space x ∈ℜn that satisfies all the constraints as shown in - 113 -

Ph.D. dissertation of Fuzhou University

Equation (6.2). This allows one to define the corresponding feasible region for the objective function space Π as shown in Equation (6.3).The performance vector, f ( x ) , maps parameter space into the objective function space as shown in Figure 6.1 for two dimensional case.

Figure 6.1: Mapping from parameter space into objective function space

A noninferior solutions point can now be defined. A point x* ∈ Ω is a noninferior

solution if for some neighborhood of x* there does not exist a ∆x such that ( x* + ∆x ) ∈ Ω and: fi ( x* + ∆x ) ≤ f i ( x* ) i = 1,..., m f j ( x* + ∆x ) < f j ( x* ) for some j

(6.4)

In the two-dimensional representation of Figure 6.2, the set of noninferior solutions lies on the curve between C and D . Points A and B represent specific noninferior points. A and B are clearly noninferior solution points because an improvement in one objective, f1 , requires a degradation in the other objective, f 2 , i.e., f1B < f1 A , f 2 B < f 2 A . Since any point in Ω that is not a noninferior point, represents a point in which improvement can be attained in all the objectives. It is clear that such a point is of no value. Multi-objective optimization is, therefore, concerned with the generation and selection of noninferior solution points.

Figure 6.2: Set of inferior solutions

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Finite element model updating of civil engineering structures under operational conditions

In this work, the goal attainment method of Gembicki [169], with some improvement in original algorithm is used to solve the above multi-objective problem of Equation (6.1).

{

}

This method expresses a set of design goals called soft constraints f * = f1* , f 2* ,...., f m*o ,

{

}

which are associated with a set of objectives f ( x ) = f1 ( x ) , f 2 ( x ) ,...., f mo ( x ) . The

problem formulation allows the objectives to be under or over achieved enabling the designer to be relatively imprecise about initial design goals. The relative degree of under or over achievement of the goals is controlled by a vector of weighting

{

}

coefficient w = w1 , w2 ,...., wmo . It can be expressed as a standard optimization problem using the formulation as shown in Equation (6.5).

γ

minimize

γ ∈ ℜ, x ∈ Ω

such that fi ( x ) − wiγ ≤ f i

(6.5) *

i = 1,...., mo

where Ω is the feasible region in the parameter space x ∈ℜ n that satisfies all the

constraints, i.e., Ω = { x ∈ℜn } , the term wiγ introduces an element of slackness into the problem, which otherwise imposes that the goals be rigidly met. The slack variable γ is a

dummy argument to minimize the vector of objectives f ( x ) simultaneously. The goal attainment method provides a convenient intuitive interpretation of the design problem, which is solvable using standard optimization procedure. For two-dimensional problem, the optimization problem can be cast as shown in Equation (6.6). minimize

γ , x∈Ω

γ

subject to f1 ( x ) − w1γ ≤ f1*

(6.6)

f 2 ( x ) − w2γ ≤ f 2*

Figure 6.3: Geometrical interpretation of goal attainment problem for 2D problems

The geometrical interpretation of goal attainment method for two-dimensional problem

is shown in Figure 6.3. Specification of the goals { f1* , f 2*} , defines the goal point P . The - 115 -

Ph.D. dissertation of Fuzhou University

weighting vector defines the direction of search from P to the feasible space Π ( γ ) . During the optimization, γ is varied which changes the size of the feasible region. The constraint boundaries converge to the unique solution point f1s , f 2 s . The goal attainment method has the advantage that it can be posed as a nonlinear programming problem. The algorithm uses SQP method, which is presented in Chapter 4. In SQP the choice of merit function for the line search is an important issue. The merit function shown in Equation (4.59) and (4.60) proposed by Han and Powel [149] is used in this work. In the goal attainment method, more appropriate merit function can be achieved by posing Equation (6.5) as the minimax problem as shown in Equation (6.7). Following the argument of Brayton et al.[172] for the minimax optimization using SQP and applying the merit function of Equations (4.59) and (4.60) for the goal attainment problem of Equation (6.7), gives the condition as shown in Equation (6.8).

{Π i }

minimize max x ∈ ℜn i

Πi =

(6.7)

fi ( x ) − fi wi

*

i = 1,...., mo

mo

Ψ ( x, γ ) = γ + ∑ ri max {0, fi ( x ) − wiγ − f i *}

(6.8)

i =1

where ri is the penalty parameters. When the merit function of Equation (6.8) is used as the

basis of line search procedure, although Ψ ( x, γ ) may decrease for a step in a given search

direction, the function max Λ i may paradoxically increase. This is accepting degradation in the worst case objective. Since the worst case objective is responsible for the value of the objective function γ , this is accepting a step that ultimately increases the objective function to be minimized. Following the argument of Brayton et al.[172], a solution is therefore to set Ψ ( x ) equal to the worst case objective, i.e., Ψ ( x ) = max Π i i

(6.9)

In the goal attainment method, the weighting coefficient is generally set equal to zero to incorporate hard constraints. The merit function of Equation (6.9) then becomes infinite for arbitrary violations of the constraints. To overcome this problem while still retaining the features of Equation (6.9), the merit function is combined with that of Equation (4.59) giving the following Equation (6.10): ⎧γ max {0, f i ( x ) − wiγ − f i *} ; if wi = 0 ⎪ i Ψ ( x, γ ) = ∑ ⎨ max Π i , i = 1,.............m ; otherwise i =1 ⎪ ⎩ i mo

- 116 -

(6.10)

Finite element model updating of civil engineering structures under operational conditions

Further, some modifications are made to the line search and Hessian in the algorithm. In general case, the approximation to the Hessian of the Lagrangian H should have zeros in the rows and columns associated with the variable γ . By initializing H as the identity matrix, this property does not appear. H is therefore initialized and maintained to have zeros in the rows and columns associated with γ . These changes make the Hessian H indefinite, therefore H is set to have zeros in the rows and columns associated with γ , except for the diagonal element, which is set to a small positive number. This allows use of the fast converging positive definite QP method which makes the algorithm more robust.

Figure 6.4: Plot of different solution points

It is important to check whether the solution obtained from the goal attainment method are Pareto optimal points or not. In general, two methods are investigated to handle this issue. In first method, the optimization is carried out with different goals and weights to the objective functions. Each time the algorithm gives the vector of design variables and value of objective functions. Each point is plotted as shown in Figure 6.4 (hollow and solid points). From these points, the Pareto solution is selected according to the Equation (6.4). Hence, the solid points as shown in Figure 6.4 are selected. These points are called the Pareto dominant solution. From these points, one point is selected which possess the least value of f1 and f 2 . In the second method, one could try to perform one-dimensional optimization on each of the components of the result that have been received from the goal attainment method (keeping the other components the same) to determine if one can do better by changing that one component, using the definition of a Pareto point given in Equation (6.4). This method is applied for the work in this thesis.

6.3 Theoretical Procedure for Multi-objective Optimization 6.3.1 Formulation of Objective Functions and Constraints It is concluded in literatures that the algorithm that use modal strain energy is more - 117 -

Ph.D. dissertation of Fuzhou University

sensitive to local damage [57,58]. It is the main motive, to investigate the modal strain energy for FE model updating in this thesis. As explained in earlier chapters, the eigenfrequencies provide global information of the structure and they can be accurately identified through the dynamic measurements. Hence, the eigenfrequencies are indispensable quantities to be used in the updating process. In this study, both modal strain energy and eigenfrequency residuals are used as two independent objective functions in proposed multi-objective optimization technique. The objective function related to the modal strain energy residual can be defined from Equation (4.27) as: ⎛ φajT Kφaj ⎞ − 1⎟ f1 ( a ) = ∑ ⎜ T ⎜ ⎟ j =1 ⎝ φej K φej ⎠ m

2

(6.11)

It is observed from Equation (6.11) that the normalization of both analytical and experimental mode shapes should be consistent and the modal expansion of measured mode shapes to FE model DOFs is necessary. In this work, the measured incomplete mode shapes are normalized by multiplying it with the modal scale factor (MSF) which is defined in Equation (3.3). The measured incomplete mode shapes are then expanded to the complete mode shapes using the mode shape expansion method. Finally, the complete mode shapes are mass normalized with the analytical mass matrix so that the measured and analytical mode shapes are normalized with the consistent criterion. The objective function in terms of the discrepancy between FE and experimental eigenfrequencies can be posed as:

⎛ λaj ⎞ f2 ( a ) = ∑ ⎜ − 1⎟ ⎜ ⎟ j =1 ⎝ λej ⎠ mf

2

(6.12)

The objective of FE model updating procedure using multi-objective optimization technique is, to find the value of vector of updating parameter a which minimizes the error between the measured and analytical quantities of Equations (6.11) and (6.12) simultaneously. To avoid the numerical problems during minimization, the objective functions are divided by the function values at the initial parameter estimation: T ⎞ 1 ms ⎛ φaj Kφaj ⎜ ⎟ f1 ( a ) = − 1 ∑ ⎟ f1 ( a0 ) j =1 ⎜⎝ φejT Kφej ⎠

- 118 -

2

(6.13)

Finite element model updating of civil engineering structures under operational conditions mf ⎛ λaj ⎞ 1 1 f2 ( a ) = − ⎜ ⎟⎟ ∑ f 2 ( a0 ) j =1 ⎜⎝ λej ⎠

2

(6.14)

As a result, after using the constraints ai and ai as lower and upper bounds respectively for updating parameters, the minimization problem can be finally posed as: minimize f ( a ) = { f1 ( a ) , f 2 ( a )} a i = 1, 2,3,......, N subject to ai ≤ ai ≤ ai

(6.15)

6.3.2 Objective Function Gradient The goal attainment method discussed above as implemented in the optimization toolbox of Matlab [67] is used to solve the minimization of Equation (6.15). To this end, the objective function gradient is needed which makes the optimization more robust. The gradients are derived by taking the derivatives of f1 and f 2 in Equations (6.13) and (6.14) with respect to ai :

⎛ φajT Kφaj ⎞⎤ ∂f1 1 ms ⎡ ⎢ ⎜ ⎟⎥ 2* * 1 = − C ∑ 1 ⎜ φejT Kφej ⎟⎥ ∂ai f1 ( a0 ) j =1 ⎢ ⎝ ⎠⎦ ⎣ ⎡ T ⎧⎪ T ⎛ T ∂K ⎞ ⎫⎪⎤ φej ⎟ ⎬⎥ ⎢ (φej Kφej ) * C2 − ⎨(φaj Kφaj ) ⎜ φej ⎪⎩ ⎢⎣ ⎝ ∂ai ⎠ ⎭⎪⎥⎦ C1 = 2 (φejT Kφej )

{

(6.16a)

}

⎡ ∂φ T ∂φ ⎤ ∂K C2 = ⎢ aj Kφaj + φajT φaj + φajT K aj ⎥ ∂ai ∂ai ⎥⎦ ⎢⎣ ∂ai ⎡ ⎛ λ ⎤ mf ⎞ 1 ∂f 2 ⎢ 2 ⎜ aj − 1 ⎟ ∂λaj ⎥ = ∑ ∂a i f 2 ( a0 ) j =1 ⎢ ⎜ ( λ )2 λej ⎟ ∂ai ⎥ ⎠ ⎣ ⎝ ej ⎦

(6.16b)

(6.16c)

(6.17)

It is recognized that, the derivatives of FE eigenfrequencies and eigenvectors have to be evaluated in order to calculate the gradients of both objective functions. The calculation of eigenvalue and eigenvector derivatives has been presented in Equations (4.32) and (4.33) respectively. It can be seen that the derivatives of the structural stiffness and mass matrices, with respect to the design variables, are required which can be calculated analytically using Equation (4.35).

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Ph.D. dissertation of Fuzhou University

6.4 Simulated Simply Supported Beam The standard simulated simply supported beam as shown in Figure 3.2 and explained in section 3.4.4, without damage and with several assumed damage elements are considered to demonstrate the multi-objective optimization procedure proposed in this topic. Several damages are introduced by reducing the stiffness of assumed elements. The modal parameters of the beam before and after damages are shown in Table 3.1. The FE model of the undamaged beam is updated to achieve the best correlation with the modal parameters of the damaged beam. The modal strain energy and eigenfrequency are used as two objective functions. The FE model updating procedure using the multi-objective technique, as explained in above theoretical background is implemented in Matlab environment. In this simulated case study, it is assumed that the first ten bending modes are available and measurements are available at all DOFs of the model. Hence, modal expansion and normalization to calculate strain energy is not necessary. Experimental modal strain energy is calculated using the damage induced mode shape and analytical stiffness matrix defined by Equation (6.13). The objective functions and their gradients are calculated using Equations (6.13)-(6.14) and Equations (6.16a)-(6.17) respectively. The elastic modulus of each element is selected as updating parameters, which results the total of 15 updating parameters. 60 50

50

damage %

40 29.99

30 20

20 10 0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 element location Figure 6.5: Location and severity of damage in simulated beam after FE model updating

The pairing of each mode during optimization is ensured with the help of the MAC criteria between FE (undamaged) mode shapes and experimental (damaged) mode shapes. The goals and the weighting vectors to control the relative under-attainment or over-attainment of the objectives are defined. Two objective functions are simultaneously - 120 -

Finite element model updating of civil engineering structures under operational conditions

minimized by the multi-objective optimization using the goal attainment method. The selected updating parameters are estimated during an iterative process. After some iteration, the procedure is converged with the value of updated parameters. One-dimensional optimization on each of the components of the updated parameters, thus obtained, are carried out to see if one can do better by changing that one component, using the definition of a Pareto point given in Equation (6.4). Table 6.1: Comparison of experimental (assumed damage) and analytical modal properties of simulated beam after updating Mode 1 2 3 4 5 6 7 8 9 10

Damaged beam 8.245 34.920 75.080 137.508 209.028 313.581 405.839 547.260 671.483 836.938

Natural frequency (Hz) Model after Updating 8.252 34.936 75.071 137.563 208.984 313.122 405.732 545.741 670.607 834.069

Error (%) 0.084 0.045 -0.011 0.039 -0.021 -0.146 -0.026 -0.277 -0.130 -0.342

MAC % 100.00 100.00 100.00 99.99 100.00 99.99 99.99 100.00 99.99 99.99

The procedure is repeated with different values of weights and goals, until the updated parameter obtained from goal attainment method satisfies the characteristics of the Pareto solution. After some trials, the points are obtained with excellent detection of the assumed damaged location and severity. The detected damage pattern is shown in Figure. 6.5. It is clearly seen that, the detection of damage on element 3 and 8 is exact with negligible error on element 10. There is negligible error on other elements also. The excellent tuning on modal parameters is shown in Table 6.1. Hence, this case study shows that, the methodology of FE model updating presented in this work is successful to correct the uncertainties and to identify the parameters in the model due to damage. In the next case study, this procedure is applied for the FE model updating of a real bridge structure.

6.5 Precast Continuous Box Girder Bridge 6.5.1 Bridge Description and Finite Element Modeling The target Hongtang Bridge over the Ming River, located at Fuzhou city, the capital of Fujian Province, China, is a multi-span continuous-deck precast concrete motorway bridge with box girders. The construction was completed in Dec.1990 and it was the longest highway bridge across the Ming River at that time. The total length of the bridge is 1843 m, with (16m+27m+4×30m+60m+120m+60m+31×40m+8×25m) different spans. It includes

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Ph.D. dissertation of Fuzhou University

three types of spans. They are simply supported precast spans, precast truss supported spans and precast continuous girder spans. The photograph of the bridge at present condition is shown in Figure 6.6.

Figure 6.6: Photo of Hongtang bridge

The submerged portions of some of the piers of Hongtang Bridge were found to be badly scoured and eroded. So, the bridge was closed for repairing and retrofitting. In this study, one portion of bridge with 6 continuous spans, which was tested on field by ambient vibration measurements, are considered for the study. The deck of the considered spans has the form of hollow-core precast concrete girder. The considered bridge has six identical straight spans of 40m each as shown in Figure 6.7 with overall width of 11m. The deck was connected to the supporting pier and abutment by neoprene bearings as shown in Figures 6.7(b) and 6.7(c). The developed FE model is aimed at simulating the dynamic behavior of the bridge in vertical direction. The deck and piles of the bridge are modeled by two dimensional beam elements. Equivalent values for the cross sectional area and inertia moment of the box girder and piers are precalculated and given as input for the beam elements. Nodes are placed at the abutments, at the pier locations, at the points of attachment of the additional masses as well as at the points where there is change in the cross section of the girders. The concrete is considered to be homogeneous with an initial value for the Young’s modulus of 3.50 ×104 MPa and density of 2,500 kg / m3 corresponding to C50 grade of concrete.

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Finite element model updating of civil engineering structures under operational conditions

1

25

17

(a) Plan and longitudinal elevation of bridge showing sensor placement for ambient vibration test 350 42

2.5 5

42 2 1900

950

2200 350

300 350

400

All dimensions are in mm

(b) Details near the connection of the abutment and piers with deck showing the bearing (c) Details of the bearings used in supprot

Figure 6.7: Details of Hongtang bridge and bearings with measurement points for ambient vibration test

- 123 -

Ph.D. dissertation of Fuzhou University

The neoprene support is normally modeled by means of the translational and rotational springs, since this type of supports allow both vertical and rotational displacements of the bridge girder. To simulate the behavior of the translational and rotational springs, each neoprene support is modeled by means of additional connected beam elements. These elements consisted of small length (0.01m) beams that connect the deck to abutments or piers. The roller supports can be simulated in the numerical model by choosing a cross section of the connected beam element with high sectional area and low inertia moment, whereas the rigid supports can be simulated in the numerical model by choosing a cross section of the connected beam element with high sectional area and inertia moment. In this work, the translational and rotational behavior of the neoprene supports are simulated by choosing the suitable values of the sectional areas and inertia moments of these connected elements. It is demonstrated in Figure 6.7(c) that each neoprene bearing has 7 layers reinforced with steel plates in between them. The equivalent Young’s modulus of a single-layered composite element is calculated with the following formula [173]:

Ecomp = 0.1(530Sn − 418) Sn =

(MPa)

An 2tn (bn + cn )

(6.18) (6.19)

where Sn is the shape coefficient of the neoprene bearing; bn , cn , tn are the width, length and thickness of one layer of neoprene bearing respectively; and An is the area of bearing. The vertical spring stiffness of the multi-layered composite bearing is then equal to:

Kv =

Ecomp An

∑t

=

Ecomp bn cn

∑ tn

N /m

L

M1 ∆2

∆1

υ

K1

K2

Figure 6.8: Diagram to calculate the equivalent rotational stiffness of support - 124 -

(6.20)

Finite element model updating of civil engineering structures under operational conditions

One bridge support consists of several neoprene bearings arranged as shown in Figure 6.7(b). The equivalent spring stiffness in vertical direction at one support is given by the summation of all individual bearing springs stiffness obtained from Equation (6.20) at that support. The equivalent rotational spring stiffness of the support around the transverse direction of the bridge is derived with reference to Figure 6.8 by using the value of vertical stiffness. Due to the application of bending moment M 1 at support, both vertical springs with stiffness K1 and K 2 , which are separated Ld distances apart, are compressed with deflection ∆1 and ∆ 2 respectively. Using simple static, it can be obtained that:

θ=

M1 ⎛ 1 1 ⎞ + ⎟ 2 ⎜ Ld ⎝ K1 K 2 ⎠

(6.21)

When K1 = K 2 = K v , the equivalent rotational stiffness of the support can be written as:

K rot

K v Ld 2 = = Nm / rad θ 2 M1

(6.22)

In this work, the rotational stiffness obtained by Equation (6.22) is also checked with the approach shown in Equation (6.23).

K rot =

Ecomp I cg

∑t

Nm / rad

(6.23)

n

where I cg is the moment of inertia of the individual bearings as shown in Figure 6.7 (b) about the centroidal axis. The values of vertical and rotational stiffness of supports are calculated in this way. The equivalent sectional area and inertia moment of the connected beam elements calculated from these vertical and rotational stiffness are used as the initial values to simulate the behavior of neoprene supports.

Figure 6.9: Finite element model of the Hongtang bridge

The FE model is developed in MBMAT [116]. In the FE model, additional masses are lumped at the corresponding nodes to take care the extra masses due to the non-structural components of the bridges. Totally 87 nodes, 159 elements (79 beam elements, 73 mass - 125 -

Ph.D. dissertation of Fuzhou University

elements, 7 connected beam elements) are recognized in the developed FE model. The FE model of the bridge is shown in Figure 6.9.

6.5.2 Ambient Vibration Measurements, Modal Parameter Identification and Model Correlation •

Field Ambient Vibration Measurements and Modal Parameter Identification

For the purpose of the evaluation of the bridge, the field dynamic testing on the Hongtang Bridge was carried out under operational conditions by the Bridge Stability and Dynamics Lab on July 11-14, 2004. The equipments used for the ambient vibration measurements include accelerometers, signal cables, and a 32-channel data acquisition system with signal amplifier and conditioner. The force-balance (891-IV type) accelerometers and INV306 data acquisition system were used.

Figure 6.10: The arrangement of accelerometers in vertical direction of Hongtang bridge

Totally, 49 measurement points were chosen at one side of the bridge as shown in Figure 6.7 (a). The accelerometers were directly installed on the surface of the bridge deck in the vertical direction. The accelerometers placed for vertical acceleration measurements are shown in Figure 6.10. Five test setups were conceived to cover the planned testing locations of the bridge. One reference location was selected near one side of the abutment. - 126 -

Finite element model updating of civil engineering structures under operational conditions

Setups 1 to 4 consisted of ten moveable accelerometers and one fixed reference accelerometer and setup 5 consisted of 9 moveable accelerometers and one fixed reference accelerometer. The sampling frequency on site was 300 Hz.

Figure 6.11: Raw measurement data of Point 29 for vertical direction of Hongtang bridge

Figure 6.12: Re-sampled data and modified power spectral density of point 29 for vertical direction of Hongtang bridge

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Ph.D. dissertation of Fuzhou University

Figure 6.13: Average normalized power spectral densities for full data in vertical direction

Figure 6.14: Typical stabilization diagram for vertical data of Hongtang bridge

The data processing and modal parameter identification were carried out by MACEC, a

- 128 -

Finite element model updating of civil engineering structures under operational conditions

modal analysis for civil engineering construction [166]. The raw measurement data of point 29 visualized in both time and frequency domain for vertical direction is shown in Figure 6.11. For vertical data, a re-sampling and filtering from 300Hz to 30Hz was carried out which leads (=3,60,448/10) 36,045 data points with a frequency range from 0 to 15Hz. Re-sampled data and modified PSD of point 29 in vertical direction is shown in Figure 6.12. Two complementary modal parameter identification techniques are implemented in this work. They are simple peak picking (PP) method in frequency-domain and more advanced stochastic subspace identification (SSI) method in time-domain. For peak picking method, the average normalized power spectral densities (ANPSDs) for all measurement data were obtained. The ANPSDs for full data in vertical direction is shown in Figure 6.13. The peak points could be clearly seen and then the frequencies were picked up. The stochastic subspace identification in time domain was then applied to the re-sampled data. The typical stabilization diagram is shown in Figure 6.14 for the re-sampled data. It is found that, the frequencies identified from peak picking and SSI methods are very near, but the mode shapes obtained from SSI are better than peak picking. So, in this work, the modal parameters obtained from SSI method are used for FE model updating which are shown in Figure 6.15 and Table 6.2. It is clearly seen from the visual inspection of mode shapes in Figure 6.15 that, relatively good mode shapes of the bridge are extracted by the stochastic subspace identification from operational modal tests and they are paired well in each considered modes.

•

Model Correlation

The most common and simplest approach to correlate two modal models is the direct comparison of the natural frequencies. The frequencies and mode shapes obtained by analytical modal analysis are shown in Table 6.2 and Figure 6.15 respectively. The initial correlation of frequencies between FE model and test are shown in Table 6.2. It shows that the correlation of identified frequencies between two methods have maximum difference of about 22.51% in the fifth vertical bending. Table 6.2:Comparison of experimental and analytical modal properties of Hongtang bridge before updating Natural frequency (Hz) Mode First vertical Second vertical Third vertical Fourth vertical Fifth vertical

Experiment 3.072 3.291 3.542 4.149 4.611

Initial FE model 3.200 3.352 3.957 4.780 5.649 - 129 -

Error(%)

MAC %

4.166 1.853 11.716 15.208 22.511

97.1 93.4 91.5 90.1 89.6

Ph.D. dissertation of Fuzhou University

Mode

Mode shapes obtained from FE analysis

Mode shapes identified from ambient test

First vertical

Second vertical

Third vertical

Fourth vertical

Fifth vertical

Figure 6.15: First five vertical mode shapes obtained from FE analysis and test of Hongtang bridge

Frequency obtained from ambient vibration test(Hz)

6 Mode pair 5.5 5 4.5 4 3.5 3 2.5 2.5

3

3.5 4 4.5 5 5.5 Frequency obtained from FEM(Hz)

6

Figure 6.16: Frequency correlation of Hongtang bridge before updating - 130 -

Finite element model updating of civil engineering structures under operational conditions

It is clearly seen from Figure 6.16 that the pairing of frequencies between initial FE model and tests emphasizing errors as departure from a diagonal line with unit slope. It can be seen that the point representing the fifth bending mode with a maximum difference has the largest departure from diagonal line, whereas the second mode with a least error of 1.85% is near diagonal line showing well matching of that frequency. The comparison of mode shapes from FE and tests (SSI) is carried out in terms of MAC values as shown in Table 6.2. It is observed that the correlation of mode shapes is good with minimum MAC of 89.6% in fifth vertical mode. Considering the fact that only limited number of modes could be excited using ambient vibration, in this work, the first five modes are considered for updating purpose.

6.5.3 Parameters Selection for Finite Element Model Updating In general, material properties of FE model such as Young’s modulus, mass density, geometric properties such as area and inertia moment of the cross section, and stiffness of the connection are normally chosen as the updating parameters. In some cases, the parameters corresponding to several elements are expected to have similar values. In these cases one super element parameter is selected rather than individual element parameters [174]. In this study, only five important parameters are selected for updating on the basis of prior knowledge about the dynamic behavior of such kind of bridges. These are vertical and rotational spring stiffness of abutment bearings and interior pier bearings, and the bending stiffness of bridge girders. The stiffness of the neoprene supports is selected, because a relatively significant displacement at the supports could be observed in the experimental mode shapes, most clearly in the fourth and fifth modes as shown in Figure 6.15, which indicates that the supports have a finite stiffness. The angular stiffness of supports is selected due to the fact that there is more uncertainty to the initial value of this parameter. Unlike the stiffness of the concrete material, the stiffness of a structural joint is particularly difficult to estimate. As well as, in this study the simple manual eigensensitivity analysis shows that this parameter is more sensitive for the most of the eigenfrequencies considered. The initial values of selected parameters are important for the convergence of FE model updating procedure. They should be selected to be as close as possible to the actual values, so that the subsequent optimization process will find the solution quickly and the chances of finding a local minimum are reduced. As explained earlier, an initial value of elastic modulus of 3.50 ×104 MPa is adopted for the elements of the concrete bridge superstructure. - 131 -

Ph.D. dissertation of Fuzhou University

The linear and angular stiffness are represented by varying the area and inertia moment of the corresponding elements connecting the abutments or piers to the bridge girder elements as explained previously.

6.5.4 Finite Element Model Updating To carry out the FE model updating in structural dynamics, the nature and number of mode shapes to be used are first confirmed. As explained in the theoretical background, the measured incomplete mode shapes are normalized by multiplying it with the modal scale factor (MSF) which is defined in Equation (3.3). The measured incomplete mode shapes are then expanded to the complete mode shapes using the mode shape expansion method suggested by Lipkins and Vandeurzen [111], as explained in Chapter 3. In this work, the number of analytical mode shapes is set equal to the identified mode shapes, i.e., l = p . Finally, the complete mode shapes are mass normalized with the analytical mass matrix, so that the measured and analytical mode shapes are normalized with the consistent criterion. Experimental modal strain energy residual is calculated by using the measured mode shape and analytical stiffness matrix as shown in Equation (6.13). The first ten analytical modes are considered to calculate the mode shape sensitivity in Equation (4.33). The objective functions and their gradients are calculated using Equations (6.13)-(6.14) and Equations (6.16a)-(6.17) respectively. The first five identified modes of vibrations are used for current FE model updating procedure. Although it is very hard to estimate the variation bound of the parameter during updating, it is assumed according to some engineering judgment. In this work, the variation bound of ±30% is allowed for elastic modulus of deck and ±150% for spring stiffness of the bearings, since the latter has more uncertainty in the case of real structures. Once the significant parameters and their initial values have been selected, the modal strain energy and eigenfrequency objective functions are simultaneously minimized by the multi-objective optimization technique using the goal attainment method, which uses the SQP method with modification to the line search and Hessian as implemented in the optimization toolbox of Matlab [67]. The goal values that the objectives attempt to attain are approximated by knowing before hand the minima of objective functions independently, sometimes using other optimization algorithm or from the nature of objective functions. A weighting vector to control the relative under-attainment or over-attainment of the objectives is then defined. When the values of goal are all nonzero, to ensure the same percentage of under- or over-attainment of the active objectives, the weighting factor is set - 132 -

Finite element model updating of civil engineering structures under operational conditions

to the absolute value of the goal. Subsequently, an iterative procedure for model tuning is carried out. The correct mode pairing during each iteration is confirmed with the help of the MAC criterion between FE mode shapes and experimental mode shapes. The selected updating parameters are estimated during an iterative process. After some iteration, the procedure is converged with the value of updated parameters. One dimensional optimization on each of the components of the updated parameters, thus obtained, are carried out to see if one can do better by changing that one component, using the definition of a Pareto point, given in Equation (6.4). The procedure is repeated with different values of weights and goals until the updated parameter obtained from goal attainment method satisfies the characteristics of the Pareto solution. After some trials, the points are obtained with value of the updating parameters. Table 6.3: Comparison of experimental and analytical modal properties of Hongtang bridge after updating Natural frequency (Hz) Mode

Experiment

First vertical Second vertical Third vertical Fourth vertical Fifth vertical

3.072 3.291 3.542 4.149 4.611

Updated FE model 3.078 3.235 3.627 4.143 4.588

Error (%) 0.195 -1.701 2.399 -0.144 -0.498

Frequency obtained from ambient vibration test(Hz)

5 Mode pair 4.5

4

3.5

3

2.5 2.5

3 3.5 4 4.5 Frequency obtained from FEM(Hz)

5

Figure 6.17: Frequency correlation of Hongtang bridge after updating

- 133 -

MAC % 97.9 94.3 92.6 92.2 91.8

Ph.D. dissertation of Fuzhou University

The final correlations of frequencies and mode shapes after FE model updating are shown in Table 6.3. This shows that the difference between FE and experimental frequencies are reduced below 2.4% , which is a significant improvement comparing to the initial FE model result as shown in Table 6.2. The excellent tuning of frequencies can be further illustrated in Figure 6.17. It is clearly seen that, all pair points are close to the diagonal. Careful inspection of MAC in Table 6.3 shows that, there is an improvement on the MAC values, since every mode considered has the magnitude more than the initial values before updating. After updating, the changes in the selected 5 updating parameters of the bridge are shown in Table 6.4. By updating the FE model using the experimental modal data, a significant correction on the initial estimation of the stiffness of the neoprene turns out. The equivalent Young’s modulus of a single-layered composite element is calculated using Equation (6.18) which uses the nominal value of Young’s modulus of neoprene. This value depends much on the hardness of the rubber and it is therefore difficult to determine exactly. Hence, using the updating procedures as explained, these values of linear and angular stiffness are identified well. Table 6.4: Value of updating parameters of Hongtang bridge before and after updating Parameters updated Elastic modulus of material of deck (MPa)

Initial Values

Updated values

Change (%)

3.50×104

3.46×104

-1.09

10

9

Linear stiffness of support at abutment (N/m)

1.14×10

8.95×10

-21.41

Linear stiffness of support at pier top (N/m)

3.54×1010

1.11×109

-96.90

7

8

28.71

10

-45.20

Angular stiffness of support at abutment(Nm/rad) Angular stiffness of support at pier top(Nm/rad)

8.55×10

10

3.19×10

1.10×10 1.75×10

The bending stiffness of the girder elements decreases only about 1.09%. The initial value for the Young’s modulus of the concrete is quite good such that only a small correction is needed. It is demonstrated that the simulation of support conditions in the FE model updating of a real bridge is very important, and replacing the support simply with rollers cannot actually simulate the dynamic behaviors of the bridge. The neoprene supports in general provide a significant value of the rotational stiffness that can not be neglected for the analysis of such kind of bridge.

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Finite element model updating of civil engineering structures under operational conditions

6.5.5 Conclusions from Updating of a Continuous Girder Bridge The following conclusions are drawn from the study:

•

The modal strain energy and eigenfrequency residuals are proposed as two independent objective functions to carry out the FE model updating of bridges in structural dynamics. It is demonstrated that the modal strain energy residual, a new objective function, is effective and efficient in updating of structural FE models.

•

To overcome the difficulty of weighing the individual objective function of more objectives in conventional FE model updating procedure, a multi-objective optimization technique is used to extremise two objective functions simultaneously and is successfully applied to the updating of a real case study of six span continuous bridge that was tested under operational conditions.

•

Only a few updating parameters are selected on the basis of the prior knowledge about the dynamic behavior of the structure and with the help of sensitivity study. With a few number of updating parameters selected in view of sound engineering intuition, the FE model updating procedure can be effectively performed.

•

One-dimensional optimization on each of the components of the updated parameters thus obtained, are carried out to see if one can do better by changing that one component, using the definition of a Pareto point. The procedure is repeated with different values of weights and goals until the updated parameter obtained from goal attainment method satisfies the characteristics of the Pareto solution.

•

The FE model updating of a real continuous bridge has demonstrated that the accurate simulation of bridge support conditions is very important, and replacing the support simply with rollers cannot actually simulate the dynamic behaviors of such kind of bridge.

6.6 Chapter Conclusions FE model updating procedure using multi-objective optimization technique is proposed for civil engineering structures. The weighting factor for each objective function is not necessary in this method. The implementation of dynamically measured modal strain energy is investigated and proposed for model updating. Analytical expressions are developed for modal strain energy error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The eigenfrequencies and modal strain energies are used as the two independent objective functions in the multi-objective optimization.

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Ph.D. dissertation of Fuzhou University

The multi-objective optimization method, called the goal attainment method is used to solve the optimization problem. The SQP algorithm is used in the goal attainment method. In MOP, the notion of optimality is not obvious since in general, a solution vector that minimizes all individual objectives simultaneously does not exist. Hence, the concept of Pareto optimality must be used to characterize the objectives. In goal attainment problem, one-dimensional optimization on each of the components of the updated parameters obtained after optimization are carried out to see if one can do better by changing that one component, using the definition of a Pareto point. The procedure is repeated with different values of weights and goals until the updated parameter obtained from goal attainment method satisfies the characteristics of the Pareto solution. The proposed FE model updating procedure is demonstrated with the help of simulated simply supported beam. As the real case study, the elastic modulus of girders and spring stiffness of neoprene support of a precast continuous bridge are estimated using the multi-objective optimization method. Only a few updating parameters are selected on the basis of the prior knowledge about the dynamic behavior of such type of structure and with the help of sensitivity study. The updated FE model of the bridge is able to produce natural frequencies in close agreement with the experiment results with enough improvement on the frequencies and MAC values of the concerned modes and still preserve the physical meaning of updating parameters.

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Finite element model updating of civil engineering structures under operational conditions

CHAPTER 7 DAMAGE DETECTION BY FINITE ELEMENT MODEL UPDATING USING MODAL FLEXIBILITY

CHAPTER SUMMARY

As an application of FE model updating in structural dynamics, a damage detection algorithm is developed from FE model updating using modal flexibility in this chapter. The Guyan reduced mass matrix of analytical model is used for mass normalization of operational vibration mode shapes, to calculate the modal flexibility. The objective function is formulated in terms of difference between analytical and experimental modal flexibility. Analytical expressions are developed for the flexibility matrix error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The optimization algorithm to minimize the objective function is realized by using trust region strategy that makes the algorithm more robust to reduce ill-conditioning problem. The procedure of damage detection is demonstrated with the help of the simulated example of simply supported beam. The effect of noise on the updating algorithm is studied using the simulated case study. The procedure is thereafter successfully applied for the damage detection of laboratory tested reinforced concrete beam with known damage pattern.

7.1 Theoretical Background 7.1.1 Objective Function and Minimization Problem It is found that modal flexibilities are more likely to indicate damage than either natural frequencies or mode shapes and modal flexibilities are sensitive to local damage [64,129,159]. The introduction and definition of modal flexibility is presented in chapter 4. The modal flexibility error is given by the expression:

G (a) = Gexp − Gana

(7.1)

in which Gexp is the measured modal flexibility matrix obtained at the measurement DOFs;

Gana is the analytical flexibility matrix corresponding to the measured DOFs; both Gexp and Gana are calculated using Equation (4.19) and a is the vector of normalized - 137 -

Ph.D. dissertation of Fuzhou University

updating parameters, which is a column matrix. The updating parameters are the uncertain physical properties of the numerical model. The objective of FE model updating problem is to find the value of vector a , which minimizes the error between the measured and analytical modal flexibility matrices. Hence, Equation (7.1) can be posed as an optimization problem: min a

G (a)

(7.2)

G (a ) = Gexp − Φ Λ −1 ΦT

(7.3)

In Equation (7.3), Φ indicates the analytical mode shape matrix corresponding to the experimental DOFs and Λ denotes the frequency matrix containing the square of circular natural frequency. Equation (7.2) represents the function to be minimized which is obviously in matrix form. To carry out the minimization of matrix in least square sense, the norm of matrix called Frobenius norm (F-norm) is utilized in this work. Hence, the minimization problem using Frobenius norm can be presented as: min a

G (a)

2

(7.4)

F

The most frequently used matrix norm is the F-norm and it is used to provide the least-squares solution of an exact or over-determined system of equations. It is a norm of nd × nd matrix A defined as the square root of the absolute square sum of its elements. Mathematically, for [ A] ∈ℜnd ×nd :

A

F

nd nd

∑∑ a jk

=

2

(7.5)

j =1 k =1

Hence, substituting Equation (7.5) into Equation (7.4), the function to be minimized becomes:

G (a)

2 F

nd nd

(

= ∑∑ G jk ( a ) j =1 k =1

)

2

(7.6)

To avoid the numerical problems during minimization, this function is divided by the function value at the initial parameter estimate.

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Finite element model updating of civil engineering structures under operational conditions

f (a) =

G (a) G ( a0 )

2 F 2

(7.7)

F

As a result, after using the constraints ai and ai as lower and upper bounds respectively for updating parameters, the minimization problem can be finally posed as: f (a)

min a

(7.8)

such that ai ≤ ai ≤ ai ,

i = 1, 2,3,......, N

7.1.2 Objective Function Gradient The trust region Newton algorithm as implemented in the optimization toolbox of Matlab [67], is used to solve the minimization problem of Equation (7.8). To this end, the objective function gradient is needed. The gradient is found by taking the derivatives of f in Equation (7.7) with respect to ai , which is shown in Equation (7.9). In this Equation (7.9), partial derivative of G ( a ) is calculated by taking the partial derivative of Equation (7.3) with respect to ai , which gives Equation (7.10) and (7.11). ∂f 1 = ∂a i G ( a0 )

2 F

nd nd

⎛

j =1 k =1

⎝

⎞

∂ ∑∑ 2 G jk ( a ) ⎜ ∂a G ( a ) ⎟ i

(7.9)

⎠ jk

∂G ∂ =− Φ Λ −1 ΦT ) ( ∂ai ∂ai ⎡ ∂Φ −1 T ⎤ ∂G ∂ = −⎢ Λ Φ )+Φ Λ −1ΦT ) ⎥ ( ( ∂ai ∂ai ⎣ ∂ai ⎦ ⎡ ∂Φ −1 T ⎤ ∂G ∂Λ −1 T ∂ = −⎢ Λ Φ )+Φ Φ ) + ΦΛ −1 ΦT ) ⎥ ( ( ( ∂ai ∂ai ∂ai ⎣ ∂ai ⎦

(7.10)

(7.11)

In matrix algebra, the partial derivative of the inverse of a matrix can be written as:

∂Λ −1 ∂Λ −1 = −Λ −1 Λ ∂ai ∂ai

(7.12)

Hence, substituting Equation (7.12) into Equation (7.11) yields: ⎡ ∂Φ −1 T ⎤ ∂G ∂Λ −1 T ∂ = −⎢ Λ Φ ) − Φ Λ −1 Λ Φ + ΦΛ −1 ΦT ) ⎥ ( ( ∂ai ∂ai ∂ai ⎣ ∂ai ⎦ - 139 -

(7.13)

Ph.D. dissertation of Fuzhou University

It is recognized from Equation (7.13) that the derivatives of FE eigenfrequencies and eigenvectors have to be evaluated in order to calculate the gradients of both objective functions. The calculation of eigenvalue and eigenvector derivatives has been presented in Equations (4.32) and (4.33) respectively. From expressions of eigenvalue and eigenvector derivatives, it can be seen that the derivatives of the structural stiffness and mass matrices, with respect to the design variables are required which can be calculated analytically using Equation (4.35).

7.1.3 Optimization Algorithm A large scale algorithm with the trust region concept is utilized to solve the optimization problem of Equation (7.8), which can be taken as a special form of Equation (4.39). These methods are more elegant, and more powerful than the line search methods. They are also less straightforward and occasionally more expensive. For both Newton and quasi-Newton 1 methods, let us recall a quadratic model at iterate xk : M k ( d ) = f k + ∇f kT d + d T H k d , 2 which is stated in Equation (4.43a). Then a minimization of M k ( d ) gives

d = d k = − H k−1∇f k and the iteration proceed with Equation (4.47). Line search methods to find α k , as discussed in Chapter 4, are introduced because this quadratic model does not yield descent, so it is not trusted. The response is to reduce α k d k . From this point of view, then, there is something inherently unsatisfying in line search methods. If the model is not trusted which yields d k , then why keep this direction and only play with its step length. This is the motivation for trust region method. In trust region methods, it is directly defined or tried to control the region in which the quadratic model is trusted. So, there is no step size α k , but rather, the entire search direction d k is chosen by controlling its maximum size. Hence, the approximate solution of the special constrained problem is considered in More and Sorensen [175] which is stated as:

1 min M k ( d ) = f k + ∇f kT d + d T H k d 2 d

(7.14)

such that d ≤ ∆ where ∆ = ∆ k represents the bound on (or, radius of) the region in which the quadratic model is trusted. Let us assume that, it is known ∆ = ∆ k . The problem of Equation (7.14) is a constrained optimization problem in a special form, having a quadratic objective function and only one constraint that can be written as quadratic d T d − ∆ 2 ≤ 0 . The theoretical - 140 -

Finite element model updating of civil engineering structures under operational conditions

interpretation yields that: •

Either d B = − H k−1∇f k satisfies d B ≤ ∆, in which case d k = d B is the solution of Equation (7.14), i.e., the constraint is not binding.

•

Or d B > ∆, and the solution of Equation (7.14) satisfies: d k = − ( H k + ϒI ) ∇f k , −1

(7.15)

d k = ∆,

i.e., there is an additional unknown ϒ (the Lagrange multiplier) such that Equation (7.15) is satisfied. In the trust region method, then, it can be imagined as a process in which it is started with ϒ = 0 in Equation (7.15) and d k = d B > ∆ . Then ϒ is gradually increased until d k = ∆ . Thus, the steepest descent component is increased in the mix that makes up d k . Indeed, if ∆ is very small, then M k ( d ) can be approximated by f k + ∇f kT d , for which the constrained minimizer is: d =−

∆ ∇f k ∇f k

(7.16)

i.e., a steepest descent direction with a known step length. In fact, the optimal step length for the steepest descent direction is known for mk :

d = −α s ∇f k , α s =

∇f k

s

2

∇f kT H k ∇f k

(7.17)

More generally, then, it has a mix of d B and d s . To solve the nonlinear Equation (7.15), just in order to find a direction d k for the current value of ∆ is not realistic. Instead, the dogleg method composes the direction d k directly from the two directions d B and d s . •

If ∆ is so large that d B ≤ ∆ then, it is set d k = d B .

•

Otherwise, if ∆ is so small that d s ≥ ∆ , then it is taken as shown in Equation (7.18). dk =

•

∆ s d ds

(7.18)

Otherwise, it is considered as shown in Equation (7.19). d = d s +ν ( d B − d s ) , - 141 -

d

2

= ∆2

(7.19)

Ph.D. dissertation of Fuzhou University

Then, expanding Equation (7.19), one gets: ∆ 2 = d s + 2ν ( d s )

T

2

(d

B

− d s ) +ν 2 d B − d s

2

This is the quadratic equation for τ yielding:

−(d s )

T

νk =

(d

(

T − d s ) + ⎡( d s ) ( d B − d s ) ⎤ + ∆ 2 − d s ⎢⎣ ⎥⎦

2

B

dB −d

2

)

dB −ds

2

s 2

Taking positive root for ν k and putting in Equation (7.19), one gets: d k = ν k d B + (1 −ν k ) d s

(7.20)

The most important question remaining in the trust region approach is to answer how to determine ∆ = ∆ k . Consider the ratio of the actual reduction in the objective function over the predicted reduction:

ρk =

f k − f ( xk + d k )

M k ( 0) − M k ( dk )

(7.21)

The denominator is always non-negative, because H k is always positive definite. If ρ k < 0 , the step is rejected. Indeed, it is demanded that ρ k should be above some positive value. So, ∆ is decreased. On the other hand, if ρ k is close to 1, then trust in the model increases, so ∆ is increased. In the large scale problems, the value of n in Equation (4.39) is large. So, some special techniques should be used to solve the problem effectively. It is often tried to hold an assumption or condition that for large scale problems, to form the product of ∇ 2 f ( x ) with a vector, takes much fewer than n 2 operations and storage locations. A rich source of problems satisfying the above conditions is sparse matrices. A matrix is sparse, if it has a high proportion of zero entries. The structure of the nonzero entries in the matrix is also very important. In case of steepest descent method, it is assumed that in all cases under consideration, the gradient ∇f k is required and is available at a reasonable cost, say n operations. Thus, the steepest descent direction d k = −∇f k is also available, unaffected by the size of the - 142 -

Finite element model updating of civil engineering structures under operational conditions

problem. The simplicity of evaluating the steepest descent direction increases its attraction for large problems. The trouble with the steepest descent remains that the iteration converges slowly, and often this only gets worse for large problems. In case of Newton’s method, one has to solve at each iteration a linear system of equations given by Equation (4.43b). If the assumption for large scale problem stated above does not hold, then there are very few alternatives to handle the problem. There are generally two approaches for solving system of equations of the form shown in Equation (4.43b). They are direct and iterative methods. Direct methods are normally based on variants of Gaussian elimination. For really large sparse matrices and for problems satisfying the assumption of large scale problems, which are not directly sparse, iterative methods are the only viable alternatives. Iterative methods for Equation (7.14) generate a sequence of iterates, just like usual nonlinear algorithms. There are various iterative methods for the inner iteration, both for the case where H k is positive definite and when it’s not. One of such methods is the linear conjugate gradient method, which has a standard algorithm to solve the problem, quite popular in mathematics.

7.2 Simulated Simply Supported Beam The standard simulated simply supported beam as shown in Figure 3.2 and explained in section 3.4.4, without damage and with several assumed damage elements are considered to demonstrate the damage detection algorithm proposed in this topic. Several damages are introduced by reducing the stiffness of assumed elements. The modal parameters of the beam before and after damages are shown in Table 3.1. The FE model updating procedure explained in theoretical background is implemented in Matlab environment. In this case study, it is assumed that the first ten bending modes are available and measurements are obtained at all DOFs of the model. Experimental modal flexibility matrix is calculated by using the damage induced mode shape and frequency information as shown in Equation (4.19). In this study, the first ten modes are used to calculate the mode shape sensitivity in Equation (4.33). The elastic modulus of each element is used as updating parameters. Thus, there are 15 updating parameters. The tolerances of objective functions and other parameters are set. An iterative procedure for model tuning was then carried out. The pairing of each mode during optimization is ensured with the help of MAC criteria between FE mode shapes and experimental mode shapes. The selected updating parameters are estimated during an iterative process. After some iteration, the procedure is converged with excellent detection of damaged location and severity. The bar diagram, corresponding to no noise case represents the detected damage - 143 -

Ph.D. dissertation of Fuzhou University

pattern in Figure 7.1. It is clearly seen that the detection of damage on element 8 and 10 is exact with small error on element 3. There is a negligible error on other elements. The excellent tuning on modal parameters is shown in Table 7.1. 60 No noise 0.5% noise 3% noise

50

damage %

40 30 20 10 0 -10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 element location

Figure 7.1: Location and severity of damage in simulated beam after FE model updating for different cases Table 7.1: Comparison of experimental (assumed damage) and analytical modal properties of simulated beam after updating Natural frequency (Hz) Mode 1 2 3 4 5 6 7 8 9 10

Damaged beam

Model after Updating

Error(%)

MAC %

8.245 34.920 75.080 137.508 209.028 313.581 405.839 547.260 671.483 836.938

8.245 34.916 75.044 137.420 208.884 313.280 405.283 546.611 670.491 835.731

0.007 -0.009 -0.046 -0.063 -0.068 -0.095 -0.136 -0.118 -0.147 -0.144

99.999 99.999 99.999 99.998 99.997 99.994 99.995 99.992 99.991 99.992

An important aspect in the development of any model update and damage detection algorithm is its sensitivity to uncertainty in the measurements. Experimental modal testing is always associated with some kinds of measurement noise or error. This case study presents the results from numerical simulations of noise to study the effects of measurement noise on updating and damage detection procedure. To study the effect of measurement noise, it is assumed that the first ten bending modes are available and measurements are obtained at all DOFs of the model. Measurement noise is simulated by adding proportional noise to each of the simulated measured mode shapes. The noise added to the simulated measured mode - 144 -

Finite element model updating of civil engineering structures under operational conditions

shape data was obtained from the Matlab [67] RANDS function. Multiplying the eigenvectors produced by simulated measured mode shapes with a fraction of the random data set (RANDS(.) x noise percentage/100) created a set of noise data, which are added to the simulated measured mode shapes to form a set of noise contaminated eigenvector data. The noise percentage factor remains the same for each mode during a given simulation. Two cases are considered with 0.5% and 3% noise level. The experimental modal flexibility matrix is constructed using Equation (4.19) with the noisy mode shapes. Again, the elastic modulus of all 15 beam elements are used as updating parameters. The similar optimization procedure as explained above is carried out. The damage pattern identified after FE model updating for two noise cases are compared in Figure 7.1. In case of 0.5% noise, the damage detection in damaged elements is good with the values of 20.52%, 49.73% and 29.6% in elements 3, 8 and 10 respectively. Some negligible values of damages are appeared on the undamaged elements. When the noise percentage is increased to 3% , it is observed that the damaged detection error is increased with values of damage detection 23%, 51.8% and 30.5% on elements 3, 8 and 10 respectively. As well as comparatively more value of damage, for example, 9.6% in element 11 is noticed in undamaged elements also. The tuning on modal parameters in case of 3% noise is shown in Table 7.2. The table shows that tuning in frequencies and MAC values are still good. Table 7.2: Comparison of experimental (assumed damage) and analytical modal properties of simulated beam with 3% noise after updating Natural frequency (Hz) Mode 1 2 3 4 5 6 7 8 9 10

Damaged beam 8.245 34.920 75.080 137.508 209.028 313.581 405.839 547.260 671.483 836.938

Model after Updating 8.085 34.354 73.913 135.333 205.242 308.366 400.675 537.816 661.384 825.920

Error(%)

MAC %

-1.940 -1.620 -1.554 -1.581 -1.811 -1.663 -1.272 -1.725 -1.503 -1.316

99.993 99.992 99.970 99.923 99.952 99.942 99.963 99.941 99.930 99.851

This example shows that the noise has an adverse affect on damage detection capability of algorithm. In these simulations, it is assumed that the measurement noise is uniform for all of the measured modes. Actually, the lower experimental modes have less error since the measurement noise is composed of higher frequency components. As seen from the - 145 -

Ph.D. dissertation of Fuzhou University

definition of modal flexibility in Equation (4.19), a measured modal flexibility matrix has more contributions from lower modes, the procedure is less affected by the relatively higher errors that appear in high frequency modal measurements.

7.3 Experimental Beam 7.3.1 Description of Experimental Beam and Modal Parameter Identification The purpose of this experimental study is to identify the damage pattern of the damaged beam using the FE model updating procedure explained in the theoretical background. An experimental program is set up to establish the relationship between progressive damage and changes of the dynamic system characteristics by Peeters et al. [176]8 in which, more details of the testing procedure can be found. The cross section of the tested concrete beam of 6 m length is shown in Figure 7.2. The reinforcement ratio in a beam is considered to be within a realistic range. By a proper choice of steel quality, the interval between the onset of cracking and beam failure can be made large enough to allow modal analysis at well separated levels of cracking. To avoid any coupling effect between horizontal and vertical bending modes, the width is chosen to be different from the height of the beam. There are six reinforcement bars of 16 mm diameter, equally distributed over the tension and compression sides, corresponding to reinforcement ratio of 1.4%. Shear reinforcement consists of vertical stirrups of 8 mm diameter at every 200 mm. The total mass of 750 kg results the density of 2500 kg/m3. Load

2m

2m

0.20m

2m

Load

0.25m

Figure 7.2: Static test arrangement and cross section of experimental beam 8

We are grateful to Prof. De. Roeck for giving permission to use the test data of such precise experiment to verify the algorithm developed in this chapter. - 146 -

Finite element model updating of civil engineering structures under operational conditions

Six step loaded static tests as shown in Table 7.3 were conducted to produce the successive damage to the beam. After each static load step, the dynamic measurements were followed up to obtain the dynamic characteristics of damaged beam. In static setup of testing, the beam was loaded by two symmetric point loads at a distance of 2 m as shown in Figure 7.2. This test setup produces a central zone of almost uniform damage intensity. At the end of each static load step, before the dynamic test was carried out, the beam surface was visually inspected to locate and quantify the cracks. Figure 7.3 shows the observed crack pattern and damage for each static load step. In this study, the load step 5 (24 kN) is aimed to demonstrate the proposed damage identification procedure. Table 7.3: Static load steps for experimental beam Load step No. Load (kN)

1 4.0

2 6.0

3 12.0

4 18.0

5 24.0

6 26.0

Figure 7.3: Observed cracks of the experimental beam in each load step

The dynamic testing was carried out on the free-free boundary condition of the beam as shown in Figure 7.4. The free-free boundary condition avoids the influence of poor defined boundary conditions on the modal parameters. After static load step, the beam was unloaded, the supports were removed and the beam was supported on flexible springs. Acceleration time-histories were vertically measured at every 0.2 m on both sides of the beam with accelerometers. No rotational and longitudinal DOFs were measured. As a result, a total of 62 responses in the vertical direction were recorded in one series. A dynamic force was generated by means of an impulse hammer but the input was not measured. Dynamic measurement was first performed for the reference (undamaged) state of the test beam. The dynamic characteristics of the reference state serve as an initial value of parameters for current FE model updating.

- 147 -

Ph.D. dissertation of Fuzhou University

Before the system identification procedure, the original measurement data often need to be pre-processed. The electrical signals (V) were scaled according to the accelerometer sensitivities to obtain accelerations m/s2, the DC components were removed, and the data were resampled by a digital low-pass filter. During dynamic testing, the measurement data were sampled at sampling frequency of 5,000 Hz. There were 12,288 data points for each channel. The measurement data were resampled at a lower rate of 2,500 Hz.

Figure 7.4: Dynamic test setup of experimental beam Table 7.4: Bending frequencies of beam (Hz) Load steps Ref. Step1 Step2 Step3 Step4 Step5 Step6

Mode 1 Freq. 21.90 20.01 19.47 19.18 18.73 18.00 16.08

Decr.(%) 8.63 11.10 12.41 14.48 17.80 26.59

Mode 2 Freq. 60.32 56.24 54.92 53.19 51.70 50.20 47.49

Decr.(%) 6.76 8.96 11.83 14.30 16.78 21.28

Mode 3 Freq. 117.02 110.85 108.61 104.36 101.18 98.21 93.72

Decr.(%) 5.27 7.18 10.82 13.53 16.07 19.91

Mode 4 Freq. 192.02 181.44 177.99 171.67 166.73 161.87 150.84

Decr.(%) 7.58 9.33 12.56 15.07 17.55 23.16

The first four bending frequencies of the beam for each damaged state, obtained from the stochastic subspace identification, are listed in Table 7.4. The relative drop of frequencies with respect to the reference state is also given. It is found that the first bending eigenfrequency was most influenced by damage. A decrease of 26.59% was observed. From the second load step to the fifth, the change of eigenfrequencies is rather small although almost all cracks were already present as shown in Figure 7.3. However, at the ultimate damage state (last load step) the first bending frequency was decreasing much - 148 -

Finite element model updating of civil engineering structures under operational conditions

more spectacular than in the previous state because of the formation of a plastic hinge. 0.06

0.08 0.06 Modal displacement

Modal displacement

0.04 0.02 0

-0.02

0.02 0

-0.02 -0.04

-0.04

damaged reference

-0.06 -0.08

0.04

-0.06

damaged reference

-0.08 0

1

2 3 4 Position on beam(m) (a) First

5

6

0

mode shape

1

2 3 4 Position on beam(m)

5

6

5

6

(b) Second mode shape

0.06 0.08 0.06 Modal displacement

Modal displacement

0.04 0.02 0

-0.02

0.02 0

-0.02 -0.04

-0.04

-0.06

-0.06 -0.08

0.04

damaged reference 0

1

2 3 4 Position on beam(m)

damaged reference

-0.08 5

6

(C) Third mode shape

0

1

2 3 4 Position on beam(m)

(d) Fourth mode shape

Figure 7.5: Identified mode shapes of experimental beam

The stochastic subspace identification, a time-domain technique, is used for system identification without using the input measurements. The frequencies and mode shape ordinates are identified at both edges of the beam. The average value from two sides is taken to extract the mode shapes of beam which results 31 measurement points along the length of beam. As explained in theoretical background, the identified four vertical mode shapes are normalized with respect to the initial mass matrix and are shown in Figure 7.5.

7.3.2 Model Updating and Damage Detection The tested beam is analytically modeled with 30 beam elements as shown in Figure 7.6.

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Ph.D. dissertation of Fuzhou University

The elastic modulus and inertia moment implemented in the original FE model are 38 GPa and 1.66 ×10−4 m 4 respectively. The recognized modal parameters for the reference and damage state from system identification and its correlation with initial FE model are shown in Table 7.5. In the reference state, the maximum difference in frequency is 2.18% in the fourth mode and minimum MAC is 99.75% in the same mode. In the damaged case, however, there is a significant difference in the frequencies values in all four modes with maximum difference 21.46% in the first mode.

L=6m

Figure 7.6: Descritization of experimental beam

A good correlation in MAC with minimum of 99.345% in fourth mode is observed. In carrying out FE model updating, the first 4 bending modes in vertical direction are used in optimization. The experimental modal flexibility matrix is calculated using the experimental mass normalized mode shape and frequency information using Equation (4.19). The objective function and gradient are calculated with the help of Equations (7.7) and (7.9) respectively. The first fifteen FE mode shapes are used to calculate the mode shape sensitivity of Equation (4.33). Table 7.5: Comparison of experimental and analytical modal properties of experimental beam before updating Natural frequency (Hz) Mode

Experiment

1 2 3 4

21.904 60.329 117.022 192.026

1 2 3 4

18.005 50.204 98.219 161.876

Initial FE Model Reference state 22.213 61.065 119.287 196.320 Damaged state 21.870 60.956 118.218 194.176

Error (%)

MAC%

1.410 1.219 1.898 2.187

99.977 99.939 99.857 99.754

21.466 21.416 20.361 19.953

99.881 99.850 99.796 99.345

The right pairing of experimental and corresponding analytical mode during iteration are confirmed by using MAC values. The elastic modulus of individual elements is used as updating parameters. As a result, there are 30 updating parameters. Suitable tolerance of objective function and other parameters are set. The selected updating parameters were - 150 -

Finite element model updating of civil engineering structures under operational conditions

estimated during an iterative process. The updating is first carried out for the reference state to recognize the damage distribution before static load is applied. After some iterations, the procedure is converged with the detection of damage pattern coefficient a i defined in Equation (4.3a). Figure 7.7 shows the stiffness distribution of the beam in reference state after updating. The distribution has random nature with decrease and increase in stiffness along the length. The maximum decrease in stiffness is 10.37% in element 17 and maximum increase in stiffness is 5.92% in element 28. The real pattern of distribution to compare with the updated results is difficult to know. The real pattern depends on the properties of concrete and other uncertainties.

50 damage %

40 30 20 10 0 -10

1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 element location Figure 7.7: Location and severity of damage after FE model updating (reference state)

Elastic modulus of each element is corrected using a i according to Equation (4.3b). This corrected value of elastic modulus is used for the updating of damage case. The whole optimization procedure is repeated for the damage case. The detected damage distribution is shown in Figure 7.8, without assumed damage pattern as presented in Ren and Roeck [50] or using damage function as shown in Maeck et al. [177].

50

damage %

40 30 20 10 0 -10

1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 element location

Figure 7.8: Location and severity of damage after FE model updating (damaged state) - 151 -

Ph.D. dissertation of Fuzhou University

It is clearly seen that the detected damage pattern is almost symmetrical in nature. The maximum value of damage is within element 10 to 20 and the damage goes on decreasing towards the both ends of the beam. The damage distribution value of element 10 and 20 are 32.18% and 34.76% respectively with maximum value 46.97% for element 18. Even though the damaged values of elements 10 to 20 are not perfectly uniform as expected, except element 16, 17 and 18 other elements in this range have almost similar values. Table 7.6: Comparison of experimental and analytical modal properties of experimental beam after updating Natural frequency (Hz) Mode

Experiment

1 2 3 4

21.904 60.329 117.022 192.026

1 2 3 4

18.005 50.204 98.219 161.876

After updating Reference state 21.870 60.956 118.218 194.176 Damaged state 17.799 52.676 104.951 172.429

Error (%)

MAC %

-0.155 1.039 1.022 1.119

99.982 99.940 99.861 99.768

-1.144 4.923 6.854 6.519

99.900 99.881 99.801 99.670

The obtained values of the frequency and MAC after updating for the reference state and damage state are summarized in Table 7.6. The comparison of Tables 7.5 and 7.6 shows that there is significant improvement in tuning in natural frequencies and also increase in MAC values. In the damaged case, the initial difference of 21.46% in the first mode is decreased to 1.14% after updating. There is also significant improvement in remaining three modes. The maximum difference in frequency is found to be 6.85% in third mode The damage detection of the same tested beam is reported in literature [50, 177]. It is found that, the identified damage distribution obtained in this work is comparable with those reported in literatures, despite all the elements in the FE model are used as updating parameters in these case studies which is the extreme adverse condition in FE model updating. Hence, the procedure of FE updating explained in this work using modal flexibility residual can be successful for the detection of damaged elements.

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7.3.3 Conclusions from the Experimental Beam The following conclusions are drawn from the study: •

A sensitivity-based FE model updating is carried out for damage detection. The proposed procedure is verified by both simulated beam with assumed noise and a real tested reinforced concrete beam in laboratory.

•

Without the assumption of the damage pattern or damage function of the tested beam, the identified damage distribution is comparable with those from the tests and reported in literatures.

•

Despite all the elements in the FE model are used as updating parameters, which is the extreme adverse condition in FE model updating, the damage detection is still acceptable. It is demonstrated that the proposed FE model updating using the modal flexibility residual is promising for the detection of damage elements.

7.4 Chapter Conclusions This chapter deals with the damage detection application of FE model updating procedure using modal flexibility. The Guyan reduced mass matrix of analytical model is used for mass normalization of operational mode shapes to calculate the modal flexibility. The objective function is formulated in terms of difference between analytical and experimental modal flexibility. Analytical expressions are developed for the flexibility matrix error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The optimization algorithm to minimize the objective function is realized by using trust region strategy that makes the algorithm more robust to reduce ill-conditioning problem. The procedure of damage detection is demonstrated with the help of simulated examples of simply supported beam. The effect of noise on the updating algorithm is studied using the simulated case study. It is demonstrated that the behavior of proposed algorithm on noise is satisfactory and the identified damage patterns are correct. The procedure is thereafter applied for the damage detection of laboratory tested reinforced concrete beam with known damage pattern. Despite all the elements in the FE model are used as updating parameters without assuming the pattern of damage, which is considered as the extreme adverse condition in FE model updating, the identified damage pattern is comparable with those obtained from the tests. It is verified that the modal flexibility is sensitive to damage and the proposed procedure of FE model updating using the modal flexibility residual is promising for the detection of damage elements. - 153 -

Ph.D. dissertation of Fuzhou University

CHAPTER 8 CONCLUSIONS AND FUTURE WORK

CHAPTER SUMMARY

Based on the research carried out in this thesis, a list of conclusions is formulated and discussed. Whereas extensive research work on updating of analytical structural dynamics models has been carried out in this thesis, the study undertaken has revealed that some further development may be necessary and of interest.

8.1 Conclusions The following conclusions are drawn from the studies carried out in this thesis: 1. A simple toolbox is developed in Matlab environment (MBMAT) for analytical modal analysis of engineering structures. The sparse function of Matlab is used to deal with the kinds of global stiffness and mass matrices generated by FE problems to improve efficiency and capacity of FE program. Two options, namely Lanczos method and function provided by Matlab for the eigensolution are provided. The input file is first created in some specified format, and the program will read and carry out modal analysis with frequency and mode shape as output. Two well known examples are solved using the program MBMAT to demonstrate the accuracy of the program. It is observed that MBMAT results show an acceptable match with the independent solution reported in the literatures. 2. Various available techniques for correlating analytical and experimental data and expanding experimental mode shapes for successful FE model updating are investigated. Frequency and MAC correlations are recognized as the best technique to use in FE model updating application. MAC has the potential to pair the correct analytical and experimental mode shapes automatically at each iteration during FE model updating. Two new methods for modal expansion are proposed using modal flexibility and normalized modal difference. Defining objective function and constraints, selecting updating parameters and using robust optimization algorithm are recognized as three critical issues of FE model updating. 3. The FE model updating frame work is developed for civil engineering structures under - 154 -

Finite element model updating of civil engineering structures under operational conditions

operational conditions that are excited by ambient vibration. Two new residuals, namely modal flexibility and modal strain energy are proposed and formulated to use in FE model updating. Many related issues, including the objective function, the gradients of the objective function, different residuals and their weighting and possible parameters for model updating are investigated. The physical parameters, geometrical parameters and boundary conditions of FE model are probable updating parameters in real civil engineering structures. Analytical algorithm is developed to calculate the modal sensitivities using the formula of Fox and Kapoor. The issue related to the mass normalization of mode shapes obtained from ambient vibration test is investigated and applied to use the modal flexibility for FE model updating. The Guyan reduced mass matrix of analytical model is used for mass normalization of operational mode shapes to calculate the modal flexibility. 4. The success of FE model updating depends heavily on the selection of updating parameters. The updating parameter selection should be made with the aim of correcting uncertainties in the model. Moreover, the objective function which represents differences between analytical and experimental results need to be sensitive to such selected parameters. Otherwise, the parameters deviate far from their initial values and lose their physical foundation in order to give acceptable correlations. To avoid the ill-conditioned numerical problem, the number of parameters should be kept as low as possible. Thus, the parameter selection requires considerable physical insight into the target structure, and trial-and-error approaches are used with different set of selected parameters. In this study, the eigenvalue sensitivity of the different possible parameters is calculated and then the most sensitive parameters with some engineering intuition are elaborately selected as the candidate parameters for updating. 5. FE model updating procedure using single-objective optimization is established and implemented. The use of dynamically measured flexibility matrices is proposed and investigated for model updating. In single-objective optimization, different residuals are combined into a single objective function using weighting factor for each residual. A necessary approach is required to solve the problem repeatedly by varying the values of weighting factors until a satisfactory solution is obtained since there is no rigid rule for selecting the weighting factors. Appropriate weights can be identified in an iterative way. The algorithms of penalty function methods, namely subproblem approximation method and first-order optimization method are explored, which are then used for FE model updating. 6. The FE model updating is carried out using eigenfrequecy residual, mode shape - 155 -

Ph.D. dissertation of Fuzhou University

related function, modal flexibility residual and different combinations of possible residuals in the objective functions with the help of simulated case study. It is demonstrated that the combination that consists of three residuals, namely eignevalue, mode shape related function and modal flexibility with weighting factor assigned to each of them is recognized as the best objective function. The single-objective optimization with eigenfrequecy residual, mode shape related function and modal flexibility residual is thereafter applied for the FE model updating of a full-size concrete filled steel tubular arch bridge that was dynamically measured under operational conditions. The updated FE model of true bridge is able to produce natural frequencies in close agreement with the experiment results with enough improvement on the frequencies and MAC values of the concerned modes and still preserve the physical meaning of updating parameters. Successful updating of the real bridge demonstrates that, even for the big model, the cost of calculation is not too high and this method is practical for daily use of engineers. 7. FE model updating procedure using multi-objective optimization technique is proposed. The weighting factor for each objective function is not necessary in this method. The implementation of dynamically measured modal strain energy is investigated and proposed for model updating. Analytical expressions are developed for modal strain energy error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The eigenfrequencies and modal strain energies are used as the two independent objective functions in the multi-objective optimization technique. The goal attainment method is used to solve the optimization problem where the Sequential Quadratic Programming algorithm is implemented. In multi-objective optimization, the notion of optimality is not obvious since in general, a solution vector that minimizes all individual objectives simultaneously does not exist. Hence, the concept of Pareto optimality is used to characterize the objectives. In the goal attainment problem, one-dimensional optimization on each of the components of the updated parameters obtained after optimization is carried out to see if one can do better by changing that one component, using the definition of a Pareto point. The procedure is repeated with different values of weights and goals until the updated parameter satisfies the characteristics of the Pareto solution. 8. The multi-objective optimization based FE model updating procedure is demonstrated by using the eigenfrequency and modal strain energy residuals with the help of simulated simply supported beam, which demonstrates that the proposed procedure is robust and excellent for the detection of assumed damage pattern. As a real case study, - 156 -

Finite element model updating of civil engineering structures under operational conditions

the elastic modulus of bridge girder and spring stiffness of neoprene support of a precast continuous box girder bridge that was tested on field under operational conditions are estimated using the multi-objective optimization method. Only a few updating parameters are selected on the basis of the prior knowledge about the dynamic behavior of such type of structure and with the help of sensitivity study. The updated FE model of the bridge is able to produce natural frequencies in close agreement with the experiment results with enough improvement on the frequencies and MAC values of the concerned modes and still preserve the physical meaning of updating parameters. 9. As an application of FE model updating in structural dynamics, a FE model updating based damage detection algorithm is proposed using modal flexibility. The Guyan reduced mass matrix of analytical model is used for mass normalization of operational mode shapes to calculate the modal flexibility. The objective function is formulated in terms of difference between analytical and experimental modal flexibility. Analytical expressions are developed for the flexibility matrix error residual gradient in terms of modal sensitivities found via method of Fox and Kapoor. The optimization algorithm to minimize the objective function is realized by using trust region strategy that makes the algorithm more robust to reduce ill-conditioning problem. 10. The procedure of damage detection is demonstrated with the help of simulated example of simply supported beam. The effect of noise on the updating algorithm is studied using the simulated case study. It is demonstrated that the behavior of proposed algorithm on noise is satisfactory and the identified damage patterns are correct. The procedure is thereafter applied for the damage detection of laboratory tested reinforced concrete beam with known damage pattern. Despite all the elements in the FE model are used as updating parameters, which is considered as the extreme adverse condition in FE model updating, the identified damage pattern is comparable with those obtained from the tests. It is verified that the modal flexibility is sensitive to damage and the proposed procedure of FE model updating using the modal flexibility residual is promising for the detection of damaged elements.

8.2 Significance of the Study FE model updating with application to civil engineering structures (CES) based on operational modal analysis (OMA) has significant theoretical importance and application potentials. Field testing and modeling, FE model updating as well as damage detection are gaining popularity as a tool for better management of bridges. The following achievements - 157 -

Ph.D. dissertation of Fuzhou University

and contributions show the originality and significance of Ph.D. work to the existing state of knowledge. 1. This research proposes two new residuals, namely modal flexibility and modal strain energy for FE model updating of CES under operational condition. The relevant gradients are derived, and successfully applied to the real bridges. 2. FE model updating procedure using single-objective optimization is investigated. The model updating is carried out using different combinations of possible residuals in the objective functions and the best combination is recognized with the help of simulated case study. This objective function is applied successfully for the FE model updating of a real concrete filled steel tubular arch bridge in which eigensensitivity method with engineering judgment is used for updating parameter selection. 3. A multi-objective optimization technique is proposed for FE model updating of civil engineering structures, which eliminates the need of weighting factor for each objective function and therefore is advantageous compared to traditional single-objective counterpart. The FE model updating procedure using the multi-objective optimization method is illustrated with the examples of both simulated simply supported beam and a real case study. The latter is used to estimate the elastic modulus of deck of bridge and spring stiffness of neoprene support of precast continuous box girder bridge. 4. Vibration-based structural damage identification based on FE model updating is explored with newly proposed modal flexibility residual. 5. Matlab version of FE analysis of CES as a toolbox is developed for further model updating/optimization study. 6. Many FE model updating related issues, including the selection of different residuals, determination of objective function, derivation of the gradients of the objective function with selection of their weighting, system matrix condensation and mode shape expansion, etc. are discussed with emphasizing the model updating of CES based on OMA. Setting-up of an objective function, selecting updating parameters and using robust optimization algorithm are three crucial steps in model updating. These three issues are thoroughly investigated in this work theoretically and applied them in real bridges. Objective function part is studied in detail and proposed new objective functions. Updating parameters are selected using eigensensitvity criteria with engineering intuition. The robust and elegant optimization algorithms are used. The basic procedure of FE model updating is improved using the findings known from the mathematical optimization study. - 158 -

Finite element model updating of civil engineering structures under operational conditions

8.3 Future Research Areas for possible further studies are summarized below. 1. The proposed FE model updating method using modal strain energy and damage detection algorithm using modal flexibility, requires global mass and stiffness matrices of an analytical model of a structure to calculate the objective function gradient matrix, and eigenproblem must be solved in every iteration. In this thesis, Matlab toolbox is developed to solve the issue. For the method to be more practical or more flexible, interfacing the updating program with existing commercial FE package may be advantageous for future use. 2. The FE model updating is an inverse problem that is used to identify the unknown physical parameters of the structures. The methodology can be applied for other inverse application of parameter estimation. For example, the procedure can be applied to estimate the distribution and history of wind pressures on structure, based on strain and acceleration measurements. Broadly, the loads acting on a structure can be known from the observation of its response. Hence, the monitoring of the structures is possible because once the loads acting in the structure is known the stresses and other information can be found. 3. One of the possibilities to FE model updating is the use of static measurements in the objective function during optimization. For example, the static strain measurement can be used for this purpose. Strain measurements do not require a frame of reference. This makes them superior to static displacement measurements. Strain measurements are also more accurate than ordinary displacement measurement and can easily be used on bridges, buildings, and space structures. Hence, one objective may be to investigate the local feature of measured strain in structural static compared with frequency or mode shape (global) in structural dynamics. 4. Other possibility for FE model updating is to combine the dynamic modal data with static strain or displacement data. The static deflection is determined absolutely but requires also the measurement of the loading force. A difficulty of the approach is that the static and the dynamic stiffness differ. Small displacements occur during the dynamic vibration measurements and a linear behavior can be assumed, which is not the case for the relatively large static displacements. 5. For model updating, it is necessary to know which regions of the structure are poorly modeled. Mottershead et al [27] recognized joints or boundaries as the sources and proposed a new parametrization. However, there are usually many alternatives for complex structures. In theses cases, systematic approaches to locate suspicious regions - 159 -

Ph.D. dissertation of Fuzhou University

are useful. Location methods address such problems. Research in this area has shown that detecting these regions presents a considerable challenge. The simplest error location techniques are the degree of freedom correlation techniques like COMAC. But these methods have no physical basis. Among the reference-based methods, the force balance method calculates a residual force vector using system matrices. And the large residual components are taken to indicate regions of errors in this method. Hence, there is still lack of effective modeling error localization means. Further work is suggested in this direction. 6. In the FE model updating procedure a minimization problem is solved, which is formulated as a least squares problem. So the convergence issues to find the global minimum point is important and difficult since FE model updating is an inverse problem. The global search methods, such as genetic algorithms (GA) and simulated annealing (SA) are in general more robust. The main drawback of such algorithms is that they require a large number of function evaluations since they are based on probabilistic searching without the use of any gradient information. Hence, further work is suggested to develop and use some new global optimization routines. 7. The recondite nature of nonlinearity has made development of correct analytical models of nonlinear systems a difficult task. So, the FE model updating is still limited in linear system with low frequency range etc. Although analytical methods and numerical tools are available for modeling specific types of nonlinearity a systematic investigation of the formulation and resolution of increase problems for nonlinear dynamics has not been reported in the literature. An investigation about the issues of nonlinear FE model updating is useful for future work.

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Ph.D. dissertation of Fuzhou University

APPENDICES APPENDIX A Mathematical Background •

Let B ∈ℜn×n be symmetric, i.e., BT = B .Then, if one considers the quadratic form yT By , for any y ∈ ℜn , B is said to be positive definite if: yT By > 0, ∀y ∈ℜn ,

•

y≠0

(A.1)

Convergence rate of algorithm

Let xk be a sequence in ℜn that converges to x* , with subscript k denoting the iteration number. A method is said to be convergent if: ∈k +1 ∈k

≤1

(A.2)

holds at each iteration, where ∈k = xk − x* . Similarly, an algorithm has r order (or rate) of convergence if:

lim k →∞

∈k +1 ∈k

r

=C

(A.3)

where C is the error constant. From the above it is seen that, C ≤ 1 when r = 1 . In this case, when r = 1 , the algorithm exhibits a linear convergence rate (which corresponds to slow convergence). When r = 2 , the rate is quadratic (which corresponds to fast convergence). When r = 1 and C = 0 , the algorithm has a superlinear rate of convergence (which also corresponds to fast convergence). In view of stability, there should be no intolerable magnification of round off errors by the algorithm. •

The l2 norm of a vector y ∈ℜn is:

- 172 -

Finite element model updating of civil engineering structures under operational conditions

n

∑y

y2= y =

•

2 i

i =1

= ( yT y )

12

=< y, y >1 2

(A.4)

Frobenius Norm

The most frequently used matrix norms are the F-norm (Frobenius norm, it is also called Euclidean or Shur norm), for [ A] ∈ ℜ N ×m :

A

•

F

N

2

m

∑∑ a

=

i =1 j =1

ij

(A.5)

Let f : ℜn → ℜ be twice continuously differentiable f ∈ G 2 ( ℜ ) . The Hessian matrix is: ⎛ ∂2 f ⎜ 2 ⎜ ∂x1 ⎜ ∂2 f ⎜ H ( x ) = ∇ 2 f ( x ) = ⎜ ∂x2 ∂x1 ⎜ . ⎜ ⎜ ∂2 f ⎜ ∂x ∂x ⎝ n 1

∂2 f ∂x1 ∂x2

.

∂2 f ∂x22

.

.

.

∂2 f ∂xn ∂x2

- 173 -

∂2 f ⎞ ⎟ ∂x1 ∂xn ⎟ ∂2 f ⎟ ⎟ ∂x2 ∂xn ⎟ . ⎟ ⎟ ∂2 f ⎟ ∂xn2 ⎟⎠

(A.6)

Ph.D. dissertation of Fuzhou University

APPENDIX B MODAL SENSITIVITIES This appendix gives the derivation of the formulas to calculate the sensitivities of the eigenvalues and mode shapes with respect to changes in the correction factors of the physical parameters. The formulas were first developed by Fox and Kapoor [132].

B.1 Eigenvalue Derivatives Let us assume that λ j and φ j be a solution of the undamped eigenvalue problem:

Kφ j = λ jM φ j

(B.1)

Premultiplying Equation (B.1) by φ j gives: T

φTj ⎣⎡ K − λ j M ⎦⎤ φ j = 0

(B.2)

Differentiating Equation (B.2) with respect to the set of correction factors that are assigned to the physical parameters ai gives: ∂φTj

⎡⎣ K − λ j M ⎤⎦ φ j + φTj ∂ai

∂ ⎡⎣ K − λ j M ⎤⎦ ∂ai

∂φ φ j + φTj ⎡⎣ K − λ j M ⎤⎦ j = 0 ∂ai

(B.3)

Due to Equation (B.1) the first and third term of Equation (B.3) are equal to zero and thus:

φTj

∂ ⎡⎣ K − λ j M ⎤⎦ ∂a i

φj = 0

(B.4)

The derivative of term in the middle of Equation (B.4) gives: ∂ ⎣⎡ K − λ j M ⎦⎤ ∂ai

=

∂K ∂λ j ∂M − M −λj ∂ai ∂ai ∂ai - 174 -

(B.5)

Finite element model updating of civil engineering structures under operational conditions

Putting this value in Equation (B.4),

φTj

∂λ j ∂K ∂M φ j − φTj φj = 0 M φ j − φTj λ j ∂ai ∂ai ∂ai

(B.6)

Applying the orthogonality condition to the second term of Equation (B.6)

φTj

∂λ ∂K ∂M φ j − j − φTj λ j φj = 0 ∂ai ∂ai ∂ai ∂λ j

⎡ ∂K ∂M ⎤ = φTj ⎢ −λj ⎥φj ∂ai ∂ai ⎦ ⎣ ∂ai

(B.7)

B.2 Eigenvector Derivatives For an undamped equation of motion, one obtains,

([ K ] − λ [ M ]) j

∂ {φ } j ∂ai

⎛ ∂[K ] ⎞ ∂ [ M ] ∂λ j +⎜ − λj − [ M ] ⎟ {φ} j = {0} ∂ai ∂ai ⎝ ∂ai ⎠

(B.8)

which includes the eigenvector and eigenvalue sensitivities of mode j with respect to a selected design parameter correction factor a . Unlike the eigenvalue sensitivity as shown in appendix B.1, Equation (B.8) cannot be directly solved for the eigenvector sensitivity as

([ K ] − λ [ M ]) is singular. Fox and Kapoor [132] proposed therefore to assume that: j

∂φ j ∂ai

d

= ∑ β jq φq

(B.9)

q =1

i.e., the eigenvector derivative is a linear combination of the eigenvectors itself. Although this assumption is reasonable, the number of available modes is often limited and d is usually smaller, namely m , and therefore, the number of included modes m directly affects the accuracy of the eigenvector sensitivities. By differentiating Equation (B.1) it follows that

- 175 -

Ph.D. dissertation of Fuzhou University

∂ ⎡⎣ K − λ j M ⎤⎦ ∂a

φ j + ⎣⎡ K − λ j M ⎦⎤

∂φ j ∂a

=0

(B.10)

Substituting Equation (B.9) into Equation (B.10) ∂ ⎡⎣ K − λ j M ⎤⎦ ∂a

(B.11)

∂ ⎡ K − λ j M ⎤⎦ ⎡⎣ K − λ j M ⎤⎦ φq = − ⎣ φj ∂a

(B.12)

q =1

d

∑β

or,

d

φ j + ⎣⎡ K − λ j M ⎦⎤ ∑ β jq φq = 0

q =1

jq

Premultiplying Equation (B.12) by φsT ,with s ≠ j;

d

∑β q =1

jq

∂ ⎡ K − λ j M ⎤⎦ φsT ⎡⎣ K − λ j M ⎤⎦ φq = −φsT ⎣ φj ∂a

(B.13)

Due to orthogonality properties, the left hand side of Equation (B.13) is equal to zero except for s = q .

∂ ⎡ K − λ j M ⎤⎦ β jq ( λq − λ j ) = −φqT ⎣ φj ∂a

for q ≠ j

(B.14)

Expanding the right side,

β jq ( λq − λ j ) = −φqT

∂λ ∂K ∂M φ j + j φqT M φ j +λ jφqT φ ∂a ∂a ∂a

j

(B.15)

As q ≠ j , the second term of right side of Equation (B.15) is zero. So,

β jq ( λq − λ j ) = −φqT

β jq =

∂K ∂M φ j +λ jφqT φ ∂a ∂a

1 φT λq − λ j q

j

∂M ⎤ ⎡ ∂K ⎢⎣ ∂a − λ j ∂a ⎥⎦ φ j

- 176 -

for q ≠ j

(B.16)

Finite element model updating of civil engineering structures under operational conditions

It is clear from Equations (B.4) and (B.14) that, for q = j , the coefficients β jq have to calculated separately. By differentiating φ Tj M φ j = 1 ,it follows that:

∂φ Tj ∂a

M φ j + φ Tj

∂φ ∂M φ j + φ Tj M j = 0 ∂a ∂a

2φ Tj M

∂φ j ∂a

= −φ Tj

∂M φj ∂a

(B.17)

Substituting Equation (B.9) into Equation (B.17) d

2φ Tj M ∑ β jq φq = −φ Tj q =1

∂M φj ∂a

d

2∑ β jq φ Tj M φq = −φ Tj q =1

(B.18)

∂M φj ∂a

and due to orthogonality condition, one obtains 1 2

β jq = − φ Tj

∂M φj ∂a

(B.19)

Hence from Equations (B.16) and (B.19) ,

⎧ T ⎡⎛ ∂K ∂Μ ⎞ − λj ⎪φq ⎢⎜ ⎟ ∂ai ⎠ ⎪ ⎣⎝ ∂ai β jq = ⎨ ⎪− 1 φ T ∂Μ φ ⎪ 2 j ∂a j , i ⎩

- 177 -

(λ

j

⎤ − λq ) ⎥ φ j , q ≠ j ⎦ q= j

(B.20)

Ph.D. dissertation of Fuzhou University

ACKNOWLEDGEMENTS First of all, I would like to acknowledge the support and contributions of my supervisor, Professor Wei Xin Ren. His enthusiasm for taking creative and different approaches to problems and the ideas have made this research an interesting and fruitful experience. In addition, his commitment to performing relevant and high quality research has kept me focused throughout my doctoral studies. I would like to acknowledge Associate Professor Zhouhong Zong for his beneficial suggestions and warm help during my staying in China. I also would like to thank Professor Baochun Chen and Professor Lin-Hai Han for their constant inspiration and best wishes for my study. I address my sincere gratitude to Professor Michael I. Friswell and Associate Professor Dionisio Bernal, for their kind reply and through discussion of my each queries just beginning from the research work. I would also like to thank all my colleagues from Bridge Stability and Dynamics Lab, for valuable scientific discussions about the subject described in this thesis. Similarly, special thanks go to Professor Prem Nath Maskey for his exceptional instruction during my Master’s studies at Tribuhuvan University, Nepal as well as his extreme care and attention during my staying in China. I am also grateful to the National Science Foundation of China (NSFC) for providing the financial support for this project under Research Grant No. 50378021 to Fuzhou University. Lastly, I owe my deepest and most sincere thanks to my wonderful parents, Badri Nath Jaishi and Devi Jaishi, my dear brother Bikas and lovely sisters Shanti, Sila and Silu. They shared my successes and disappointments at every moment of time. Their unwavering support and selfless attitudes are an endless source of inspiration and confidence. Words could never express the thanks they deserve. Bijaya Jaishi Fuzhou, China

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Finite element model updating of civil engineering structures under operational conditions

CURRICULUM VITAE Name: Bijaya Jaishi Date of Birth: February 5, 1974 Education:

2002-2005 Ph.D. student at the Department of Civil Engineering and Architecture, Fuzhou University, People’s Republic of China 1999-2001 M.Sc. in Structural Engineering, Institute of Engineering, Tribhuvan University, Nepal 1994-1998 B.E. in Civil Engineering, Institute of Engineering, Tribhuvan University, Nepal Work Experience:

2001-2002 Assistant Professor, Institute of Engineering, Tribhuvan University, Kathmandu, Nepal 1998-1999 Structural Engineer, Nepal Engineering Consultancy, Kathmandu, Nepal Publications: (a) Thesis: Seismic capacity evaluation of multi-tiered temples of Nepal, M.Sc. thesis, Department of Civil Engineering, M.Sc. program in structural engineering, Kathmandu, Nepal, Dec 2001. (b) Publications in international journals [1] Bijaya Jaishi, Wei-Xin Ren: Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimization technique, Finite Elements in Analysis and Design, 2004. (Temporarily accepted) [2] Bijaya Jaishi, Wei-Xin Ren: Damage detection by finite element model updating using modal flexibility residual, Journal of sound and vibration, 2005. (in press). [3] Bijaya Jaishi, Wei-Xin Ren: Structural finite element model updating using ambient vibration test results, Journal of Structural Engineering, ASCE, Vol.131, No.4, pp.617-628, 2005. [4] Zhou-Hong Zong, Bijaya Jaishi, Ji-Ping Ge, Wei-Xin Ren: Dynamic analysis of a half-through concrete-filled steel tubular arch bridge, Engineering Structures, Vol. 27, pp.3 -15, 2005

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Ph.D. dissertation of Fuzhou University [5] Bijaya Jaishi, Wei-Xin Ren, Zhou-Hong Zong, Prem Nath Maskey: Dynamic and seismic performance of old multi-tiered temples in Nepal, Engineering Structures, Vol. 25, pp.1827-1839, 2003

(c) Conference proceedings [1] Wei-Xin Ren and Bijaya Jaishi: Multiobjecive optimization based finite element model updating of bridges through operational identification, First International Operational Modal Analysis Conference – IOMAC, Denmark, 2005 (Accepted). [2] Wei-Xin Ren, Bijaya Jaishi and Zhou-Hong Zong: Concrete-filled steel tubular arch bridge: Dynamic testing and FE model updating, Arch Bridges IV: Advances in Assessment, Structural Design and Construction, P. Roca and C. Molins (Eds.), © CIMNE, Barcelona, pp.703-715, 2004. [3] Wei-Xin Ren and Bijaya Jaishi: Finite element model updating of bridges by using ambient vibration testing, Progress in Structural Engineering, Mechanics and Computation, Edited by Alphose Zingoni, A.A. Balkema Publishers, pp.709-714, 2004. [4] Bijaya Jaishi and Wei-Xin Ren: Objective functions for finite element model updating in structural dynamics, Proceedings of Eighth International Symposium on Structural Engineering for Young Experts, August 20-23, Xi’an, China, Science Press, pp.50-55, 2004. [5] Bijaya Jaishi, Wei-Xin Ren, Zhouhong Zong and Prem Nath Maskey: Dynamic analysis of old multi-tiered temples of Nepal, IMAC-XXI: A Conference on Structural Dynamics, February 3-6, 2003, Kissimmee, Florida, USA, 2003. [6] Zhou-Hong Zong, Bijaya Jaishi, Youqin Lin, and Weixin Ren: Experimental modal analysis of a CFT arch bridge, IMAC-XXI: A Conference on Structural Dynamics, Kissimmee, Florida, USA, 2003. [7] Bijaya Jaishi,Wei-Xin Ren and Prem Nath Maskey: Seismic capacity evaluation of old multi-tiered temples. China-Japan Workshop on Vibration Control and Health Monitoring of Structures and Third Chinese Symposium on Structural Vibration Control, Shanghai, China, 2002. [8] Zhou-Hong Zong, Bijaya Jaishi, Youqin Lin, Wei-xin Ren: Experimental and analytical modal analysis of CFT arch bridge. China-Japan Workshop on Vibration Control and Health Monitoring of Structures and Third Chinese Symposium on Structural Vibration Control, Shanghai, China, 2002.

(d) Papers in Chinese journals and internal reports [1] Bijaya Jaishi and Wei-Xin Ren: Use of modal flexibility and NMD for mode shape expansion. A research report. Department of Civil Engineering, Fuzhou University, P.R. of China, 2004. [2] Wei-Xin Ren, Bijaya Jaishi, Zhou-Hong Zong and Prem Nath Maskey: Ambient vibration measurements and dynamic study of three old temples of Nepal. A research report. Department of Civil Engineering, Fuzhou University P.R. of China, 2003. [3] Zhou-Hong Zong, Bijaya Jaishi, Lin You-Qin, Wei-Xin Ren: Experimental and analytical modal analysis of a concrete-filled tube arch bridge over Xining Beichuan River, Journal of China Railway Society, Vol.25, pp89-96, 2003.

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