55525700 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

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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：

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

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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。

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。

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。

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)。

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

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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，

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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。

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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

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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.

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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

ix

CONTENTS

Abstract in Chinese i

Abstract vi

Contents ix

List of Symbols xiii

List of Figures xx

List of Tables xxii

1. Introduction 1

1.1 Background……………………………………………………………………... 11.2 Objectives and Scopes of the Research………………………………………… 31.3 Contributions of the Research………………………………………………….. 31.4 Literature Survey……………………………………………………………….. 5 1.4.1 Direct Methods…………………………………………………………. 5 1.4.2 Sensitivity Based Methods……………………………………………... 6 1.4.2.1 Parameter Identification of Structures………………………... 7 1.4.2.2 Damage Detection of Structures……………………………… 81.5 Organization of Dissertation……………………………………………………. 11

2 Finite Element Modal Analysis Toolbox Development 14

2.1 Introduction………….……………………………………………………..…… 142.2 Shape Functions……….………………………………………………………... 152.3 Finite Element Mass and Stiffness Matrices……………………………………. 162.4 Governing Equation and Solution……………………….……………………... 202.5 Element Types…………………………..……………….……………………… 222.6 Program Realization in Matlab…………..……………….…………………….. 242.7 Program Verification…..………………………………….…………………….. 25 2.7.1 Time Period of Simply Supported Beam………….……………………. 25 2.7.2 Plane Frame – Bathe and Wilson Eigenvalue Problem…………….…... 272.8 Chapter Conclusions…………………….……………….……………………... 29

3 Finite Element Model Updating in Structural Dynamics 30

3.1 Finite Element Modeling, Modal Testing, and System Identification for Model Updating………………………………………………………………………... 30

3.1.1 Finite Element Modeling…….…………………………………………. 30 3.1.2 Modal Testing and System Identification….…………………………… 31

x

3.2 Techniques for Comparison and Correlation for Model updating.……………... 34 3.2.1 Direct Natural Frequency Correlation….………………………………. 34 3.2.2 Visual Comparison of Mode Shapes…………………………………… 34 3.2.3 Direct Mode Shape Correlation………………………………………… 35 3.2.4 Modal Assurance Criterion……………………………………………... 35 3.2.5 Normalized Modal Difference………………………………………….. 36 3.2.6 Coordinate Modal Assurance Criterion………………………………… 37 3.2.7 Orthogonality Methods…………………………………………………. 37 3.2.8 Energy Comparison and Force Balance………………………………... 383.3 Incompatibility in Measured and Finite Element Data…………………………. 38 3.3.1 Model Reduction.………………………………………………………. 38 3.3.2 Mode Shape Expansion………………………………………………… 40 3.3.2.1 Kidder Dynamic Expansion………………………………….. 41 3.3.2.2 Modal Expansion Method……………………………………. 423.4 Two Proposed Methods for Mode Shape Expansion…………………………... 43 3.4.1 Modal Flexibility Method……………………………………………… 43 3.4.2 Normalized Modal Difference Method………………………………… 44 3.4.3 Performance Metrics…………………………………………………… 44 3.4.4 Simulated Case Study………………………………………………….. 443.5 Three Key Issues of Finite Element Model Updating…………………………. 473.6 Chapter Conclusions…………………….……………….…………………….. 49

4 Finite Element Model Updating Procedure 50

4.1 Theoretical Procedure.………………………………………………………….. 50 4.1.1 Objective Function……………………………………………………... 52 4.1.1.1 Eigenfrequencies……………………………………………... 53 4.1.1.2 Mode Shapes…………………………………………………. 54 4.1.1.3 Modal Flexibility…………………………………………….. 55 4.1.1.4 Modal Strain Energy…………………………………………. 58 4.1.2 Weighting………………………………………………………………. 60 4.1.3 Gradient of Objective Function………………………………………… 614.2 Finite Element Model Updating Parameters……………………………………. 63 4.2.1 Physical Parameters……………………………………………………. 63 4.2.2 Substructure Parameters……………………………………………….. 644.3 Selection of Updating Parameters……………………………………………… 64 4.3.1 Empirically Based Selection of Updating Parameters…………………. 65 4.3.2 Sensitivity Based Selection of Updating Parameters………………….. 664.4 Optimization algorithm………………………………………………………… 67 4.4.1 Search Direction……………………………………………………….. 69 4.4.2 Line Search and Trust Region Strategies………………………………. 71 4.4.3 Sequential Quadratic Programming.………..………………………….. 72 4.4.3.1 Updating the Hessian Matrix of the Lagrange Function…...… 72 4.4.3.2 Solution of Quadratic Programming Problem………………... 73 4.4.3.3 Line Search and Merit Function……………………………… 754.5 Chapter Conclusions…. ………………………………………………………... 76

xi

5 Finite Element Model Updating Using Single-Objective Optimization 77

5.1 Mass Normalization of Operational Mode Shapes. ……………………………. 77 5.1.1 Sensitivity Based Method………………………………………………. 78 5.1.2 Using Orthogonality of Modes with Mass Matrix……………………… 78 5.1.3 Finite Element Model Approach………………………………………... 795.2 Objective Functions and Constraints. ………………………………………….. 805.3 Optimization Techniques……………………………………………………….. 81 5.3.1 Subproblem Approximation Method…………………………………… 82 5.3.2 First-Order Optimization Method………………………………….…… 855.4 Simulated Simply Supported Beam. …………………………………………… 885.5 Concrete Filled Steel Tubular Arch Bridge…………………………………….. 93 5.5.1 Bridge Description and Finite Element Modeling……………………... 93 5.5.2 Ambient Vibration Testing, Modal Parameter Identification and Model

Correlation.. ……………………………………………………………. 96 5.5.3 Parameters Selection for Finite Element model Updating……………... 104 5.5.4 Finite Element Model Updating………………………………………... 105 5.5.5 Physical Meaning of Updated Parameters……………………………… 109 5.5.6 Conclusions from the Updating of Full Scale Arch Bridge……………. 1105.6 Chapter Conclusions……………………………………………………………. 111

6 Finite Element Model Updating Using Multi-Objective Optimization 112

6.1 Introduction……………………………………………………………………... 1126.2 Multi-objective Optimization…………………………………………………... 1136.3 Theoretical Procedure for Multi-objective Optimization………………………. 117 6.3.1 Formulation of Objective Functions and Constraints…...……………… 117 6.3.2 Objective Function Gradient…………………………………………… 1196.4 Simulated Simply Supported Beam…. ………………………………………… 1206.5 Precast Continuous Box Girder Bridge. ……………………………………….. 121 6.5.1 Bridge Description and Finite Element Modeling……………………… 121 6.5.2 Ambient Vibration Measurements, Modal Parameter Identification and

Model Correlation……………………………………………………… 126 6.5.3 Parameters Selection for Finite Element Model Updating……... ……... 131 6.5.4 Finite Element Model Updating... ……………………………………... 132 6.5.5 Conclusions from Updating of a Continuous Girder Bridge.. …………. 1356.6 Chapter Conclusions……………………………………………………………. 135

7 Damage Detection by Finite Element Model Updating Using Modal Flexibility………………………………………………………………………. 137

7.1 Theoretical Background………………………………………………………… 137 7.1.1 Objective Function and Minimization Problem………………………... 137 7.1.2 Objective Function Gradient…………………………………………… 139 7.1.3 Optimization algorithm………………………………………………… 1407.2 Simulated Simply Supported Beam…………………………………………….. 143

xii

7.3 Experimental Beam…………………………………………………………….. 146 7.3.1 Description of Experimental Beam and Modal Parameter Identification 146 7.3.2 Model Updating and Damage Detection……………………………….. 149 7.3.3 Conclusions from Experimental Beam…………………………………. 1537.4 Chapter Conclusions……………………………………………………………. 153

8 Conclusions and Future Work 154

8.1 Conclusions……………………………………………………………………... 1548.2 Significance of the Study……..………………………………………………… 1578.3 Future Research………………………………………………………...….…… 159

References 161

Appendices 172

Acknowledgements 178

Curriculum Vitae 179

xiii

LIST OF SYMBOLS

Acronyms

COMAC Coordinate modal assurance criterion DOFs Degrees of freedom FE Finite element FRF Frequency response function IRF Impulse response function MAC Modal assurance criterion MOP Multi-objective optimization MSE Modal strain energy MSF Modal scale factor NMD Normalized modal difference SQP Sequential quadratic programming

General conventions

( )i

fa

∂ •∂

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

( )T• Transpose of •

( ) 1−• Inverse of matrix •

• Absolute value of •

( ) 1+• Pseudo inverse of matrix •

2• 2l norm of vector •

F• Frobenius norm of matrix •

k• Quantity • evaluated at iteration k

∑ Sum operator

∫ Integration operator

xiv

General symbols

a Vector of normalized updating parameters

ia , ia Upper and lower limit for the design variable a

iji baa ,,0 Coefficients used in optimization

,,i j kA B C State variables related parameters during optimization

kA Active constraints at the solution point

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

A Cross sectional area of an element

nA Area of the neoprene bearing

ia , ib Coefficients of updating parameters

nb Width of neoprene bearing

ib Vector that shows right side of linear system of equations

[ ]B Derivative of the shape function matrix

c Constant vector

nc Length of neoprene bearing

C Constants that are internally chosen between 0 and 1

1 4,..,c c Constants associated with objective function

[ ]D Matrix of material constants

kd Search direction at k -th iteration ˆkd Search direction for internal loop at k -th iteration

1 4,..,d d Constants associated with penalty function W )( jd Parameter showing the direction to the line search

2nE Weighted least square error norm for the objective function

E Modulus of elasticity of material of an element

compE Modulus of elasticity of composite material

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

jfr Frequency corresponding to j -th mode

ejfr , ajfr Experimental and analytical frequencies of j -th mode

( )f a , f Objective function

f Function approximation

0f Reference objective function value

),( kPxF Unconstrained objective function

( )1 0f a Initial values of objective function 1f when a =0

( )xf Vector of objective functions

xv

*f Vector of design goals i.e., * * * *1 2, ,...., mf f f f=

( )F t Dynamic force in a system

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

aF Vector of applied static loads

( ) F ta

Dynamic forces corresponding to master DOFs.

sG Shear modulus

[ ]G Modal flexibility matrix

[ ]nG Modal flexibility, formed from the measured modes

[ ]rG Residual flexibility

[ ] expmmG G⎡ ⎤= ⎣ ⎦ Measured flexibility matrix

anaG Analytical flexibility matrix corresponding to the measured DOFs

fG⎡ ⎤⎣ ⎦ Flexibility matrix due to flexible modes

( )0G a Flexibility estimate at initial parameters values

( )ig x Inequality constraints

, ,i j kg h w State variables(equality and inequality constraints)

ig , jh Upper and lower bounds of state variables

, ,G H W Penalty functions for state variable constraints

g∧

, h∧

, w∧

State variables approximations 2

k kH f= ∇ Hessian matrix at k -th iteration

zzI , yyI Z and Y moment of inertia

cgI Moment of inertia of the individual bearings

kji ,, Indices

[ ]J Jacobian matrix between the global and a local coordinate

[ ]K Global stiffness matrix

[ ]eK , 'eK⎡ ⎤⎣ ⎦ Elemental stiffness matrix in original and transformed system

[ ]aK , [ ]NK Reduced and full stiffness matrix

K•⎡ ⎤⎣ ⎦ Stiffness corresponding to •DOFs when unit displacement at DOFs.

vK Vertical stiffness of neoprene support

rotK Rotational stiffness of bearing

1 2,K K Spring stiffness of two springs

1L Lower limit to constrain the MAC

el Length of element

l Number of analytical modes considered in modal expansion

cl Number of active constraints

xvi

dL Distance between two springs in bearing

kM Quadratic model at k -th iteration

om Number of objective functions

M Global mass matrix of system

1M Bending moment

em Number of equality constraints

m Total number of constraints

sm Number of mode shapes considered

fm Number of eigenfrequency residual

km Component of diagonal mass matrix

m Mass per unit length

321 mmm ++ Number of state variables

jMSE Modal strain energy corresponding to j -th mode

jMSF Modal scale factor corresponding to j -th mode

iMAC MAC number for the thi mode pair

[ ]aM Analytical global mass matrix reduced to measurement DOfs

[ ]M , [ ]NM Global mass matrix

[ ]eM , '[ ]eM Elemental mass matrix in original and transformed system

[ ]M• Mass corresponding to •DOFs when unit displacement is at DOFs

elemN Number of finite elements in system

mN Number of elements in substructures

N Total number of finite element DOFs

rn Number of residuals

n Number of updating parameters

iN Shape function corresponding to the i -th node

dn Number of the measurement points

sn Number of subproblem iterations

sN Maximum number of iterations

siN Maximum number of sequential infeasible design sets

cn Current number of design sets

p Number of experimental modes considered in modal expansion

vp Some vector

whgx PPPP ,,, Penalties applied to the constrained design and state variables

Q Unitary matrix so that *kA Q R=

xvii

),( qxQ Unconstrained objective function with response surface parameter ( )q

,f pQ Q Function related to objective function and penalty constraint respectively

kq ( ) ( )1k kf x f x+∇ −∇

( )q d Objective function which is function of search direction

R Upper triangular matrix of the same dimension as kA

r Radius of gyration ( )z zzr sqrt I A= , ( )y yyr sqrt I A=

nr Number of nodes

fr Eigenfrequency residual

sr Mode shape residual

ir Penalty parameter

jr Residual corresponding to j -th mode

jji

i

rS

a∂

=∂

Sensitivity matrix

ks 1k kx x+ −

jS Line search parameter *js Largest possible step size for the line search of the current iteration

maxS Maximum (percent) line search step size

nS Shape factor of neoprene bearing

t Time period

nt Thickness of bearing

T Transformation matrix

ppT Transformation matrix for modal expansion

u ,u Displacement and acceleration of system

, ,u v w Displacements of a point

1 2 nu u u Degrees of freedom

iu Deflection coefficient at point i

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

aju Analytical uniform load surface

eju Experimental uniform load surface

u Vector of resulting static responses

v Any vector

kν Constant at k -th iteration

jw Square root of weighting factor of residual jr

cw Constant scalar

w Vector of weighting functions, i.e., 1 2, ,...., mw w w w=

xviii

, ,x y z Coordinates of a point

ax , ox Master ( )a and slave ( )o degrees of freedom

x Updating parameters or design variables

ix , ix Upper and lower bounds on the design variables *x Optimal design set, noninferior solution

X Penalty function used to enforce design variable constraints jx Design variable at j -th iteration

ky Mass normalized trial vector at k -th iteration

jz Aanalytical modal parameters at j -th iteration

z Experimental modal parameters kZ Matrix formed from the QR decomposition of the matrix kA

kα Step length during k -th iteration

jα Weighting factor for j -th eigenvalue

sα Line search parameter

jβ Weighting factor for j -th modeshape

χ Tolerances for state variable ig

iδ 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

( )*e jrφ * indicates the complex conjugate of element in mode shape

ajφ , ejφ j -th analytical and experimental mode shape vectors

eΦ Experimental mode shape matrix

Φ Analytical mode shape matrix corresponding to the experimental DOFs

a jφ j -th measured mode shape vector

,n rm m=ΦΦ Φ Measured and unmeasured mode shapes at the measured DOFs

iφ Mode shape vector obtained from ambient vibration test

jφ j -th mode shape vector

ijΦ Normalized i -th component of the j -th modal vector

ikΦ i -th coefficient of unit mass normalized modal vector for mode k

fΦ Flexible mode shapes

, i jϕ ϕ Any two modal vectors

xix

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

γ Dummy argument to minimize the vector of objectives

κ Large integer

jλ j -th eigenvalue

ajλ , ejλ j -th finite element and experimental eigenvalue

ρ Mass density

iρ Design variable tolerance

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

τ Objective function tolerance

ω Angular frequency

jω j -th measured natural frequency

1 2 3, ,ξ ξ ξ Local coordinates jψ Weight associated with design set j

( )xΨ ( ),x γΨ 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 2ωΛ = Matrix of eigenvalue

,n rΛ = Λ Λ Eigenvalues matrix of measured and unmeasured modes.

fΛ Eigenvalues matrix corresponding to the flexible modes

( )γΠ Feasible region for the objective function space

iϒ Lagrange’s multiplier

Ω Feasible region in parameter space nx∈ℜ nℜ Parameter space which represents real n-vectors ( n 1× matrices)

xx

LIST OF FIGURES

2.1 Hermite shape functions of plane beam element…………………………….. 162.2 Local and global coordinate system…………………………………………. 182.3 Degrees of freedom…………………………………………………………… 192.4 Simply supported beam to verify MBMAT……………………………………. 252.5 First bending mode of simply supported beam as seen in the output window

of MBMAT…………………………………………………………………… 262.6 Bending modes of simply supported beam obtained from MBMAT………….. 272.7 Nine storey, ten bay plane frame to verify MBMAT…………………………. 282.8 First bending mode of frame as seen in the output window of MBMAT……... 282.9 Bending modes of plane frame obtained from MBMAT……………………… 293.1 Relationship between FE modelling, testing and system identification for

FE model updating……………………………………………………………. 333.2 Standard simulated simply supported beam before and

after introducing damage……………………………………………………… 453.3 MAC values between the actual and expanded mode shapes…………………. 463.4 Norm errors for different expansion methods…………………………….…… 463.5 Norm of eigenvector differences……………………………………………… 473.6 Schematic diagram to show the main issues of model updating

(a) poor selection of updating parameters (b) poor setting up of objective function………………………………………. 48

4.1 The general procedure of the FE model updating method…………………….. 514.2 Graphical interpretation of quasi-Newton method…………………………….. 705.1 A simulated simply supported beam…………………………………………... 895.2 Correlation of simulated beam after updating with frequency residual,

MAC function and flexibility residual in objective function………………….. 905.3 Photo of Beichuan river concrete-filled steel tubular arch bridge…………….. 935.4 Elevation and plan of arch bridge……………………………………………... 945.5 Cross section of arch rib and deck beam connection of arch bridge…………... 955.6 Three dimensional FE model of arch bridge………………………………….. 965.7 Details of measurement points of arch bridge…………………………………. 975.8 Data acquisition system and arrangement of accelerometers in vertical

direction of arch bridge………………………………………………………... 985.9 Raw measurement data of Point 9 for vertical direction of arch bridge……….. 995.10 Re-sampled data and modified power spectral density of point 9 for

vertical direction of arch bridge……………………………………………….. 995.11 Average normalized power spectral densities for full data in vertical

direction of arch bridge………………………………………………………... 1005.12 Typical stabilization diagram for vertical data of arch bridge…………………. 1005.13 First six mode shapes obtained from FE analysis and test of arch bridge…..…. 1015.14 Frequency and MAC correlation of arch bridge before updating……………... 1035.15 Eigenvalues sensitivity of arch bridge to potential parameters………………... 104

xxi

5.16 Convergence of six FE eigenvalues during updating of arch bridge………….. 1075.17 Frequency and MAC correlation of arch bridge after updating……………….. 1086.1 Mapping from parameter space into objective function space………………… 1146.2 Set of inferior solutions……………………………………………………….. 1146.3 Geometrical interpretation of goal attainment problem for 2D problems……... 1156.4 Plot of different solution points……………………………………………….. 1176.5 Location and severity of damage in simulated beam after FE model updating... 1206.6 Photo of Hongtang bridge…………………………………………………….. 1226.7 Details of Hongtang bridge and bearings with measurement points for

ambient vibration test…………………………………………………………. 123

6.8 Diagram to calculate the equivalent rotational stiffness of support…………… 1246.9 Finite element model of the Hongtang bridge…………………………………. 1256.10 The arrangement of accelerometers in vertical direction of Hongtang bridge… 1266.11 Raw measurement data of Point 29 for vertical direction of Hongtang bridge... 1276.12 Re-sampled data and modified power spectral density of point 29 for

vertical direction of Hongtang bridge………………………………………… 127

6.13 Average normalized power spectral densities for full data in vertical direction. 1286.14 Typical stabilization diagram for vertical data of Hongtang bridge…………… 1286.15 First five vertical mode shapes obtained from FE analysis and

Test of Hongtang bridge………………………………………………………. 130

6.16 Frequency correlation of Hongtang bridge before updating…………………... 1306.17 Frequency correlation of Hongtang bridge after updating…………………….. 1337.1 Location and severity of damage in simulated beam after FE model

updating for different cases…………………………………………………… 144

7.2 Static test arrangement and cross section of experimental beam……………… 1467.3 Observed cracks of the experimental beam in each load step…………………. 1477.4 Dynamic test setup of experimental beam…………………………………….. 1487.5 Identified mode shapes of experimental beam………………………………… 1497.6 Descritization of experimental beam………………………………………….. 1507.7 Location and severity of damage after FE model updating (reference state) …. 1517.8 Location and severity of damage after FE model updating (damaged state) ….. 151

xxii

LIST OF TABLES

2.1 Comparison of time period (sec) with Clough and Penzien and Mario Paz…… 272.2 Comparison of eigenvalues with Bathe and Wilson 1972 and Peterson 1981… 293.1 Comparison of experimental (assumed damage) and initial analytical

modal properties of simulated simply supported beam………………………… 455.1 Comparison of experimental (assumed damage) and analytical modal

properties of simulated beam before updating………………………………….. 895.2 Results of simulated beam after updating with different residuals in objective

function……………………………………………………………………………….….. 915.3 Error detection after updating when frequency, MAC related function

and flexibility residual are used in objective function………………………….. 925.4 Test setup in vertical direction of arch bridge……………………………………… 975.5 Comparison of experimental and analytical modal properties of arch

bridge before updating…………………………………………………………. 102

5.6 Comparison of experimental and analytical modal properties of arch bridge after updating…………………………………………………………….

108

5.7 Value of updating parameters of arch bridge before and after updating………... 1096.1 Comparison of experimental (assumed damage) and analytical

modal properties of simulated beam after updating…………………………….. 121

6.2 Comparison of experimental and analytical modal properties of Hongtang bridge before updating……………………………………………

129

6.3 Comparison of experimental and analytical modal properties of Hongtang bridge after updating………………………………………………

133

6.4 Value of updating parameters of Hongtang bridge before and after updating….. 1347.1 Comparison of experimental (assumed damage) and analytical

modal properties of simulated beam after updating…………………………….. 144

7.2 Comparison of experimental (assumed damage) and analytical modal properties of simulated beam with 3% noise after updating……………..

145

7.3 Static load steps for experimental beam………………………………………... 1477.4 Bending frequencies of beam (Hz) …………………………………………….. 1487.5 Comparison of experimental and analytical modal properties of

experimental beam before updating……………………………………………. 150

7.6 Comparison of experimental and analytical modal properties of experimental beam after updating………………………………………………

152

Finite element model updating of civil engineering structures under operational conditions

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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

Ph.D. dissertation of Fuzhou University

- 2 -

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.


- 3 -

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


- 4 -

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.


- 5 -

• 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


- 6 -

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].


- 7 -

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


- 8 -

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,


- 9 -

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


- 10 -

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.


- 11 -

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


- 12 -

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


- 13 -

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.


- 14 -

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


- 15 -

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:

1

1

(2 .1)

(2.2)

n

n

r

i iir

i ii

x N x

u N u

=

=

=

=

∑

∑

where ix is the x coordinate of the i -th node and iu 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 nr nodes and iN is the shape function corresponding to the i -th node.

The shape functions iN are functions of position and for reasons of generality are given

in terms of the local coordinates ( )1 2 3, ,ξ ξ ξ such that the boundaries of the element

describe a cube ( )2 2 2× × in the 1 2 3, ,ξ ξ ξ 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].


- 16 -

Figure 2.1: Hermite shape functions of plane beam element

( ) ( )

( ) ( )

( ) ( )

( ) ( )

21

22

23

24

1 1 241 1 141 1 241 1 14

N

N

N

N

ξ ξ

ξ ξ

ξ ξ

ξ ξ

= − +

= − +

= + −

= + −

(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:

[ ] [ ] [ ] ( )1 1 1

1 2 31 1 1detT

eM N N J d d dρ ξ ξ ξ− − −

= ∫ ∫ ∫ (2.4)

[ ] [ ] [ ][ ] ( )1 1 1

1 2 31 1 1detT

eK B D B J d d dξ ξ ξ− − −

= ∫ ∫ ∫ (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:


- 17 -

[ ]

[ ]

1

111

1

111

Te

Te

dxM A N N dd

dxK EI B B dd

ρ ξξ

ξξ

−

−

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞

= ⎜ ⎟⎝ ⎠

∫

∫ (2.6)

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:

[ ][ ]

'

'

Te e

Te e

M T M T

K T K T

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2.7)

where 'eM⎡ ⎤⎣ ⎦ and '

eK⎡ ⎤⎣ ⎦ 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).

[ ]

[ ]

'

1

'

1

(2 .8)

(2.9)

elem

elem

N

ei

N

ei

M M

K K

=

=

⎡ ⎤= ⎣ ⎦

⎡ ⎤= ⎣ ⎦

∑

∑

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


- 18 -

other elements at nodes, which are shared between those elements at each degree of

freedom in turn.

(a) Truss3d element 1 0 0 1 0 00 0 0 0 0 00 0 0 0 0 01 0 0 1 0 0

0 0 0 0 0 00 0 0 0 0 0

e e e

e

E AKl

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2 0 0 1 0 00 2 0 0 1 00 0 2 0 0 11 0 0 2 0 060 1 0 0 2 00 0 1 0 0 2

e e eA lm ρ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Figure 2.2: Local and global coordinate system

(b) Beam3d

3 2 3 2

3 2 3 2

2 2

2 2

0 0 0 0 0 0 0 0 0 0

12 6 12 60 0 0 0 0 0 0 0

12 6 12 60 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

6 4 6 20 0 0 0 0 0 0 0

6 4 6 20 0 0 0 0 0 0 0

0 0

e e

z z z z

e e e e

y y y y

e e e e

x x

e e

y y y y

e ee e

z z z z

e ee ee

e

EA EAl l

EI EI EI EIl l l l

EI EI EI EI

l l l lGI GIl l

EI EI EI EIl ll l

EI EI EI EIl ll l

KEAl

−

−

− − −

−

−

−

=−

3 2 3 2

3 2 3 2

2 2

2 2

0 0 0 0 0 0 0 0

12 6 12 60 0 0 0 0 0 0 0

12 6 12 60 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

6 2 6 40 0 0 0 0 0 0 0

6 2 6 40 0 0 0 0 0 0 0

e

z z z z

e e e e

y y y y

e e e e

x x

e e

y y y y

e ee e

z z z z

e ee e

EAl

EI EI EI EIl l l l

EI EI EI EI

l l l lGI GIl l

EI EI EI EIl ll l

EI EI EI EIl ll l

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

− − −

−

−

−

−⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎦⎢ ⎥


- 19 -

Figure 2.3: Degrees of freedom

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

1 10 0 0 0 0 0 0 0 0 03 6

6 613 11 9 130 0 0 0 0 0 0 065 210 70 42010 5 10

6 613 11 9 130 0 0 0 0 0 0 065 210 70 42010 5 10

0 0 0 0 0 0 0 03

z z z ze

e e e

z z z ze e

e e e e

yy zz

ee e

r r r rl ll l l l

r r r rl ll l l l

I I IA

m A lρ

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟+ + − − −

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟+ − + − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+

=

2 2 222 2

2 2 2 3

2 222

2 2 2

0 03

211 1 13 10 0 0 0 0 0 0 0210 105 420 14010 15 10 30

211 1 130 0 0 0 0 0 0 0210 105 42010 15 10

yy zz

y y yze e e e

e e e e

y yze e e

e e

IA

r r rrl l l ll l l l

r rrl l l ll l l

+

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟− + + − − − +

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟+ + − −

⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

22

3

22 2 2

2 2 2 2

22 2 2

2 2 2 2

1140 30

1 10 0 0 0 0 0 0 0 0 06 3

6 69 13 13 110 0 0 0 0 0 0 070 420 65 2105 10 5 10

6 69 13 13 110 0 0 0 0 0 0 070 420 65 2105 10 5 10

0 0

y

e

yz z ze

e e e

yz z ze e

e e e e

r

l

rr r rl ll l l l

rr r rl ll l l l

⎛ ⎞⎜ ⎟+⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− − + − +

⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− − − + +

⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

2 22 22 2

2 3 2 2

22 22

2 3

0 0 0 0 0 0 0 03 3

213 1 11 10 0 0 0 0 0 0 0420 140 210 10510 30 10 15

13 1 110 0 0 0 0420 140 21010 30 1

yy zz yy zz

y yz ze e e e

e e e e

yz ze e e

e e

I I I IA A

r rr rl l l ll l l l

rr rl l ll l

+ +

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− − + + +

⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟− − − + − +

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

22

2 2210 0 0

1050 15z

ee e

rll l

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟+⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎢ ⎥

⎢ ⎥


- 20 -

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:

[ ] [ ]( ) 2 0K Mω φ− = (2.11)

The mass normalized mode shapes satisfy the orthogonality conditions with respect to

mass and stiffness matrices as defined in the following expressions:

[ ] [ ][ ] [ ][ ] [ ][ ] [ ]

(2.12a)

(2.12b)

T

T

M I

K

Φ Φ =

Φ Φ = Λ

where 2NωΛ = 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


- 21 -

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 1y , a sequence of vectors , 2,3...ky k = 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 [ ]1 2, ,...., kY y y y= satisfies the relationship as shown in Equation

(2.15).

.

1

1 1

1 1 1 2 1, 0

k k

k k k

k k k k k k

Tk k k

kk

k

K y M yy M y

y y y y

y M yyy

αα β β

β

β

−

− −

− − − −

=== − − =

=

=

(2.13)

1 2

2 2 3

3 3

1

r

k k

k k

T

α ββ α β

β α

α ββ α

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2.14)

( )1TrY MK M Y T− = (2.15)

j jYφ φ= (2.16)

Thus, the eigenvectors jφ of rT 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 ( )rT k n= are the

reciprocals of , 1,....,j j nλ = ,where jλ is the eigenvalue. The analysis above relates strictly

to the case when the stiffness matrix is positive definite.


- 22 -

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:

1c = 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:

1c = Cross-sectional area

2c = Young's modulus of elasticity

3c = 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:



3c = 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


- 23 -

element has six DOFs.

• A beam2d element requires four constants:



3c = Moment of Inertia

4c = 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:



3c = Moment of Inertia ( ZZI )

4c = Mass density

• The following four constants are optional

5c = axial angle, which defaults to zero

6c = shear modulus, which is calculated assuming Poisson's ratio 0.3.

7c = moment of inertia ( YYI ), which is assumed equal to ( ZZI ).

8c = moment of inertia ( xxI ) which is assumed equal to ( YYI + ZZI ).

• 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 1c , 2c , 3c X-mass = 1c , Y-mass = 3c , Z-mass = 2c

• 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.


- 24 -

• A mass3d element may have between one and six constants which define the mass as

follows:

Constants 1c - 6c X= 1c , Y= 2c , Z= 3c , Xrot= 4c , Yrot= 5c , Zrot= 6c

• 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.


- 25 -

(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].

b=0.25m

L=6m

h=0.2m

2 2

3 4

Material properties Typical sectional properties

E=3.2 10 / 0.05

Density 2500 / 1.6666 04

E N m A m

kg m I E m

+ =

= = −

Figure 2.4: Simply supported beam to verify MBMAT


- 26 -

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).

2 2

2nn EI

L mπω ×

= (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.


- 27 -

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 MBMAT Independent Difference (%) 1 0.111225 0.111175 0.0449 2 0.027843 0.027793 0.1799 3 0.012402 0.012352 0.4047 4 0.006996 0.006948 0.6908 5 0.004493 0.004447 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.


- 28 -

10 @ 20' = 200'

9 @

10'

= 9

0'

2 2

2 4

Material properties Typical section properties

E=432000 3

Mass per unit length 3 sec 1

k ft A ft

k ft I ft

=

= − =

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


- 29 -

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 Bathe and Wilson 1972 Peterson 1981 Independent Difference (%) Independent Difference (%)

1 0.589814 0.589541 0.0463 0.589541 0.0463 2 5.550460 5.526950 0.4253 5.526960 0.4251 3 16.775700 16.587800 1.1327 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.


- 30 -

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.


- 31 -

• 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


- 32 -

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.


- 33 -

• Some modes of the structure may not be excited during the test or, even if excited,

some modes may not be identified.

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

Analytical modeling process

• Estimate geometric & material properties

• Simplifying assumptions • Boundary conditions • Judgment of modeler

• Parameter error • Model form error • Discretization error

FE model [ ][ ]K M

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

Testing

Raw data

System identification

Experimental model [ ] [ ]Λ Φ

• Random errors • Systematic errors • Modal and spatial

incompleteness

• Model reduction/eigenvector expansion

• Choose error residual • Choose Updating parameters • Choose optimization algorithm

Validation/Correlation

Refined model and updated parametersnew newK M⎡ ⎤ ⎡ ⎤

⎣ ⎦ ⎣ ⎦

Model updating process


- 34 -

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).

100ej ajf j

ej

fr frfr−

∈ = × (3.1)

( )1

2

1

100

f

f

m

ej ajj

f m

ejj

fr fr

fr

=

=

⎡ ⎤−⎢ ⎥

⎢ ⎥∈ = ×⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑

∑ (3.2)

where ejfr and ajfr are the experimental and analytical frequencies of j -th mode

respectively and fm 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


- 35 -

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]:

( )Taj ej

j j Tej ej

MSFφ φ

γφ φ

= (3.3)

in which, ajφ and ejφ 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:


- 36 -

( )( ) ( )

2Taj ej

j T Taj aj ej ej

MACφ φ

φ φ φ φ= (3.4)

where ajφ is the analytical eigenmode that has been paired with the j -th experimental

mode ejφ . 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].

( )2

1

11 100sm

js j

MACmφ

=

⎡ ⎤⎢ ⎥∈ = − ×⎢ ⎥⎣ ⎦

∑ (3.5)

An overall mode shape error indicator may be calculated from Equation (3.5), in

which sm 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 ejφ and analytical ajφ mode

shape is defined as:

( )

2

2

,aj j ej

j aj ejj ej

NMDφ γ φ

φ φγ φ

−= (3.6)

where Modal Scale Factor ( )jγ is given in Equation (3.3) and 2

is the 2l norm of a

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

formula:


- 37 -

j

j

1 MACMACjNMD−

= (3.7)

In practice, the NMD is a close estimate of the average difference between the

components of both vectors andaj ejφ φ 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 sm mode pairs via MAC, COMAC calculates the amount of

correlation at each coordinate over all correlated mode pairs as:

( ) ( )

( ) ( )

2*

1

2 2

1 1

s

s s

m

a ejr jrr

j m m

a ejr jrr r

COMAC =

= =

φ φ=

φ φ

∑

∑ ∑ (3.8)

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).

[ ] [ ][ ]Tortho e a eM⎡ ⎤∈ = Φ Φ⎣ ⎦ (3.9)

in which eφ is the experimental mode shape and aM 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.


- 38 -

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].

Kinetic energy = [ ] 12

T

j jMφ φ (3.10)

Potential energy = [ ] 12

T

j jKφ φ (3.11)

The kinetic and potential energies can be compared as:

[ ] [ ]

[ ] [ ]

k

p

1 12 21 12 2

T T

ej a ej aj a aj

T T

ej a ej aj a aj

M M

K K

∈ = φ φ − φ φ

∈ = φ φ − φ φ (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:

[ ] [ ]

ej

aj

F

F

a ej

a aj

K

K

= φ

= φ (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


- 39 -

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:

[ ] a

N ao

xx T x

x⎧ ⎫⎪ ⎪= =⎨ ⎬⎪ ⎪⎩ ⎭

(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:

[ ] [ ] [ ][ ]Ta NM T M T= (3.15)

[ ] [ ] [ ][ ]Ta NK T K T= (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).

[ ] [ ][ ] [ ]

( ) ( )

aaa ao a

oa oo o o

F tK K xK K x F t

⎧ ⎫⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦ ⎩ ⎭ ⎪ ⎪⎩ ⎭

(3.17)

[ ] [ ] [ ] ( ) 1 1s oo oa a oo ox K K x K F t− −= − + (3.18)

Assuming that there are no external forces at the slave DOFs, the Guyan reduction

transformation matrix can be obtained as:

[ ][ ]

[ ] [ ]1oo oa

IT

K K−

⎡ ⎤= ⎢ ⎥

−⎢ ⎥⎣ ⎦ (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.


- 40 -

• 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


- 41 -

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,

2 00

a a ao aa ao a jj

oa oo oa oo o j

K K M MK K M M

φω

φ⎧ ⎫⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎧ ⎫⎪ ⎪− =⎜ ⎟⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎩ ⎭⎪ ⎪⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎩ ⎭

(3.20)

where jω and a jφ represents j -th measured natural frequency and corresponding mode

shape at the measured coordinates and o jφ 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).

( ) ( ) 2 2 0a jaa j aa ao j a o o jK M K Mω φ ω φ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + − =⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.21)

( ) ( ) 2 2 0a joa j o a oo j oo o jK M K Mω φ ω φ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + − =⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.22)

( ) ( ) 12 2a jo j oo j oo o a j o aK M K Mφ ω ω φ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.23)

( ) ( ) 2 2a jo j a o j a o a a j a aK M K Mφ ω ω φ

+⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.24)

where + denotes the pseudo inverse. To use third method, from Equation (3.21) and (3.22)

one can define two matrices, 1A and 2A as shown in Equation (3.25). Hence, Equation

(3.21) can be written as shown in Equations (3.26) and (3.27).

2 2

1 22 2

a a j a a ao j ao

oa j oa oo j oo

K M K MA A

K M K M

ω ω

ω ω=

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(3.25)

[ ] [ ] 1 2 0a j o jA Aφ φ+ = (3.26)

[ ] [ ] 2 1 a jo j A Aφ φ+= (3.27)

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


- 42 -

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 [ ]2A used in

Equation (3.27) of the third method in general, satisfies the relationship [ ] [ ] [ ]2 2A A I+ = .

Hence, the vector o jφ 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 ppT is defined by:

a p a p p pT⎡ ⎤ ⎡ ⎤ ⎡ ⎤Φ = Φ⎣ ⎦ ⎣ ⎦⎣ ⎦ (3.28)

where p is the number of modes considered, a pΦ is the measured mode shape and a pΦ is

the analytical mode shape corresponding to measurement DOFs. Applying pseudo inverse

to Equation (3.28):

a pp p a pT+⎡ ⎤⎡ ⎤ ⎡ ⎤= Φ Φ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.29)

This transformation matrix is then used to expand the measured mode shape to

unmeasured DOFs according to Equation (3.30).

o p o p p pT⎡ ⎤ ⎡ ⎤ ⎡ ⎤Φ = Φ⎣ ⎦ ⎣ ⎦⎣ ⎦ (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:

( ) ( )

1 1

2 2

d d

d d

e a

n p n ll pe a

N n p N n l

T× ×

×

− × − ×

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤Φ Φ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤Φ Φ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(3.31)

in which e and a represent the experimental and analytical quantities respectively, N is

the number of DOFs, dn 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).

( ) 1

1 1 1 1

T Ta a a ET−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= Φ Φ Φ Φ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (3.32)


- 43 -

The mode shapes at the unmeasured DOFs, i.e., 2e⎡ ⎤Φ⎣ ⎦ , 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 [ ]n nG × is defined

as [115]:

[ ] [ ] [ ]1 TG −⎡ ⎤= Φ Λ Φ⎣ ⎦ (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 ppT⎡ ⎤⎣ ⎦ 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:

1 1 min ( ) ( )T Ta p p p a p a p p p p p a p p p

Fpp

A AT

− −Φ Λ Φ − Φ Λ Φ Such that p p p pΛ = Λ (3.34)

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

1 The detail discussion of modal flexibility is presented in chapter 4. 2 Frobenius Norm is defined in appendix A.


- 44 -

into unconstrained form which can be easily solved to get the transformation matrix p pT⎡ ⎤⎣ ⎦ .

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, ( )( )1 1C diag diag NMD= − .

[ ]1o p o p C⎡ ⎤ ⎡ ⎤Φ = Φ⎣ ⎦⎣ ⎦ (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):

[ ] [ ][ ]

exp1(%) *100exact anded

exact

ErrorΦ − Φ

=Φ

(3.36)

[ ] [ ][ ]

exp2 (%) *100exact anded

exact

ErrorΦ − Φ

=Φ

(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


- 45 -

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 32500 /kg m and 23.2 10 /E N m+ respectively. Similarly area of cross section and

moment of inertia of simulated beam are 20.05m and 41.66 04E m− respectively.

DAM

1 2 3

b=0.25m

12

L=6mDAM

4 5 6 7

h=0.2m

DAM

108 9 11 1413 15

2 2

3 4

Before damageMaterial properties Typical section properties

E=3.2 10 / 0.05

Density 2500 / 1.6666 04Damage applied

Reduce E of element 3 = 20 %

E N m A m

kg m I E m

+ =

= = −

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

Natural frequency (Hz) Mode Damaged beam Undamaged beam Error (%) MAC %

1 8.245 8.990 9.035 99.918 2 34.920 35.914 2.846 99.869 3 75.080 80.632 7.394 99.216 4 137.508 142.930 3.943 99.588 5 209.028 222.532 6.460 97.497 6 313.581 319.160 1.779 99.528 7 405.839 432.532 6.577 97.444 8 547.260 562.405 2.767 99.107 9 671.483 708.677 5.539 98.424 10 836.938 871.146 4.087 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%.


- 46 -

1 2 3 4 5 6 7 8 9 10

0.75

0.8

0.85

0.9

0.95

1

Modes

MAC

Kidder Dynamic ExpansionModal ExpansionExpansion using modal FlexibilityExpansion using NMD

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

1 2 3 40

5

10

15

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

Erro

r1(%

)

Mode 1 to 10 respectively

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 ppT⎡ ⎤⎣ ⎦ 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.


- 47 -

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.

1 2 3 40

5

10

15

20

25

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

Erro

r2 (%

)

Mode 1 to 10 respectively

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


- 48 -

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.

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].

1ℜ

2ℜ

3ℜ

optFEM

upFEM

initFEM

(b)

1ℜ

2ℜ

3ℜinitFEM

upFEM

optFEM

(a)


- 49 -

• Region 1ℜ contains all the possible FE models of a structure. • Region 2ℜ contains all the FE models that correlate well with experimental results.

One of these models, optFEM gives the best possible description of dynamic behavior of the structure.

• Region 3ℜ is a set of models that can be derived from the initial FE model, initFEM , by varying the selected updating parameters. Both the initial FE model, initFEM , and the updated model, upFEM , are the members of 3ℜ . The dimension of 3ℜ 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 3ℜ and 2ℜ (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 optFEM within the common space of 2ℜ and 3ℜ (Figure 3.6(b)). Then, whether upFEM will converge to optFEM depends mainly on used optimization algorithm. A poor set-up of an objective function will never allow upFEM to move toward optFEM .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.


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.


- 50 -

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


- 51 -

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

Select updating parameters, Sensitivity analysis, Initialization

Initial values 0; 0x j =

FE analysis Computation of analytical modal parameters

( )i jz z x=

Correlation Automatic pairing of mode shapes using MAC

criteria

Ambient vibration test planning

Ambient vibration test

System identification

Experimental modal parameters: z

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

2

jf z z= −

Optimization step Updated values 1jx +

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

1j j= +

No

Yes

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

Convergence ?


- 52 -

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≤ x∀ (4.1)

, 1, 2,3,......i i ix x x i n≤ ≤ =

where the upper ( )ix and lower ( )ix 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

2

1 1

r rn n

j j jj j

f x z x z r x= =

⎡ ⎤= − =⎣ ⎦∑ ∑ (4.2)

where each ( )jz x represents an analytical modal quantity which is a nonlinear function of

the optimization or design variables nx∈ℜ and z refers to the measured modal parameters.

In order to obtain a unique solution, the number of residuals rn 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 0x is chosen as dimensionless updating parameter a . By using the normalized

parameters a , problems of numerical ill-conditioning due to large relative differences in


- 53 -

parameter magnitudes can be avoided.

0

0

i ii

i

x xax−

= − (4.3a)

0 (1 )i i ix x a= − (4.3b)

The objective of FE model updating problem is to find the value of vector ia of

Equation (4.3) which minimizes the error between the measured and analytical modal

parameters. Hence, Equation (4.2) becomes:

( ) ( )2

1

rn

jj

f a r a=

= ∑ (4.4)

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 fr is formulated as:

( ) 1,.........,aj ejf f

ej

r a j mλ λλ−

= ∈ (4.5)


- 54 -

with eigenvalue ( )22j jfrλ π= ∗ ∗ where jfr is the eigenfrequency corresponding to j -th

mode. ajλ and ejλ are analytical and corresponding experimental eigenvalue, respectively.

fm refers to the number of identified eigenfrequencies that are used in the updating

process. Relative differences are taken in fr 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:

( ) ( ) ( ) 1,.........,s aj j ej sr a x MSF x j mφ φ= − × ∈ (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


- 55 -

Equation (4.7). Unlike in the previous approach, the authors do not square the residuals sr

when including them in the objective function.

( ) ( ) , 1,.........,s aj ej sr x NMD j mφ φ= ∈ (4.7)

After trying several expressions, Moller and Friberg [124] proposed the following

residual expression related to mode shapes.

( )( )

2

21

1,.........,j

s sj

MACr a j m

MAC

−= ∈ (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:

[ ] au G F= (4.9)

where aF 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 aF is 1 and all other

entries in aF 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


- 56 -

theorem [125] that the flexibility influence coefficient matrix is symmetric for linear

systems, such that:

ij jiG G= (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 2kdiag ω 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:

[ ] [ ] [ ][ ] [ ][ ] [ ]( ) 11 1T TK−− − −= Φ Λ Φ = Φ Λ Φ (4.11)

The flexibility is defined as the inverse of the stiffness matrix as shown in Equation

(4.12).

[ ] [ ] 1G K −≡ (4.12)

Substituting Equation (4.12) into Equation (4.11) yields the inverse vibration

representation of the flexibility matrix as:

[ ] [ ][ ] [ ]1 TG −≡ Φ Λ Φ (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:


- 57 -

1 T

f f f fG−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤≡ Φ Λ Φ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (4.14)

where f⎡ ⎤Λ⎣ ⎦ contains only those eigenvalues corresponding to the flexible modes of the

system and f⎡ ⎤Φ⎣ ⎦ 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:

[ ] [ ] [ ]n rG G G= + (4.15)

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

[ ] [ ][ ] [ ]1 Tn n n nG −= Φ Λ Φ (4.16)

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

[ ] [ ][ ] [ ]1 Tr r r rG −= Φ Λ Φ (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:

[ ] [ ] [ ]1 1T Tmm n n n r r rm m m mG − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= Φ Λ Φ + Φ Λ Φ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (4.18)

where [ ]mmG is called the measured flexibility matrix, andn rm mΦ Φ are respectively the

measured and unmeasured mode shapes of the structure at the measured DOFs,

and andn rΛ Λ 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


- 58 -

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,

[ ] [ ] 1 Tmm n n nm mG −⎡ ⎤ ⎡ ⎤= Φ Λ Φ⎣ ⎦ ⎣ ⎦ (4.19)

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

[ ] 1

1

NT

j j jj

G φ φ −

=

= Λ∑ (4.20)

For the sake of clarity, the measured flexibility matrix [ ]mmG will be referred to simply

as expG⎡ ⎤⎣ ⎦ , nmΦ will be referred as Φ , and nΛ 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:

( ) 22flex exp anar a G G= − (4.21)

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

anaG 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,


- 59 -

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,

1

1 1strain energy2 2

NT

j jj

F u F u=

= =∑ (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:

1strain energy2

TF GF= (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:

1strain energy2

Tu Ku= (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:

j1MSE2

Tj jKφ φ= (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).

( )2

2energy

11

s Tmaj ajT

j ej ej

Kr x

Kφ φφ φ=

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠∑ (4.27)

in which ajφ and ejφ 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


- 60 -

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:

( )2

1

minrn

j jj

w r x=

⎡ ⎤⎣ ⎦∑ (4.28)

where jw is the square root of weighting factor of residual jr and rn 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


- 61 -

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


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 ,

( ) ( ) ( )1

2rn

jj

ji i

r af ar a

a a=

∂∂=

∂ ∂∑ (4.29)

In matrix form, this can be expressed as,

( ) ( ) ( )2 Tji j

i

f aS a r a

a∂

=∂

(4.30)

where the matrix( )j

i

r aa

∂

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


- 62 -

the residual vector with respect to each correction parameter ia is called the sensitivity

matrix. This matrix can be expressed as:

( )( )

.1,......., no. of residuals considered

. .1,......, no. of updating parameters

.

j rji

i

r j nS

i na

⎡ ⎤⎢ ⎥∂ =⎧⎪⎢ ⎥= ⎨⎢ ⎥ =∂ ⎪⎩⎢ ⎥⎢ ⎥⎣ ⎦

(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:

j Tj j j

i i i

Ka a aλ

φ λ φ∂ ⎡ ⎤∂ ∂Μ

= −⎢ ⎥∂ ∂ ∂⎣ ⎦ (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.,

1

dj

jq qqia

φβ φ

=

∂=

∂ ∑ (4.33)

where the coefficients jqβ 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

( )

,

,

12

Tq j j q j

i ijq

Tj j

i

K q ja a

q ja

φ λ λ λ φβ

φ φ

⎧ ⎡ ⎤⎛ ⎞∂ ∂Μ− − ≠⎪ ⎢ ⎥⎜ ⎟∂ ∂⎪ ⎝ ⎠⎣ ⎦= ⎨

∂Μ⎪− =⎪ ∂⎩

(4.34)

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 d<< where d is the analytical


- 63 -

model order. The value of this number should be high enough in view of condition of

gradient matrix. If the residual vector ( )r x is additionally weighted, as expressed by (4.28),

the gradient vector should be multiplied with the corresponding weighting

value 2jw where jw is the square root of weighting factor of residual jr .

From Equations (4.32) and (4.34), it can be seen that the derivatives of the structural

stiffness and mass matrices, with respect to the design variables, are required. Naturally,

the analytical expressions for these entities may be developed, but a new programming

effort would be required each time when a new type of design variable is introduced. By

adopting the linearized matrix derivatives, viz., first-order Taylor approximation at the

current design point, these limitations can be avoided. Hence it follows that,

( ) ( )

( ) ( )

i i i

i

i i i

i

K a ae K aKa a

a ae aa a

+ ∆ −∂=

∂ ∆

Μ + ∆ −Μ∂Μ=

∂ ∆

(4.35)

where a∆ is the step length and ie is the vector with i -th element equal to 1, and zero

elsewhere. It is observed that the expressions are exact in case the matrices are linear with

respect to the i -th design variable. It should be noted that the evaluation of approximate

matrix derivatives according to Equation (4.35) does not involve any additional problem

solution, but it suffices to assemble and to save the appropriate system matrix for each design

variable increment, which is a minor computational effort.

4.2 Finite Element Model Updating Parameters

An updating parameter is a value in the definition of the FE model or in the system

matrix that is changed in the subsequent correction step. Two parametrization methods

presented below are the most widely used.

4.2.1 Physical Parameters

Physical parameters of an FE model such as material properties, geometrical properties

are used as updating parameters. Correction methods, which use the physical parameters of

an FE model as updating parameters, must exchange data with the FE model code at

various steps in the updating process. They provide updated matrix models that can be

easily interpreted in terms of physical meaningful parameters.


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4.2.2 Substructure Parameters

Another possibility is to update parameters which are associated with individual finite

elements or groups of finite elements. The individual terms in the mass and stiffness

matrices may depend upon one or many finite elements according to the connectivity of the

FE mesh. Thus, the updated mass and stiffness may be written as:

1

mN

i ii

K a K=

∆ =∑ (4.36a)

1

mN

i ii

M b M=

∆ =∑ (4.36b)

where iK and iM is the stiffness and mass matrices of the i -th substructure, mN is the

number of elements and the coefficients ia and ib are the updating parameters. If there is

no error in the i -th element, ia and ib should be unity whereas ia or

1(or 1)ib << >> 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


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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


- 66 -

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


- 67 -

this approach, the eigenvalue sensitivity matrix is approximated using the forward

difference of the function with respect to each parameter considered.

( ) ( )j i

i i

a a e aa a

∂ + ∆ −=

∂ ∆

λ λ λ (4.37)

( )100i ii

Da a a∆∆ = − (4.38)

where D∆ is a forward difference step size (in %), taken 0.2 general and ia , ia 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, 1 2 3 nx x x x x= …… , 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, ( ) ( )0 1,......,i eg x i m= = ,


- 68 -

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

,x x respectively. The general optimization problem is stated as:

( )

( )( )

nminimize

subject to 0, 1,....,

0, 1,....,i e

i e

f xx

g x i m

g x i m mx x x

∈ℜ

= =

≤ = +

≤ ≤

(4.39)

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

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

the equality and inequality constraints evaluated at x ( )( ): n mg x ℜ →ℜ . The special form

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

( )( ) ( )

( ) ( )( ) ( )

1

2

3

Minimize

subject to 1, 2,3,...

1, 2,3,....

1, 2,3....

( 1, 2,3,... )

i i

jj

kkk

f f x

g x g i m

h h x j m

w w x w k m

x x x i nii i

=

≤ =

≤ =

≤ ≤ =

≤ ≤ =

(4.40)

where x is the vector of design variable with parameter bounds ,x x and ig , jh , kw represent

the state variables (equality and inequality constraints) containing with under bar and over

bar representing lower and upper bounds respectively and 1 2 3m m m+ + = 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


- 69 -

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:

1k k k kx x dα+ = + (4.41)

where kd is a search direction and 0kα > is chosen so that 1k kf f+ < .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 kd 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:

k kd f= −∇ (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:

( ) ( )12

T Tk k k k kf x d f d f d H d M d+ ≈ + ∇ + = (4.43a)

If matrix of second derivative of objective function kH is positive definite4, then the

minimum of ( )kM d is at its critical point, 0k k kf H d∇ + = . Thus, the Newton's direction is

given as:

1k k kd H f−= − ∇ (4.43b)

The Newton’s direction of Equation (4.43b) has two drawbacks. One is that the computed direction kd is not necessarily a descent direction unless 2

kf∇ is positive

3 Hessian is defined in appendix A. 4 Positive definite property is defined in appendix A.


- 70 -

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 kH which is symmetric positive definite, easily computable and invertible, and somehow approximates the action of 2

kf∇ . kH is not computed from scratch at each iteration but updated using gradient information from the most recent step.

k k kH d f= −∇ (4.44)

Given the current iterate kx and the approximate Hessian matrix kH at kx , the linear system is solved to generate the direction kd . Updates of kH 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 1kH + 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

( ) ( )

1

1

1

k k k

k k k

k k k

H s qs x x

q f x f x

+

+

+

== −

= ∇ −∇

(4.45)

Typically, some additional requirements on 1kH + (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


- 71 -

method. The formula is given by

1

T T Tk k k k k k

k k T Tk k k k k

q q H s s HH Hq s s H s+ = + − (4.46)

As a starting point, 0H 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 1H − .

4.4.2 Line Search and Trust Region Strategies

The term line search refers to a procedure for choosing kα 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 0x is close

enough to the minimum solution *x .

1k k kx x d+ = + (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 1kf + as

compared to kf . If this is not achieved by k kx d+ then in Equation (4.41), kα is found by

line searching, or modifying the direction kd 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 kd is hold fixed and searching for a step

length kα 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 k kf x dα

α+ is not solved due to the solution being expensive. Rather, a weak line

search is performed, accepting as kα 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 quadractic convergence is defined in appendix A. 6 superlinear convergence is defined in appendix A.


- 72 -

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:

( ) ( )( )

* * *

1*

*

. 0

0 1,...,

0 1,...,

m

i ii

i e

i e

f x g x

g x i m

i m m

=+ λ ∇ =

∇ = =

λ ≥ = +

∑

(4.48)

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


- 73 -

of the Lagrangian function, H , is calculated using the BFGS method shown in Equation

(4.46) in which,

( ) ( ) ( ) ( )1 11 1

. .n n

k k i i k k i i ki i

q f x g x f x g x+ += =

⎛ ⎞= ∇ + ϒ ∇ − ∇ + ϒ ∇⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑ (4.49)

where ( ), 1,.....,i i mϒ = 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 Tk kq s is positive at each update and that H is initialized with a

positive definite matrix, where ks is defined in Equation (4.45). When Tk kq s is not

positive, kq is modified on an element by element basis so that 0Tk kq s > . The general aim

of this modification is to distort the elements of kq , which contribute to a positive definite

update, as little as possible. Therefore, in the initial phase of the modification, the most

negative element of *k kq s is repeatedly halved. This procedure is continued until Tk kq s is

greater than or equal to 1e-5 in this application. If after this procedure, Tk kq s is still not

positive, kq is modified by adding a vector v multiplied by a constant scalar cw , as shown

in Equation (4.50) and w is systematically increased until Tk kq s becomes positive.

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 1

k k

where, . .

if q . 0 and q . 0 1,...

0 otherwise

k k c

i i k i k i k i k

c ki i i

i

q q w vv g x g x g x g x

w s i m

v

+ +

= +

= ∇ −∇

< < =

=

(4.50)

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 iA refers to the i -th row of the m-by-n matrix A .

( )n

1minimize2

1,...1,...

T T

i i e

i i e

q d d Hd c dd

A d b i mA d b i m m

= +∈ℜ

= =≤ = +

(4.51)

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, kA , which is an estimate of the


- 74 -

active constraints at the solution point.

kA is updated at each iteration, k , and this is used to form a basis for a search

direction ˆkd . Equality constraints always remain in the active set, kA . The notation for the

variable, ˆkd ,is used here to distinguish it from kd in the major iterations of the SQP

method. The search direction, ˆkd , is calculated and minimizes the objective function

while remaining on any active constraint boundaries. The feasible subspace for ˆkd is

formed from a basis kZ , whose columns are orthogonal to the estimate of the active

set kA (i.e., 0k kA Z = ). Thus, a search direction, which is formed from a linear

summation of any combination of the columns of kZ , is guaranteed to remain on the

boundaries of the active constraints.

The matrix kZ is formed from the last cm l− columns of the QR decomposition of the

matrix kA , where cl is the number of active constraints and cl m< . That is, kZ is given

by:

[ ]:, 1:

where0

k c

T Tk

Z Q l m

RQ A

= +

⎡ ⎤= ⎢ ⎥⎣ ⎦

(4.52)

where R is an upper triangular matrix of the same dimension as kA and Q is a unitary

matrix so that *kA Q R= . Having found kZ , a new search direction ˆkd is sought that

minimizes ( )q d where ˆkd is in the null space of the active constraints, that is, ˆ

kd is a

linear combination of the columns of ˆ:k k kZ d Z p= for some vector vp .Then, if quadratic

is considered as a function of p , by substituting for ˆkd ,one gets Equation (4.53).

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

( ) 12

T T Tv v k k v k vq p p Z HZ p c Z p= + (4.53)

( ) T Tv k k v kq p Z HZ p Z c∇ = + (4.54)

where ( )vq p∇ is referred to as the projected gradient of the quadratic function because it

is the gradient projected in the subspace defined by kZ . The term Tk kz HZ is called the

projected Hessian. Assuming the Hessian matrix H is positive definite, then the minimum

of the function ( )vq p in the subspace defined by kZ occurs when ( ) 0vq p∇ = , 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).

T Tk k v kZ HZ p Z c= − (4.55)


- 75 -

1ˆ ˆwhere T

k k k k kx x d d Z 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 ˆkd is the exact step to the

minimum of the function restricted to the null space of kA . 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 ˆkd 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 ˆkd is given by:

( ) ( )min 1,....,î k

i k

A x bi m

A di

⎧ ⎫− −⎪ ⎪α = =⎨ ⎬⎪ ⎪⎩ ⎭

(4.57)

which is defined for constraints not in the active set, and where the direction ˆkd is towards

the constraint boundary, i.e., ˆ 0, 1,....,i kA d i m> = . When n independent constraints are

included in the active set, without location of the minimum, Lagrange multipliers, kϒ are

calculated that satisfy the nonsingular set of linear equations

Tk kA cϒ = (4.58)

If all elements of kϒ are positive, kx is the optimal solution of QP (Equation (4.52)).

However, if any component of kϒ 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 kd , which is used to form a

new iteration 1k k k kx x d+ = +α . 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 kd 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


- 76 -

implementation. They also recommend setting the penalty parameter as shown in Equation

(4.60).

( ) ( ) ( ) ( ) 1 1

. .max 0,m me

i i iii i me

x f x r g x r g x= = +

Ψ = + +∑ ∑ (4.59)

( ) ( )( )11max , , 1,...,2i k i k ii ir r r i m

i+

⎧ ⎫= = ϒ + ϒ =⎨ ⎬⎩ ⎭

(4.60)

This allows positive contribution form constraints that are inactive in the QP solution

but were recently active. In this implementation, initially the penalty parameter ir is set to:

( )( )i

i

f xr

g x∇

=∇

(4.61)

where . represents the 2l 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.


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 2l norm is defined in appendix A.


- 77 -

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.


- 78 -

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:

i i iφ φδ= (5.1)

where iφ is the mode shape obtained from operational measurements and iδ 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 iδ can be divided into three broad categories as explained below.

5.1.1 Sensitivity Based Method

A recently proposed approach to extract the constants iδ 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 iδ 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:

1 2

1

NT

i i ii

M φφ δ−

=

=∑ (5.2)

The possibility is to use any information that may be available regarding 1M − to set up

equations to compute iδ . It may be reasonable to assume that 1M − is diagonal and in this

case one can use the off diagonal zeros as a constraints [154,155,156]. Hence, one can


- 79 -

compute iδ 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.

][ iT

i

ijij

M ϕϕ

ϕ=Φ (5.3a)

2

1

ijij n

k kjk

m

ϕ

ϕ=

Φ =

∑ (5.3b)


- 80 -

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:

( )2

11

0 1fm

aj ejj j

j ej

f xλ λ

α αλ=

⎛ ⎞−= ≤ ≤⎜ ⎟⎜ ⎟

⎝ ⎠∑ (5.4)

( )( )2

21

10 1

smj

j jj j

MACf x

MACβ β

=

−= ≤ ≤∑ (5.5)

where jα and jβ 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. ajλ and ejλ are the FE and experimental eigenvalue of the j -th

mode respectively and jMAC 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 iu under uniformly distributed unit load,

called the uniform load surface (ULS), is defined in Equation (5.6), the objective

function ( )3f x considering the modal flexibility residual can be presented as shown in

Equation (5.7).

( ) ( )12

1

d

s

n

ik kjmj

ik k

uω=

=

Φ Φ=

∑∑ (5.6)

( )2

31

*dn

aj ejs

jd ej

u umf xn u=

⎡ ⎤−= ⎢ ⎥

⎢ ⎥⎣ ⎦∑ (5.7)

where aju and eju are analytical and experimental uniform load surface respectively.

Similarly, dn and sm 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).


- 81 -

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.

1 2 3( ) ( ) ( ) ( )f x f x f x f x= + + (5.8)

0 aj ej ULλ λ≤ − ≤ (5.9)

11 ≤≤ MACL (5.10)

where UL is the upper limit whose value can be set as absolute error of the j -th eigenvalue

and 1L 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.

( )( ) ( )

( ) ( )( ) ( )

1

2

3

Minimize

subject to 1, 2,3,...

1, 2,3,....

1, 2,3....

( 1, 2,3,... )

i i

jj

kkk

f f x

g x g i m

h h x j m

w w x w k m

x x x i nii i

=

≤ =

≤ =

≤ ≤ =

≤ ≤ =

(5.11)

where x is the vector of design variable with parameter bounds ,x x and ig , jh , kw represent

the state variables(equality and inequality constraints) containing under and over bar

representing lower and upper bounds respectively and 1 2 3m m m+ + = number of state


- 82 -

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.

ˆ ( ) ( )f x f x error= + (5.12a)

( ) ( )

( ) ( )

( ) ( ) errorxwxw

errorxhxh

errorxgxg

+=

+=

+=

∧

∧

∧

(5.12b)

The most complex form that the approximations can take on is a fully quadratic

representation with cross terms. Using the example of objective function,

0

n n n

i i ij i ji i j

f a a x b x x∧

= + +∑ ∑∑ (5.13)

where ia and ijb 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:


- 83 -

( )( ) ( )( )2

2

1

ˆcjn

j jn

j

E f fψ=

= −∑ (5.14)

where, jψ is the weight associated with design set j and cn is the current number of

design sets. Similar, 2nE norms are formed for each state variable. The coefficients in

Equation (5.14) are determined by minimizing 2nE with respect to the coefficients.

• Minimizing the Subproblem Approximations

With the function approximation available, the constrained minimization problem can

be recast as follows:

ˆMinimize ( )subject to

f f x= (5.15a)

( )

( ) ( )

( ) ( )

( )

1

2

3

1, 2,3....,

1, 2,3....,

1, 2,3....,

( 1, 2,3,..., )

ii i

i ii

jj j

k kk k k

x x x i n

g x g A i m

h B h x j m

w C w x w C k m

∧

∧

∧

≤ ≤ =

≤ + =

− ≤ =

− ≤ ≤ + =

(5.15b)

where iA , jB , kC 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:

( ) ( )31 2

01 1 1 1

Minimize,

mm mn

k k i i j ki i j k

F x P f f P X x G g H h W w∧ ∧ ∧ ∧

= = = =

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

∑ ∑ ∑ ∑ (5.16)

where ( ), kF x p represents the unconstrained objective function that varies with the design

variables and parameter kP , 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 0f is introduced to achieve consistence unit. It is clear that

the unconstrained objective function (also termed a response surface) ( ), kF x p vary with

the design variables and the quantity kP ,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:


- 84 -

( ) ( ) ( )( ) ( ) ( )1 2 i

3 4 i

if x1,2,3,...,

if xi

ii

c c x x x x xX x i n

c c x x x x x⎧ ⎫+ − < −∈ −⎪ ⎪= =⎨ ⎬+ − ≥ −∈ −⎪ ⎪⎩ ⎭

(5.17)

where, 1 2 3 4, , ,c c c c 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:

( )( ) ( )( ) ( )

( )1 2 i1

3 4 i

îfW w 1,2,3,...,îf

i i i i ii

i i i i i

d d w w w w w wi m

d d w w w w w w

⎧ ⎫+ − < −∈ −⎪ ⎪= =⎨ ⎬+ − ≥ −∈ −⎪ ⎪⎩ ⎭

(5.18)

where, 1 2 3 4, , ,d d d d are constants that are internally calculated and similar form can be

written for andG H . A Sequential Unconstrained Minimization Technique (SUMT) is

employed to reach the minimum unconstrained objective function ( )jf at design iteration j

that is, ( ) ( ) ( ) ( )jj j jx x f f→ → where( )j

x is the design variable vector corresponding

to ( )jf . The final step performed at each design iteration is the determination of the design

variable vector to be used in the next iteration ( )j+1 . Vector ( )j+1x is determined according

to the following equation.

( ) ( ) ( ) ( )( )1 jj b bx x C x x+ = + − (5.19)

where ( )bx 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 ( )jx , or the previous design set ( )1jx − or the best design set ( )bx is feasible and

one of the following conditions is satisfied.

( ) ( )1j jf f τ−− ≤ (5.20a)

( ) ( )j bf f τ− ≤ (5.20b)

( ) ( ) ( )1 1, 2,3,...,j ji i ix x i nρ−− ≤ = (5.20c)

( ) ( ) ( )1,2,3,...,j bi i ix x i nρ− ≤ = (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


- 85 -

conditions are reached.

s sn N= (5.21)

si sin N= (5.22)

where sn is the number of subproblem iterations, sin is the number of sequential

infeasible design set, sN is the maximum number of iterations and siN 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.

31 2

1 1 1 10

( , ) ( ) ( ) ( ) ( )mm mn

x i g i h j w ki i j k

fQ x q P x q P g P h P wf = = = =

⎡ ⎤= + + + +⎢ ⎥

⎣ ⎦∑ ∑ ∑ ∑ (5.23)

where ( , )Q x q is the dimensionless unconstrained objective function; xP , gP , hP , wP are the

penalties applied to the constrained design and state variables and 0f 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 xP are applied to the design variables. State variables constraints are represented

by extended interior penalty functions , ,g h wP P P . For example, for state variables

constrained by an upper limit, the penalty function is written as:


- 86 -

( )2

ig i

i i

gP gg

κ⎛ ⎞

= ⎜ ⎟+ χ⎝ ⎠ (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 ig . 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 ( )fQ x and ( , )pQ x q as defined in Equations (5.25) and (5.26). Then, Equation

(5.23) takes the form as shown in Equation (5.27).

( )0

ffQ xf

= (5.25)

31 2

1 1 1 1( , ) ( ) ( ) ( ) ( )

mm mn

p x i g i h j w ki i j k

Q x q P x q P g P h P w= = = =

⎡ ⎤= + + +⎢ ⎥

⎣ ⎦∑ ∑ ∑ ∑ (5.26)

( ) ( ) ( ), ,f pQ x q Q x Q x q= + (5.27)

where functions fQ and pQ 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 ( )jd is devised. The next iteration ( )1j + is obtained from Equation (5.28).

In this equation, measured from ( )jx ,the line search parameter jS corresponds to the

minimum value of Q in the direction ( )jd .

( ) ( ) ( )jj

jj dSxx +=+1 (5.28)

The solution for jS uses a combination of a golden-section algorithm and local quadratic

fitting technique. The range for jS is limited to:

*max0100j jSs s≤ ≤ (5.29)

where *js is the largest possible step size for the line search of the current iteration (internally

computed) and maxS 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


- 87 -

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

the unconstrained objective function.

( ) ( )( ) ( ) ( )00 0 0, pfd Q x q d d= −∇ = + (5.30)

in which 1q = and

( ) ( )( ) ( ) ( )( )0 0 0 0andf p pfd Q x d Q x= −∇ = −∇ (5.31)

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

subsequent iterations ( )0j > ,conjugate directions are formed according to the

Polak-Ribiererecursion formula [164].

( ) ( )( ) ( )11,j j j

k jd Q x q r d −−= −∇ + (5.32)

( )( ) ( )( ) ( )( )( )( )

1

1 21

, , ,

,

Tj j j

jj

Q x q Q x q Q x qr

Q x q

−

−−

⎡ ⎤∇ −∇ ∇⎢ ⎥⎣ ⎦=∇

(5.33)

It should be noticed that when all design variable constraints are satisfied, ( ) 0x iP x = .

This means that q can be factored out of pQ , and can be written as:

( )( ) ( )( ) ( ), if 1, 2,3,...j jp p i i iQ x q q Q x x x x i 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).

( ) ( ) ( )jj jpfd d d= + (5.35)

( ) ( )( ) ( )11

jj jf j ffd Q x r d

−−= −∇ + (5.36)

( ) ( )( ) ( )11

jj jp p j pd q Q x r d

−−= − ∇ + (5.37)


- 88 -

The algorithm is occasionally restarted by setting 1 0jr − = , 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:

( )( ) ( )( ) ( )( )j j ji

i i

Q x Q x x e Q x

x x

∂ + ∆ −≈

∂ ∆ (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

( )100i i i

Dx x x∆∆ = − (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 ( )1j + set and the best ( )b set as shown in

Equation (5.40), in which τ is the objective function tolerance.

( ) ( )1j jf f −− ≤τ and ( ) ( )j bf 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

1in N= (5.41)

where in is the number of iterations, 1N 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 32500 /kg m 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.


- 89 -

DAM

1 2 3

b=0.25m

12

L=6mDAM

4 5 6 7

h=0.2m

DAM

108 9 11 1413 15

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

Natural frequency (Hz) Mode Damaged beam Initial FE model Error (%) MAC %

1 7.257 8.990 23.80 99.5 2 33.683 35.914 6.62 99.4 3 68.706 80.632 17.35 96.2 4 131.100 142.930 9.02 97.7 5 141.200 149.140 5.62 99.8 6 193.560 222.530 14.96 90.3 7 305.210 319.160 4.57 97.7 8 382.460 432.530 13.09 92.4 9 420.960 449.050 6.67 99.3 10 522.300 562.420 7.68 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.


- 90 -

• 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

locations are not known.

0 100 200 300 400 500 6000

100

200

300

400

500

600

Updated frequncies(Hz)

Freq

uenc

ies

obta

ined

from

ass

umed

dam

aged

bea

m(H

z)

UPs=6UPs=10UPs=30

(a) Frequencies correlation

1 2 3 4 5 6 7 8 9 100.88

0.9

0.92

0.94

0.96

0.98

1

Mode Number

MA

C

UPs=6UPs=10UPs=30

(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


- 91 -

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

MAC Tuning Residuals in objective function

Number of updating

parameters

Max. error infrequency tuning (%) Min. Max.

Max. error in damage

detection (%)6 0.10 0.99 1.00 3.5

10 0.30 0.98 0.99 18.5 Frequency

Only 30 3.50 0.97 0.99 26.7 6 0.18 0.99 1.00 21.1

10 0.20 0.99 1.00 22.5 MAC related

function Only 30 0.42 0.99 0.99 23.1

6 0.40 0.99 0.99 7.6 10 0.37 0.99 0.99 23.0

Frequency + MAC related function

30 3.05 0.97 0.99 31.4 6 0.29 0.99 1.00 3.2

10 0.40 0.99 1.00 14.9 Flexibility

Only 30 2.36 0.97 0.99 24.5 6 0.26 0.99 1.00 3.1

10 0.27 0.99 0.99 12.3 Frequency + MAC related function +

flexibility 30 1.29 0.97 0.99 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).


- 92 -

Table 5.3: Error detection after updating when frequency, MAC related function and flexibility residual are used in objective function

UPs =6 UPs =10 UPs =30 Elem. Error (%)

E I Error (%)

E I Error (%)

E I 1 1.6 0.2 2 3.3 0.6 3 0.1 1.5 1.2 12.3 10.0 12.7 4 0.6 0.7 5 3.2 2.5 2.7 2.0 6 13.3 13.1 7 3.4 5.1 8 0.8 0.7 5.5 5.3 0.2 4.0 9 0.0 3.4 10 2.8 3.1 0.8 0.8 13.2 18.3 11 6.8 3.4 12 3.7 0.6 13 6.9 5.1 12.5 10.5 14 1.6 0.5 15 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.


- 93 -

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


- 94 -

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.

Pingan

2212.20

old bridge

15×500

9000

Tied bar

750

central line of bridge2237.20

2244.64

750

Huangyuan

(a) Elevation

50

1850

850

50

850

50

500500500500500500500500

transverse supporting steel-truss

river side

9000

(b) Plan

Figure 5.4: Elevation and plan of arch bridge


- 95 -

(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.


- 96 -

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


- 97 -

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 Measurement points Reference point (at same location)1 1，2，3，4，5，6，7，8 R1 2 9，10，11，12，13，14，15，16 R2 3 17，18，19，20，21，22，23，24 R3 4 25，26，27，28，29，30，31，32 R4

4(20) 6(22)15(31)13(29)11(17)

16(32)

12(28)10(16) 14(30)

Accelerometer

9(25)

8(24)

5(21)3(19) 7(23)

1(17)

Reference Point

2(18)

Tied bar

17

32

31

29

30

27

26

28

24

22

23

20

19

21

18

25

10

Upstream

2

Reference Ponit

1

Bridge Surface

6

4

3

5

8

7

9

12

11

13

14

15

16

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.


- 98 -

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) INV306 data acquisition system (b) Arrangement of accelerometers

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


- 99 -

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


- 100 -

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


- 101 -

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


- 102 -

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

Natural frequency (Hz) Mode Experiment Initial FE model Error(%) MAC %

First vertical 2.002 1.743 -12.93 93.0 Second vertical 2.511 2.210 -11.98 96.0 First torsion 2.827 2.391 -15.42 96.8 First transverse 2.780 2.669 -3.99 62.1 Third vertical 3.473 2.778 -20.01 75.1 Fourth vertical 3.864 3.541 -8.35 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


- 103 -

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 near zero.

1.5 2 2.5 3 3.5 4 4.51.5

2

2.5

3

3.5

4

4.5

Frequency obtained from FEM (Hz)

Freq

uenc

y ob

tain

ed fr

om a

mbi

ent v

ibra

tion

test

(Hz)

Mode pair



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


- 104 -

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 V 2 V 1 Tor 1 Tra 3 V 4 V-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Modes

Eige

nval

ue s

ensi

tivity

E of concrete filled tubular arch ribE of material of deckThickness of deckDensity of concrete filled tubular arch ribDensity of deck material I of concrete filled tubular arch ribA of concrete filled tubular arch rib

1 V 2 V 1 Tor 1 Tra 3 V 4 V-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Modes

Eige

nval

ue s

ensi

tivity

E of steel used in hollow tubeE of cross girderI of cross girder about major axisArea of suspender Spring stiffness in transverse directionE of wall above deckDensity of wall above deckA of prestressed cable

Figure 5.15: Eigenvalues sensitivity of arch bridge to potential parameters


- 105 -

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


- 106 -

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 iα , the

difference between analytical and the measured eigenvalues of the lower modes can be

further minimized. In this work, based on the iterative procedures, iα 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


- 107 -

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

to find an exact solution based on the given tolerance.

70 80 90 100 110 120 130

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

No of iterations

Rat

io o

f FE

to

EX e

igen

valu

e in

eac

h ite

ratio

n

First bendingSecond bendingFirst torsionFirst transverseThird bendingFourth bending

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.


- 108 -

Table 5.6: Comparison of experimental and analytical modal properties of arch bridge after updating

Natural frequency (Hz) Mode Experiment After updating Error(%) MAC %

First vertical 2.002 1.962 -1.99 93.7 Second vertical 2.511 2.493 -0.69 96.5 First torsion 2.827 2.815 -0.42 97.3 First transverse 2.780 2.770 -0.35 76.9 Third vertical 3.473 3.256 -6.23 90.9 Fourth vertical 3.864 4.027 4.18 80.9

1.5 2 2.5 3 3.5 4 4.51.5

2

2.5

3

3.5

4

4.5

Frequency obtained from updated FEM (Hz)

Freq

uenc

y ob

tain

ed fr

om a

mbi

ent v

ibra

tion

test

(Hz)

Mode pair



Figure 5.17: Frequency and MAC correlation of arch bridge after updating


- 109 -

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

Updated values Change (%)

Elastic modulus of arch (Pa) 4.56×1010 5.30×1010 16.3 Elastic modulus of cross girders (Pa) 3.45×1010 4.26×1010 23.5 Elastic modulus of deck (Pa) 3.00×1010 3.90×1010 30.0 moment of Inertia of cross girder (m4) 0.0756 0.0972 28.5 Thickness of bridge deck (m) 0.25 0.246 -1.6 Mass density of arch (kg/m3) 2871.0 2010.0 -30.0 Mass density of deck (kg/m3) 2500.0 2144.0 -14.2 Sectional area of arch (m2) 0.4311 0.3384 -21.5 Sectional area of suspender (m2) 0.0025 0.0021 -16.0 Spring stiffness in lateral direction(N/m) 500000 516550 3.3


- 110 -

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.


- 111 -


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


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.


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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


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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 ( )xf subject to a number of constraints or bounds. A multi-objective

optimization problem is formulated as follows:

( ) ( ) ( ) ( )

( )( )

1 1n

minimize , ,.....,

subject to 0 1,....,

0 1,...

om

i e

i e

f x f x f x f xx

g x i m

g x i m mx x x

=∈ℜ

= =

≤ = +

≤ ≤

(6.1)

where x is the vector of design parameters, i.e., 1, 2.... nx x x x= , 1, 2, ......, omf f f are the

correlations or objective functions which are to be minimized and vector

function ( )g x returns the values of the equality and inequality constraints, which are also

called hard constraints and andx 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 ( )if x simultaneously does not exist. Instead, the

concept of noninferiority [170] (also called Pareto optimality [171]) must be used to

characterize the objectives.

( )( )

n

subject to 0 1,....,

0 1,...i e

i e

x

g x i m

g x i m mx x x

Ω = ∈ℜ

= =

≤ = +

≤ ≤

(6.2)

( )n where subject toy y f x xΠ = ∈ℜ = ∈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 nx∈ℜ that satisfies all the constraints as shown in


- 114 -

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:

( ) ( )( ) ( )

* *

* *

1,...,

for some

i i

j j

f x x f x i m

f x x f x 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, 1f , requires a degradation in the other objective, 2f , i.e., 1 1 2 2,B A B Af f f f< < .

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


- 115 -

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 * * * *1 2, ,....,

omf f f f= ,

which are associated with a set of objectives ( ) ( ) ( ) ( ) 1 2, ,....,omf x f x f x f 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 1 2, ,....,omw w w w= . It can be expressed as a standard optimization problem

using the formulation as shown in Equation (6.5).

( ) *

minimize,

such that 1,....,i i i o

xf x w f i m

γγ

γ

∈ℜ ∈Ω

− ≤ =

(6.5)

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

constraints, i.e., nxΩ = ∈ℜ , the term iwγ 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).

( )( )

*1 1 1

*2 2 2

minimize,

subject tox

f x w f

f x w f

γγ

γ

γ

∈Ω

− ≤

− ≤

(6.6)

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 * *1 2,f f , defines the goal point P . The


- 116 -

weighting vector defines the direction of search from P to the feasible space ( )γΠ .

During the optimization, γ is varied which changes the size of the feasible region. The

constraint boundaries converge to the unique solution point 1 , 2s sf f . 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).

( ) *

minimize max

1,....,

in

i ii o

i

x if x f

i mw

Π∈ℜ

−Π = =

(6.7)

( ) ( ) *1

, max 0,om

i i i ii

x r f x w fγ γ γ=

Ψ = + − −∑ (6.8)

where ir is the penalty parameters. When the merit function of Equation (6.8) is used as the basis of line search procedure, although ( ),x γΨ may decrease for a step in a given search direction, the function imax Λ 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 ( )xΨ equal to the worst case objective, i.e.,

( ) max ixi

Ψ = Π (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):

( )( ) *

1

max 0, ; if 0, max , 1,............. ; otherwise

om i i i i i

ii

f x w f wx i m

i

γ γγ

=

⎧ − − =⎪Ψ = ⎨ Π =⎪⎩∑ (6.10)


- 117 -

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 1f and 2f . 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


- 118 -

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:

( )2

11

1Tmaj ajT

j ej ej

Kf a

Kφ φφ φ=

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠∑ (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:

( )2

21

1fm

aj

ejjf a

λλ=

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠∑ (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:

( ) ( )

2

11 0 1

1 1s

Tmaj ajT

j ej ej

Kf a

f a K

φ φ

φ φ=

⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠

∑ (6.13)


- 119 -

( ) ( )

2

22 0 1

1 1fm

aj

ejjf a

f aλλ=

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠∑ (6.14)

As a result, after using the constraints ia and ia as lower and upper bounds respectively

for updating parameters, the minimization problem can be finally posed as:

( ) ( ) ( ) 1 2minimize ,

subject to 1, 2,3,......,i i i

f a f a f aa

a a a i N

=

≤ ≤ = (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 1f and 2f in Equations (6.13) and (6.14)

with respect to ia :

( )

11

1 0 1

1 2* * 1s

Tmaj ajT

i j ej ej

Kf Ca f a K

φ φ

φ φ=

⎡ ⎤⎛ ⎞∂ ⎢ ⎥⎜ ⎟= −⎜ ⎟∂ ⎢ ⎥⎝ ⎠⎣ ⎦

∑ (6.16a)

( ) ( )

( )

2

1 2

*T T Tej ej aj aj ej ej

i

Tej ej

KK C Ka

CK

φ φ φ φ φ φ

φ φ

⎡ ⎤⎧ ⎫⎛ ⎞∂⎪ ⎪−⎢ ⎥⎨ ⎬⎜ ⎟∂⎪ ⎪⎢ ⎥⎝ ⎠⎩ ⎭⎣ ⎦= (6.16b)

2

Taj ajT T

aj aj aj aji i i

KC K Ka a aφ φ

φ φ φ φ⎡ ⎤∂ ∂∂

= + +⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

(6.16c)

( ) ( )

22

12 0

1 12fm

aj aj

ji ej iej

fa f a a

λ λλλ=

⎡ ⎤⎛ ⎞ ∂∂ ⎢ ⎥⎜ ⎟= −⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎣ ⎦

∑ (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).


- 120 -


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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

10

20

30

40

50

60

element location

dam

age

%

20

50

29.99

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


- 121 -

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

Natural frequency (Hz) Mode Damaged beam Model after Updating Error (%) MAC %

1 8.245 8.252 0.084 100.00 2 34.920 34.936 0.045 100.00 3 75.080 75.071 -0.011 100.00 4 137.508 137.563 0.039 99.99 5 209.028 208.984 -0.021 100.00 6 313.581 313.122 -0.146 99.99 7 405.839 405.732 -0.026 99.99 8 547.260 545.741 -0.277 100.00 9 671.483 670.607 -0.130 99.99 10 836.938 834.069 -0.342 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


- 122 -

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 43.50 10× MPa and density of 32,500 /kg m corresponding to C50 grade of concrete.


- 123 -

400350

350300

2200

1900950

2.5

5

242

350

42

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

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

All dimensions are in mm

1 17 25

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


- 124 -

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]:

0.1(530 418) ( ) (6.18)

(6.19)2 ( )

comp n

nn

n n n

E S MPa

ASt b c

= −

=+

where nS is the shape coefficient of the neoprene bearing; , ,n n nb c t are the width, length and thickness of one layer of neoprene bearing respectively; and nA is the area of bearing. The vertical spring stiffness of the multi-layered composite bearing is then equal to:

/comp n comp n nv

n

E A E b cK N m

t t= =∑ ∑

(6.20)

∆1∆2

M1

υ

K2 K1

L

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


- 125 -

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 1M at support, both vertical springs

with stiffness 1K and 2K , which are separated dL distances apart, are compressed with

deflection 1∆ and 2∆ respectively. Using simple static, it can be obtained that:

12

1 2

1 1

d

ML K K

θ⎛ ⎞

= +⎜ ⎟⎝ ⎠

(6.21)

When 1 2 vK K K= = , the equivalent rotational stiffness of the support can be written as:

21 /

2v d

rotK LMK Nm rad

θ= = (6.22)

In this work, the rotational stiffness obtained by Equation (6.22) is also checked with

the approach shown in Equation (6.23).

/comp cgrot

n

E IK Nm rad

t=∑

(6.23)

where cgI 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


- 126 -

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.


- 127 -

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


- 128 -

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


- 129 -

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 Experiment Initial FE model Error(%) MAC %

First vertical 3.072 3.200 4.166 97.1 Second vertical 3.291 3.352 1.853 93.4 Third vertical 3.542 3.957 11.716 91.5 Fourth vertical 4.149 4.780 15.208 90.1 Fifth vertical 4.611 5.649 22.511 89.6


- 130 -

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

2.5 3 3.5 4 4.5 5 5.5 62.5

3

3.5

4

4.5

5

5.5

6

Frequency obtained from FEM(Hz)

Freq

uenc

y ob

tain

ed fr

om a

mbi

ent v

ibra

tion

test(

Hz) Mode pair

Figure 6.16: Frequency correlation of Hongtang bridge before updating


- 131 -

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 43.50 10× MPa is adopted for the elements of the concrete bridge superstructure.


- 132 -

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


- 133 -

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 Updated FE model Error (%) MAC %

First vertical 3.072 3.078 0.195 97.9 Second vertical 3.291 3.235 -1.701 94.3 Third vertical 3.542 3.627 2.399 92.6 Fourth vertical 4.149 4.143 -0.144 92.2 Fifth vertical 4.611 4.588 -0.498 91.8

2.5 3 3.5 4 4.5 52.5

3

3.5

4

4.5

5

Frequency obtained from FEM(Hz)

Freq

uenc

y ob

tain

ed fr

om a

mbi

ent v

ibra

tion

test(

Hz) Mode pair

Figure 6.17: Frequency correlation of Hongtang bridge after updating


- 134 -

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 Initial Values

Updated values

Change (%)

Elastic modulus of material of deck (MPa) 3.50×104 3.46×104 -1.09

Linear stiffness of support at abutment (N/m) 1.14×1010 8.95×109 -21.41

Linear stiffness of support at pier top (N/m) 3.54×1010 1.11×109 -96.90

Angular stiffness of support at abutment(Nm/rad) 8.55×107 1.10×108 28.71 Angular stiffness of support at pier top(Nm/rad) 3.19×1010 1.75×1010 -45.20

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.


- 135 -

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.


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.


- 136 -

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.


- 137 -

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:

exp( ) anaG a G G= − (7.1)

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

anaG is the analytical flexibility matrix corresponding to the measured DOFs;

both expG and anaG are calculated using Equation (4.19) and a is the vector of normalized


- 138 -

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 G aa (7.2)

1exp( ) TG a G −= −Φ Λ Φ (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:

( ) 2min

FG a

a (7.4)

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 d dn n× matrix A defined as the square root of the absolute square sum of its elements.

Mathematically, for [ ] d dn nA ×∈ℜ :

2

1 1

d dn n

jkFj k

A a= =

= ∑∑ (7.5)

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

becomes:

( ) ( )( )2

2

1 1

d dn n

jkFj k

G a G a= =

=∑∑ (7.6)

To avoid the numerical problems during minimization, this function is divided by the

function value at the initial parameter estimate.


- 139 -

( )( )( )

2

20

F

F

G af a

G a= (7.7)

As a result, after using the constraints ia and ia as lower and upper bounds respectively

for updating parameters, the minimization problem can be finally posed as:

( )min

such that , 1, 2,3,......,i i i

f aaa a a i N≤ ≤ =

(7.8)

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 ia , 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 ia , which gives Equation (7.10) and (7.11).

( )( ) ( )2

1 10

1 2d dn n

jki ij k jkF

f G a G aa aG a = =

⎛ ⎞∂ ∂= ⎜ ⎟∂ ∂⎝ ⎠

∑∑ (7.9)

( )

( ) ( )

1

1 1

T

i i

T T

i i i

Ga a

Ga a a

−

− −

∂ ∂= − Φ Λ Φ

∂ ∂

⎡ ⎤∂ ∂Φ ∂= − Λ Φ +Φ Λ Φ⎢ ⎥∂ ∂ ∂⎣ ⎦

(7.10)

( ) ( ) ( )1

1 1T T T

i i i i

Ga a a a

−− −⎡ ⎤∂ ∂Φ ∂Λ ∂

= − Λ Φ +Φ Φ +ΦΛ Φ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ (7.11)

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

11 1

i ia a

−− −∂Λ ∂Λ

= −Λ Λ∂ ∂

(7.12)

Hence, substituting Equation (7.12) into Equation (7.11) yields:

( ) ( )1 1 1 1T T T

i i i i

Ga a a a

− − − −⎡ ⎤∂ ∂Φ ∂Λ ∂= − Λ Φ −Φ Λ Λ Φ +ΦΛ Φ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

(7.13)


- 140 -

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

methods, let us recall a quadratic model at iterate kx : ( ) 12

T Tk k k kM d f f d d H d= +∇ + ,

which is stated in Equation (4.43a). Then a minimization of ( )kM d gives 1

k k kd d H f−= = − ∇ and the iteration proceed with Equation (4.47). Line search methods to

find kα , 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 k kdα . From this point of view,

then, there is something inherently unsatisfying in line search methods. If the model is not

trusted which yields kd , 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 kα , but rather, the entire search

direction kd 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:

( ) 1min2

such that

T Tk k k kM d f f d d H d

dd

= +∇ +

≤ ∆

(7.14)

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 2 0Td d −∆ ≤ . The theoretical


- 141 -

interpretation yields that:

• Either 1Bk kd H f−= − ∇ satisfies ,Bd ≤ ∆ in which case B

kd d= is the solution of

Equation (7.14), i.e., the constraint is not binding.

• Or ,Bd > ∆ and the solution of Equation (7.14) satisfies:

( ) 1,

,k k k

k

d H I f

d

−= − + ϒ ∇

= ∆ (7.15)

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 0ϒ = in Equation (7.15) and Bkd d= > ∆ . Then ϒ is gradually increased

until kd = ∆ . Thus, the steepest descent component is increased in the mix that makes

up kd . Indeed, if ∆ is very small, then ( )kM d can be approximated by Tk kf f d+∇ , for

which the constrained minimizer is:

kk

d ff∆

= − ∇∇

(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 km :

2

, kss k s T

k k k

fd f

f H fα α

∇= − ∇ =

∇ ∇ (7.17)

More generally, then, it has a mix of andB sd d . To solve the nonlinear Equation (7.15),

just in order to find a direction kd for the current value of ∆ is not realistic. Instead, the

dogleg method composes the direction kd directly from the two directions andB sd d .

• If ∆ is so large that Bd ≤ ∆ then, it is set Bkd d= .

• Otherwise, if ∆ is so small that sd ≥ ∆ , then it is taken as shown in Equation (7.18).

sk s

d dd∆

= (7.18)

• Otherwise, it is considered as shown in Equation (7.19).

( ) 2 2,s B sd d d d dν= + − = ∆ (7.19)


- 142 -

Then, expanding Equation (7.19), one gets:

( ) ( )2 22 22Ts s B s B sd d d d d dν ν∆ = + − + −

This is the quadratic equation for τ yielding:

( ) ( ) ( ) ( ) ( )2 2 22

2

T Ts B s s B s s B s

k B s

d d d d d d d d d

d dν

⎡ ⎤− − + − + ∆ − −⎢ ⎥⎣ ⎦=−

Taking positive root for kν and putting in Equation (7.19), one gets:

( )1B sk k kd d dν ν= + − (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:

( )( ) ( )0

k k kk

k k k

f f x dM M d

ρ− +

=−

(7.21)

The denominator is always non-negative, because kH is always positive definite.

If 0kρ < , the step is rejected. Indeed, it is demanded that kρ should be above some

positive value. So, ∆ is decreased. On the other hand, if kρ 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 2n 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 kf∇ is required and is available at a reasonable cost, say n operations. Thus,

the steepest descent direction k kd f= −∇ is also available, unaffected by the size of the


- 143 -

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 kH 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.


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


- 144 -

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.

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

0

10

20

30

40

50

60

element location

dam

age

%No noise0.5% noise3% noise

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 Damaged beam Model after Updating Error(%) MAC %

1 8.245 8.245 0.007 99.9992 34.920 34.916 -0.009 99.9993 75.080 75.044 -0.046 99.9994 137.508 137.420 -0.063 99.9985 209.028 208.884 -0.068 99.9976 313.581 313.280 -0.095 99.9947 405.839 405.283 -0.136 99.9958 547.260 546.611 -0.118 99.9929 671.483 670.491 -0.147 99.99110 836.938 835.731 -0.144 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


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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 Damaged beam Model after Updating Error(%) MAC %

1 8.245 8.085 -1.940 99.9932 34.920 34.354 -1.620 99.9923 75.080 73.913 -1.554 99.9704 137.508 135.333 -1.581 99.9235 209.028 205.242 -1.811 99.9526 313.581 308.366 -1.663 99.9427 405.839 400.675 -1.272 99.9638 547.260 537.816 -1.725 99.9419 671.483 661.384 -1.503 99.93010 836.938 825.920 -1.316 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


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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.

2m 2m2m

LoadLoad

0.25m

0.20

m

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.


- 147 -

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. 1 2 3 4 5 6 Load (kN) 4.0 6.0 12.0 18.0 24.0 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.


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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)

Mode 1 Mode 2 Mode 3 Mode 4 Load steps

Freq. Decr.(%) Freq. Decr.(%) Freq. Decr.(%) Freq. Decr.(%)

Ref. 21.90 - 60.32 - 117.02 - 192.02 - Step1 20.01 8.63 56.24 6.76 110.85 5.27 181.44 7.58 Step2 19.47 11.10 54.92 8.96 108.61 7.18 177.99 9.33 Step3 19.18 12.41 53.19 11.83 104.36 10.82 171.67 12.56 Step4 18.73 14.48 51.70 14.30 101.18 13.53 166.73 15.07 Step5 18.00 17.80 50.20 16.78 98.21 16.07 161.87 17.55 Step6 16.08 26.59 47.49 21.28 93.72 19.91 150.84 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


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

0 1 2 3 4 5 6-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Position on beam(m)

Mod

al d

ispla

cem

ent

damagedreference

(a) First mode shape

0 1 2 3 4 5 6

-0.08-0.06-0.04-0.02

00.020.040.060.08

Position on beam(m)

Mod

al d

ispla

cem

ent

damagedreference

(b) Second mode shape

0 1 2 3 4 5 6-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Position on beam(m)

Mod

al d

ispla

cem

ent

damagedreference

(C) Third mode shape

0 1 2 3 4 5 6

-0.08-0.06-0.04-0.02

00.020.040.060.08

Position on beam(m)

Mod

al d

ispla

cem

ent

damagedreference

(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.


- 150 -

The elastic modulus and inertia moment implemented in the original FE model are 38 GPa

and 4 41.66 10 m−× 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 Initial FE Model Error (%) MAC%

Reference state 1 21.904 22.213 1.410 99.977 2 60.329 61.065 1.219 99.939 3 117.022 119.287 1.898 99.857 4 192.026 196.320 2.187 99.754

Damaged state 1 18.005 21.870 21.466 99.881 2 50.204 60.956 21.416 99.850 3 98.219 118.218 20.361 99.796 4 161.876 194.176 19.953 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


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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 ia 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.

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29-10

0

10

20

30

40

50

element location

dam

age

%

Figure 7.7: Location and severity of damage after FE model updating (reference state)

Elastic modulus of each element is corrected using ia 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].

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29-10

0

10

20

30

40

50

element location

dam

age

%

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


- 152 -

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 After updating Error (%) MAC %

Reference state 1 21.904 21.870 -0.155 99.982 2 60.329 60.956 1.039 99.940 3 117.022 118.218 1.022 99.861 4 192.026 194.176 1.119 99.768

Damaged state 1 18.005 17.799 -1.144 99.900 2 50.204 52.676 4.923 99.881 3 98.219 104.951 6.854 99.801 4 161.876 172.429 6.519 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.


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.


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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


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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


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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,


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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


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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.


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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


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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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APPENDICES

APPENDIX A

Mathematical Background

• Let n nB ×∈ℜ be symmetric, i.e., TB B= .Then, if one considers the quadratic

form Ty By , for any ny∈ℜ , B is said to be positive definite if:

0, , 0T ny By y y> ∀ ∈ℜ ≠ (A.1)

• Convergence rate of algorithm

Let kx be a sequence in nℜ that converges to *x , with subscript k denoting the iteration

number. A method is said to be convergent if:

1 1k

k

+∈≤

∈ (A.2)

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

convergence if:

1lim kr

kC

k

+∈=

∈→∞ (A.3)

where C is the error constant. From the above it is seen that, 1C ≤ when 1r = . In this case,

when 1r = , the algorithm exhibits a linear convergence rate (which corresponds to slow

convergence). When 2r = , the rate is quadratic (which corresponds to fast convergence).

When 1and 0r C= = , 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 2l norm of a vector ny∈ℜ is:


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( )1 22 1 22

1

,n

Ti

i

y y y y y y y=

= = = =< >∑ (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 [ ] N mA ×∈ℜ :

2

1 1

N m

ijFi j

A a= =

= ∑∑ (A.5)

• Let : nf ℜ →ℜ be twice continuously differentiable ( )2f G∈ ℜ . The Hessian matrix

is:

( ) ( )

2 2 2

21 1 2 1

2 2 2

2 22 1 2 2

2 2 2

21 2

.

.

. . . .

n

n

n n n

f f fx x x x x

f f fH x f x x x x x x

f f fx x x x x

⎛ ⎞∂ ∂ ∂⎜ ⎟∂ ∂ ∂ ∂ ∂⎜ ⎟⎜ ⎟∂ ∂ ∂⎜ ⎟

= ∇ = ∂ ∂ ∂ ∂ ∂⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(A.6)


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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 jλ and jφ be a solution of the undamped eigenvalue problem:

j j jK Mφ = λ φ (B.1)

Premultiplying Equation (B.1) by Tjφ gives:

0Tj j jK M⎡ ⎤φ −λ φ =⎣ ⎦ (B.2)

Differentiating Equation (B.2) with respect to the set of correction factors that are assigned

to the physical parameters ia gives:

0T

jj jT Tj j j j j j

i i i

K MK M K M

a a a

⎡ ⎤∂ −λ∂φ ∂φ⎣ ⎦⎡ ⎤ ⎡ ⎤− λ φ + φ φ + φ −λ =⎣ ⎦ ⎣ ⎦∂ ∂ ∂ (B.3)

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

0jTj j

i

K Ma

⎡ ⎤∂ −λ⎣ ⎦φ φ =∂

(B.4)

The derivative of term in the middle of Equation (B.4) gives:

j jj

i i i i

K M K MMa a a a

⎡ ⎤∂ −λ ∂λ∂ ∂⎣ ⎦ = − −λ∂ ∂ ∂ ∂

(B.5)


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Putting this value in Equation (B.4),

0jT T Tj j j j j j j

i i i

K MMa a a

∂λ∂ ∂φ φ −φ φ −φ λ φ =

∂ ∂ ∂ (B.6)

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

0jT Tj j j j j

i i i

K Ma a a

∂λ∂ ∂φ φ − −φ λ φ =

∂ ∂ ∂

j Tj j j

i i i

K Ma a a

∂λ ⎡ ⎤∂ ∂= φ −λ φ⎢ ⎥∂ ∂ ∂⎣ ⎦

(B.7)

B.2 Eigenvector Derivatives

For an undamped equation of motion, one obtains,

[ ] [ ]( ) [ ] [ ] [ ] 0j jj j j

i i i i

K MK M M

a a a a

∂ ∂⎛ ⎞∂ ∂− + − − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

φ λλ λ φ (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 [ ] [ ]( )jK Mλ− is singular. Fox and Kapoor [132] proposed therefore to assume that:

1

dj

jq qqia

φβ φ

=

∂=

∂ ∑ (B.9)

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


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0j jj j

K MK M

a aλ φ

φ λ⎡ ⎤∂ − ∂⎣ ⎦ ⎡ ⎤+ − =⎣ ⎦∂ ∂

(B.10)

Substituting Equation (B.9) into Equation (B.10)

1

0d

jj j jq q

q

K MK M

aλ

φ λ β φ=

⎡ ⎤∂ −⎣ ⎦ ⎡ ⎤+ − =⎣ ⎦∂ ∑ (B.11)

or, 1

dj

jq j q jq

K MK M

aλ

β λ φ φ=

⎡ ⎤∂ −⎣ ⎦⎡ ⎤− = −⎣ ⎦ ∂∑ (B.12)

Premultiplying Equation (B.12) by Tsφ ,with ;s j≠

1

djT T

jq s j q s jq

K MK M

aλ

β φ λ φ φ φ=

⎡ ⎤∂ −⎣ ⎦⎡ ⎤− = −⎣ ⎦ ∂∑ (B.13)

Due to orthogonality properties, the left hand side of Equation (B.13) is equal to zero

except for s q= .

( ) forjTjq q j q j

K Mq j

aλ

β λ λ φ φ⎡ ⎤∂ −⎣ ⎦− = − ≠

∂ (B.14)

Expanding the right side,

( ) jT T Tjq q j q j q j j q j

K MMa a a

λβ λ λ φ φ φ φ λ φ φ

∂∂ ∂− = − + +

∂ ∂ ∂ (B.15)

As q j≠ , the second term of right side of Equation (B.15) is zero. So,

( ) T Tjq q j q j j q j

K Ma a

β λ λ φ φ λ φ φ∂ ∂− = − +

∂ ∂

1 forTjq q j j

q j

K M q ja a

β φ λ φλ λ

∂ ∂⎡ ⎤= − ≠⎢ ⎥− ∂ ∂⎣ ⎦ (B.16)


- 177 -

It is clear from Equations (B.4) and (B.14) that, for q j= , the coefficients jqβ have to

calculated separately. By differentiating 1Tj jMφ φ = ,it follows that:

0Tj jT T

j j j jMM M

a a aφ φ

φ φ φ φ∂ ∂∂

+ + =∂ ∂ ∂

2 jT Tj j j

MMa aφ

φ φ φ∂ ∂

= −∂ ∂

(B.17)

Substituting Equation (B.9) into Equation (B.17)

12

dT Tj jq q j j

q

MMa

φ β φ φ φ=

∂= −

∂∑ (B.18)

12

dT T

jq j q j jq

MMa

β φ φ φ φ=

∂= −

∂∑

and due to orthogonality condition, one obtains

12

Tjq j j

Ma

β φ φ∂= −

∂ (B.19)

Hence from Equations (B.16) and (B.19) ,

( )

,

,

12

Tq j j q j

i ijq

Tj j

i

K q ja a

q ja

φ λ λ λ φβ

φ φ

⎧ ⎡ ⎤⎛ ⎞∂ ∂Μ− − ≠⎪ ⎢ ⎥⎜ ⎟∂ ∂⎪ ⎝ ⎠⎣ ⎦= ⎨

∂Μ⎪− =⎪ ∂⎩

(B.20)


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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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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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[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.

Documents

55525700 Finite Element Model Updating of Civil Engineering Structures Under Operational Conditions