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BRIDGE DAMAGE DETECTION USING A SYSTEM IDENTIFICATION METHOD

By

WIRAT LERTPAITOONPAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2000

Copyright 2000

by

Wirat Lertpaitoonpan

The author dedicates this dissertation to his parents, Arnong and RoongrotLertpaitoonpan

iv

ACKNOWLEDGMENTS

The author very sincerely acknowledges the tremendous contribution of his

advisor, Dr. Marc I. Hoit, who is not only the chair of supervisory committee for this

research, but who also provided the author with ideas, guidance, support, and

encouragement throughout the research. This research would not have succeeded without

his advice. The author is greatly indebted for his kindness. The author would also like to

thank to Dr. Clifford O. Hays, Jr., Dr. John M. Lybas, and Dr. Duane S. Ellifritt, as well

as Dr. Fernando E. Fagundo for their friendly help and advice during the author’s

graduate studies. Their knowledge and kindness is highly appreciated. The author also

would like to thank to Dr. Ian Flood for his advice and his kindness in being a member of

the supervisory committee.

The author also would like to thank to his colleagues at the civil engineering

workstation lab, especially M. Williams, for their friendship and support.

Finally, the author would also like to thank his family for their love,

encouragement, and concern and P. Srisaichua for her understanding and support.

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TABLE OF CONTENTSpage

ACKNOWLEDGMENTS ..................................................................................................iv

LIST OF TABLES........................................................................................................... viii

LIST OF FIGURES ............................................................................................................ix

ABSTRACT ......................................................................................................................xv

1 INTRODUCTION .........................................................................................................1

1.1 Background.............................................................................................................11.2 Objective of Dissertation ........................................................................................31.3 Literature Review ...................................................................................................31.4 Organization of Dissertation...................................................................................8

2 MODELING OF BRIDGE STRUCTURES................................................................10

2.1 Introduction...........................................................................................................102.2 Two-Dimensional Model......................................................................................112.3 Three-Dimensional Model....................................................................................14

2.3.1 Modeling of Girders ...................................................................................15 2.3.2 Modeling of Slabs.......................................................................................17 2.3.3 Modeling of Diaphragms............................................................................18 2.3.4 Modeling of Secondary Structures .............................................................19 2.3.5 Modeling of Supports .................................................................................19 2.3.6 Modeling of Bridges ...................................................................................20

2.4 AASHTO LRFD Live Load Specification ...........................................................23 2.4.1 Design Truck ..............................................................................................23 2.4.2 Design Tandem...........................................................................................24 2.4.3 Design Lane Load.......................................................................................25 2.4.4 Application of Design Vehicular Live Loads .............................................25

2.5 Two-Dimensional Live Load Modeling ...............................................................26

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2.5.1 Longitudinal Distribution ...........................................................................26 2.5.2 Transverse Distribution ..............................................................................27

2.6 Comparing Models ...............................................................................................29 2.6.1 Load Case 1 (two trucks) ............................................................................30 2.6.2 Load Case 2 (one truck)..............................................................................31

3 OVERVIEW OF OPTIMIZATION TECHNIQUES...................................................33

3.1 Introduction...........................................................................................................333.2 Basic Concept .......................................................................................................343.3 General Procedure of Optimization Technique ....................................................36

3.3.1 Optimality of Unconstrained Problems ......................................................38 3.3.2 Optimality of Constrained Problems ..........................................................40

3.4 Procedures of the Unconstrained Optimization Technique ...................................43 3.4.1 Search Direction .........................................................................................45 3.4.2 Finding Step Length....................................................................................59 3.4.3 Condition of Convergence..........................................................................65

4 BRIDGE DAMAGE DETECTION.............................................................................69

4.1 Introduction...........................................................................................................694.2 Basic Concept .......................................................................................................704.3 Characteristics of Bridge Structure.......................................................................72

4.3.1 Eigenvalues and Eigenvectors ....................................................................72 4.3.2 Ritz Vectors ................................................................................................75 4.3.3 Observed Characteristics ............................................................................77 4.3.4 Simulating the Observed Characteristics....................................................77 4.3.5 Model Characteristics .................................................................................80

4.4 Finite Element Model of Bridge Structure ...........................................................804.5 Finite Element Analysis Program.........................................................................814.6 Optimization Routine ...........................................................................................824.7 Damage Detection Routine...................................................................................834.8 Parameters of Damage Detection Routine............................................................83

4.8.1 Objective Function......................................................................................83 4.8.2 Design Variables.........................................................................................86

4.9 Improvement of the Damage Detection Routine ..................................................884.10 Damage Detection Testing on Structural Elements............................................89

4.10.1 Test of Spring Element .............................................................................91 4.10.2 Test of Truss Element...............................................................................93 4.10.3 Test of Beam Element ..............................................................................95 4.10.4 Test of Shell Element ...............................................................................97 4.10.5 Test of Combining Element of Beam and Shell Elements .......................99

4.11 Damage Detection Testing on Bridge Structure ...............................................103 4.11.1 Corrosion in Reinforcing Steel ...............................................................105 4.11.2 Weakening of Material Properties in Girders .........................................110

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4.11.3 Weakening of Material Properties in Slabs ............................................115 4.11.4 Damage in Supports................................................................................120 4.11.5 Cracking of bridge girder........................................................................125

5 PARAMETRIC STUDY ...........................................................................................131

5.1 Introduction.........................................................................................................1315.2 Detection Techniques .........................................................................................1325.3 Magnitude of Perturbations ................................................................................1335.4 Responses ...........................................................................................................1375.5 Objective Functions ............................................................................................1405.6 Noise...................................................................................................................143

6 CONCLUSION..........................................................................................................149

REFERENCES ................................................................................................................152

BIOGRAPHICAL SKETCH ...........................................................................................155

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LIST OF TABLES

Table page

2.1 Results of Load Case 1............................................................................................... 31

2.2 Results of Load Case 2............................................................................................... 31

5.1 Noise Effect.............................................................................................................. 144

ix

LIST OF FIGURES

Figure page

2.1 Two-Dimensional Beam Element and Its Degrees of Freedom .................................11

2.2 Longitudinal of Bridge Structure 2-D Modeling ........................................................12

2.3 Transverse Direction of Bridge Structure...................................................................12

2.4 Simple Supports Between Girders..............................................................................13

2.5 Continuous Slab and Hinge on top of Girders............................................................13

2.6 Continuous Slab and Frame Action............................................................................14

2.7 Three-Dimensional Beam Element and Its Degree of Freedoms. ..............................15

2.8 Three-Dimensional Truss Element and Its Degree of Freedoms................................16

2.9 Three-Dimensional Girder Model with Rigid End Offset. .........................................16

2.10 Four-Node Shell Element and Its Degree of Freedom..............................................17

2.11 Slab Model with Rigid End Offset ...........................................................................18

2.12 Girder-Slab Bridge Structures with Diaphragms and Parapets.................................21

2.13 Girder-Slab Bridge Model with Diaphragms and Parapets ......................................22

2.14 LRFD Design Truck .................................................................................................24

2.15 LRFD Design Tandem..............................................................................................25

2.16 Longitudinal Load Distribution in 2-D Model .........................................................27

2.17 Moving Load in Transverse Direction......................................................................28

2.18 Load Distribution on Simple Supports between Girders ..........................................28

x

2.19 Load Distribution on Continuous Slab and Hinge at top of Girders ........................29

2.20 Load Distribution on Continuous Slab and Frame Action .......................................29

2.21 Longitudinal Load Position ......................................................................................30

2.22 Transverse Load Position of Load Case 1 ................................................................30

2.23 Transverse Load Position of Load Case 2 ................................................................31

3.1 G lobal, Local Optimum and Stationary Points. .........................................................38

3.2 Geometrical Interpretation of Kuhn-Tucker Conditions ............................................42

3.3 Flow Chart of Unconstrained Optimization Procedures.............................................44

3.4 Flow Chart of Random Search Optimization Procedures...........................................46

3.5 Flow Chart of Steepest Decent Optimization Procedures ..........................................48

3.6 Example of Graphical Movement of Steepest Decent Optimization..........................49

3.7 Flow Chart of Fletcher-Reeves Conjugate Gradient Optimization Procedures..........52

3.8 Example of Movement of Conjugate Gradient Optimization.....................................53

3.9 Flow Chart of Quasi-Newton Optimization Procedures.............................................58

3.10 Flow Chart of Bracket One-dimensional search Method .........................................60

3.11 Flow Chart of Golden Section One-dimensional search Method.............................66

3.12 Flow Chart of Terminating Optimization Process....................................................68

4.1 Concept of Bridge Damage Detection Using System Identification ..........................71

4.2 Diagram of Impact Hammer Vibration Testing..........................................................78

4.4 Flow Chart of General Procedure of Damage Detection Routine ..............................84

4.5 Flow Chart of Damage Detection Routine with Screening Technique.......................90

4.6 Testing Spring Element Model...................................................................................91

4.7 Damaged Indicators of Simulated Damaged Spring...................................................92

xi

4.8 Predicted Damaged Indicators from Damage Detection Program..............................92

4.9 Testing Truss Element Model.....................................................................................93

4.10 Damaged Indicators of Simulated Damaged Truss ..................................................94

4.11 Predicted Damaged Indicators from Damage Detection Program............................94

4.12 % Extent Error of Predicted Damaged Indicators.....................................................95

4.13 Testing Beam Element Model ..................................................................................95

4.14 Damaged Indicators of Simulated Damaged Beam ..................................................96



4.17 Testing Shell Element Model ...................................................................................97

4.18 Damaged Indicators of Simulated Damaged Shell Structure ...................................98



4.21 Testing Model of Combine Beam and Shell Elements...........................................100

4.22 Damaged Indicators of Simulated Damaged Shell Structure .................................101

4.23 Predicted Damaged Indicators from Standard Technique ......................................101

4.24 % Extent Error of Predicted Damaged Indicators from Standard Technique.........102

4.25 Predicted Damaged Indicators from Screening Technique.....................................102

4.26 % Extent Error of Predicted Damaged Indicators from Screening Technique .......102

4.27 Testing Model of a Bridge Structure ......................................................................105

4.28 Damaged Indicators of Simulated Corrosion in Reinforcing Steel ........................105

4.29 Predicted Damaged Indicators from Eigen Properties Characteristics ...................106

4.30 %Extent Error from Eigen Properties Characteristics ............................................106

xii

4.31 Predicted Damaged Indicators from Ritz Vector Characteristics ...........................107

4.32 %Extent Error from Ritz Vector Characteristics ....................................................107





4.37 Damaged Indicators of Simulated Weakening in Material of Bridge Girders........110



4.40 Predicted Damaged Indicators from Ritz Vectors Characteristics .........................112

4.41 %Extent Error from Ritz Vectors Characteristics ..................................................112





4.46 Damaged Indicators of Simulated Weakening in Material of Bridge Slab.............115








xiii

4.54 %Extent Error from Dynamic Response Characteristics........................................119

4.55 Damaged Indicators of Simulated Damage in a Girder Support ............................120









4.64 Damaged Indicators of Simulated Damage of Cracking in a Girder ......................125









5.1 Standard Algorithm and Screening Algorithm .........................................................133

5.2 Magnitude of Perturbation........................................................................................135

5.3 Damage Indicators of Simulated Damage Structure.................................................136

5.4 Damage Indicators of Predicted Damage from 10% Perturbation Rate ...................136

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5.5 Damage Indicators of Predicted Damage from 5% Perturbation Rate .....................136

5.6 Damage Indicators of Predicted Damage from 0.5% Perturbation Rate ..................137

5.7 Performance of Ritz Vectors and Eigen Properties ..................................................138

5.8 Damage Indicators of Simulated Damage Structure.................................................139

5.9 Damage Indicators of Predicted Damage from Eigen Properties .............................139

5.10 Damage Indicators of Predicted Damage from Ritz Vectors..................................139

5.11 Performance of Objective Functions ......................................................................141

5.12 Damage Indicators of Simulated Damage Structure...............................................142

5.13 Damage Indicators of Predicted Damage from Least Squared Errors ....................142

5.14 Damage Indicators of Predicted Damage from Relative Errors .............................142

5.15 Damaged Indicators of Simulated Damage in a Girder Support ............................145

5.16 Predicted Damaged Indicators without Noise ........................................................146

5.17 Predicted Damaged Indicators for 5% Noise Indicators.........................................146

5.18 Predicted Damaged Indicators for 10% Noise Indicators.......................................146



xv

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

BRIDGE DAMAGE DETECTION USING A SYSTEM IDENTIFICATION METHOD

By

Wirat Lertpaitoonpan

May 2000

Chairman: Marc I. HoitMajor Department: Civil Engineering

Highway bridges are large and expensive structures. Failure in such a structure

causes a huge impact on human life and the economy. Many studies show that

approximately 250,000 of the more than existing 570,000 highway bridges in the United

States are deficient and in need of rehabilitation. It is necessary to know the condition of

bridges in order to prevent an abrupt failure. Visual investigation is normally used to

monitor bridge structures but is not sufficient. Sophisticated procedures like x-ray,

acoustic emission, and magnetic resonance can provide great detail and a reliable

investigation, but those procedures are expensive and time consuming. An alternative

method for monitoring bridge structures is damage detection using system identification

methods. These methods are in a group of nondestructive damage detection techniques.

System identification is the process of matching a mathematical model to an existing

structure. This is based on the fact that when structure is damaged, its characteristic

response is also changed. This research studies the possibility of using system

xvi

identification methods to detect damage in bridge structures using the information of the

change in structural characteristics. A three-dimensional bridge finite element model was

used as the mathematical model. Eigen properties (eigenvectors and eigenvalues) and

Ritz vectors are used as structural characteristics. This research proposes a screening

algorithm for finding the damage by reducing the number of design variables and

adjusting the amount of perturbation during the optimization as well as using a relative

error as an objective function. This research also studies the sensitivity of major

parameters that affect the damage detection using the system identification method. The

research yields a damage detection tool that successfully identifies location and extent of

simulated damage in bridge structures.

1

CHAPTER 1INTRODUCTION

1.1 Background

Highway bridges accumulate damage through out their service life. The damage

comes not only from the usual weakening in material properties during service life but

also from unexpected extreme events like earthquakes, storms, or ship-impact. Damage

can be denoted by cracking in the structure, corrosion, deterioration of material

properties, or loss of pre-stressing. For many of these causes, the defects in highway

bridges are not always visual or detected easily. Regardless, it is necessary to know if a

bridge has any types of defects or damage. It is important to identify the damage, since a

failure in a highway bridge may cause a catastrophic failure resulting in loss of human life

and a large economic impact.

There are an estimated 577,710 bridges in the United States: 42 percent of them,

approximately 250,000 bridges, are deficient and in need of rehabilitation (Gordon 1989).

A similar study by Better Roads (1994) presented the bridge inventory in the United

States based on a sufficiency rating (SR) assigned by FHWA. The report shows 30

percent of the highway bridges in the United States are substandard (sufficient rating of

80 or less). With the importance of damage in highway bridges and the high amount of

current deficient highway bridges, proper damage detection methods are needed.

2

Because the cost of large-scale destructive testing is prohibitive, nondestructive

structure damage detection is most suitable for bridges. Several nondestructive structure

methods of monitoring damage in highway bridges include visual inspection

supplemented with X-rays, acoustic emission, magnetic resonance, and ultrasonic testing.

Quite often these experimental methods are costly, time consuming and difficult to

perform on inaccessible structural components. Alternatively, analytical methods are

sometimes used to ascertain the extent of damage. A damage detection technique known

as system identification can be used to determine the extent of the damage using a finite

element model and the characteristic responses of the investigated bridge. The system

identification method applied to bridges can be faster and cheaper at performing damage

detection.

Monitoring damage in the structures can be classified into 4 levels:

1. Detect the changes of the structure2. Detect the location of the damage3. Detect the extent of the damages4. Detect the type of damages.

System identification is the process of constructing a mathematical model of a

dynamical system from observations and prior knowledge (Norton 1986). Theoretically,

damage detection using system identification methods can perform the detection for all

four levels. Damage detection with system identification uses the fact that all structural

damage changes the structural properties, stiffness, mass or damping ratio, which affect

the overall response, natural frequency, mode shapes, displacements from loading. Using

system identification, a finite element (FE) model can be created to represent an

undamaged bridge, which can then be used to compute the response of the bridge model

3

for comparison to the measured response from the damaged bridge. If the responses are

not matched, the finite element model is updated using optimization techniques until the

responses match to a minimum error. This model then represents the current properties of

the damaged bridge.

1.2 Objective of Dissertation

Over the last decade, theories of system identification and highway bridge damage

detection have been the main focus of much research. But in practice, there have been

lack of an application of this damage detection method to general bridge structures.

Currently using system identification as a damage detection tool on bridge structures

requires an in-depth analysis of the particular bridge. However, damage detection using

system identification methods is very effective and cheap if the system is applied to the

bridge structure when the bridge is built. This research performs parametric studies for

general bridges and develops a procedure for damage detection using system

identification technique, which is built into an application that can be used to design

highway bridge structures. In doing so, an engineer using this application to design the

bridge structures, will automatically have the monitoring application in place.

1.3 Literature Review

Nondestructive structure damage detection techniques have been studied widely in

recent years, especially the system identification method. Several techniques have been

developed for nondestructive structure damage detection, as well as field experiments to

confirm the possibility of using the system identification to detect the damage in real

4

structures. The following literatures are some of the examples, which are the baseline of

present study.

Chen and Garba (1988) study the damage detection using the dynamic response

eigenmodes as the characteristic of the structure and using the structural stiffness matrix

as the variable. The analytical stiffness matrix was perturbed to match the damaged

structure. The minimum deviation approach was used as the identification. The study was

fairly successful in determining the damaged stiffness matrix. However the system could

not clearly determine the location of the damage since each element of the structure

stiffness matrix could have contributions from many members of the structure.

Soeio (1990) investigated the output error and equation error method of damage

assessment using system identification. The output error and equation error technique

determines the changes in the analytical model necessary to minimize the differences

between the measured and predicted response. The study used static displacement and

eigenmodes as the measured response. The planar (2-D) truss and its finite element model

were built and tested. The output error method was able to detect damage in the structure

if the errors in the measurements are kept within certain bounds. However there is

difficulty in reducing the design variables in the system. The present research adopts his

basic idea of this output error approach in the optimization process.

Muhammad, Halling, and Womack (1998) presented the results of forced

vibration testing of a full-scale reinforced concrete bridge span. A nine-span three-lane

freeway overpass structure was demolished, leaving an isolated single span supported by

two bents. These two bents were subjected to lateral load capacity testing and retrofitting

using carbon fiber composite wrapping. Testing performed forced vibration dynamic

5

testing between each episode of damage or retrofit. The project tested seven different

conditions of damage and retrofit. Data was collected with an array of accelerometers.

The dynamic characteristics, natural frequencies corresponding to the first three mode

shapes, were measured. The results show that the natural frequencies of the structure

decreased significantly due to the damage of the structure. Changes in both the amplitude

and the shapes of the modes were noted in the experimental results. This study indicates

the possibility of using dynamic responses as parameters in the system identification

damage detection.

Doebling and Farrar (1997) studied the effect of measurement statistics on the

detection of damage in the Almosa Canyon Bridge. The paper presents a comparison of

statistics on the measured modal parameters of a bridge structure to the expected changes

in those parameters caused by damaged. The work considers the most commonly used

modal parameters for indication of damage: modal frequency, mode shape, and mode

shape curvature. The study performed a test on the existing bridge, Alamosa Canyon

Bridge. The bridge is a concrete slab on steel girders with a span of 50 feet. The test also

simulated the damages in the finite element model by reducing the modulus of elasticity

in the damaged area to match the damage in the field. The results of this test show that

modal frequency undergoes a statistically significant change as a result of simulated

damage, as well as the individual components of the mode shape and mode shape

curvature.

Zimmerman, James, and Cao (1999) presented an experimental study of damage

detection using Ritz vectors. The Ritz vector is not new from the theoretical standpoint,

but in practice, to extract the higher modes, the use of Ritz vectors needs more study. The

6

first mode of a Ritz vector is the static deformation shape of the structure due to a

particular applied load. This study proposes an approach to extract the higher mode Ritz

vectors experimentally and compare the results of damage detection using Ritz vector and

traditional modal parameters obtained from both accelerometer and strain sensors. The

approach has been tested experimentally by using 35 inch cantilever beam. Seven

accelerometers were placed uniformly on the beam and the force was applied by a

calibrated impact hammer. The study concludes that both Ritz vector and dynamic

characteristics can be used as parameters in the damage detection but the Ritz vector

properties provided a better solution. This study shows the applicable of the Ritz vector to

the damage detection using system identification method.

Bolton, Stubbs, Park, Choi, and Sikorsky (1998) presented the measuring of

bridge modal parameters for use in non-destructive damage detection. This project

measures the dynamic characteristic (frequency, mode shape and damping ratio) of an

existing reinforced concrete bridge. Portable equipment including a relatively small drop

hammer and accelerometers were used to acquire modal properties of the structure. The

results of the field testing indicate that the collection of modal data using the portable

instrumentation performed well in providing the baseline modal data needed to detect

damage in medium-sized reinforced concrete highway bridge structures. The report also

shows that the results of modal analysis were very good. Some complex modes were

found, but most fundamental mode shapes and frequencies were in good agreement with

analytical results. Recorded data had good signal to noise ratio. Response is well above

ambient noise levels created by nearby traffic. Results from this testing indicate it is

7

possible to acquire baseline modal data on highway bridge structures without impeding

the usability of the bridge during testing.

Masri, Nakamura, Chassiakos, and Caughey (1996) developed a neural network

approach detecting changes in structure parameters. Their approach relies on the use of

vibration measurement from an undamaged structure to train a neural network for

identification purposes. The trained network is fed comparable vibration measurements

from the same structure under different types of response in order to monitor the damage

of the structure. It was shown that, through simulation studies with linear as well as

nonlinear models typically encountered in the applied mechanics fields, the proposed

damage detection methodology is capable of detecting relatively small changes in

structural parameters, even when the vibration measurements are noise-polluted.

However this neural network approach needs a training database and also works with

only a certain type of a structure. In the case of highway bridge structures, these problems

would cause some difficulties from the practical standpoint.

Juneja, Haftka, and Cudney (1998) presented the damage detection technique

using system identification based on combining frequency signature with contrast

maximization approach. Contrast maximization is used to find the excitation forces that

create maximum differences in the response of the damaged structure and the analytical

response of the undamaged structure. The optimal excitations for the damage structure are

then matched against a database of optimal excitations to locate the damage. The

technique was then tested with 36 degree of freedom space truss. The space truss and its

finite element model were built. The finite element model has been corrected using the

experimental data. The technique was applied to locate the damage in several members.

8

The experimental results indicate that this technique can identify the damage in the

structure. However this method also needs a database to store characteristics of particular

damaged scenario of the structure, which is difficult to obtain for bridge structures unlike

some other structures that can be built and then intentionally damaged and then the

needed characteristics measured. This approach shows the successful and improved

features of damage detection using the system identification method, but it is not yet

suitable for highway bridge structures.

1.4 Organization of Dissertation

This dissertation is separated into six chapters. Chapter 1 presents an introduction

which provides the background and objective of this research as well as a review of some

of the previous work in this area. Chapter 2 presents structural modeling, starting with a

simple two-dimensional model, and then presents a more complex three-dimensional

model. The standard specification of live loading according to AASHTO LRFD

specifications is presented along with its application to finite element models of highway

bridge structures. The load modeling of the two models is then described, and a

comparison of the results of live load distribution between the two models is given.

Chapter 3 contains an overview of optimization theory and its applications. The chapter

starts with the basic idea of the optimization and then presents the algorithm used in this

study. Chapter 4 presents the bridge damage detection procedure using the system

identification method. The potential parameters for the bridge damage detection are also

introduced. Finally the test of damage detection procedure is performed. The parametric

9

study of the bridge damage detection using the system identification method is reported in

Chapter 5. Chapter 6 includes summary, conclusion, and suggestions for future study.

10

CHAPTER 2MODELING OF BRIDGE STRUCTURES

2.1 Introduction

To design, analyze, or monitor the health of a bridge, a mathematical model of the

real bridge structure is needed. Several mathematical models can be used for these

purposes, depending on the desired degree of accuracy and capability of available

resources. Since this research seeks to develop an application that can analyze and

monitor a bridge structure, this chapter will investigate such a mathematical model. The

long-term goal of this research is to be a part of a complete application for bridge

structures, which will be able to design both superstructures and substructures of bridges.

This investigation will also consider the accuracy of live load distribution of the models

using AASHTO LRFD standard loads from superstructures onto substructures. This

chapter starts with modeling a simple 2-dimensional finite element model (2-D FEM),

and then describes the more complex 3-dimensional model (3-D FEM). For the

investigation of the live load distribution of the model, the AASHTO LRFD specification

for live loading is described as well as the load distribution modeling of the two structural

models. The load distribution comparison of two models is presented last.

11

2.2 Two-Dimensional Model

A two-dimensional model of bridge structures is the most common simple model,

but its accuracy is in doubt. In the two-dimensional model, the structures are simply

modeled by two-dimensional beam elements, which are resistant to a bending moment

about out of plane axis and in plane axial forces. The degree of freedom of a 2-D beam

element is shown in Figure 2.1.

Figure 2.1 Two-Dimensional Beam Element and Its Degrees of Freedom.

Bridge structures are in reality three-dimensional structures. To analyze the forces

in the structures, two models are needed for forces in both longitudinal and transverse

directions. In the longitudinal model, the 2-D beam elements sitting on supports at the

end of spans can be used as shown in Figure 2.2.

The bridge structures in the transverse direction can be modeled differently

depending on the connection of the bridge slab and girders. Figure 2.3 shows the bridge

structures in transverse direction.

This study considers three types of transverse bridge structure models:

1. Simple supports between girders.2. Continuous slab and hinge on top of girders.3. Continuous slab with frame action.

12

Figure 2.2 Longitudinal of Bridge Structure 2-D Modeling.

Figure 2.3 Transverse Direction of Bridge Structure.

The first model is the simplest that has been used in conventional design. This

model assumes that there are simple supports between the adjacent girders as shown in

Figure 2.4.

(a) Longtitudinal Direction of Bridge Structure Model.

Slab

Girder

Pier

(b) Longtitudinal Direction of Bridge Structure 2-D FEM.

2-D Beam Elements

Bridge Slab

Bridge Girder

Pier Cap

13

Figure 2.4 Simple Supports Between Girders.

The second model considers the continuous effect of the slab but no transfer of

bending moment between the slab and girder. Figure 2.5 shows a simplified structural

model of this type.

Figure 2.5 Continuous Slab and Hinge on Top of Girders.

The last model considers not only the continuous effect of the slab but also the

transfer of bending moment between the slab and girder by rigid frame action. Figure 2.6

shows a simplified structural model of this type.

Bridge Slab

Bridge GirderPier Cap

Hinges

Bridge

BridgePier Cap

Hinge

14

Figure 2.6 Continuous Slab and Frame Action.

The 2-D beam elements are the only element that is needed in order to use the 2-D

bridge structure models, which is very simple compared to the 3-D models.

2.3 Three-Dimensional Model

Three-dimensional (3-D) models are more complicated models and need more

resources for analysis. In the 3-D model, each component of the bridges will be modeled

by different types of elements to more accurately model each component’s behavior.

Because of that, the analytical result from 3-D models is more accurate than the 2-D

model. The main components of bridge structures are the girders, deck slab, diaphragms,

supports, and possible composite parapet. However, the bridge structures can have

additional components that have an effect on structure behaviors, and they should be

added to the structure model. This research will focus mainly on the common components

stated above.

Bridge Slab


15

2.3.1 Modeling of Girders

The bridge girders are modeled with 3-D beam elements. These beam elements

take into account shear deformation, axial, and flexural deformations using standard

beam theory. To model the beam element, we need to know the cross-sectional properties

(modulus of elasticity, sectional area, and moment of inertia) as well as its geometry. The

degrees of freedom of a 3-D beam element are shown in Figure 2.7.

Figure 2.7 Three-Dimensional Beam Element and Its Degree of Freedoms.

If the bridge has reinforcing steel or prestressed tendons in the girders, the girders

will have the attachment of 3-D truss elements, which represent the reinforcing steel or

prestressed steel. In the case of prestressed steel the truss element will have an initial

force prior to the application of the loads to account for the prestressed forces. In both

cases, the truss elements will use the same node as the beam element but with a rigid-end

offset. The offset from the beam elements creates the eccentricity as it is in the real

girders. The truss elements will resist the elongation and contraction along the element.

The degrees of freedom of the 3-D truss element are shown in Figure 2.8. Figure 2.9

shows a typical prestressed concrete girder modeling with the rigid end offset.

XZ Y

16

Figure 2.8 Three-Dimensional Truss Element and Its Degree of Freedoms.

Figure 2.9 Three-Dimensional Girder Model with Rigid End Offset.

XZ Y

Rigid EndOffset

GirderCentroid

(a) Cross Section of Girder with Rigid End Offset

SteelCentroid

Rigid EndOffset

BeamElement

TrussElement

(b) Finite Element Model of Girder with Rigid End Offset

17

2.3.2 Modeling of Slabs

The typical bridge slab has a deflection along the vertical axis, curvature, and

displacement in longitudinal and transverse axes. The flat shell element or plate bending

element with in-plane displacements has the degrees of freedom or behavior to match the

slab characteristics. It is reasonable to model the bridge slab structure with flat shell

elements. There are two common theories for plate elements. The Kirchhoff plate theory

assumes that normal to the surface remain normal, thereby ignoring the shear

deformations. The Kirchhoff plate type is suitable for thin plates, where the shear

deformations are very small or negligible. The other theory is Mindlin plate theory. This

theory accounts for shear deformations in the element so that the normal vector does not

remain normal after being loaded. The Mindlin plate element is good for the thicker

plates which may experience shear deformation. The highway bridge slab is the later

condition so in this study, the Mindlin plate element with in-plane translation is used for

the bridge slab component. This element requires a modulus of elasticity of the material,

thickness of slab, and a coordinate. The degrees of freedom of four-node shell element are

shown in Figure 2.10.

Figure 2.10 Four-Node Shell Element and Its Degree of Freedom.

X

Z Y

18

If there is reinforcing steel or prestressed tendons in the slab, the slab will have

attached 3-D truss elements in a similar fashion to the girders. Figure 2.11 shows the slab

modeling with the rigid end offset.

Figure 2.11 Slab Model with Rigid End Offset.

2.3.3 Modeling of Diaphragms

Diaphragms are sometimes called cross beams because they lie transversely to the

main girder. The diaphragms themselves behave like girders, so in this study the three-

dimensional beam elements are used to model the diaphragms. The diaphragms link the

longitudinal girders together. Bridge structures need the diaphragms to provide stability

Rigid EndOffset

SlabCentroid

(a) Cross Section of Slab with Rigid End Offset.

SteelCentroid

Rigid EndOffset

ShellElement

TrussElement

(b) Finite Element Model of Slab with Rigid End Offset.

19

and lateral resistance for the girders. The diaphragms also help to distribute loads on the

slab to the girders. According to the AASHTO LRFD specification for steel structures,

the diaphragms may be placed at the end of the structure, across interior supports, and

intermittently along the span. At the end of the bridge and intermediate point where the

continuity of the slab is broken, diaphragms shall support the edges of the slab. For

concrete structures, the diaphragms shall be provided at abutments, piers and hinge joints

to resist lateral forces and transmit loads to point of support. Intermediate diaphragms

may be used to provide torsional resistance and to support the deck at the point of

discontinuity or at an angle point in the girders. Diaphragms may be omitted where tests

or structural analysis show them to be unnecessary.

2.3.4 Modeling of Secondary Structures

The secondary structures are the components that do not directly support the

applied loads. Parapets, curbs, sidewalks and railing are the examples of secondary

structures. If those secondary structures are designed to act compositely with the main

structure, they provide additional stiffness to the bridge structure. Secondary structures

need to be modeled as parts of the bridge structure since they will effect to the

characteristic response of the structure. The 3-D beam elements with rigid end offset to

the main structures can be used to model these structures.

2.3.5 Modeling of Supports

One of the big problems in modal analysis is the boundary condition of the

structural model. Having bad boundary conditions usually creates a large error in the

finite element analysis responses. The general support conditions (fixed, hinge, or roller

supports) do not really exist in the real structures. Because of friction, elastic properties

20

and imperfection of material, the supports condition tend to be in between the ideal

condition. Using elastic spring elements to model supports of the girders can reduce the

boundary condition problems. Hays, Consolazio, Hoit and Kakhandiki (1994) proposed a

reasonable value for the support stiffness of 1000 kip/in at the end supports and 3000

kip/in at the interior supports. However if the information of support stiffness is provided,

the model will be more accurate.

2.3.6 Modeling of Bridges

From the bridge component models presented above, the structural model of the

bridge can be built by connecting those component together according to the their

geometry and behaviors. Starting with the bridge slab, four-node shell elements are

created as a grid over the bridge. The girders then are added to the model by using 3-D

beam element with rigid-end offset from the centroid of slab to centroid of girders. The

spring elements are attached to the girder as elastic supports for the girders at both ends.

Next the diaphragms and secondary structures, if present, are connected to the girders and

slab according to their geometry by using the beam element in the same fashion as

girders. The complete 3-D finite element model of bridge structures is shown in Figure

2.12 and Figure 2.13.

This bridge structure model accounts for the effect of composite sections of the

slab and girders by those rigid links from centroid of the slab to the centroid of girders.

Therefore the girders properties are modeled using the girder gross section properties.

21

Figure 2.12 Girder-Slab Bridge Structures with Diaphragms and Parapets.

SpanLength

Slab

(a) Plan View of the Bridge Structures.

Diaphragm

Support Girder

Deck slab Parapet

(b) Cross-Section View of the Bridge Structures.

22

Figure 2.13 Girder-Slab Bridge Model with Diaphragms and Parapets.

Spring element (support)

Beam rigid link

Beam element (girder)

Beam element (diaphragm)

Shell element (slab)Beam element (parapet)

(b) Cross-Section View of the Bridge Structures Model.

(a) Overall View of the Bridge Structures Model.

23

2.4 AASHTO LRFD Live Load Specification

To design bridges, the designers have to follow the AASHTO specifications

(American Association of State Highway and Transportation Official). The Specification

provides a new system of loading, LRFD (Load and Resistant Factor Design), in the

AASHTO LRFD BRIDGE DESIGN SPECIFICATIONS (1994). The LRFD specification

separates loads on bridges as two categories: permanent loads and transient loads. A

correct finite element model of a highway bridge must consider all the essential

components that contribute to the structural response, but one of the most important loads

is vehicular live load. The LRFD specification states that the vehicular live loading on the

roadways of bridges or incidental structures, designated HL-93 (Highways Loading

adopted in 1993), shall consist of a combination of the design truck or design tandem, and

design lane load. Each design lane under consideration shall be occupied by either the

design truck or tandem, coincident with the lane load, where applicable. The loads shall

be assumed to occupy 10.0 ft transversely with in a design lane. This is a departure from

the previous AASHTO specifications which did not consider the contribution of truck

and lane loads together.

2.4.1 Design Truck

The AASHTO LRFD code provides a specification for a standard design truck,

base on the magnitude and position of concentrated loads from an actual truck. The

previous standard design truck called HS-20 is used in this case. The standard design

truck consists of three axles. The first axle has 8 kip weight and the last two axles have 32

24

kip weights. The first two axles are separated by 14 feet and the last two axles are spaced

between 14.0 feet and 30.0 feet to produce extreme force effects. The weights and

spacing of axles and wheels for the design truck are shown in Figure 2.14

Figure 2.14 LRFD Design Truck.

2.4.2 Design Tandem

The design tandem load is a group of two heavy axle loads with a close spacing.

This load represents the special vehicle type like a military vehicle. The design tandem

consists of a pair of 25.0 kip axles spaced 4.0 ft apart. The transverse spacing of wheels is

taken as 6.0 ft. The weights and spacing of axles and wheels for the design tandem are

shown in Figure 2.15

14 to 30 ft14 ft

32.0 kip32.0 kip8.0 kip

6 ft

(a) Side View of the Design Truck.

(b) Rear View of the Design Truck.

25

Figure 2.15 LRFD Desi

2.4.3 Design Lane Load

The design lane

condition due to traffic

of 0.64 klf, uniformly d

lane load is assumed to

2.4.4 Application of De

The AASHTO L

the requirement of load

25.0 kip25.0 kip

4 ft

(a) Side View of the Design Tandem.

gn Tandem.

load in AASHTO specification is intended to simulate the load

congestion on the bridge. The design lane load consists of a load

istributed in the longitudinal direction. Transversely, the design

be uniformly distributed over a 10.0 ft width on the deck.

sign Vehicular Live Loads

RFD specification provides not only the standard loading but also

position to produce the maximum force effect for design. As

6 ft

(b) Rear View of the Design Tandem.

26

stated in the specification, the extreme force effect is taken as the larger of the following

three cases:

1. The effect of the design tandem specified in Figure 2.15 combined with theeffect of the design lane load.2. The effect of one design truck with variable axle spacing specified in Figure2.14 combined with the effect of the design lane load.3. For both negative moment between points of dead load contraflexure, andreaction at the interior piers, only 90% of the effect of two design trucks spaced aminimum of 50 ft between the lead axle of one truck and rear axle of the othertruck, combined with 90% of the effect of the design lane load; the distancebetween the 32.0 kip axles of each truck shall be taken as 14.0 ft.

Both the design lanes and the position of the 10.0 ft loaded width in each lane

shall be positioned to produce extreme force effects. The lengths of the design lanes, or

parts thereof, which contribute to the extreme force effect under consideration, shall be

loaded with the design lane load.

2.5 Two-Dimensional Live Load Modeling

Finding the load distribution from the superstructures onto the substructures can

not be done accurately in a single 2-D model. A designer has to separate the longitudinal

distribution and transverse distribution into two separated models. Each direction is

analyzed using a two-dimensional analysis.

2.5.1 Longitudinal Distribution

In the longitudinal load distribution models, truckloads will be modeled as a series

of concentrated loads, which have the magnitude equal to the weight of the axles. The

lane loads are modeled as a uniformly distributed load. The structure, in the longitudinal

direction, is subjected to the simplified loads in order to compute the maximum reaction

27

at the middle support (pier) by moving the series of loads in the longitudinal direction

according to the AASHTO LRFD specification as shown in the Figure 2.16.

Figure 2.16 Longitudinal Load Distribution in 2-D Model.

2.5.2 Transverse Distribution

The transverse distribution load models the structure in transverse direction at the

middle support (pier). The model uses the reaction, Rt, which is computed from the

longitudinal direction as equivalent wheel loads. To calculate the reaction at each girder

(a) Longitudinal Moving Loads.

32 kip 32 kip8 kip

(b) Simplified Load and Structure Model.

Rt

28

for all possible load cases, the equivalent load is moved along the transverse direction

according to the AASHTO LRFD specification as shown in Figure 2.17.

Figure 2.17 Moving Load in Transverse Direction.

The transverse distribution load model has been studied with three general types

of structural mathematical models. These are shown below:

Figure 2.18 Load Distribution on Simple Supports between Girders.

Bridge Slab

Bridge Girder

Pier Cap

Rt/2Rt/26 ft.Bridge Slab


Hinges

Rt/2Rt/2

6 ft.Rt/2Rt/2

6 ft.

29

Figure 2.19 Load Distribution on Continuous Slab and Hinge at Top of Girders.

Figure 2.20 Load Distribution on Continuous Slab and Frame Action.

2.6 Comparing Models

As stated before, this study is interested in using a model that is suitable for both

design and monitoring bridges structures. We have presented two general models: simple

2-D model and the more complex 3-D model. The 2-D model is easier to analyze, but the

accuracy is doubtful when compared to the 3-D model. The test of live load distribution

from bridge superstructures into bridge substructures was performed to compare the

results of 2-D model to 3-D model. The structural models of the bridge were created

using both 2-D and 3-D modeling techniques. Both models were analyzed with the bridge

Rt/2Rt/2

6 ft.

Rt/2Rt/2

6 ft.Rt/2Rt/2

6 ft.Bridge Slab


Hinges

Rt/2Rt/26 ft.

Rt/2Rt/2

6 ft.Rt/2Rt/2

6 ft.Bridge Slab


30

structures subjected to several load cases. The 2-D model was analyzed by using SSTAN

(Static Structural Analysis) written by Hoit (1995) and the 3-D model is analyzed by

LiveGen (Live Load Generation) developed for this study base on BRUFEM (Hays et al.

1994). The following are a couple of examples from those analyses.

Figure 2.21 Longitudinal Load Position.

2.6.1 Load Case 1 (two trucks)

The first load case considers two trucks placed on the structure simultaneously.

Figure 2.22 Transverse Load Position of Load Case 1.

32.0 K32.0 K8.0 K

100 ft.100 ft.

6 ft.4 ft.Bridge Slab

Bridge Girder

Pier Cap

Rt/26 ft.

Rt/2 Rt/2 Rt/2

31

Table 2.1 Results of Load Case 1.

2-D % Forced Difference from the 3-D ModelModel Girder#1 Girder#2 Girder#3 Girder#4

Model 1 -1.3% 3.0% 7.1% -7.9%Model 2 9.5% -16.0% 23.3% -18.3%Model 3 3.3% -3.5% 12.4% -12.7%

2.6.2 Load Case 2 (one truck)

The second load case considers only one truck placed on the structure.

Figure 2.23 Transverse Load Position of Load Case 2.

Table 2.2 Results of Load Case 2.

2-D % Forced Difference from the 3-D ModelModel Girder#1 Girder#2 Girder#3 Girder#4

Model 1 -6.2% 27.8% -50.0% -50.0%Model 2 5.2% 10.4% -57.6% -53.6%Model 3 -2.4% 28.6% -55.7% -45.3%

The 2-D models analysis provides uncertain results. For the full span load, the

results of 2-D model analysis are not much different than the 3-D model analysis. If the

Bridge Slab


Rt/2

6 ft.

Rt/2

32

loads do not cover the whole span, the results of 2-D model analysis are a lot different

than the 3-D model. With the uncertain accuracy, using 2-D model analysis might not

cover the worst case of loading. This study suggests that to design bridge structures the 3-

D model is more reasonable and reliable. The study of live load distribution from bridge

superstructures into bridge substructure has been studied more intensively by Williams

(2000) using neural networks to position live loads on bridge pier.

The accurate mathematical model is very important in the damage detection using

system identification methods. The model has to be able to provide accurate behavior of a

real structure in order to use response data from finite element analysis to identify an

existing structure. The accurate 3-D finite element model is chosen to be a model of

bridge structure in this study.

33

CHAPTER 3OVERVIEW OF OPTIMIZATION TECHNIQUES

3.1 Introduction

Optimization techniques are implemented as one of the main components in

damage detection using the system identification method. In system identification,

optimization techniques are used to match the finite element model responses to the

damaged structure responses. This chapter will describe the background and basic theory

of optimization techniques as well as the optimizing algorithm that is used in the damage

detection procedure in this research.

Gottfried and Weisman (1973) presented an interesting point of view about

optimization techniques:

In a classical sense, optimization can be defined as the art of obtaining the bestpolicies to satisfy certain objective, at the same time satisfying fixed requirements.It would be presumptuous at this time to suggest that optimization has attained thestatus of science rather than art; however, recent advances in applied mathematics,operations research, and digital-computer technology enable many complexindustrial problems in engineering and economics to be optimized successfully byapplication of logical and systematic techniques. (p.4)

One of the optimization techniques called mathematical programming has a

relatively short history, approximately fifty years of development starting in the 1950s.

However, in the last decade, optimization techniques have been widely developed as

automated designing tools for most of the engineering fields. Many optimizing algorithms

34

have been developed. Each of the algorithms has their own advantages and disadvantages.

Although optimization techniques try to find the best solution for the problem, it is not

always possible for the solution to reach the optimum point depending on the

characteristics of the problem.

3.2 Basic Concept

Optimization techniques seek the best solution while satisfying certain constraints.

This concept comes from the human intuition that seeks to have the best solution with the

least scarcity under certain rules. The concept of optimization can be translated into a

numerical form as follows:

Minimize: F(X) (3.1)

Such that: gj(X) ≤ 0 j = 1,m (3.2)

hk(X) = 0 k = 1,l (3.3)

XLi ≤ Xi ≤ XU

i i = 1, n (3.4)

Where X is a vector of design variables

=

nX

XXX

.

.3

2

1

X

gj(X) is an inequality constraint

35

hk(X) is an equality constraint

XLi is a lower limit of Xi

XUi is an upper limit of Xi

The system of equations (3.1) to (3.4) comprise the “General Problem Statement”

or “Standard Formula for Optimization”. There are three main components in the system

of standard formulas for optimization: variables, constraints, and an objective function.

The variables are designated by X in the formulas and represent a vector of variables Xi.

The design variables are the parameters that are changed or updated by the algorithm in

order to improve the solution. In structural design, the design variables can be the cross-

sectional area, width, thickness, and weight, for example.

The constraints are the restrictions for the problem. The constraints can be

classified into three categories: inequality constraints, equality constraints, and side

constraints. The inequality constraints are shown in the Equation (3.2). The inequality

constraints are not only restrict to “less than or equal” as shown in (3.2) but also “greater

than or equal,” however most of the optimization researchers prefer to use the form as

shown in (3.2). An example of inequality constraints could be “a total weight of a design

girder that is less than two hundred pounds.” Equation (3.3) represents the equality

constraints. Some algorithms can not handle equality constraints. Modifications have to

be done by using two inequality constraints; one that is greater than or equal and another

one that is less than or equal. An example of equality constraints could be “a perimeter of

a design girder that has to be equal to fifty inches.” The last constraints are side

constraints or boundary constraints. These constraints define the upper and lower bound

36

of the design variables as shown in (3.4). These constraints are treated differently from

the first two. Problems that have only side constraints, no equality constraints and

inequality constraints, are considered by some algorithms to be an unconstrained

problem. An example of side constraints could be “each side of a design girder has to be

greater than ten inches and less than fifteen inches.”

The last component is the objective function, shown in Equation (3.1). The

objective function represents the goal that the optimization searches for. The minimum

cost of production or maximum load capacities of a girder are examples of objective

functions. Once again, the objective function (3.1) is not restricted to minimization but

can also be used for maximization. However, the minimization of function is more

common among optimization researchers.

3.3 General Procedure of Optimization Technique

The optimization technique that is used in this research and in many other

engineering fields relies on a numerical search. These techniques rely on a search

direction to improve the solution. The techniques start with known initial design variables

and an objective function. Changes to the design variables are made gradually to improve

the objective function without violating the constraints using the search direction

technique. The change procedures are repeated until the objective function can no longer

be improved or the necessary optimality conditions called Kuhn-Tucker conditions are

met (Haftka and Gurdal 1996). The optimization procedure using numerical search

techniques can be written in numerical form as:

37

Xq = Xq-1+αSq (3.9)

Where Xq is an updated vector of design variables.

Xq-1 is current vector of design variables.

α is a scalar value of step length (move parameter).

Sq is a vector of updated search direction.

The goal of this optimization technique is to get to the optimum solution for the

problem. However there are two types of optimality in the optimization: local optimality

and global optimality. The global minimum is defined by having the lowest possible value

of the objective function. The local minimum is defined by having the lowest value of the

objective function in a specific domain.

Global optimum:

F(X*) ≤ F(X) for all X (3.10)

Local optimum:

F(X*) ≤ F(X) if ||X-X*|| < R for some R (3.11)

Where X* is the vector of design variables at optimum.

Figure 3.1 illustrates graphically the local and global minimum. In the figure point

C is a local minimum and point E is the global minimum of this problem.

38

Figure 3.1 Global, Local Optimum, and Stationary Points.

The general procedure of optimization techniques can be classified into two

categories: unconstrained problems and constrained problems.

3.3.1 Optimality of Unconstrained Problems

The unconstrained problems are problems that try to find the minimum of

objective function (F(X)) without constraints (gj(X) or hk(X)). For unconstrained

problems the Kuhn-Tucker conditions are simply where the gradient of objective function

vanishes (Vanderplaats 1999).

∇ F(X) = 0 (3.12)

X

F(X)

E

A

B

C

D

39

where

∂∂

∂∂∂

∂∂

∂

=∇

nXF

XF

XF

XF

F

)(....

)(

)(

)(

)(3

2

1

X

X

X

X

X .

The Kuhn-Tucker condition is a necessary condition or a condition for a stationary

point. It is not sufficient to indicate the optimality of the problem by only satisfying the

Kuhn-Tucker condition. The sufficient condition for the optimality is the positive definite

of a Hessian matrix. The Hessian matrix (H(X)) is a second derivative of objective

function with respect to the design variables:

∂∂∂

∂∂∂

∂∂∂

∂∂∂

∂∂

∂∂∂

∂∂∂

∂∂∂

∂∂

=

nnnn

n

n

XXF

XXF

XXF

XXF

XF

XXF

XXF

XXF

XF

H

)(...)()(............

)(...)()(

)(...)()(

)(

2

1

2

1

2

2

2

22

2

12

21

2

21

2

21

2

XXX

XXX

XXX

X . (3.13)

The term positive definite means that all the eigenvalues of the matrix are

positive. The condition for the positive definite Hessian matrix can be defined as the

following:

40

∆XTH(X*)∆X > 0 (3.14)

Note that even a positive definite Hessian Matrix guarantees only a local

minimum. All the points shown in the Figure 3.1 (A, B, C, D, and E) satisfy the Kuhn-

Tucker necessary condition (∇ F(X) = 0) but only point C and E satisfy the sufficient

condition so that only points C and E are the minimum points.

3.3.2 Optimality of Constrained Problems

The constrained problems try to find the minimum of objective function (F(X))

with at least one constraint (gj(X) or hk(X)). For this case the optimality conditions are not

as simple as before. First of all, the Lagrangian function is introduced as:

∑∑=

+=

++=h

g

g n

kkkn

n

jjj hgFL

11)()()(),( XXXX λλλ (3.15)

where λ is a Lagrangian multiplier,

F(X) is an objective function,

gj(X) is an inequality constraint function,

hk(X) is an equality constraint function,

ng is the number of inequality constraints, and

nh is the number of equality constraints.

The governing equation for the optimality necessary condition in the constrained

problem is the stationary condition of the Lagrangian function.

41

Assuming the objective function (F(X)), and all constraints functions (gj(X) and

hk(X)) are differentiable. The Kuhn-Tucker conditions for the necessary condition of

optimality consists of three requirements as following:

1. All the design variables need to be in the feasible domain:

All gj(X*) ≤ 0 (3.16)

2. The product of λ j and gj(X*) must be zero:

λ j gj(X*) = 0 j = 1, ng (3.17)

3. The gradient of the Lagrangian function must vanish:

0),( * =∇ λXL

0)()()(1

*

1

** =∇+∇+∇ ∑∑=

+=

h

g

g n

kkkn

n

jjj hgF XXX λλ (3.18)

λ j ≥ 0 j = 1, ng

where

∂∂

∂∂∂

∂∂

∂

=∇

n

j

j

j

j

j

Xg

Xg

Xg

Xg

g

)(....

)(

)(

)(

)(3

2

1

X

X

X

X

X .

42

The physical meaning of the Kuhn-Tucker conditions can be shown in two-

dimensional space as Figure 3.2.

0)()()( 1111 =∇+∇+∇ XXX ggF λλ

Figure 3.2 Geometrical Interpretation of Kuhn-Tucker Conditions.

The feasible domain denotes all the possible design spaces that satisfy all the

constraints. The infeasible domain is the spaces that violate at least one of the constraints.

The constraints that are the boundaries of the feasible and infeasible domain, (at gj = 0)

are called active constraints and all others constraints are called inactive constraints.

∇ F

∇ g1∇ g2

g1 = 0

g2 = 0

F=k

∇ Fλ1∇ g1

λ2∇ g2

X1

X2

Feasible domain

Infeasible domain

43

3.4 Procedures of the Unconstrained Optimization Technique

The unconstrained optimization problem in this study is defined as a problem that

does not have any inequality constraints (gj(X)) or equality constraints (hk(X)). However,

the problem can have side constraints (boundary constraints).

The standard formulation is:

Minimize: F(X)

Such that: XLi ≤ Xi ≤ XU

i i = 1, n

With the known initial design variables, the optimization technique will update

the design variables such that the objective is improved while the design variables are in

the feasible domain using the equation search technique as stated in equation (3.9), which

is repeated here:

X q = Xq-1+αSq (3.19)

The general procedures of this optimization technique can be shown as a flow

chart in Figure 3.3.

The main issues in the optimization procedure for the unconstrained problems

using search methods consist of the following three parts:

1. Determine the useable-feasible search direction, S.2. Compute the scalar of step length or move parameter, α.3. Determine if the problem has converged to an acceptable solution.

44

Figure 3.3 Flow Chart of Unconstrained Optimization Procedures.

Start, known Xinitial

q=q+1

Evaluate F(Xq-1)

Compute ∇ F(Xq-1)

Determine search direction Sq

Compute α

Compute F(Xq)= F(Xq-1)+ α Sq

Check for Convergence to optimum.

Stop

q=0 and X0=Xinitial

Yes

No

45

3.4.1 Search Direction

The search direction is one of the most important tasks of the optimization

technique. The name of the optimization algorithm usually comes from its search

direction technique such as Steepest Descent, Conjugate Gradient, BFGS, etc. Here the

search direction techniques will be classified upon their degree of derivative in the

objective function required. There are many search direction algorithms which have been

developed in the last fifty years. This chapter will present only the basic algorithms that

lead to the algorithm used in this research.

3.4.1.1 Zero Order Method

Zero order search direction methods do not require any derivative of the objective

function. They employ the optimum solution by using the value of objective function.

The examples of zero order search direction technique are Random Search, Sequential

Simplex Method, and Powell’s Conjugate Direction Method. The procedure of Random

Search will be described by an example.

The Random Search is the simplest and easiest search method but it is also the

most inefficient search method (Vanderplaats 1999). The Random Search method

chooses the next set of design variables (Xq) randomly in the feasible domain (within the

boundary constraints) and then evaluates the objective function with these new design

variables. The new value of the objective function will be compared to the previous

value. If the new value is lower, the new set of design variables is kept. The procedure is

repeated until the iteration number reaches the maximum number of iterations, then the

process is terminated. The flow chart of this method is shown in the Figure 3.4.

46

Figure 3.4 Flow Chart of Random Search Optimization Procedures.

The Random Search technique, like the other zero order methods, is usually easy

to implement and requires very small computer storage. The problem of zero order search

methods is that they require a very high number of function evaluations in order to

converge to the optimum, even for a simple problem. These zero order search techniques

F*= F(X*)

q=q+1

Select Xq, XL < Xq < XU

Evaluate F(Xq)

X* = Xq

Stop

Yes

No


q=0 , X0=Xinitial, X* = X0

F*= F(X*)

No

Yes

Check F(Xq)<F(X*)

Check q > qmax

47

are not suitable for the complicated problems which have a computationally expensive

function evaluation.

3.4.1.2 First Order Method

The first order search direction methods require a first derivative of the objective

function in order to compute the search direction for the optimization procedure. The

Steepest Decent, and the Fletcher-Reeves Conjugate Gradient Method are well-known

techniques among the first order methods. This section will present the procedures of

these two methods.

3.4.1.2.1 Steepest Decent MethodThe steepest decent technique is one of the oldest methods for optimization with

multiple design variables and also the simplest method among the first order methods.

This method was first developed by Cauchy in 1847 for solving a system of linear

equations. The steepest decent method computes the search direction from the negative of

the first order derivative of the objective function. The search direction usually is

normalized such that the magnitude of the search direction vector equals to one (unit

vector). The steepest decent search direction may be shown in the following form.

)( 1−−∇= qq F XS (3.20)

After the search direction is found, the one-dimensional search is performed to

find a step length for a minimum of the objective function. The flow chart of the steepest

decent method is presented in the Figure 3.5.

48

Figure 3.5 Flow Chart of Steepest Decent Optimization Procedures.


q=q+1

Evaluate F(Xq-1)

Compute ∇ F(Xq-1)

Sq = -∇ F(Xq-1)

Compute α

Compute F(Xq)= F(Xq-1)+ α Sq

Check for Convergence to optimum.

Stop

q=0 and X0=Xinitial

Yes

No

49

The steepest decent technique normally makes a large improvement at the very

first iterations. After that, the improvement is very small with a zigzag pattern called

“hemstitching” which can cause a poor convergent rate. The poor performance of this

method arises because it does not make use of the information from the previous

iterations. The steepest decent movement pattern is demonstrated in the Figure 3.6.

Figure 3.6 Example of Graphical Movement of Steepest Decent Optimization.

The steepest decent method has only a linear rate of convergence. The

performance of the steepest decent method can be improved by re-scaling the design

X2

X1

50

variables. Unfortunately, the procedure to re-scale the large problem requires a lot of

work including calculation of the Hessian matrix and an eigenvalue analysis.

3.4.1.2.2 Fletcher-Reeves Conjugate Gradient MethodThe Fletcher-Reeves conjugate gradient method modifies the steep decent method

by making use of the information from the previous iteration. The method uses the

conjugate gradient to determine the search direction. The conjugate gradient condition is

shown in Equation (3.21).

( ) 0)( =jTi SHS (3.21)

Where Si and Sj are the search directions.

H is the Hessian Matrix.

The Fletcher-Reeves conjugate gradient method starts with the same procedure as

the steepest decent in the first iteration by using the negative of the objective function

gradient as a first search direction. From the second iteration onward the search directions

are selected such that they are conjugate to the previous search direction by using the

following equation.

Sq = -∇ F(Xq-1)+βqSq-1 (3.22)

Where( )( )22

21

−

−

∇

∇=

q

qq

XF

XFβ . (3.23)

51

The Fletcher-Reeves conjugate gradient method theoretically will converge a

quadratic objective function to the minimum by “n” or less iteration, with “n” equal to the

number of design variables. Most engineering problems are not quadratic functions and

some numerical error exists. As a result, the Fletcher-Reeves conjugate gradient method

needs to be restarted periodically to ensure a good optimization. The flow chart of the

Fletcher-Reeves conjugate gradient method is presented in the Figure 3.7.

The Fletcher-Reeves conjugate gradient method has a quadratic convergent rate,

which is a lot more efficient than the linear convergence rate in steepest decent method,

with only small amount of modification. This method is also easy to manipulate and

needs only a small amount of computer storage. The Fletcher-Reeves conjugate gradient

movement pattern compared to the steepest decent is shown in the Figure 3.8.

Even though the Fletcher-Reeves conjugate method offers a significant

improvement over the steepest decent method, the performance of this method lags

behind the second order search methods.

52

Figure 3.7 Flow Ch

initial

No

StopYe


art of Fletcher-Reeves Conjugate Gradient Optimization Procedures.

S = -∇ F(X)

Compute α

Compute α

Compute X= X+ α S

Stop

Evaluate ∇ F=∇ F(X)

A=∇ FT∇ F

q=0 and X=X

Yes

Check α=0s

No


B=∇ FT∇ F

β=B/A

S = -∇ F +βS

A=B

θ = S ∇ F

θ ≥ 0Yes

No

Compute α

Compute X= X+ α S

Converge

53

Figure 3.8 Example of Movement of Conjugate Gradient Optimization.

3.4.1.3 Second Order Method

Second order search direction methods improve the efficiency of the optimization

by using first and second derivative in addition to the value of objective function. The

Newton second order method is the most common and straightforward technique for the

second order search direction method (Fletcher 1980).

The Newton second order method is derived from a truncated Taylor series

expansion of F(X) about Xq to the second order term.

X2

X1

54

[ ] XXXXXXXX δδδδ )(21)()()( qTTqqq HFFF +∇+=+ (3.24)

where δX=Xq+1-Xq. (3.25)

The correction δX is defined such that the derivative of the objective function

(3.24) vanishes. Then the correction δX can be written as:

[ ] )()( 1 qq FH XXX ∇−= −δ (3.26)

Rewriting the equation (3.25) for Xq+1 and substituting equation (3.26) into

equation (3.27) yields.

Xq+1= Xq + δX (3.27)

[ ] )()( 11 qqqq FH XXXX ∇−= −+ (3.28)

From the basic formula of updating the design variables (3.9), repeated here as

(3.29). If the scalar parameter is equal to one (α =1.0), using equation (3.28) the search

direction vector can be described in terms of the gradient and Hessian matrix of the

objective function as shown in equation (3.30).

Xq = Xq-1+αSq (3.29)

[ ] )()(1 qqq FH XXS ∇−= − (3.30)

55

Equation (3.30) provides the search direction for the general Newton second-

order method. This method can converge a quadratic function to an optimum with only

one iteration. Again in practice, most of the problems are not truly quadratic. A

modification strategy is needed for a particular type of problem to improve the

convergence efficiency. Regardless of the efficiency, the second order method needs a

real Hessian matrix, which is very expensive to calculate in a practical engineering

problem. This problem leads to newer algorithms that have the equivalent convergence

rate but do not need the real Hessian matrix. These algorithms are grouped into a Quasi-

Newton method.

3.4.1.4 Quasi-Newton Method

The quasi-Newton search direction method combines the idea of the Fletcher-

Reeves conjugate gradient and the Newton second order method. The conjugate gradient

method uses the information of the last iteration to compute a scalar parameter β. The

quasi-Newton method also uses the information of previous iterations. Instead of the very

last iteration, the quasi-Newton method keeps all the previous information in a matrix

form. This method is sometimes called the variable metric method. The quasi-Newton

method is derived in the same way as Newton second order method, from the truncated

Taylor series expansion of objective gradient about Xq. This time the real Hessian matrix

will be approximated by G(Xq).

[ ] XXXX δ)()()( 1 qqq GFF +∇=∇ + (3.31)

Assuming G(Xq) as an approximated Hessian matrix (H(Xq)) at the qth iteration.

The equation (3.31) can be rewritten as:

56

[ ] XXXX δ)()()( 1 qqq GFF =∇−∇ + (3.32)

and

[ ] XXXX δ=∇−∇ ++ )()()( 11 qqq FFB (3.33)

where B(Xq+1) is the approximated inverse matrix of the Hessian, [H(Xq)]-1

Many books have a compact form of equation (3.32) and (3.33) as follows:

yq = Gqpq (3.34)

Bq+1yq = pq (3.35)

where pq = δX = Xq+1-Xq.

yq = ∇ F(Xq+1) - ∇ F(Xq)

Equation (3.35) is called the quasi-Newton or secant relation (Halfka and Gurdal.

1996). This equation condition must be satisfied in order to update the matrix Gq or Bq.

The quasi-Newton method procedure starts by assigning the identity matrix to the

approximated inverse Hessian matrix (B0 = I) and computes the search direction from the

following equation.

Sq = -Bq ∇ F(Xq) (3.36)

The first search direction is indeed the search direction of steepest decent. Having

the search direction, the one-dimensional search is computed for the step length following

the steepest decent method. During subsequent iterations, the approximated inverse

Hessian matrix is updated such that it satisfies equation (3.35). The most common way to

57

update the Bq matrix and satisfy the quasi-Newton condition is by adding a symmetric

matrix to the previous Bq as shown in the following equation.

Bq+1 = Bq + Eq (3.37)

Eq is called a symmetric update matrix. This update matrix is available in many

forms. The following form of update matrix is very popular among the quasi-Newton

methods.

[ ]TqqTqqTqqqqTq yBppyByByBpp )()(12 +−−++=

σθ

τθ

σθτσE (3.38)

where σ = (pq)Tyq.

τ = (yq)T Hq yq

There are two popular quasi-Newton methods that are based on the equation

(3.38): the Broydon-Fletcher-Goldfarb-Shanno (BFGS) method and the Davidon-

Fletcher-Powell (DFP) method. The parameter θ in the equation (3.38) determines the

two methods.

If θ=0, then Eq in the equation (3.38) results in the DFP method. If θ=1, then the

Eq in the equation results in the BFGS method.

Both DFP and BFGS are very efficient methods. Based on many numerical

experiments (Fletcher 1980), the BFGS method provides an excellent efficiency among

the quasi-Newton methods. Because of the performance of the BFGS method, this

research chose the BFGS method as an optimization technique for the damage detection.

The flow chart of the quasi-Newton method is shown in Figure 3.8.

58

Figure 3.9 Flow Chart of Quasi-Newton Optimization Procedures.


X=Xinitial

and B=I

S = -B∇ F(X)

Compute α

Compute X= X+ α S

Compute E from (3.38)


B= B + E


StopYes

No

Converge?

59

3.4.2 Finding Step Length

The procedure of finding the step length for each search direction is sometimes

called one-dimensional search or line search. Having the search direction means knowing

what direction to go but the question still exist: How far to go in this direction? The

scalar parameter α, called step length, is introduced into the update variable equation

(3.9) as the length to go in the direction. The step length can be computed to minimize the

objective function. The step length is the only design variable that exists in this sub

optimization and the search direction is already known. That is the reason of the name

one-dimensional search. The necessary condition for the minimum of the objective

function for one-dimensional search is the vanishing of the first derivative.

0)( =αα

ddF (3.39)

Many techniques are developed for one-dimensional search directions such as

bracket method, golden section search, Fibonnaci section search, quadratic interpolation,

and cubic interpolation methods. In this chapter, the bracket method and golden section

method will be described.

The bracket method is the simplest and most straightforward method. This

method assumes a starting point and evaluates the objective function at the point. It then

gradually moves to a new point and evaluates the objective function again and compares

to the previous objective value. If the new point provides a lower objective value, then the

point is kept and movement continues along the same direction. If the objective value is

higher, the point is ignored and movement continues in the opposite direction. The move

60

will be stopped when the accuracy of the result is in an acceptable range. Figure 3.10

shows the typical flowchart of bracket method.

Figure 3.10 Flow Chart of Bracket One-dimensional search Method.

Evaluate Fq=F(αq)

Start, assume αinitial, βinitial

StopYes

No

Converge?

Evaluate Ft=F(αq+βq)

Ft < Fq

Yes

No

q=0, αq = αinitial

βq = βinitial

αq+1 = αq + βq

βq+1= φβq

Yes

No

Ft < Fq+1q = q+1

αq+1 = αq

βq+1= -ηβq

Yes

NoFt < Fq+1

q = q+1

αq+1 = αq + βq

βq+1= φβq

Fq+1 = F(αq+1)Ft=F(αq+1+βq+1)


Yes

No

Ft < Fq+1

αL = α0 - ηβ0

αU = α0 + β0

αL = αq-2

αU = αq

q=0, αq = αL or αU

βq = µ β0


61

Where β is the step size of movement.

φ >1.0

0 < η < 1/φ

0 < µ < 1.0

The constant parameters in the bracket method are important to the rate of

convergence or efficiency of the method. The bracket method is a reliable method but the

proper values of their constant parameters need to be chosen for a fast convergence.

The golden section search method is one of the most popular one-dimensional

search methods. The golden section method uses the same idea as the bracket method but

the parameters of the movement are defined to provide a highly efficient rate of

convergence.

The golden section search method assumes that the objective function is a

unimodal function. The unimodal function has to satisfy the following conditions:

The F(α) is unimodal in the interval of I0 if there is an α* that minimizes the F(α)

in the interval I0 and for any two points αa, αb in the interval I0 where αa < αb , if αb < α*

then F(αb) < F(αa) and if αa > α* then F(αa) < F(αb). The unimodal function does not need

to be a continuous nor continuous in the first derivative.

The golden section search starts from knowing a bracket of the solution, αL and

αU. The method will try to narrow down the bracket by introducing the new two

intermediate points and evaluating the objective function to set a new boundary. The new

points, αa and αb have specific proportion conditions of symmetry about the center of the

62

interval. It also has a constant ratio of the distance of the new points and the total length

of the interval. The numerical formula of the conditions can be described as follows.

αa < αb (3.40)

αU - αb = αa - αL (3.41)

aU

ab

LU

La

αααα

αααα

−−=

−− (3.42)

Equation (3.41) represents the symmetry and equation (3.42) represents the

constant ratio. To simplify the formulation, the upper bound and lower bound (αU and αL)

can be normalized to one and zero respectively. After the normalization, equation (3.42)

can be written as:

a

aba

αααα

−−=

1(3.43)

from (3.41) αb = 1 - αa (3.44)

substituting (3.44) into (3.43) givves

a

aa

ααα

−−=

121 (3.45)

rearranging the form in (3.45)

(αa)2- 3αa + 1 = 0 (3.46)

63

solving the quadratic equation (3.46) for the roots of αa gives

618034.2,381966.02

53 =±=aα (3.47)

There are two roots in equation (3.47) but one of them is higher than the upper

bound (αU=1.0) so that there is only one feasible solution:

αa = 0.381966 (3.48)

back substituting αa of (3.48) into (3.44)

αb = 1-0.381966 = 0.618034 (3.48)

dividing (3.48) by (3.47) gives

618034.1=a

b

αα (3.49)

The ratio in equation (3.49) is called golden section number. This number also has

other special conditions that are:

1618034.0 −=== a

bb

b

a

ααα

αα (3.50)

and ( )2ba αα = (3.51)

64

From the concept of the golden section number, the golden section search

algorithm can be developed using this information. Vanderplaats (1999) presented the

following golden section search procedure:

The algorithm starts with an initial boundary interval (αU - αL) and specifies the

relative tolerance (ε) and the number of function evaluation (N).

LU αααε−∆= (3.52)

By specifying the desired total tolerance, ∆α, the relative tolerance can be

computed from equation (3.52). The length of the new interval based on golden section

number theory can be computed using the following equation.

αU - αa = αb - αL = 1-τ (3.53)

where τ = αa = 0.381966.

The relative tolerance can be written in the form of the maximum number of

function evaluations based on the reduction of interval by the golden section number

theory.

( )( )31 −−= Nτε (3.54)

where N is the maximum number of function evaluations, including the first three

initial evaluations (F(αL), F(αU), and F(αa)).

65

Solving the equation (3.54) for N:

( ) 3)ln(078.23)1ln(

ln +−=+−

= ετεN (3.55)

From equation (3.55), the maximum number of function evaluation can be defined

by having the desired relative tolerance. This number is used as a convergence criterion of

this algorithm.

The two new points can be written in term of τ as follows:

αa = (1-τ) αL + τ αU (3.56)

αb = (τ) αL + (1-τ) αU (3.57)

The flow chart of this algorithm is shown in the Figure (3.11).

3.4.3 Condition of Convergence

This section will discuss the conditions to stop the iteration in the optimization

process. The conditions of stopping the optimization iteration are not only when the

solution reach the optimum, but also when the solution will never reach the optimum.

Termination of the optimization process at the optimum solution uses the Khun-Tucker

necessary conditions for optimality as stated in equation (3.12). In practice, the gradient

of the objective function is not need to be zero but close to zero within the acceptable

tolerance (εK). In some problems, if the solution is close to the optimum, the

improvement of the objective function is very slow. The optimization process should be

stopped here too.

66

Figure 3.11 Flow Chart of Golden Section One-dimensional search Method.

Start, known αU, αL, τ, ε

K=3

K > NYes

No

Evaluate FU, FL, N

Yes NoFa > Fb

αa = (1-τ)αL+τ αU

Fa= F(αa)

αb = τ αL+(1-τ) αU

Fb= F(αb)

K=K+1

Stop

αU = αb

FU= Fb

αb = αa

Fb= Fa

αa = (1-τ)αL+τ αU

Fa= F(αa)

αL = αa

FL= Fa

αa = αb

Fa= Fb

αb = τ αL+(1-τ) αU

Fb= F(αb)

67

The conditions to identify this situation are the change of objective function, both

absolute and relative change. If the changes are smaller than an acceptable tolerance, the

process should be terminated. The formulation of these conditions are defined as follows:

Absolute change

Aqq XFXF ε≤− − )()( 1 (3.58)

Relative change

Rq

qq

XF

XFXFε≤

− −

)(

)()( 1

(3.59)

where εA is tolerance of absolute change.

εR is tolerance of relative change.

In some cases, the solution of the optimization will never converge to the

optimum because of numerical problems or error in the processes. The maximum number

of iterations needs to be checked to avoid the infinite loop problem. The condition may be

written as:

q < qmax (3.60)

where q is the iteration number.

qmax is the maximum allowable iteration number.

The general flow chart of the convergence condition is shown in Figure 3.11.

68

Figure 3.12 Flow Chart of Terminating Optimization Process.

The problem of bridge damage detection using the system identification method

falls into the category of this unconstrained optimization problem. The optimization

algorithm that used in this research is based on the BFGS method and the golden section

one-dimensional search technique.

Enter, known qmax, εK, εA, εR

DFA= |F(Xq)- F(Xq-1)|

q > qmaxYes

No

Stop

DFA < εAYes

No

Stop

DFR= DFA/|F(Xq)|

DFR < εRYes

No

Stop

SFK= ∇ F(Xq)

SFK < εKYes

No

Stop

Continue the optimizing

69

CHAPTER 4BRIDGE DAMAGE DETECTION

4.1 Introduction

Bride damage detection is a tool for investigating the health of bridge structures

subjected to service loads. There are many detection algorithms that have been developed

for bridges. Material testing in the field likes x-rays, acoustic emission and ultrasonic

testing have been successfully used to detect damage in a bridge. However these methods

are costly, time consuming and can have difficulty in examining hidden areas. These

methods may be considered as localized techniques because the procedures need to be

done point by point or element by element. The alternative techniques for damage

detection rely on system identification. These techniques are considered to be global

methods because they use the overall characteristics of bridge structures to evaluate

damage.

System identification techniques are the methods of matching or finding the

mathematical model that identifies the investigated structure. The main advantages of

these techniques are their relatively cheap cost and speed when used for real time health

monitoring. However these methods can only assist the damage detection process by

predicting the location and/or extent of the damage. After that an investigation in that area

needs to be done.

70

System identification itself can be classified into two categories: complete

identification and partial identification, based on a priori knowledge of the system. The

complete identification describes the system that has very little information about the

investigated system. This type of identification may not yield a good physical

identification. On the other hand, the partial identification already knows a lot of

information of the investigated system and tries to adjust the model from the known

information. This method yields a better solution. The damage detection of bridge

structures using system identification falls into the partial identification category because

all the main components of the structure and their behaviors are known. The system

instead looks for changes in the bridge structures. This chapter will describe the basic

idea and procedures of damage detection using system identification.

4.2 Basic Concept

Bridge damage detection using the system identification technique identifies the

damage of bridge structures by matching the characteristics of the damaged bridge and

the characteristics of the finite element model that represents the damaged bridge. The

process starts with measuring the characteristics of the investigated bridge in the field and

creating a finite element model of the undamaged bridge structure. Next comes analyzing

the finite element model for its characteristics. An optimization technique is used to

minimize the different characteristics from the damaged bridge structure and its finite

element model. The optimization technique will modify the finite element model until the

difference is minimized. After the optimization converges to a minimum error, the finite

element model of the bridge structure will represent the damaged bridge structure.

71

The basic concept of this damage detection technique can be shown as a diagram

in Figure 4.1.

Figure 4.1 Concept of Bridge Damage Detection Using System Identification

The bridge damage detection using the system identification method consists of

four main components as follows:

1. Characteristics of bridge structure from both the damaged bridge and FEM.2. A finite element model of bridge structure.3. A finite element analysis program.4. Optimization routines.

Change FEM usingoptimization

technique to minimizethe difference of

responses

FE analysis

The FEM represents thedamaged bridge

Yes

No

Observed responsesfrom damaged

bridge

Responses fromFE analysis

Compare theresponses if converge

to the minimum

72

4.3 Characteristics of Bridge Structure

There are many characteristics of bridge structure that have been used in damage

detection with system identification methods. The characteristics that are suitable for the

structural damage detection show an obvious change when the structure properties

change. This study investigates both static and dynamic characteristics of bridge

structures. The dynamics characteristics of the bridge structure that are used most often

are the modal responses: eigenvalues and eigenvectors (frequencies and mode shapes).

The static characteristics used in the study are Ritz vectors (displacement shapes) of the

structures subjected to a particular static loading. Both characteristics have been tested

experimentally and have shown success in matching the finite element model to the

damaged bridge.

4.3.1 Eigenvalues and Eigenvectors

The eigen properties, eigenvalues and eigenvectors, are unique characteristics of a

structure with a certain stiffness, mass, and damping. The eigenvalues represent vibration

frequencies and eigenvalues represent mode shapes of the structure under free vibration.

These properties change if the structure is damaged. From the unique characteristics for a

particular structure of the eigen properties, when the properties of the bridge structure are

changed, the eigen properties of the structure are also changed. The derivation of the

structure properties and eigen characteristics of a structure is shown as follows:

The free vibration-governing equation is given by:

0)()()( =++•••

ttt XKXCXM (4.1)

73

where M is the mass matrix of the structure.

C is the damping matrix of the structure.

X is a displacement vector and the dot over X represent the derivative with

respected to time (t).

•X is a vector of velocity.

••X is a vector of acceleration.

The general solution of the equation (4.1) can be written in harmonic form as:

X(t) = ΦΦΦΦeλt (4.2)

where ΦΦΦΦ is a constant vector.

λ is a scalar value.

substituting equation (4.2) into (4.1) yields.

( ) 02 =Φ+λ+λ KCM (4.3)

The non-trivial solution of the equation (4.3) yields to complex conjugate pairs of

eigenvalues (λ i) and eigenvectors (Φi). The eigenvalues are computed from the following

equation.

21 iiiii i ζωωζλ −+−= (4.4)

74

where ζ is a damping ratio , kmc

2=ζ .

ω is the circular frequency, mk=ω .

Equation (4.4) is a complex solution, which does not yield an obvious physical

meaning. However in most structural modeling, the damping is small and negligible

(Kaouk 1993). If the damping ratio is negligible, the free vibration equation becomes.

0)()( =+••

tt XKXM (4.5)

The general solution of the equation (4.5) can be written in harmonic form as:

X(t) = ΦΦΦΦ sin(ωt) (4.6)

and so

)()( 2 tt XX ω−=••

(4.7)

substituting equation (4.7) into (4.5) yields

02 =Φ−Φ MK ω (4.8)

The non-trivial solution of the homogeneous equation (4.8) has to satisfy the

condition for the characteristic determinant to vanish:

02 =− MK ω (4.9)

75

Solving the equation (4.9) yields ‘n’ values of ω which are the eigenvalues of the

problem. And the corresponding eigenvectors can be computed by back substituting the

eigenvalues into equation (4.8).

where λ i = ωi2

πω2

= Natural frequency of a structure

Φ = Mode shapes of the structure

4.3.2 Ritz Vectors

Ritz vectors have become alternative characteristics for many modal analyses

because experimental studies show the outstanding identification performance of the Ritz

vectors. Zimmerman and Cao (1997) present four advantages of the Ritz vectors over the

eigenvectors as follows:

1. Ritz vectors automatically include the static correction.2. Ritz vectors are computational less expensive.3. Ritz vectors are generated by a load will be excited by that load.4. Ritz vectors require fewer modes than eigenvectors for response prediction at the same accuracy.

The Ritz vectors are load dependent characteristics of a structure subjected to

particular loads. The first mode of Ritz vectors is simply a displaced shape of the

structure subjected to statically applied loads. The successive Ritz vectors are functions of

the previous Ritz vector and the mass and stiffness matrices. The formulations of the Ritz

vectors are described as follows:

76

From the dynamic governing equation:

FXKXCXM =++•••

)()()( ttt (4.10)

where F is the vector of applied forces.

The first mode of Ritz vector is computed from the static displacement as follows:

sFKR 1*1

−= (4.11)

where R1* is the first mode of Ritz vector.

Fs is the applied static loads.

Mass normalizing the first Ritz vector such that

( ) ( )*1

T*1

*1

1

M RR

RR =

where R1 is the mass normalized Ritz vector.

The successive mode of Ritz vectors can be computed from the previous Ritz

vector as follows:

11*

−−= ii MRKR (4.13)

The new Ritz vector has to be orthogonal to the previous Ritz vectors and mass

normalized as shown in the following formulas.

( ) j

1i

1j

*i

Tj

*ii RMRRRR ∑

−

=−= (4.14)

77

1=iTi MRR (4.15)

4.3.3 Observed Characteristics

The observed characteristics of an investigated bridge are needed in order to use

the system identification methods to detect the damage in the bridge structures. The

characteristics (eigenvalues, eigenvector, and Ritz vectors) can be extracted from the field

measurements by using a vibration test. A vibration test usually consists of four major

parts of hardware: a mounting system, exciting source, transducers, and data analysis and

recorder (Friswell and Mottershead 1995).

The mounting system is used to set up a suitable site for the test such as a tower

frame for locking up the transducers. The mounting system will vary from test to test

depending on the conditions that are needed in the test. The exciting source such as a

shaker or impact hammer is used to vibrate the structures. The shaker and impact hammer

basically applies loads to the structures in a sufficient amount for the vibration to occur.

Transducers are used to measure the applied forces and also the responses of the

structures. The information from the test is collected and analyzed by a machine like an

ADCs (Analogue to Digital Converters). A simple setup of an impact hammer test is

shown in the Figure 4.2.

4.3.4 Simulating the Observed Characteristics

This research concentrates on improving the techniques of damage detection using

system identification and presents a parametric study for general bridges. For the purpose

of this study many different characteristic responses are needed.

78

Figure 4.2 Diagram of Impact Hammer Vibration Testing.

The observed characteristics of the damaged structures in this work are therefore

simulated by finite element analyses. From the simulated damage, the location, extent and

type of damages are exactly known. The comparison of the prediction from the damage

detection and the damaged structures is obvious.

The procedures of bridge damage simulation are described here, starting with the

creation of a bridge finite element model as it is built for a healthy bridge model. The

structure is then intentionally damaged by reducing or taking off the structural properties

AccelerometerForce transducer

Impact hammer

Investigated Structure

Signal Conditioning

ADCs

Computer

Anti-aliasing Filter

79

of some structural elements, which simulates damage in those elements. The damages

considered in this study are corrosion in reinforcing steel, weakening of material,

cracking in bridge girders and damage in supports. The damaged bridge model is then

analyzed by a finite element analysis program to yield the characteristics of the damaged

bridge.

The corrosion in reinforcing steel is simulated by modeling the bridge girders

using beam elements with eccentric truss elements. The eccentric truss elements represent

the reinforcing steel in the girders. The model is intentionally damaged by reducing the

area of the truss elements at the point of interest.

The weakening of structural material can be simulated by reducing the modulus of

elasticity of that material at the desired location of damage.

The cracking of the bridge girders is simulated by reducing the moment of inertia

and cross-sectional area of the beam elements, which represent the bridge girders in the

model, at the damage locations.

The supports of bridge structures in this study are modeled as elastic spring

elements. The damage in supports can be simulated by reducing the stiffness of the spring

elements at the damaged location.

The measurement of characteristics in the field may not provide the perfect

responses when compared to a simulation. Noise usually comes with the response data.

The magnitudes of noise depend upon the measuring procedures and accuracy of the

hardware that is used in the measurement. Even though complete data noise is out side

the scope of this study, this research also presents the effect of noise in the damaged

response in the parametric study. The noise is generated by a random function with a

80

normal distribution with zero mean value. The simulated noise is imposed into the

damaged responses using the following formula:

Φ* = Φ + Φ (γ/100) random (-1,1) (4.16)

where Φ* is the response data with noise.

Φ is the response data without noise.

γ is the percentage of noise level to the response magnitude.

random(-1,1) is a random value from –1.0 to 1.0 with a normal

distributed and zero mean value.

4.3.5 Model Characteristics

The characteristics of the mathematical model are computed from a finite element

analysis of the bridge structure starting with the undamaged model; the same model that

is created before the intentional damage in the previous section. After the analysis is

done, the output is the characteristics of the structural model. The optimization technique

will then evaluate the error and modify repeatedly until the system converges to a

minimum error.

4.4 Finite Element Model of Bridge Structure

The finite element model of the bridge structures has to be able to represent the

real bridge behaviors. The model should contain all of the structural elements that affect

the characteristics of the investigated bridge. In practice, the model should have been

calibrated to the real bridge structure. The procedure of calibration or refinement of the

model is similar the damage detection in many ways except calibration should have been

81

done immediately after the bridge was built. Since the model of the healthy bridge

structure is supposed to be very close to the real structure, the calibration is looking for a

small change in the model rather than a big change like in damaged structures. The three-

dimensional finite element model used has shell elements to represent the bridge slab,

beam elements represent girders and secondary structures, truss elements to represent

reinforcing steel, and elastic spring elements to represent supports. The detail of this

model is described in the Chapter 3. The figure of the general model is shown again in

Figure 4.3.

4.5 Finite Element Analysis Program

The bridge damage detection using system identification needs a finite element

program that can analyze a bridge structure model and provide an output of the

characteristics for that system. In this study the author modified the existing finite

element analysis program SIMPAL (SIMPle AnaLysis), which was developed by Dr.

Marc Hoit (1983). There are three main reasons that the author used SIMPAL as a base

analysis program in this study. First, the program has the capability of analyzing the

bridge structures and provides the characteristics that are needed in this research

(eigenvalues, eigenvector, and Ritz vectors). Second, the source code of this program was

available. Finally, the program is also a base program of Florida Pier program, which

could extend the capability of the program to be able to design the entire bridge structure

including damage detection capability.

82

Figure 4.3 General Girder-Slab Bridge Model.

4.6 Optimization Routine

An optimization technique is one of the main components of damage detection

using system identification methods. Damage detection needs a reliable optimization

routine to minimize the differences of the characteristics of a finite element model and the

characteristics of a damaged bridge. Design Optimization Tools (DOT) and Design

Spring element (support)

Beam rigid link

Beam element (girder)

Beam element (diaphragm)Shell element (slab)

Beam element (parapet)

(b) Cross-Section View of the Bridge Structures Model.

(a) Overall View of the Bridge Structures Model.

83

Optimization Control (DOC) which are developed by Vanderplaats Research &

Development (1995) are used as a base routine for the optimization technique in this

research. These routines are selected because they have capability of optimizing a large

problem with the BFGS algorithm that will be used in this study. The other main reason

is that the source code of this routine is available.

4.7 Damage Detection Routine

From the concept of the damage detection using system identification, the author

developed a damage detection routine using the base routine of finite element analysis

program, SIMPAL and the base routines of optimization programs provided by DOT and

DOC. The general flow chart of the damage detection routine is shown in Figure 4.4.

4.8 Parameters of Damage Detection Routine

There are two parameters that need to be discussed before performing the damage

detection. The parameters are the response error or the objective function to be minimized

and the design variables, value to be changed, of the system.

4.8.1 Objective Function

The objective function is the difference between the model responses and the

observed responses. However the difference or the error of the responses may be

computed in two ways: absolute error and relative error. Each error is defined as follows:

Absolute error:

eA = Φo - Φm (4.17)

84

Figure 4.4 Flow Chart of General Procedure of D

Start

ComputeResponse Error

Optimizer

Minimum Error?

Stop

Adjust FEModel

Initial FEModel

ObservedResponses

AnalyticalResponses

FE Analysis

s
Ye
No

amage Detection Routine.

85

Relative error:

eR = (Φo - Φm)/ (Φo) (4.18)

where Φo is an observed response.

Φm is an analytical response form FEM.

One of the most popular schemes to find a minimum error is the least square

errors method. The least square error method minimizes the sum of squared errors. The

objective function of this scheme can be written as follows:

∑=

Φ−Φ=N

imioisF

1

2)()(X (4.19)

where Fs(X) is an objective function in least square sense.

N is the number of response data.

The lease square errors method has two advantages that make it popular. First, a

large error is attached with a large penalty because the square will magnify the errors.

Second, the computation of this method is very simple so that the computing time is very

small when compared to the analysis time. However, the least square method uses an

absolute error to find the minimum objective which is biased to the magnitude of the

responses such that the higher magnitude has more control of the objective.

The author proposes an objective function that uses the relative error instead of

absolute error. With this objective function, the error will be normalized to the magnitude

of the response. The formulation of this objective function is shown as follows:

86

∑= Φ

Φ−Φ=

N

i oi

mioiRSF

1

2)()(X (4.20)

where FRS(X) is an objective function for sum of square relative error.

4.8.2 Design Variables

The design variables of the damage detection are the parameters that will be

adjusted if the minimum error does not converge. The design variables in system

identification can be classified into two types: global and local parameters. In the case of

bridge damage detection, the global parameters can be each element in the structural

stiffness matrix and the local parameters can be the structural properties in each element.

The global parameters can yield the correct solution but do not necessarily yield an

obvious physical meaning. On the other hand, the local parameters yield the location and

extent of change in the structural element.

Based on the fact that when a structure is damaged, the stiffness of that structure

is changed as well. Changes in the modulus of elasticity will change the stiffness of the

element. The relationships of the modulus of elasticity and structural element stiffness are

shown as follows:

Beam element:

LEIkB ∝ (4.21)

87

Truss element:

LEAkT ∝ (4.22)

Plate element:

LEbtkP

3

∝ (4.23)

where E is the modulus of elasticity.

I is the moment of inertia.

L is the length of the element.

b is the width of plate element.

t is the thickness of plate element.

The modulus of elasticity in each element directly effects the response, it is

chosen as the design variables in this study. In order to increase the efficiency of the

optimization technique, the normalized modulus of elasticity called, the multiplier, is

used as the design variables. The normalized design variables vary from zero to one.

Accordingly, the design variable equal to one indicates no change in that element and the

design variable equal to zero indicates full damage or the element does not exist. The

damaged indicator is defined as a ratio of the damage extent in an element. The damaged

indicator can be written as the following formula:

ρi = 1-Xi (4.24)

where ρi is a damage indicator of element, i.

88

Xi is a normalized design variable of element, i.

4.9 Improvement of the Damage Detection Routine

Most of the time, bridge damage detection problems have a large number of

design variables because of the size and complexity of bridge structures. Unfortunately,

the optimization performance varies inversely with the number of design variables. This

research therefore proposes a screening technique to reduce the number of design

variables. Instead of using a standard technique with a single optimization loop and one

set of design variables with one magnitude of perturbation, the screening algorithm uses

multi-optimization loops with reduced design variables and perturbation magnitude. The

first loop of optimization is performed starting with a full domain of design variables and

a relatively large perturbation. After the first loop, the design variables that have no effect

on the change or the effect is smaller than a cut off tolerance are eliminated from the

design variables domain. The next loop of the optimization uses the new domain of

design variables, which is smaller than the first loop and the magnitude of perturbation is

also decreased. The change in the magnitude of perturbation helps to refine the effect of

each design variable and also reduces the chance of looping in a local minimum. The

condition of terminating the outer loop of the optimization consists of two criteria. The

first criterion is maximum number of loops, Nmax. This is to prevent an infinite

optimization loop. The maximum number of loops is chosen by user. The second

termination condition is the non-improvement of the objective function. If the objective

function is not improved in successive ‘Ns’ loops, the detection processes will be stopped.

Ns is the number of successive loops of non-improvement. Sometimes the optimization

89

may slow down or stop the improvement for a couple loops and then improve the solution

again after that. The general flow chart of this algorithm is shown in Figure 4.5.

The optimization performance also can be improved if more information of the

structure is known. For example if an investigated structure has a suspect of damage

element, the damage detection routine can scope the design variables to only the

suspected element which reduce the number of design variable and also focus on the

potential damaged element. This procedure can dramatically improve the damage

detection performance.

4.10 Damage Detection Testing on Structural Elements

The damage detection program is tested starting with the individual basic

elements that will be used in the bridge structure model: spring, truss, beam, and shell

elements. After the individual elements are tested, the composite structures are then built

and tested. The test uses the first mode dynamic response as the structural characteristics.

The eigenvalue and eigenvector are combined as the dynamic response. The least squared

error is used as the objective function in the test. At the beginning, the test used only the

standard optimization to find the damage in the elements. The screening technique was

used later on in the composite structures. The results of testing show the plot of the

damage indicator between the simulated damage-structures and the prediction of damages

from the damage detection program. The damage indicators of simulated damage-

structures are known prior to the test because the structures are intentionally damaged.

The damage detection program will find the location and extent of the damage, which are

shown by the damaged indicators.

90

Figure 4.5 Flow Chart of Damage Detection Routine with Screening Technique.

Start, N=0,N*=0

ComputeResponse Error

Optimizer

Stop

ObservedResponses

Analytical Responses

FE Analysis

Initial FE Model

Yes

No

N=N+1

Minimum Error?

YesN > Nmax

No

Improve?No

N*=N*+1

N* > Ns

Stop

Yes

NoYes

Screen for New DomainVariables

Reduce Magnitude ofPerturbation

Adjust FEModel

91

A damage indicator equal to zero means that there is no damage in the element

and conversely a damage indicator equals to one means that the element is fully damage.

If the damage indicator is in-between zero and one, it represents the ratio of damage in

the element, for example if the damage indicator is equal to 0.3 that means the element

lost 30% of its stiffness.

4.10.1 Test of Spring Element

Three-dimensional axial springs with three degrees of freedom at a node attached

to the ends of a beam element are shown in the Figure 4.6. Initially all the spring elements

have the same properties with a spring stiffness of 12,000 kip/ft.

Figure 4.6 Testing Spring Element Model.

Damage was simulated by reducing the spring stiffness in the element #4 (DOF

#4) by 50% (6,000 kip/ft) from the original stiffness. SIMPAL was used to obtain the

response of the simulated damage-structures. These responses are used as the observed

responses in the damaged detection program. The initial structural model is then used as a

healthy model and the damaged detection program investigates and adjusts the initial

model until the minimum error converges. The simulated damage indicators are shown in

92

Figure 4.7 and the results of the damage prediction from the damage detection program

are shown in Figure 4.8.

Figure 4.7 Damaged Indicators of Simulated Damaged Spring.

Figure 4.8 Predicted Damaged Indicators from Damage Detection Program.

Predicted Damage

0 0 0

0.5

0 00

0.10.20.30.40.50.6

1 2 3 4 5 6

Element #

Dam

aged

Indi

cato

rSimulated Damage

0 0 0

0.5

0 00

0.10.20.30.40.50.6

1 2 3 4 5 6

Element #

Dam

aged

Indi

cato

r

93

From the results of this test, the damage detection program predicts that the

damage occurred at the element #4 with the damaged extent of 50%. The test indicated a

correct damage location as well as the extent of the damage.

4.10.2 Test of Truss Element

The truss-testing model uses 13 steel truss elements connected to 8 nodes with

simple supports at both ends as shown in Figure 4.9. Each truss element originally has the

same material properties, modulus of elasticity = 5000000 k/ft2, cross sectional area =

0.10 ft2, and mass = 0.001 kip-s2/ft2.

Figure 4.9 Testing Truss Element Model.

The damage in a truss element is simulated by reducing the cross sectional area of

truss in element #5 by 50% (0.05 ft2) from the original area. Using the same testing

procedures as describe in the Section 4.10.1 the damaged indicators of the simulated

damaged truss are shown in Figure 4.10 and the results of the damage prediction of the

damage detection program are shown in Figure 4.11. The error of the prediction is a

difference between the predicted and simulated damage indicators and described as a

percentage of the damage. The formulation of percentage of damage is shown as:

9 1011

1 2 3

4 5 6

78

12

13

94

% error = (ρo - ρp)*100 (4.26)

where ρo is the damaged indicator of the simulated damage.

ρp is the predicted damaged indicator of the program.

The extent error of the program prediction is presented in Figure 4.12.

Figure 4.10 Damaged Indicators of Simulated Damaged Truss.


Simulated Damage

0 0 0 0

0.5

0 0 0 0 0 0 0 00

0.10.20.30.40.50.6

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

Element #

Dam

aged

Indi

cato

r

Predicted Damage

0.02 0 0 0

0.513

0 0.01 0 0 0 0 0 0.040

0.10.20.30.40.50.6

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

Element #

Dam

aged

Indi

cato

r

95

Figure 4.12 % Extent Error of Predicted Damaged Indicators.

The results from the prediction show that 51.3% damage occurred in element #5

and very small damage in other three elements. The error of damaged extent is maximum

at 4%. The results can be concluded that the damage detection program predicts the

correct damaged location and extent with small errors.

4.10.3 Test of Beam Element

The beam-element testing model uses 10 beam-elements with simple supports at

the end of span as shown in Figure 4.13. All elements originally have the same properties,

modulus of elasticity = 600000 k/ft2, cross-sectional area = 7.035 ft2 torsional moment of

inertia = 2.144 ft4, bending moment of inertia = 25.0 ft4, and mass = 0.033 kip-s2/ft2.

Figure 4.13 Testing Beam Element Model.

%Extent Error

2%

0% 0% 0%

1%

0%1%

0% 0% 0% 0% 0%

4%

0%1%2%3%4%5%

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

Element #

Erro

r

101 32 4 5 6 7 8 9

96

The damage is simulated by reducing the bending moment of inertia in element #

6 by 50% (12.5 ft4) from original, using the same testing procedures as describe in the

Section 4.10.1. The simulated damage indicators of the girder are shown in Figure 4.14

and the results of the damage prediction are shown in Figure 4.15. The extent of the

prediction error is presented in Figure 4.16.

Figure 4.14 Damaged Indicators of Simulated Damaged Beam.


Simulated Damage

0 0 0 0 0

0.5

0 0 0 00

0.2

0.4

0.6

1 2 3 4 5 6 7 8 9 10

Element #

Dam

aged

In

dica

tor

Predicted Damage

0 0.01 0.02 0.04 0.02

0.52

0.01 0.01 0 00

0.2

0.4

0.6

1 2 3 4 5 6 7 8 9 10

Element #

Dam

aged

In

dica

tor

97


The results from the prediction indicate the 52.0% damage in element #6 and

small damage scatter along the beam with a maximum of 4% damage. The maximum

error of the damaged extent is 4% at the element #4. The results show some more small

errors over the beam, however the damage detection program still predicts an obvious

correct damaged location.

4.10.4 Test of Shell Element

The shell-element testing model uses 10 shell-elements with simple supports at

the end of the span as shown in the Figure 4.17. All elements originally have the same

properties of, modulus of elasticity = 500000 k/ft2, thickness = 8 in., and mass = 0.00347

kip-s2/ft3.

Figure 4.17 Testing Shell Element Model.

%Extent Error

0%1%

2%

4%2% 2%

1% 1%0% 0%

0%1%2%3%4%

1 2 3 4 5 6 7 8 9 10

Element #Er

ror

101 32 4 5 6 7 8 9

98

The damage is simulated by reducing the modulus of elasticity of the element # 5

by 50% (250,000 k/ft2) from the original, using the same testing procedures as describe in

the Section 4.10.1. The simulated damage indicators of the shell structure are shown in

Figure 4.18 and the results of the damage prediction are shown in Figure 4.19. The extent

error of the prediction is presented in Figure 4.20.

Figure 4.18 Damaged Indicators of Simulated Damaged Shell Structure.


Simulated Damage

0 0 0 0

0.5

0 0 0 0 00

0.2

0.40.6

1 2 3 4 5 6 7 8 9 10

Element #

Dam

aged

In

dica

tor

Predicted Damage

0.010.030.020.06

0.45

0.050.010.030.010.010.00

0.20

0.40

0.60

1 2 3 4 5 6 7 8 9 10

Element #

Dam

aged

In

dica

tor

99


The shell structure is predicted to have a 45.0% damage in element #5 and small

damage of 1% to 6% along the shell structure. The maximum error of the damaged extent

is 6% at the shell element #4. The prediction shows more errors in the shell structure.

Regardless, the damage detection program still predicts a correct damaged location with

5% error in the extent of damage.

From the results of individual element tests, the damage detection program shows

the capability of detecting damage in every type of structural element. However the

performance of detecting damage is different between the element types depending on the

complexity of the element. The damage detection presents an excellent performance in

the simplest structure, spring-element structure, and the performance gradually decreases

when the structure behaviors are more complex.

4.10.5 Test of Combining Element of Beam and Shell Elements

This model uses 10x10 shell-elements and 4x10 beam-elements with simple

support at the ends of girders as shown in Figure 4.21. All shell elements originally have

%Extent Error

1%3%

2%

6% 5% 5%

1%3%

1% 1%0%2%4%6%8%

1 2 3 4 5 6 7 8 9 10

Element #Er

ror

100

the same properties, modulus of elasticity = 500000 k/ft2, thickness = 8 in, and mass =

0.00347 kip-s2/ft3 and all beam elements also have the same properties, modulus of

elasticity = 600000 k/ft2, cross-sectional area = 7.035 ft2, torsional moment of inertia =

2.144 ft4, bending moment of inertia = 25.0 ft4, and mass = 0.033 kip-s2/ft2.

Figure 4.21 Testing Model of Combine Beam and Shell Elements.

The damage in the structure is simulated by reducing the modulus of elasticity of

shell element # 25 (shaded) by 50% (250000 k/ft2) from the original modulus of

elasticity. The test used both the standard and screening technique in this problem

because of the greater complexity of the structure. Other procedures remained the same as

described in the Section 4.10.1. The first series of results come from the standard

optimization technique. The simulated damaged indicators of the combined structures are

shown in Figure 4.22 and the results of the damage prediction of the damage detection

program are shown in Figure 4.23. The extent error of the prediction is presented in

Figure 4.24. The second series of results come from the screening optimization technique.

101

The simulated damaged indicators are not changed (Figure 4.22). The results of the

damage prediction of the second technique are shown in Figure 4.25. The extent error of

the second technique prediction is presented in Figure 4.26.

Figure 4.22 Damaged Indicators of Simulated Damaged Shell Structure.

Figure 4.23 Predicted Damaged Indicators from Standard Technique.

Simulated Damage

00.20.40.6

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

Element #

Dam

aged

In

dica

tor

Predicted Damage

0.00.20.40.6

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

Element #

Dam

aged

In

dica

tor

102

Figure 4.24 % Extent Error of Predicted Damaged Indicators from Standard Technique.

Figure 4.25 Predicted Damaged Indicators from Screening Technique.

Figure 4.26 % Extent Error of Predicted Damaged Indicators from Screening Technique.

%Extent Error

0.0%5.0%

10.0%15.0%

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

Element #

Erro

r

Predicted Damage

00.20.40.6

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

Element #

Dam

aged

In

dica

tor

%Extent Error

0%1%2%3%

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

Element #

Erro

r

103

The standard optimization predicts a 40.0% damage at element #25 and small

damage of 1% to 10% scattering over the structure. The maximum error of the damaged

extent is 12%. Even though the prediction shows the large damage at element #25, which

is correct, it also shows the damage as high as 10% in other locations where indeed there

is no damage, which is not desirable. The screening optimization technique presents a

much better performance. The prediction shows a correct location and extent of damage

at element #25 with very tiny errors in other locations. The maximum error of the

damaged extent is only 2%.

4.11 Damage Detection Testing on Bridge Structure

After the test of individual element systems and a combined beam and shell

structure in the previous section, the damaged detection program now is used to detect the

damage in a highway bridge structure model. This test creates a model for a girder-slab

bridge structure on simple supports with a 90 feet span length and 48 feet width. The

bridge has 4 equally spaced concrete girders (standard AASHTO girder type V), 8 inch

thick concrete slab, two parapets along the sides and a diaphragm at each end of the span.

The model consist of 11x10 shell-elements for the slab, 4x10 beam-elements for the

girders, 4x10 truss-elements for the reinforcing steel in the girders, 2x10 beam-elements

for the parapets, 2x11 beam-elements for the diaphragm, and 2x4 spring elements for

supports as shown in the Figure 4.27. Each structure originally has the same properties.

All slab elements have modulus of elasticity = 500000 k/ft2, thickness = 8 in, and mass =

0.00347 kip-s2/ft3. All girder elements have modulus of elasticity = 600000 k/ft2, cross-

sectional area = 7.035 ft2, torsional moment of inertia = 2.144 ft4, bending moment of

104

inertia = 25.0 ft4, and mass = 0.033 kip-s2/ft2. Parapet and diaphragm elements have

modulus of elasticity = 500000 k/ft2, cross-sectional area = 2.775 ft2, torsional moment of

inertia = 0.672 ft4, bending moment of inertia = 1.725 ft4, and mass = 0.013 kip-s2/ft2.

The reinforcing steels have modulus of elasticity = 4000000 k/ft2, cross-sectional area =

0.20 ft2, and mass = 0.00302 kip-s2/ft2, The supports have spring stiffness of 12000 kip/ft

The test uses both the standard and screening optimization algorithm. The

standard algorithm uses the single loop optimization with least squared errors objective

function. The screening algorithm uses the multi-loop optimization technique that stated

in the Section 8.7. The screening algorithm also uses relative errors as an objective

function. The test uses both dynamic and static characteristics. The eigenvalue and

eigenvector are combined for the dynamic characteristics. And Ritz vector is used for the

static characteristics. The results are shown in terms of damage indicators and the error of

predicted damaged extent.

The test considers five types of damage as follows:

1. Corrosion in reinforcing steel.2. Weakening of material properties in girders.3. Weakening of material properties in slab.4. Damage in supports.5. Cracking in bridge girders.

For the best comparison the presented test uses the same magnitude of damage in

each type of simulated damage. The following tests show the performance at 30%

damaged magnitude

105

Figure 4.27 Testing Model of a Bridge Structure.

4.11.1 Corrosion in Reinforcing Steel

The corrosion in reinforcing steel is simulated by reducing the cross-sectional area

of the truss elements. In this test, the truss element #156 and #157 are intentionally

damaged by reducing 30% of the initial cross-sectional area. The simulated damage

indicators of the bridge structures are shown in Figure 4.28

Figure 4.28 Damaged Indicators of Simulated Corrosion in Reinforcing Steel.

Simulated Damage

00.10.20.30.4

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

In

dica

tor

106

4.11.1.1 Standard Optimization Algorithm

4.11.1.1.1 Using eigen properties as characteristics. The results of the damage

prediction from the damage detection program are shown in Figure 4.29. The extent error

of the prediction is presented in Figure 4.30.

Figure 4.29 Predicted Damaged Indicators from Eigen Properties Characteristics.

Figure 4.30 %Extent Error from Eigen Properties Characteristics.

The results show large damage from element #154 to #157 which cover the

correct damaged locations at element #156 and #157 with relatively high extent errors,

%Extent Error

0.0%10.0%

20.0%

30.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

0.00.10.20.3

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

In

dica

tor

107

11.0% and 17.0% extent error at element #156 and #157 respectively. The prediction also

shows damage of 1% to 10% over other reinforcing steel where none exists.

4.11.1.1.2 Using Ritz vectors as characteristics. The results of the damage

prediction are shown in Figure 4.31. The extent error of the prediction is presented in

Figure 4.32.

Figure 4.31 Predicted Damaged Indicators from Ritz Vector Characteristics.

Figure 4.32 %Extent Error from Ritz Vector Characteristics.

The results show large damage at the element #157 and #158 which is the correct

damaged location and its adjacent element. The extent error of the prediction are

%Extent Error

0.0%

10.0%

20.0%

30.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

0.00.10.20.3

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

108

relatively high (10%-25%) at the damaged element. The prediction also shows a 17%

damage in the adjacent row elements and small damage of 2%-14% on the nearby

locations where none exists.

4.11.1.2 Screening Optimization Algorithm






%Extent Error

0.0%5.0%

10.0%15.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

0.00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

109

The results show large damage at element #156 and #157 which are correct

damaged locations with 1.6% and 9.1% extent error at element #156 and #157

respectively. However the prediction also shows small damage of 1% to 10% surrounding

the damaged locations. In the sense of damage detection, these errors are not bad because

they show the damage only the nearby the correct damaged locations.

4.11.1.2.2 Using Ritz vectors as characteristics. The second series of the results

come from using static responses as structural characteristics. The results of the damage


Figure 4.36.



%Extent Error

0.0%0.5%1.0%1.5%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

0.00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

110

The results show damage at element #156 and #157 which are correct damaged

locations with only 1.0% extent error at element #156. The prediction presents a excellent

damage detection.

4.11.2 Weakening of Material Properties in Girders

Reducing the modulus of elasticity of beam elements simulates the weakening of

material properties in girders. In this test, the beam element number 6 and 7 are

intentionally damaged by reducing 30% of the modulus of elasticity. The simulated

damaged indicators of the bridge structures are shown in Figure 4.37

Figure 4.37 Damaged Indicators of Simulated Weakening in Material of Bridge Girders.





Simulated Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

111



The results show large damage from element #4 to #7 which cover the correct

damaged locations at element #6 and #7 with relatively high extent errors, 12.0% and

18.0% at element #6 and #7 respectively. The prediction also shows a 15% damage in the

adjacent girders and small damage of 1%-2% in the bridge slab which actually does not

have any damage.

%Extent Error

0.0%10.0%

20.0%

30.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

rPredicted Damage

0.00.10.20.3

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

Indi

cato

r

112



Figure 4.41.

Figure 4.40 Predicted Damaged Indicators from Ritz Vectors Characteristics.

Figure 4.41 %Extent Error from Ritz Vectors Characteristics.

The results show large damage over the range of element #5 to #8 which cover the

correct damaged locations at element #6 and #7. The extent errors of the prediction are

6% and 10% at the damaged locations which are not very high. However the surrounding

elements are shown damaged at 5% to 15% and also 1% to 5% damage are also predicted

in the slab and reinforcing steel which actually do not have damage.

%Extent Error

0.0%5.0%

10.0%15.0%20.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190Element #

Erro

r

Predicted Damage

00.10.20.3

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

113







The results show large damage at element #6 and #7 which are correct damaged

locations with 9% and 6% extent error at element #6 and #7 respectively. However the

%Extent Error

0.0%

5.0%

10.0%15.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

0.0

0.1

0.2

0.3

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

114

prediction also shows small damage of 1% to 10% surrounding the damaged locations

and also scattered damage less than 2% in the bridge slab which actually has no damage.



Figure 4.45.



The results show a large damage at element #6 and #7 which are correct damaged

locations with less than 1% extent errors at the damaged elements. There are tiny errors

%Extent Error

0.0%

0.5%1.0%

1.5%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

0.00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

115

(less than 1%) surrounding the damaged locations. The prediction presents a excellent

damage detection.

4.11.3 Weakening of Material Properties in Slabs

The weakening of material properties in girders is simulated by reducing the

modulus of elasticity of beam elements. In the test, the shell elements #64, #65, #66, #76,

#77, and #78 are intentionally damaged by reducing 30% of their modulus of elasticity.

The simulated damaged indicators of the bridge structures are shown in Figure 4.46.

Figure 4.46 Damaged Indicators of Simulated Weakening in Material of Bridge Slab.



prediction from the damage detection program shown in Figure 4.47. The extent error of

the prediction is presented in Figure 4.48.

Simulated Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

116



The results show no large damage in the structure. The maximum damage in the

prediction is only 1.7% damage at the location that actually has no damage. At the

simulated damage location, the prediction shows less than 0.5% damage where indeed

30% damage exists. This prediction was not successful.

%Extent Error

0.0%10.0%20.0%30.0%40.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

rPredicted Damage

00.0050.01

0.0150.02

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

Indi

cato

r

117



Figure 4.50.



The results show no large damage in the structure. The maximum damage from

the prediction program is only 2.0% damage at the location that actually has no damage.

At the simulated damage locations, the prediction shows less than 0.2% damage where

actually have 30% damage. This prediction was not successful.

%Extent Error

0.0%10.0%20.0%30.0%40.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.010.020.03

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

118



prediction from the damage detection program shown in Figure 4.51. The extent error of

the prediction is presented in Figure 4.52.



The results show no obvious large damage in the structure. The maximum damage

from the prediction program is less than 11.0% damage, The prediction shows some

%Extent Error

0.0%

10.0%

20.0%30.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.050.1

0.15

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

In

dica

tor

119

damage 2% to 10.0% at the simulated damage locations but the actual damage is 30%.

This prediction was not successful.



Figure 4.54.


Figure 4.54 %Extent Error from Dynamic Response Characteristics.

The results show a large damage at element #64, #65, #66, #76, #77, and #78

where are correct damaged locations. The extent errors at the damage locations are 1% to

%Extent Error

0.0%2.0%4.0%6.0%8.0%

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Erro

r

Predicted Damage

00.10.20.30.4

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

169

176

183

190

197

Element #

Dam

aged

Indi

cato

r

120

3%. The prediction shows small damage in some other elements with extent error less

than 6%. The prediction successfully locates the damaged locations and the damaged

extent.

4.11.4 Damage in Supports

The damage in supports is simulated by reducing the stiffness of spring elements.

In this test, the spring element # 191 is intentionally damaged by reducing 30% of the

spring stiffness. The simulated damaged indicators are shown in Figure 4.55

Figure 4.55 Damaged Indicators of Simulated Damage in a Girder Support.





Simulated Damage

00.10.20.30.4

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

In

dica

tor

121




the prediction program is only 1.3% damage. At the simulated damage location, the

prediction shows 0.8% damage, but actually has 30% damage. This prediction was not

successful.

%Extent Error

0.0%10.0%20.0%30.0%40.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

rPredicted Damage

00.0050.01

0.0151 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

In

dica

tor

122



Figure 4.59.




the prediction program is less than 1.0% damage. At the simulated damage locations, the

prediction shows 0.40% damage. This prediction was not successful.

%Extent Error

0.0%10.0%20.0%30.0%40.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.0020.0040.0060.008

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

123







The results show large damage at element #191 which is a correct damaged

location with 4.6% extent error. The prediction also shows some small damage of 4% to

%Extent Error

0.0%2.0%4.0%6.0%8.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

In

dica

tor

124

6% in the girder that connect to the damaged support as well as 4% to 5% of damage in

the adjacent supports. There are also small damages less than 3% scattered in bridge slab.



Figure 4.63.



The results show a large damage at element #191 which is a correct damaged

location with less than 0.1% extent error at the damaged element. The prediction shows a

%Extent Error

0.0%0.5%1.0%1.5%2.0%

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Erro

r

Predicted Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

125

small damage of 1.7% in a girder that is connected to the damaged support. There are

some very small errors (less than 1%) scattered in the bridge slab. The prediction presents

very good damage detection.

4.11.5 Cracking of bridge girder

The cracking in bridge girder is simulated by reducing the moment of inertia and

cross-sectional area of beam elements. In the test, beam element number 6 is intentionally

damaged by reducing 30% of its moment of inertia and cross-sectional area. The

simulated damaged indicators of the bridge structures are shown in Figure 4.64

Figure 4.64 Damaged Indicators of Simulated Damage of Cracking in a Girder.





Simulated Damage

0.00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

126



The results show large damage from element #6 where is the correct damaged

location but the predicted extent is only 15.0% compared to 30.0% of the simulated

damage. The prediction also shows 1% to 7% damages in the surrounding area and in the

adjacent girder.

%Extent Error

0.0%5.0%

10.0%15.0%20.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.050.1

0.150.2

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

127



Figure 4.68.



The results show large damage at element #6 which is the correct damaged

location. The extent error of the prediction is 7% at the damaged location. The prediction

also shows 1% to 6% damages in the surrounding area and in the adjacent girder. Very

small damages are also predicted in the bridge slab.

%Extent Error

0.0%2.0%4.0%6.0%8.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.10.20.3

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

128







The results show large damage from element #6 which is the correct damaged

location but the predicted extent is only 19.0% rather than 30.0% from the simulated

%Extent Error

0.0%

5.0%

10.0%15.0%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Erro

r

Predicted Damage

00.050.1

0.150.2

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

Element #

Dam

aged

In

dica

tor

129

damage. The prediction also shows 1% to 9% damages surrounding the damaged

element.



Figure 4.72.



%Extent Error

0.0%2.0%4.0%6.0%

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Erro

r

Predicted Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

130

The results show a large damage at element #6 which is the correct damaged

location with less than 1% extent errors at the damaged elements. There are very small

errors (less than 5%) around the damaged locations and also less than 2% damages are

shown in the bridge slab. The prediction presents good damage detection.

From the bridge damaged detection testing, the damage detection program with

the screening algorithm can successfully predict both location as well as the extent of

damage. However the performance of the damage detection also depends upon the type of

damage. The damage that has more effect on the bridge structural stiffness will be easier

to detect with this system identification method. The damage that has less effect on the

structural stiffness, like the weakening in some of the slab elements, presents a difficulty

in predicting damage. The screening algorithm shows a much better detection

performance over the standard algorithm especially when the damage does not cause

much change in the structural stiffness. Ritz vectors also present more sensitivity to

damage than the traditional eigen properties.

131

CHAPTER 5PARAMETRIC STUDY

5.1 Introduction

There are many important parameters in detection of damage using system

identification method. Each parameter affects the performance of the damage detection.

In order to use this damage detection method in practice, the behavior of each parameter

needs to be studied. In this study, more than 500 simulations have been tested. This

chapter presents the influence to detection performance of 5 main parameters in the

damage detection, listed as follows:

1. Detection techniques.2. Magnitude of perturbations.3. Responses.4. Objective functions.5. Noises.

Each parameter will be tested in several different conditions and different

damages. Each test condition maintains all other parameters constant except the

investigated parameter. The results of the tests are shown in terms of the sum of the

difference in the damage indicators. The error formula is described as follows:

( )∑=

−=N

i

ip

isAbs

1

ρρµ (5.1)

132

where µ is the error indicator.

ρs is the damage indicator of simulated damage structure.

ρp is the damage indicator of the predicted damage.

N is the number of design variables.

The smaller the value of the error indicator is the better performance of the

damage detection.

5.2 Detection Techniques

This test investigates in more depth the performance of the detection techniques

between the standard algorithm and the screening algorithm presented in the Chapter 4. In

this test, both algorithms use the same objective function, either least squared errors or

relative errors. The test shows 30 different test conditions.

Performance of the standard algorithm is heavily dependent on the magnitude of

the perturbation in the optimization techniques. However, there is a high level of

uncertainty in the effect of perturbation rate, which will be shown in the next section.

Each condition of the standard algorithm was tested with fifty different perturbation rates.

The results presented here are the best results from the fifty samples for a given condition.

The results are shown in Figure 5.1.

133

Figure 5.1 Standard Algorithm and Screening Algorithm.

Figure 5.1 shows that the error indicators of the screening algorithm are always

lower or equal to the error indicators of standard algorithm for the same conditions. These

results confirm the results in the previous chapter, that the screening algorithm always has

a better performance than the standard algorithm.

5.3 Magnitude of Perturbations

The magnitude of perturbation is the amount of change applied to each element at

the first step of optimization to create a first gradient vector by the forward finite

difference method as shown below:

)()()()(

1

1

ii

ii FFFXX

XXXX

−−

≈∂

∂

+

+ (5.2)

Algorithm Performance

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25 30

Condition #

Erro

r Ind

icat

or

Screening Algorithm Standard Algorithm

134

The magnitude of perturbation is defined as the difference between the design

variables after the first change and the initial design variables. The magnitude of

perturbation may be written as follows:

01 XX −=δ (5.3)

( )100*

0

01 ×−=δX

XX(5.4)

where δ is the magnitude of perturbation.

δ* is the percentage of perturbation.

X1 is the first changed design variables.

X0 is the initial design variables.

The magnitude of perturbation is important to the success of the damage detection

using system identification methods, especially for the standard method which uses one

perturbation magnitude. A problem can occur when the system is in a deep local

minimum and the optimization may not be able to get out of it. For this case, the

optimization will not converge to the global minimum. This study seeks a reasonable

range of perturbation by testing the system with many different conditions. The other

parameters are held constant for each perturbation rate. The results are described by the

final value of objective function for each condition. The lower the value of the objective

function is, the more successful the damage detection. The results are presented in the

Figure 5.2.

135

Figure 5.2 Magnitude of Perturbation.

The results show an uncertainty of the perturbation effect in the damage detection.

There is no particular perturbation rate that significantly out performs others. In some

cases, the perturbation rate may not effect the outcome of the detection but the

perturbation rate can also cause the success or non-success of the damage detection. For

example, from test condition number 25th, damage was simulated at the element #90 by

reducing by 30% of the modulus of elasticity. The damage indicators of simulated

damage structure are shown in Figure 5.3. The predictions of the damage detection for

three different perturbation rate: 10%, 5%, and 0.5%, are presented in Figure 5.4, 5.5 and

5.6, respectively.

Magnitude of Perturbation

00.20.40.60.8

11.21.41.61.8

0 5 10 15 20 25 30Condition #

Obj

ectiv

e Fu

nctio

n

50%20%10%5%1%0.5%0.1%

Condition #

Magnitude of Perturbation

136

Figure 5.3 Damage Indicators of Simulated Damage Structure.

Figure 5.4 Damage Indicators of Predicted Damage from 10% Perturbation Rate.

Figure 5.5 Damage Indicators of Predicted Damage from 5% Perturbation Rate.

Predicted Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

rSimulated Damage

0.00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

Predicted Damage

00.10.20.30.4

1 9 17 25 33 41 49 57 65 73 81 89 97 105

113

121

129

137

145

153

161

169

177

185

193

Element #

Dam

aged

Indi

cato

r

137

Figure 5.6 Damage Indicators of Predicted Damage from 0.5% Perturbation Rate.

From the results, using a 10% or 5% perturbation rate yields an obvious damage at

element #90 but using a 0.5% perturbation rate does not give any obvious damage. The

5% perturbation rate has only less than 1% extent error. The 10% and 0.5% perturbation

rates have 5% and 15% extent error, respectively. This example shows the effect of

different perturbation rates.

If the standard algorithm is used, several perturbation rates need to be tested to

determine which one gives a minimum value of the objective function. The screening

algorithm already includes the change of perturbation rate in the procedure. However if

the problem has a deep local minimum, the change in the procedure may not be enough to

jump out of a local minimum. Changing the starting point of the perturbation rate is

always a good option to check.

5.4 Responses

The response represents the characteristics of the structure that are used to identify

structure. This study considers two types of response: eigen properties and Ritz vectors.

The eigen properties are the traditional characteristic that are widely used in system

Predicted Damage

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identification methods. The Ritz vector has become a focal point of system identification

study in the last ten years because of its high sensitivity. This study presents the effect of

both responses to the damage detection. The test used exactly the same conditions for

both responses in the damage detection. The results are presented in terms of error

indicators µ, as described in equation (5.1). Again, the lower error indicator represents a

better performance. The results of 30 different conditions are shown in Figure 5.7. The

detail example of condition #7 which simulate damage in element#65th by reducing its

moment of inertia by 20%. The simulated damaged indictor is presented in Figure 5.8.

The predicted damaged indicator from eigen properties and Ritz vectors are presented in

Figure 5.9, and 5.10 respectively.

Figure 5.7 Performance of Ritz Vectors and Eigen Properties.

Performance of Responses

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25 30

Condition Number

Erro

r Ind

icca

tor

Ritz Vectors Eigen Properties

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Figure 5.9 Damage Indicators of Predicted Damage from Eigen Properties.

Figure 5.10 Damage Indicators of Predicted Damage from Ritz Vectors.

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The Ritz vectors mostly show a lower number of error indictors, which means that

the Ritz vector response yields better performance than the eigen properties response.

However since the Ritz vector is a load dependent characteristic, another question rises.

Where is the location to apply the load? From this study, the proper location to apply load

is the location that gives large deflection to the system. Loading applied to the structure

for measuring Ritz vectors can be at more than one location. The magnitude of applied

load is not a parameter because the Ritz vector is a normalized vector. The optimum

number of loads and their locations can improve the damage detection performance

however it needs to be studied in-depth.

5.5 Objective Functions

The objective function represents the value that the optimization technique is

trying to minimize. The least squared errors method is used widely in system

identification methods as an objective function. This research proposes relative errors as

an objective function instead of the absolute errors in the traditional least squared method.

The formulas of the two methods are repeated as follows:

Least squared errors:

∑=

Φ−Φ=N

imioisF

1

2)()(X (5.5)

Sum of squared relative errors:

∑= Φ

Φ−Φ=

N

i oi

mioiRSF

1

2)()(X (5.6)

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where FS(X) is an objective function in least square sense.

FRS(X) is an objective function for sum of square relative error.

N is the number of response data.

Φo is an observed response.

Φm is an analytical response form FEM.

This section presents the effect of each objective functions to the damage

detection. The performance of the damage detection is presented by the error indicators µ

as described in equation (5.1). The results are shown in Figure 5.11. Condition #7, in

which element #41 is damaged by reducing its modulus of elasticity by 30%, is presented

in detail by Figure 12, 13 and 14.

Figure 5.11 Performance of Objective Functions.

Performance of Objective Function

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 18 20

Condition Number

Erro

r Ind

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tor

Relative Errors Least Squared Errors

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Figure 5.13 Damage Indicators of Predicted Damage from Least Squared Errors.

Figure 5.14 Damage Indicators of Predicted Damage from Relative Errors.

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The results show an obvious advantage of the sum of squared relative errors

objective function over the least squared errors objective function. In some cases, the

damage detection would not even converge with the least square errors but did converge

with the relative errors. The screening algorithm that the author proposes used the relative

errors as an objective function.

5.6 Noise

During the measurement of the observed response, poor measuring equipment and

careless measurements can cause noise in the response data. Noise has a lot of influence

on damage detection using system identification. The effect of noise depends heavily on

the type and magnitude of damage as well as the level of noise. This section presents the

effect of noise on damage detection using the system identification method. The noise

was simulated by a random function with a normal distribution and zero mean value. The

simulated noises are imposed into the damaged responses using Equation (4.16) which is

repeated here:

Φ* = Φ + Φ (γ/100) random (-1,1) (5.7)

where Φ* is the response data with noise.

Φ is the response data without noise.

γ is the percentage of noise level to the response magnitude.

random(-1,1) is a random value from –1.0 to 1.0 with a normal

distributed and zero mean value.

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Since the effect of noise on the damage detection performance also depends on the

level of damage, the magnitude of noise is described in terms of percentage of damage

level.

[ ]δψ=γ100

(5.8)

where ψ is the noise indicator in percent.

δ is the extent of damage.

The study presents the effect on five different types of damage that were shown in

Chapter 4, with different level of noise indicators: 5%, 10%, 20% and 50%. The

performance of the damage detection with noise, in terms of error indicators µ, is

presented in Table 5.1.

Table 5.1 Noise Effect

Type of Damage Error Indicatorsψ=0% ψ=5% ψ=10% ψ=20% ψ=50%

Corrosion in ReinforcingSteel

0.02 0.66 0.72 1.10 2.40

Weakening of GirdersMaterial

0.03 0.10 0.19 1.82 5.52

Weakening of Slab Material 0.40 2.54 3.05 3.94 9.31Damage in Supports 0.07 0.29 0.37 1.20 2.45Cracking in Girders 0.23 0.59 1.00 1.48 2.50

The results show the large effect of noise on the performance of damage detection

with a system identification method. The error indicators show an amount of error that

increase while the level of noises was increased. This does not show the entire behavior.

As stated before, the noise effect on the detection is heavily dependent on the type of

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damage. In this test, the corrosion in reinforcing steel has a very small change in

magnitude of characteristic response after damage. When noise was imposed on the

observed response, the new observed response varied more than the response from the

corrosion alone. The damage detection can no longer predict the damage that exists in the

structure if the noise effect is larger than the damaged effect. Other types of damage have

a similar behavior in dealing with noisy response data. The error indicators are

significantly increased when the level of noise is higher. However, if the level of noise is

not too high, the damage detection can still successfully identifying the location and

extent of the damage. The prediction may show an increase of damage in other locations

that actually have no damage. The predicted damage in the undamaged locations

increases with the level of noise and reduces the obviousness of the predicted damage in

the damaged location. If the noise level is high enough, the damage detection is unable to

identify the damage. The effect of noise is shown by plotting the damage indicators for a

support damage in different noise level. Figure 5.15 shows the damage indicators of a

simulated damage structure. The effect of noise level of 0%, 5%, 10%, 20%, and 50% are

shown in Figure 5.16, 5.17, 5.18, 5.19, and 5.20 respectively.

Figure 5.15 Damaged Indicators of Simulated Damage in a Girder Support.

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Figure 5.16 Predicted Damaged Indicators without Noise.

Figure 5.17 Predicted Damaged Indicators for 5% Noise Indicators.


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The plots show that the prediction of damage is obviously correct with the

response data that has no noise until a 10% noise indicator imposed. At a 20% noise

indicator the prediction still shows large damage at the correct location but also shows

large damage in other locations that actually have no damage and the error of damage

extent also increased at the damage location. With 50% noise indicator, the damage

detector still predicts a large damage at the damaged element but can no longer

distinguish from others because there is damage predicted in undamaged locations. The

Predicted Damage

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damage at other locations have larger or equal damage magnitude than the correct

location.

The effect of noise on other types of damage also has the same tendency as shown

above. The measurement of observed response data should be carefully done with proper

devises in order to reduce the noise in the response data, which increases the performance

of the damage detection using system identification.

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CHAPTER 6CONCLUSION

A damage detection technique for bridge structures has been developed. The

technique is based on a non-destructive damage detection method called system

identification. This research uses a three-dimensional finite element model to represent a

bridge structure. A bridge structure model is composed of four basic structural elements:

3-D beam elements to represent bridge girders, shell elements to represent the bridge slab,

truss elements to represent the reinforcing steel, and elastic spring elements to represent

girder supports. Each element is connected to form a bridge structure using a rigid link

offset to account for the composite properties of the structure. The research develops a

damage detection tool based on two software packages: SIMPAL for finite element

analysis and DOC&DOT for the optimization techniques. The study uses the BFGS

optimization technique in the damage detection tool. The BFGS algorithm has a quadratic

convergence rate without using the second derivative of the objective function. Eigen

properties (eigenvalue and eigenvectors) and Ritz vectors are used as characteristic

responses of the bridge structure.

The research proposes a damage detection technique called a screening algorithm.

This screening algorithm was implemented to reduce the number of design variables in

the optimization and thereby improve the performance of the damage detection. The

screening algorithm uses the information from previous iterations of optimization to

150

screen out the inactive or nearby inactive design variables and adjusts the magnitude of

perturbation to avoid a local minimum. The algorithm also uses relative errors as an

objective function instead of traditional least squared errors, which use absolute errors.

This research studies in detail the behaviors of each parameter that affect the

performance of this damage detection technique. Hundreds of damage detection scenarios

have been tested by the damage detection tool. Each damage condition is simulated by a

finite element analysis. Five different types of damage: corrosion in reinforcing steel,

weakening of material in the girder and slab, damage in a support, and cracking in a

girder are presented in the bridge damage detection testing in Chapter 4.

The results show that the screening algorithm outperforms the standard algorithm

and provides more capability to detect damage in a more complex structure. The Ritz

vector as a characteristic response was shown to be more sensitive to the damage than the

traditional eigen properties. Since the Ritz vector is a load dependent characteristic, the

location of applied load becomes another parameter to the damage detection. From this

study, the proper location of applied load is the location that gives a large deflection to

the system. The accuracy of this damage detection also depends on the purity of observed

response data. Noise from the measurement procedure in the observed response data can

significantly reduce the accuracy of the damage detection if the noise level is too high.

The collection of observed data must be done with care in order to use this damage

detection technique.

From the hundreds of tests, the damage detection using the screening algorithm

with the Ritz vectors as characteristic responses has the best capability of predicting the

correct location and extent of the simulated damage in bridge structures. However this

151

damage detection technique is not a complete health monitoring system for bridges but a

preliminary to the bridge-health investigation. This damage detection makes the

investigation faster, cheaper and easier.

Damage detection using system identification methods has so many parameters

that affect accuracy and capability of the damage detection. The performance of this

damage detection can be improved by further studies. Some topics that the author suggest

for future study are as follows:

1. The in field test of this damage detection with noise should be done to be moreconfident of the capability of the algorithm.2. A new algorithm of optimization techniques may be used in the damagedetection procedure. New optimization techniques have been widely developed inrecent years. It is possible to improve the damage detection with more advanceoptimization techniques.3. Other possible characteristic responses that have a high sensitivity to damagecould be investigated.4. The performance of damage detection using Ritz vectors as characteristicresponse can be improved by study in behavior of loading applied to the structurefor measuring Ritz vectors. An optimum number of applied loads and theirlocations can increase the efficiency of the damage detection.

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BIOGRAPHICAL SKETCH

Wirat Lertpaitoonpan was born in Samutsakorn, Thailand, in August 1970. He

went to middle and high school in Bangkok. He began his college study at the college of

engineering, Chulalongkorn University, Bangkok. He received his bachelor’s degree in

civil engineering in 1993. After graduation he worked in an engineering consulting firm

for 2 years. Then he became an instructor in the civil engineering department of Sripatum

University. A year later after he received a scholarship from Sripatum University to

continue his studies in the United States. He enrolled in the University of Florida in

January 1996 and received his master of engineering degree in May 1997. He continued

his work on a doctorate in civil engineering at the University of Florida and received his

Ph.D. in May 2000.

I certify that I have read this study and that in my opinion it conforms toacceptable standards of scholarly presentation and is fully adequate, in scope and quality,as a Dissertation for the degree of Doctor of Philosophy.

_________________________________Marc I. Hoit, ChairProfessor of Civil Engineering


_________________________________Duane EllifrittProfessor of Civil Engineering


_________________________________John LybasAssociate Professor of Civil Engineering


_________________________________Fernando E. FagundoProfessor of Civil Engineering


_________________________________Ian FloodAssociate Professor of Building Construction

This Dissertation was submitted to the Graduate Faculty of the College ofEngineering and to the Graduate School and was accepted as partial fulfillment of therequirements for the degree of Doctor of Philosophy.

May 2000 _________________________________M. J. OhanianDean, College of Engineering

_________________________________Winfred M. PhillipsDean, Graduate School

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