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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.
v
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
vi
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
vii
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
viii
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
4.15 Predicted Damaged Indicators from Damage Detection Program............................96
4.16 % Extent Error of Predicted Damaged Indicators.....................................................97
4.17 Testing Shell Element Model ...................................................................................97
4.18 Damaged Indicators of Simulated Damaged Shell Structure ...................................98
4.19 Predicted Damaged Indicators from Damage Detection Program............................98
4.20 % Extent Error of Predicted Damaged Indicators.....................................................99
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
4.33 Predicted Damaged Indicators from Eigen Properties Characteristics ...................108
4.34 %Extent Error from Eigen Properties Characteristics ............................................108
4.35 Predicted Damaged Indicators from Ritz Vector Characteristics ...........................109
4.36 %Extent Error from Ritz Vector Characteristics ....................................................109
4.37 Damaged Indicators of Simulated Weakening in Material of Bridge Girders........110
4.38 Predicted Damaged Indicators from Eigen Properties Characteristics ...................111
4.39 %Extent Error from Eigen Properties Characteristics ............................................111
4.40 Predicted Damaged Indicators from Ritz Vectors Characteristics .........................112
4.41 %Extent Error from Ritz Vectors Characteristics ..................................................112
4.42 Predicted Damaged Indicators from Eigen Properties Characteristics ...................113
4.43 %Extent Error from Eigen Properties Characteristics ............................................113
4.44 Predicted Damaged Indicators from Ritz Vector Characteristics ...........................114
4.45 %Extent Error from Ritz Vector Characteristics ....................................................114
4.46 Damaged Indicators of Simulated Weakening in Material of Bridge Slab.............115
4.47 Predicted Damaged Indicators from Eigen Properties Characteristics ...................116
4.48 %Extent Error from Eigen Properties Characteristics ............................................116
4.49 Predicted Damaged Indicators from Ritz Vectors Characteristics .........................117
4.50 %Extent Error from Ritz Vectors Characteristics ..................................................117
4.51 Predicted Damaged Indicators from Eigen Properties Characteristics ...................118
4.52 %Extent Error from Eigen Properties Characteristics ............................................118
4.53 Predicted Damaged Indicators from Ritz Vector Characteristics ...........................119
xiii
4.54 %Extent Error from Dynamic Response Characteristics........................................119
4.55 Damaged Indicators of Simulated Damage in a Girder Support ............................120
4.56 Predicted Damaged Indicators from Eigen Properties Characteristics ...................121
4.57 %Extent Error from Eigen Properties Characteristics ............................................121
4.58 Predicted Damaged Indicators from Ritz Vectors Characteristics .........................122
4.59 %Extent Error from Ritz Vectors Characteristics ..................................................122
4.60 Predicted Damaged Indicators from Eigen Properties Characteristics ...................123
4.61 %Extent Error from Eigen Properties Characteristics ............................................123
4.62 Predicted Damaged Indicators from Ritz Vector Characteristics ...........................124
4.63 %Extent Error from Ritz Vector Characteristics ....................................................124
4.64 Damaged Indicators of Simulated Damage of Cracking in a Girder ......................125
4.65 Predicted Damaged Indicators from Eigen Properties Characteristics ...................126
4.66 %Extent Error from Eigen Properties Characteristics ............................................126
4.67 Predicted Damaged Indicators from Ritz Vectors Characteristics .........................127
4.68 %Extent Error from Ritz Vectors Characteristics ..................................................127
4.69 Predicted Damaged Indicators from Eigen Properties Characteristics ...................128
4.70 %Extent Error from Eigen Properties Characteristics ............................................128
4.71 Predicted Damaged Indicators from Ritz Vector Characteristics ...........................129
4.72 %Extent Error from Ritz Vector Characteristics ....................................................129
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
xiv
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
5.19 Predicted Damaged Indicators for 20% Noise Indicators.......................................147
5.20 Predicted Damaged Indicators for 50% Noise Indicators.......................................147
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
Bridge GirderPier Cap
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
Bridge GirderPier Cap
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
Bridge GirderPier Cap
Hinges
Rt/2Rt/26 ft.
Rt/2Rt/2
6 ft.Rt/2Rt/2
6 ft.Bridge Slab
Bridge GirderPier Cap
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
Bridge GirderPier Cap
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
Start, known Xinitial
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.
Start, known Xinitial
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
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
Start, known Xinitial
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
Evaluate ∇ F=∇ F(X)
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.
Evaluate ∇ F=∇ F(X)
X=Xinitial
and B=I
S = -B∇ F(X)
Compute α
Compute X= X+ α S
Compute E from (3.38)
Evaluate ∇ F=∇ F(X)
B= B + E
Start, known Xinitial
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)
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
Fq+1 = F(αq+1)Ft=F(αq+1+βq+1)
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
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
YeNo
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.
Figure 4.11 Predicted Damaged Indicators from Damage Detection Program.
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.
Figure 4.15 Predicted Damaged Indicators from Damage Detection Program.
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
Figure 4.16 % Extent Error of Predicted Damaged Indicators.
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.
Figure 4.19 Predicted Damaged Indicators from Damage Detection Program.
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
Figure 4.20 % Extent Error of Predicted Damaged Indicators.
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
4.11.1.2.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.33. The extent error
of the prediction is presented in Figure 4.34.
Figure 4.33 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.34 %Extent Error from Eigen Properties Characteristics.
%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
prediction are shown in Figure 4.35. The extent error of the prediction is presented in
Figure 4.36.
Figure 4.35 Predicted Damaged Indicators from Ritz Vector Characteristics.
Figure 4.36 %Extent Error from Ritz Vector Characteristics.
%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.
4.11.2.1 Standard Optimization Algorithm
4.11.2.1.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.38. The extent error
of the prediction is presented in Figure 4.39.
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
Figure 4.38 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.39 %Extent Error from Eigen Properties Characteristics.
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
4.11.2.1.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.40. The extent error of the prediction is presented in
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
4.11.2.2 Screening Optimization Algorithm
4.11.2.2.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.42. The extent error
of the prediction is presented in Figure 4.43.
Figure 4.42 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.43 %Extent Error from Eigen Properties Characteristics.
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.
4.11.2.2.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.44. The extent error of the prediction is presented in
Figure 4.45.
Figure 4.44 Predicted Damaged Indicators from Ritz Vector Characteristics.
Figure 4.45 %Extent Error from Ritz Vector Characteristics.
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.
4.11.3.1 Standard Optimization Algorithm
4.11.3.1.1 Using eigen properties as characteristics. The results of the damage
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
Figure 4.47 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.48 %Extent Error from Eigen Properties Characteristics.
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
4.11.3.1.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.49. The extent error of the prediction is presented in
Figure 4.50.
Figure 4.49 Predicted Damaged Indicators from Ritz Vectors Characteristics.
Figure 4.50 %Extent Error from Ritz Vectors Characteristics.
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
4.11.3.2 Screening Optimization Algorithm
4.11.3.2.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program shown in Figure 4.51. The extent error of
the prediction is presented in Figure 4.52.
Figure 4.51 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.52 %Extent Error from Eigen Properties Characteristics.
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.
4.11.3.2.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.53. The extent error of the prediction is presented in
Figure 4.54.
Figure 4.53 Predicted Damaged Indicators from Ritz Vector Characteristics.
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.
4.11.4.1 Standard Optimization Algorithm
4.11.4.1.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.56. The extent error
of the prediction is presented in Figure 4.57.
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
Figure 4.56 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.57 %Extent Error from Eigen Properties Characteristics.
The results show no large damage in the structure. The maximum damage from
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
4.11.4.1.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.58. The extent error of the prediction is presented in
Figure 4.59.
Figure 4.58 Predicted Damaged Indicators from Ritz Vectors Characteristics.
Figure 4.59 %Extent Error from Ritz Vectors Characteristics.
The results show no large damage in the structure. The maximum damage from
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
4.11.4.2 Screening Optimization Algorithm
4.11.4.2.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.60. The extent error
of the prediction is presented in Figure 4.61.
Figure 4.60 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.61 %Extent Error from Eigen Properties Characteristics.
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.
4.11.4.2.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.62. The extent error of the prediction is presented in
Figure 4.63.
Figure 4.62 Predicted Damaged Indicators from Ritz Vector Characteristics.
Figure 4.63 %Extent Error from Ritz Vector Characteristics.
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.
4.11.5.1 Standard Optimization Algorithm
4.11.5.1.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.65. The extent error
of the prediction is presented in Figure 4.66.
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
Figure 4.65 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.66 %Extent Error from Eigen Properties Characteristics.
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
4.11.5.1.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.67. The extent error of the prediction is presented in
Figure 4.68.
Figure 4.67 Predicted Damaged Indicators from Ritz Vectors Characteristics.
Figure 4.68 %Extent Error from Ritz Vectors Characteristics.
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
4.11.5.2 Screening Optimization Algorithm
4.11.5.2.1 Using eigen properties as characteristics. The results of the damage
prediction from the damage detection program are shown in Figure 4.69. The extent error
of the prediction is presented in Figure 4.70.
Figure 4.69 Predicted Damaged Indicators from Eigen Properties Characteristics.
Figure 4.70 %Extent Error from Eigen Properties Characteristics.
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
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Element #
Dam
aged
In
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damage. The prediction also shows 1% to 9% damages surrounding the damaged
element.
4.11.5.2.2 Using Ritz vectors as characteristics. The results of the damage
prediction are shown in Figure 4.71. The extent error of the prediction is presented in
Figure 4.72.
Figure 4.71 Predicted Damaged Indicators from Ritz Vector Characteristics.
Figure 4.72 %Extent Error from Ritz Vector Characteristics.
%Extent Error
0.0%2.0%4.0%6.0%
1 9 17 25 33 41 49 57 65 73 81 89 97 105
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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)
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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
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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
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121
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0.00.10.20.30.4
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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
139
Figure 5.8 Damage Indicators of Simulated Damage Structure.
Figure 5.9 Damage Indicators of Predicted Damage from Eigen Properties.
Figure 5.10 Damage Indicators of Predicted Damage from Ritz Vectors.
Predicted Damage
00.10.20.3
1 9 17 25 33 41 49 57 65 73 81 89 97 105
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137
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Element #
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aged
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rSimulated Damage
0.00.1
0.2
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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)
141
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
icca
tor
Relative Errors Least Squared Errors
142
Figure 5.12 Damage Indicators of Simulated Damage Structure.
Figure 5.13 Damage Indicators of Predicted Damage from Least Squared Errors.
Figure 5.14 Damage Indicators of Predicted Damage from Relative Errors.
Predicted Damage
00.10.20.3
1 9 17 25 33 41 49 57 65 73 81 89 97 105
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aged
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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.
144
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
145
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.
Simulated Damage
00.10.20.30.4
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Figure 5.16 Predicted Damaged Indicators without Noise.
Figure 5.17 Predicted Damaged Indicators for 5% Noise Indicators.
Figure 5.18 Predicted Damaged Indicators for 10% Noise Indicators.
Predicted Damage
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Figure 5.19 Predicted Damaged Indicators for 20% Noise Indicators.
Figure 5.20 Predicted Damaged Indicators for 50% Noise Indicators.
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
00.050.1
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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.
149
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.
152
REFERENCES
American Association of State, Highway, and Transportation Officials (AASHTO).(1994). AASHTO-LRFD Bridge Design Specifications. Washington, DC:AASHTO.
Better Roads. (1994). 1994 Bridge Inventory. Better Roads, Wm. O. Dannhausen 64(11):28.
Bolton, R., Stubbs, N., Park, S., Choi, S., and Sikorsky, C. (1998). Measuring BridgeModal Parameters for Use in Non-Destructive Damage Detection andPerformance Algorithms. Proceedings of the International Modal AnalysisConference, SEM, Bethel, CT 2: 1269-1275.
Cao, T., and Zimmerman, D. (1997). Procedure to Extract Ritz Vectors from DynamicTesting Data. Proceedings of the International Modal Analysis Conference, SEM,Bethel, CT 2: 1036-1042
Cauchy, A. (1947). Methode Generale pour la Resolution des Systemes D’equationsSimultanees. Comp. Rend. l’Academie des Sciences, Paris 5:536-538.
Chen, J., and Gabar, J. (1988). On-orbit Damage Assessment for Large Space Structure.AIAA Journal, 26(9): 1119-1126.
Cook, R., Malkus, D., and Plesha, M. (1989). Concepts and Applications of FiniteElement Analysis. New York: Wiley.
Doebling, S., Farrar, C., and Goodman, R. (1997). Effects of Measurement Statistics onthe Detection of Damage in the Alamosa Canyon Bridge. Proceedings of theInternational Modal Analysis Conference, SEM, Bethel, CT 1: 919-929.
Flesch, R., Stebernjak, B., Maeck, J., and Olia, S. (1999). FE-Modelling of RC Structureswithin the Simces Project. Proceedings of the International Modal AnalysisConference, SEM, Bethel, CT 1: 1042-1048.
Fletcher, R. (1980). Practical Methods of Optimization. New York: Wiley.
153
Friswell, M., and Mottershead, J. (1995). Finite Element Model Updating in StructuralDynamics. Dordrecht, The Netherlands: Kluwer.
Gordon, S. (1989). Durability of Highway Bridges. Proceedings of IABSE Symposium,Lisbon, Potugal 1:19-31
Gottfried, B., Weisman, J. (1973). Introduction to Optimization Theory. EnglewoodCliffs, NJ: Prentice-Hall.
Haftka, R., and Gurdal, Z. (1996). Elements of Structural Optimization. Dordrecht, TheNetherlands: Kluwer.
Hays, C., Hoit, M., Consolazio, G., and Kakhandiki, A. (1994). Bridge Rating of Girder-Slab Bridges Using Automated Finite Element Technology. Structure ResearchReport No. 94-1, Department of Civil Engineering, University of Florida,Gainesville.
Hoit, M. (1983). SIMPAL Users Guide. Department of Civil Engineering, University ofFlorida, Gainesville.
Hoit, M. (1995). Computer-Assisted Structural Analysis and Modeling. Upper SaddleRiver, NJ: Prentice-Hall.
Juneja, V., Haftka, R., and Cudney, H. (1997). Damage Detection and DamageDetectability Analysis and Experiments. Journal of Aerospace Engineering 10(4):135-142.
Kaouk, M. (1993). Structural Damage Assessment and Finite Element Model RefinementUsing Measured Modal Data. Ph.D. Dissertation, University of Florida,Gainesville.
Kramer, C., Smet, C., and Peeters, B. (1999). Comparison Ambient and Forced VibrationTesting of Civil Engineering Structures. Proceeding of the International ModalAnalysis Conference, SEM, Bethel, CT 2: 1030-1034.
Lasdon, L. (1970). Optimization Theory for Large Systems. New York: Macmillan.
Masri, S., Nakamura, M., Chassikos, A., and Caughey, T. (1996). Neural NetworkApproach to Detection of Changes in Structural Parameters. Journal ofEngineering Mechanics, ASCE 122(4): 350-360.
Muhummad, I., Halling, M., and Womack, K. (1998). Force Vibration Testing of A Full-Scale Bridge Span. Department of Civil and Environmental Engineering, UtahState University, Logan, Utah.
154
Norton, J. (1986). An Introduction to Identification. London: Harcourt Brace Jovanovich.
Petro, S., Chen, S., Gangarao H., Venkatappa, S. (1997). Damage Detection UsingVibration Measurements. Proceedings of the International Modal AnalysisConference, SEM, Bethel, CT 1: 113-119.
Soeio, F. (1990). Structural Damage Assessment Using Identification Techniques. Ph.D.Dissertation, University of Florida, Gainesville.
Vanderplaats, G. (1999). Numerical Optimization Techniques for Engineering Design.Colorado Springs, CO: Vanderplaats Research & Development.
Williams, M. (2000). Using Neural Networks to Position Live Loads on Bridge Piers.Ph.D. Dissertation, University of Florida, Gainesville.
Zimmerman, D., and Cao, T. (1997). Effects of Noise on Measured Ritz Vectors.Proceedings of DETC'97, ASME, Sacramento, CA 1:1-8.
Zimmerman, D., James, G., and Cao, T. (1999). An Experimental Study of DamageDetection Using Modal, Strain, and Ritz Properties. Proceedings of theInternational Modal Analysis Conference, SEM, Bethel, CT 1: 586-592
155
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
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.
_________________________________Duane EllifrittProfessor of Civil Engineering
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.
_________________________________John LybasAssociate Professor of Civil Engineering
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.
_________________________________Fernando E. FagundoProfessor of Civil Engineering
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.
_________________________________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