BRIDGE DAMAGE DETECTION USING A SYSTEM IDENTIFICATION METHOD
By
WIRAT LERTPAITOONPAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000
Copyright 2000
by
Wirat Lertpaitoonpan
The author dedicates this dissertation to his parents, Among and Roongrot
Lertpaitoonpan
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 0. 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.
TABLE OF CONTENTS
page
A C K N O W L E D G M E N T S ..................................................................................................iv
LIST OF TABLES .......................................... .................................... viii
LIST OF FIGURES ............................. ............ ............................. ix
A B S T R A C T ...................................................................................................................... x v
1 IN TR O D U C T IO N .............. ................................................. .......... .. .......... ..... 1
1 .1 B ack g ro u n d ...................................................................................................... 1
1.2 O objective of D issertation ................. ...........................................................3......
1.3 Literature Review ........ .............................................3
1.4 O organization of D issertation ..................................... ...................... ............... 8
2 MODELING OF BRIDGE STRUCTURES.......................................................... 10
2 .1 In tro d u ctio n ........................................................................................................... 1 0
2.2 Tw oD im ensional M odel ....................................... ........................ .............. 11
2.3 ThreeD im ensional M odel ...................................... ....................... .............. 14
2 .3.1 M odeling of G irders ..................................... ....................... ............. 15
2.3.2 M odeling of Slabs ................................................................... 17
2.3.3 M odeling of D iaphragm s.................. ................................................. 18
2.3.4 M odeling of Secondary Structures ......................................... .............. 19
2.3.5 M odeling of Supports .................................... ...................... .............. 19
2 .3.6 M odeling of B ridges .............................................................. ................ 20
2.4 AASHTO LRFD Live Load Specification ......................................................23
2.4.1 D esign Truck ................ ............. ............................................ 23
2.4.2 D design Tandem ......................................................................... ............ 24
2.4.3 D esign L ane L oad ................. ......... .......... ..... ............... 25
2.4.4 Application of Design Vehicular Live Loads....................................... 25
2.5 TwoDimensional Live Load Modeling ............... ....................................26
2.5.1 L ongitudinal D distribution ...................................................... ................ 26
2.5.2 Transverse D distribution ......................................................... ................ 27
2.6 Comparing Models .......................................................................... 29
2.6.1 L oad C ase 1 (tw o trucks) ....................................................... ................ 30
2.6.2 L oad C ase 2 (one truck)......................................................... ............... 3 1
3 OVERVIEW OF OPTIMIZATION TECHNIQUES................................................ 33
3 .1 In tro d u ctio n ........................................................................................................... 3 3
3 .2 B asic C on cept ........................................................................... ...... . ... ............ 34
3.3 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 D direction ... .. .................. ............................................... 45
3.4.2 Finding Step L ength...................................... ...................... ............... 59
3.4.3 C condition of C onvergence..................................................... ............... 65
4 BRIDGE DAMAGE DETECTION........................................................................69
4 .1 In tro d u ctio n ........................................................................................................... 6 9
4 .2 B asic C o n cep t ....................................................................................................... 7 0
4.3 Characteristics of B ridge Structure.................................................. ............... 72
4.3.1 Eigenvalues and Eigenvectors ............................................... ................ 72
4.3.2 R itz V ectors ..................................................................... ............... 75
4.3.3 Observed Characteristics ................... ................77
4.3.4 Simulating the Observed Characteristics .............................................. 77
4.3.5 M odel Characteristics ......................................................... 80
4.4 Finite Element Model of Bridge Structure ......................................................80
4.5 Finite Elem ent A analysis Program .................................................... ............... 81
4 .6 O ptim ization R outine .......................................... ......................... ............... 82
4.7 Damage Detection Routine..................... ............... 83
4.8 Parameters of Damage Detection Routine...................................................... 83
4 .8.1 O objective F unction ....................................... ....................... .............. 83
4 .8.2 D esign V ariables.................................................................. ................. 86
4.9 Improvement of the Damage Detection Routine.............................................88
4.10 Damage Detection Testing on Structural Elements ................. ..................... 89
4.10.1 T est of Spring E lem ent ........................................................ ............... 91
4.10.2 Test of Truss Element................ ............... 93
4.10.3 T est of B eam E lem ent ......................................................... ............... 95
4.10.4 T est of Shell E lem ent .......................................................... ............... 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
vi
4.11.3 Weakening of Material Properties in Slabs.................. ................. 115
4.11.4 D am age in Supports...... ............ ............ .................... 120
4.11.5 Cracking of bridge girder...... ........ ...... .................... 125
5 PARA M ETRIC STU D Y ................................................................ .............. 131
5 .1 In tro d u ctio n ......................................................................................................... 13 1
5.2 D election Techniques .............................................................. .............. 132
5.3 M agnitude of Perturbations ...... .............. .............. .................... 133
5 .4 R e sp o n se s ........................................................................................................... 13 7
5.5 Objective Functions .................................................................. 140
5 .6 N o ise ............................................................................................................... . .. 1 4 3
6 CONCLUSION ........................................... .............................. 149
REFERENCES ....................................................... ............................ 152
BIOGRAPH ICAL SKETCH ................................................................ .............. 155
LIST OF TABLES
Table page
2.1 R results of Load C ase 1 .... .. ................................. .......................... .............. 31
2.2 R results of L oad C ase 2 .. .................................................................. .............. 31
5 .1 N o ise E ffect............................................................................................................. 14 4
LIST OF FIGURES
Figure page
2.1 TwoDimensional Beam Element and Its Degrees of Freedom .............................. 11
2.2 Longitudinal of Bridge Structure 2D M odeling .................................... .............. 12
2.3 Transverse D direction of Bridge Structure............................................... .............. 12
2.4 Sim ple Supports B etw een G irders.......................................................... .............. 13
2.5 Continuous Slab and Hinge on top of Girders ............... ................................... 13
2.6 Continuous Slab and Fram e A action ........................................................ .............. 14
2.7 ThreeDimensional Beam Element and Its Degree of Freedoms. ........................... 15
2.8 ThreeDimensional Truss Element and Its Degree of Freedoms............................. 16
2.9 ThreeDimensional Girder Model with Rigid End Offset.................................... 16
2.10 FourNode Shell Element and Its Degree of Freedom........................................ 17
2.11 Slab M odel w ith Rigid End O ffset ....................................................... .............. 18
2.12 GirderSlab Bridge Structures with Diaphragms and Parapets...............................21
2.13 GirderSlab Bridge Model with Diaphragms and Parapets .................................22
2.14 LRFD Design Truck ................................... ............................... 24
2.15 LR FD D esign T andem ...................................................................... ................ 25
2.16 Longitudinal Load Distribution in 2D M odel .................................... ................ 27
2.17 M oving Load in Transverse D irection................................................. ............... 28
2.18 Load Distribution on Simple Supports between Girders ................ ..................... 28
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
2.2 1 L ongitudinal L oad P 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 KuhnTucker 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 FletcherReeves Conjugate Gradient Optimization Procedures.......... 52
3.8 Example of Movement of Conjugate Gradient Optimization................................53
3.9 Flow Chart of QuasiNewton Optimization Procedures.......................................58
3.10 Flow Chart of Bracket Onedimensional search Method .................................... 60
3.11 Flow Chart of Golden Section Onedimensional 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 E lem ent M odel ..................................... ...................... ............... 91
4.7 Damaged Indicators of Simulated Damaged Spring.............................. ................ 92
4.8 Predicted Damaged Indicators from Damage Detection Program........................... 92
4.9 T testing Truss E lem ent M odel................................................................ ............... 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 B eam E lem ent M odel ............................................................. ................ 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 E lem ent M odel ..................................... ...................... ................ 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
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
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 M agnitude of Perturbation..................................... ......................... .............. 135
5.3 Damage Indicators of Simulated Damage Structure...... ................. ................. 136
5.4 Damage Indicators of Predicted Damage from 10% Perturbation Rate ................ 136
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
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BRIDGE DAMAGE DETECTION USING A SYSTEM IDENTIFICATION METHOD
By
Wirat Lertpaitoonpan
May 2000
Chairman: Marc I. Hoit
Major 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 xray,
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
identification methods to detect damage in bridge structures using the information of the
change in structural characteristics. A threedimensional 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.
CHAPTER 1
INTRODUCTION
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 shipimpact. Damage
can be denoted by cracking in the structure, corrosion, deterioration of material
properties, or loss of prestressing. 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.
Because the cost of largescale 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 Xrays, 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 structure
2. Detect the location of the damage
3. Detect the extent of the damages
4. 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
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 indepth 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
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 (2D) 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 fullscale reinforced concrete bridge span. A ninespan threelane
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
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
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 nondestructive 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 mediumsized 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
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 noisepolluted.
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.
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 twodimensional model, and then presents a more complex threedimensional
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.
CHAPTER 2
MODELING 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
longterm 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 2dimensional finite element model (2D FEM),
and then describes the more complex 3dimensional model (3D 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.
2.2 TwoDimensional Model
A twodimensional model of bridge structures is the most common simple model,
but its accuracy is in doubt. In the twodimensional model, the structures are simply
modeled by twodimensional 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 2D beam
element is shown in Figure 2.1.
Figure 2.1 TwoDimensional Beam Element and Its Degrees of Freedom.
Bridge structures are in reality threedimensional 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 2D 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.
Slab
SGirder
\i xPier
(a) Longtitudinal Direction of Bridge Structure Model.
Y 2D Beam Elements
(b) Longtitudinal Direction of Bridge Structure 2D FEM.
Figure 2.2 Longitudinal of Bridge Structure 2D Modeling.
Bridge Slab
Bridge Girder
Pier Cap
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.
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.
Bridge Hing
Bridge I
Pier Cap 
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 Girder 
Pier Cap  ,
Figure 2.6 Continuous Slab and Frame Action.
The 2D beam elements are the only element that is needed in order to use the 2D
bridge structure models, which is very simple compared to the 3D models.
2.3 ThreeDimensional Model
Threedimensional (3D) models are more complicated models and need more
resources for analysis. In the 3D 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 3D models is more accurate than the 2D
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.
2.3.1 Modeling of Girders
The bridge girders are modeled with 3D 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 crosssectional properties
(modulus of elasticity, sectional area, and moment of inertia) as well as its geometry. The
degrees of freedom of a 3D beam element are shown in Figure 2.7.
Figure 2.7 ThreeDimensional 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 3D 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 rigidend
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 3D truss element are shown in Figure 2.8. Figure 2.9
shows a typical prestressed concrete girder modeling with the rigid end offset.
Vy
Figure 2.8 ThreeDimensional Truss Element and Its Degree of Freedoms.
Girder
Centroid
Steel
Centroid
Rigid End
Offset
(a) Cross Section of Girder with Rigid End Offset
Beam
Element
Truss
Element
Rigid End
Offset
(b) Finite Element Model of Girder with Rigid End Offset
Figure 2.9 ThreeDimensional Girder Model with Rigid End Offset.
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 inplane 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 inplane 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 fournode shell element are
shown in Figure 2.10.
Z 2X
Figure 2.10 FourNode Shell Element and Its Degree of Freedom.
If there is reinforcing steel or prestressed tendons in the slab, the slab will have
attached 3D truss elements in a similar fashion to the girders. Figure 2.11 shows the slab
modeling with the rigid end offset.
Slab
Centroid
Steel 
Centroid
(a) Cross Section of Slab with Rigid End Offset.
(b) Finite Element Model of Slab with 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 End
___Offset
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 3D 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
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, fournode shell elements are
created as a grid over the bridge. The girders then are added to the model by using 3D
beam element with rigidend 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 3D 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.
Slab
Span
  Length
(a) Plan View of the Bridge Structures.
Deck slab Parapet
Support Girder
Diaphragm
(b) CrossSection View of the Bridge Structures.
Figure 2.12 GirderSlab Bridge Structures with Diaphragms and Parapets.
1 p p 1 1 i 1 i1
(a) Overall View of the Bridge Structures Model.
Beam element (diaphragm)
Beam element (parapet)
Shell element (slab)
Beam rigid link
Spring element (support) Beam element (girder)
(b) CrossSection View of the Bridge Structures Model.
Figure 2.13 GirderSlab Bridge Model with Diaphragms and Parapets.
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 HL93 (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 HS20 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
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
8.0 kip 32.0 kip 32.0 1
14 ft 14 to 30 ft
(a) Side View of the Design Truck.
(b) Rear View of the Design Truck.
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
25.0 kip 25.0 kip
(a) Side View of the Design Tandem.
(b) Rear View of the Design Tandem.
Figure 2.15 LRFD Design Tandem.
2.4.3 Design Lane Load
The design lane load in AASHTO specification is intended to simulate the load
condition due to traffic congestion on the bridge. The design lane load consists of a load
of 0.64 klf, uniformly distributed in the longitudinal direction. Transversely, the design
lane load is assumed to be uniformly distributed over a 10.0 ft width on the deck.
2.4.4 Application of Design Vehicular Live Loads
The AASHTO LRFD specification provides not only the standard loading but also
the requirement of load position to produce the maximum force effect for design. As
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 the
effect of the design lane load.
2. The effect of one design truck with variable axle spacing specified in Figure
2.14 combined with the effect of the design lane load.
3. For both negative moment between points of dead load contraflexure, and
reaction at the interior piers, only 90% of the effect of two design trucks spaced a
minimum of 50 ft between the lead axle of one truck and rear axle of the other
truck, combined with 90% of the effect of the design lane load; the distance
between 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 TwoDimensional Live Load Modeling
Finding the load distribution from the superstructures onto the substructures can
not be done accurately in a single 2D model. A designer has to separate the longitudinal
distribution and transverse distribution into two separated models. Each direction is
analyzed using a twodimensional 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
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.
"............. ........ .
\y ... ...... ... \........... \...J
(a) Longitudinal Moving Loads.
32 kip 32 kip
8 kip
i .
IRt
(b) Simplified Load and Structure Model.
Figure 2.16 Longitudinal Load Distribution in 2D 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
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.
....Bridge Slab...................................... ................
Bridge Girder
Pier Cap
_ ..1..
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:
Rt/2
Rt/2 Rt/2 R2Rt/
Figure 2.18 Load Distribution on Simple Supports between Girders.
i
m
I

Rt/2 Rt/2 Rt/2 Rt/2 Rt/2 Rt/2
Bridge Slab 6 ft. ...... .. ...... ft.
t ... .... .... ... ..........
Bridge Girder N Hinges
Pier Cap
Figure 2.19 Load Distribution on Continuous Slab and Hinge at Top of Girders.

Rt/2 Rt/2 Rt/2 Rt/2 Rt/2 Rt/2
Bridge Slab 6 ft. 6ft. 6 ft.
Bridge Girder 
Pier Cap 1
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
2D model and the more complex 3D model. The 2D model is easier to analyze, but the
accuracy is doubtful when compared to the 3D model. The test of live load distribution
from bridge superstructures into bridge substructures was performed to compare the
results of 2D model to 3D model. The structural models of the bridge were created
using both 2D and 3D modeling techniques. Both models were analyzed with the bridge
structures subjected to several load cases. The 2D model was analyzed by using SSTAN
(Static Structural Analysis) written by Hoit (1995) and the 3D 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.
32.0 K 32.0]
S++ I
S100 ft.
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.
R,/2 R,/2 R,/2 Rt/2
6 ft. 4 ft. J, 6ft.
Bridge Slab
Bridge Girder
Pier Cap bm
Figure 2.22 Transverse Load Position of Load Case 1.
7t 
.JN
100 ft. 7r
Table 2.1 Results of Load Case 1.
2D % Forced Difference from the 3D Model
Model 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.
Rt/2 Rt/2
Bridge Sftlab
Bridge Slab ^ r
Pier Cap
Figure 2.23 Transverse Load Position of Load Case 2.
Table 2.2 Results of Load Case 2.
2D % Forced Difference from the 3D Model
Model 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 2D models analysis provides uncertain results. For the full span load, the
results of 2D model analysis are not much different than the 3D model analysis. If the
loads do not cover the whole span, the results of 2D model analysis are a lot different
than the 3D model. With the uncertain accuracy, using 2D 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 3D finite element model is chosen to be a model of
bridge structure in this study.
CHAPTER 3
OVERVIEW 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 best
policies to satisfy certain objective, at the same time satisfying fixed requirements.
It would be presumptuous at this time to suggest that optimization has attained the
status of science rather than art; however, recent advances in applied mathematics,
operations research, and digitalcomputer technology enable many complex
industrial problems in engineering and economics to be optimized successfully by
application 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
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, (3.3)
XLi < Xi xUi i = 1, n (3.4)
Where X is a vector of design variables
X,
X2
X3
X=
xn
gj(X) is an inequality constraint
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
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 KuhnTucker conditions are
met (Haftka and Gurdal 1996). The optimization procedure using numerical search
techniques can be written in numerical form as:
Xq= xql+S'q (3.9)
Where Xq is an updated vector of design variables.
Xq1 is current vector of design variables.
a is a scalar value of step length (move parameter).
S' 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 XX* < 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.
F(X)
A
B
E
X
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 KuhnTucker conditions are simply where the gradient of objective function
vanishes (Vanderplaats 1999).
VF(X) = 0
(3.12)
where
VF(X)=
aF(X)
axi
aF(X)
aX2
aF(X)
aX3
aF(X)
ax
The KuhnTucker 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
KuhnTucker 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:
d2F(X)
a2F(X)
025025
a2F(X)
a2F(X)
X222
a2F(X)
iX Xi
D2F(X)
a2F(X)
S2F(X)
aX TaX
(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:
AXTH(X*)AX > 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 (VF(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:
ng nh
L(X,X ) = F(X) + J jg (X) + A +khk (X) (3.15)
y=l k=1
where k 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.
Assuming the objective function (F(X)), and all constraints functions (gj(X) and
hk(X)) are differentiable. The KuhnTucker 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 j and gj(X*) must be zero:
Sgj(X*) = 0 j = 1, ng (3.17)
3. The gradient of the Lagrangian function must vanish:
VL(X*, 1) = 0
VF(X*)+ Ij1VgJ(X*)+ ,ng+kVhk(X*)= 0 (3.18)
j=1 k=1
j 0 j = 1, ng
gJ (X)
aX1
agJ (X)
aX2
Bg (X)
where Vgi (X) = (X)
gX3
ag (X)
42
The physical meaning of the KuhnTucker conditions can be shown in two
dimensional space as Figure 3.2.
VF(X) + ,V Vg, (X) + j {Vg, (X) = 0
Feasible domain
VF /
feasible domain
F=k
Sg2= 0
Vg2P Vgi
1 r X,
Figure 3.2 Geometrical Interpretation of KuhnTucker 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.
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 < XUi 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 = Xql+S (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 useablefeasible search direction, S.
2. Compute the scalar of step length or move parameter, ca.
3. Determine if the problem has converged to an acceptable solution.
Figure 3.3 Flow Chart of Unconstrained Optimization Procedures.
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.
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
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 FletcherReeves Conjugate Gradient Method are wellknown
techniques among the first order methods. This section will present the procedures of
these two methods.
3.4.1.2.1 Steepest Decent Method
The 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.
S" = VF(X"') (3.20)
After the search direction is found, the onedimensional 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.
Figure 3.5 Flow Chart of Steepest Decent Optimization Procedures.
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
hemstitchingg" 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.
X2
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 rescaling the design
variables. Unfortunately, the procedure to rescale the large problem requires a lot of
work including calculation of the Hessian matrix and an eigenvalue analysis.
3.4.1.2.2 FletcherReeves Conjugate Gradient Method
The FletcherReeves 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).
(S H(SJ) = 0 (3.21)
Where S' and Si are the search directions.
H is the Hessian Matrix.
The FletcherReeves 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 = VF(Xq)+qSq1 (3.22)
~Where VF(Xq1 2
Where f' = 2. (3.23)
VF(Xq2)2
The FletcherReeves 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 FletcherReeves conjugate gradient method
needs to be restarted periodically to ensure a good optimization. The flow chart of the
FletcherReeves conjugate gradient method is presented in the Figure 3.7.
The FletcherReeves 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 FletcherReeves conjugate gradient
movement pattern compared to the steepest decent is shown in the Figure 3.8.
Even though the FletcherReeves conjugate method offers a significant
improvement over the steepest decent method, the performance of this method lags
behind the second order search methods.
_< Yes
No
Yes
SStop )
Figure 3.7 Flow Chart of FletcherReeves Conjugate Gradient Optimization Procedures.
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.
F(X + A) = F(Xq)+ VF(Xq)Tr X + ] T' [H(Xq)]5X (3.24)
2
where 8X=Xq+lXq. (3.25)
The correction AX is defined such that the derivative of the objective function
(3.24) vanishes. Then the correction AX can be written as:
S= [H(Xq) VF(Xq) (3.26)
Rewriting the equation (3.25) for Xq+1 and substituting equation (3.26) into
equation (3.27) yields.
Xq+l= Xq + X (3.27)
Xq+' = Xq [H(Xq)]'VF(Xq) (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 (a =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 = Xql+Sq (3.29)
(3.30)
Sq = [H(Xq) VF(Xq)
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 QuasiNewton Method
The quasiNewton 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 P. The
quasiNewton method also uses the information of previous iterations. Instead of the very
last iteration, the quasiNewton method keeps all the previous information in a matrix
form. This method is sometimes called the variable metric method. The quasiNewton
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).
VF(Xq+l) = VF(Xq)+ [G(Xq)kX (3.31)
Assuming G(Xq) as an approximated Hessian matrix (H(Xq)) at the qth iteration.
The equation (3.31) can be rewritten as:
VF(Xql) VF(Xq) = [G(Xq)kX (3.32)
and
B(Xq+I )[VF(Xq+) VF(Xq) = =X (3.33)
where B(Xq+l) 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+yq = pq (3.35)
where pq = 8X = Xq+Xq.
yq = VF(Xq+l) VF(Xq)
Equation (3.35) is called the quasiNewton or secant relation (Halfka and Gurdal.
1996). This equation condition must be satisfied in order to update the matrix Gq or Bq.
The quasiNewton method procedure starts by assigning the identity matrix to the
approximated inverse Hessian matrix (B = I) and computes the search direction from the
following equation.
Sq = B'q VF(X'q) (3.36)
The first search direction is indeed the search direction of steepest decent. Having
the search direction, the onedimensional 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
update the B' matrix and satisfy the quasiNewton condition is by adding a symmetric
matrix to the previous B' as shown in the following equation.
Bq+ = B'q + E'q (3.37)
E' 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 quasiNewton
methods.
Eq = +OppI + Bqyq(Bqyq)T [BqyqpT + p(Bqyq)T] (3.38)
where y = (pq)Tyq.
= (yq)T Lq yq
There are two popular quasiNewton methods that are based on the equation
(3.38): the BroydonFletcherGoldfarbShanno (BFGS) method and the Davidon
FletcherPowell (DFP) method. The parameter 0 in the equation (3.38) determines the
two methods.
If 0=0, then E' in the equation (3.38) results in the DFP method. If 0=1, then the
E' 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 quasiNewton 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 quasiNewton method is shown in Figure 3.8.
Figure 3.9 Flow Chart of QuasiNewton Optimization Procedures.
3.4.2 Finding Step Length
The procedure of finding the step length for each search direction is sometimes
called onedimensional 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 ct, 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
onedimensional search. The necessary condition for the minimum of the objective
function for onedimensional search is the vanishing of the first derivative.
dF(a) 0 (3.39)
da
Many techniques are developed for onedimensional 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
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 Onedimensional search Method.
Where (3 is the step size of movement.
p>1.o
0 < < 1/0
0< p < 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 onedimensional
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(x) is unimodal in the interval of I if there is an a* that minimizes the F(c)
in the interval 1 and for any two points .a, ab in the interval 1 where cDa < if ab < K*
then F(cb) < F(aDa) and if ua > a* then F(maa) < F(ab). 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, aL and
aU. 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, (a and ab have specific proportion conditions of symmetry about the center of the
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.
Xa b
au b = a L
a' _L ab _ac
au _aL au _a
(3.40)
(3.41)
(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 ((U and ,L)
can be normalized to one and zero respectively. After the normalization, equation (3.42)
can be written as:
ab a
aa = 
aa
1 a"
from (3.41) ab 1 (a
(3.43)
(3.44)
substituting (3.44) into (3.43) givves
a 1 2a
a 1a
1 a"
(3.45)
rearranging the form in (3.45)
((a)2 30D + 1 = 0
(3.46)
solving the quadratic equation (3.46) for the roots of ca gives
3 5
a = _ = 0.381966,2.618034
2
(3.47)
There are two roots in equation (3.47) but one of them is higher than the upper
bound (cU=1.0) so that there is only one feasible solution:
a = 0.381966
(3.48)
back substituting ca of (3.48) into (3.44)
ab= 10.381966 = 0.618034
(3.48)
dividing (3.48) by (3.47) gives
(3.49)
ab
=1.618034
ac
The ratio in equation (3.49) is called golden section number. This number also has
other special conditions that are:
aa b
a 0.618034 =ab = 
ab a0
(3.50)
a (aby
(3.51)
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 (aLu L) and specifies the
relative tolerance (e) and the number of function evaluation (N).
e (3.52)
By specifying the desired total tolerance, Ac, 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.
c (,a=ab L= 1 (3.53)
where T= 0a= 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.
S(l 3) (3.54)
where N is the maximum number of function evaluations, including the first three
initial evaluations (F(L), F((ou), and F(oa)).
Solving the equation (3.54) for N:
N = ln( +3=2.0781n(e)+3 (3.55)
In(1 r)
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 ofr as follows:
a= =(1T) +L + e (3.56)
ob = (C) (L + (l) (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 KhunTucker
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 (EK). 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.
Figure 3.11 Flow Chart of Golden Section Onedimensional search Method.
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
F(X) )F(Xql)
Relative change
F(Xq) F(Xq' ) R (3.59)
F(Xq)
where EA is tolerance of absolute change.
ER 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.
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
onedimensional search technique.
CHAPTER 4
BRIDGE 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 xrays, 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.
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.
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.
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 vibrationgoverning equation is given by:
M X(t) + C X(t) + K X(t) = 0
(4.1)
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) = DeXt (4.2)
where D is a constant vector.
) is a scalar value.
substituting equation (4.2) into (4.1) yields.
(MV2 + C + K = 0 (4.3)
The nontrivial solution of the equation (4.3) yields to complex conjugate pairs of
eigenvalues (ki) and eigenvectors (0Qi). The eigenvalues are computed from the following
equation.
S= '+i(, 1 2 (4.4)
where
c
is a damping ratio, c = .
2Vkm
Co is the circular frequency, o( = m .
I m
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.
MX(t)+KX(t) = 0
(4.5)
The general solution of the equation (4.5) can be written in harmonic form as:
X(t) = ( sin(cot)
(4.6)
and so
X(t) = o2 X(t)
(4.7)
substituting equation (4.7) into (4.5) yields
KcTw )2 M(I) =
(4.8)
The nontrivial solution of the homogeneous equation (4.8) has to satisfy the
condition for the characteristic determinant to vanish:
K _2M = 0
(4.9)
Solving the equation (4.9) yields 'n' values of co which are the eigenvalues of the
problem. And the corresponding eigenvectors can be computed by back substituting the
eigenvalues into equation (4.8).
where %i = COl2
Natural frequency of a structure
27r
D = 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:
M X(t) + C X(t) + K X(t)= F (4.10)
where F is the vector of applied forces.
The first mode of Ritz vector is computed from the static displacement as follows:
R1 = K1'F (4.11)
where RI* is the first mode of Ritz vector.
Fs is the applied static loads.
Mass normalizing the first Ritz vector such that
R1 R
(R, )TMv(R, )
where Ri is the mass normalized Ritz vector.
The successive mode of Ritz vectors can be computed from the previous Ritz
vector as follows:
R* = K1MR,1 (4.13)
The new Ritz vector has to be orthogonal to the previous Ritz vectors and mass
normalized as shown in the following formulas.
11
R =R* 1 (RMR (4.14)
J=1
R MR =1 (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.
1 u1 CI, IanfuuLi Accelerometer
I Investigated Structure
Signal Conditioning ,
Antialiasing Filter
ADCs
Computer
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
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 crosssectional 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
normal distribution with zero mean value. The simulated noise is imposed into the
damaged responses using the following formula:
S* = + 0 (7/100) random (1,1) (4.16)
where V* is the response data with noise.
c is the response data without noise.
y 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
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.
(a) Overall View of the Bridge Structures Model.
Beai
H
m element (diaphragm) Beam element (parapet)
Shell element (slab)
Beam rigid link
Spring element (support) Beam element (girder)
(b) CrossSection View of the Bridge Structures Model.
Figure 4.3 General GirderSlab 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
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 = (o Dm
(4.17)
Figure 4.4 Flow Chart of General Procedure of Damage Detection Routine.
