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Studies into data-driven approaches for nonlinear system identification, condition assessment, and health monitoring

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Studies into data-driven approaches for nonlinear system identification, condition assessment, and health monitoring

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Content Studies into Data-Driven Approaches for Nonlinear System
Identification, Condition Assessment, and Health Monitoring
by
Armen Derkevorkian
ADissertationPresentedtothe
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(Civil Engineering)
May 2014
Copyright 2014 Armen Derkevorkian
Dedicated to my parents.
My father, for inspiring me with what you left behind.
My mother, for your heroic e↵orts to help me achieve my goals.
ii
Acknowledgements
First and foremost, I would like to thank my dear advisor, Prof Sami F Masri, for
his invaluable guidance throughout my PhD studies. His consistent positive impact on
my academic, scientific, and professional development has been tremendous. I am ex-
tremely grateful for his strong support, valuable advice, and continuous encouragement
throughout my studies.
I would like to thank the USC Viterbi School of Engineering for the generous sup-
port through Viterbi Dean’s Doctoral Fellowship, the USC Sonny Astani Department
of Civil and Environmental Engineering for the support through teaching and research
assistantships, the NASA Dryden Flight Research Center, and the NASA University Re-
search Center (URC) at the California State University Los Angeles for their support. I
would like to thank my qualification and oral exam committee members, Prof L. Carter
Wellford, Prof Petros Ioannou, Prof Burcin Becerik-Gerber, and Prof Jiin-Jen Lee, for
their valuable advice and feedback throughout the process. I would like to thank Prof
Helen Ryaciotaki-Boussalis and Jessica Alvarenga at the NASA URC Space Center, Dr
LanceRichards,JohnBakalyar,andFranciscoPenaattheNASADrydenFlightResearch
Center, ProfHae-BumYunattheUniversityofCentralFlorida, ProfPeizhenLiatTongji
Univerity, Prof Yozo Fujino and Dr Dionysius Siringoringo at the University of Tokyo, for
valuable collaborations on various projects. I would like to thank the team members of
the structural health monitoring research group at USC and my ocemates, Dr Miguel
iii
Ricardo Hernandez-Garcia, Dr Mohammad-Reza Jahanshahi, Dr Reza Jafarkhani, Dr Ali
Bolourchi, and Dr Vahid Keshavarzzadeh, for providing a unique working experience.
iv
Table of Contents
Abstract xiv
I Introduction 1
I.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
II Nonlinear Data-Driven Computational Models for Response Prediction
and Change Detection 8
II.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
II.1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
II.2 Shaking-Table Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
II.2.1 Description of the Models . . . . . . . . . . . . . . . . . . . . . . . 12
II.2.2 Instrumentation and Measurement Locations . . . . . . . . . . . . 14
II.2.3 Loading Schedule and Network/Data ID’s . . . . . . . . . . . . . . 15
II.2.4 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II.3 Computational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II.3.1 Input-Output Configuration of the Neural Nets . . . . . . . . . . . 17
II.3.2 Ordinary Di↵erential Equation (ODE) Solver . . . . . . . . . . . . 19
II.3.3 Global Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II.3.4 Subsystem Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 24
II.3.5 Testing and Validation . . . . . . . . . . . . . . . . . . . . . . . . . 26
II.4 Change-Detection in the SFSI Systems . . . . . . . . . . . . . . . . . . . . 34
II.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
II.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
II.4.3 Visual Observations During and After the Tests. . . . . . . . . . . 40
II.4.4 Comparison of the Detection Results to Changes in Natural Fre-
quencies in the Pile Foundation (SPSI) System . . . . . . . . . . . 43
II.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
II.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
II.7 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
IIIDevelopmentandValidationofNonlinearComputationalModelsofDis-
persed Structures Under Strong Earthquake Excitation 50
III.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
III.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
III.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
III.1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
III.2 Yokohama-Bay Bridge (YBB) . . . . . . . . . . . . . . . . . . . . . . . . . 54
III.2.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
III.2.2 Sensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
III.2.3 Natural Frequencies Based on Ambient Measurements . . . . . . . 57
III.3 Great East Japan 2011 Earthquake . . . . . . . . . . . . . . . . . . . . . . 58
III.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
III.3.2 Earthquake Response of Yokohama-Bay Bridge . . . . . . . . . . . 58
III.3.3 Earthquake Response Data Sets . . . . . . . . . . . . . . . . . . . 59
III.4 System Identification Studies . . . . . . . . . . . . . . . . . . . . . . . . . 60
III.4.1 Overview of Some System Identification Methods . . . . . . . . . . 60
III.4.2 System Identification Results Based on Equivalent-Linear System . 63
III.4.3 Detection and Quantification of Nonlinearities in Response . . . . 64
III.5 Development Of Reduced-Order Nonlinear Computational Models . . . . 73
III.5.1 Equivalent Linear Model . . . . . . . . . . . . . . . . . . . . . . . . 73
III.5.2 Noninear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
III.6 Validation Of Reduced-Order Nonlinear Computational Models . . . . . . 77
III.6.1 OverviewofTime-MarchingApproachesforDynamicResponseCal-
culations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
III.6.2 Assessing the Accuracy of Linear vs Nonlinear Dynamic Response
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
III.6.3 Dependance of Bridge Damping Parameters on Nature of Compu-
tational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
III.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
III.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
III.9 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
IVEvaluation of a Strain-Based Deformation Shape Estimation Algorithm
for Control and Monitoring Applications 89
IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
IV.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
IV.1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
IV.1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
IV.2 Experimental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
IV.2.1 Test Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
IV.2.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
IV.3 Computational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
vi
IV.3.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . 97
IV.3.2 Loading of the Models . . . . . . . . . . . . . . . . . . . . . . . . . 98
IV.4 Shape Detection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 98
IV.4.1 Fiber-Optic Strain Sensing (FOSS) Algorithm . . . . . . . . . . . . 100
IV.4.2 Modal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
IV.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
IV.5.1 FOSS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
IV.5.2 Modal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
IV.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
IV.7 Conclussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
IV.8 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
V Investigation of an Operating-Load Estimation Algorithm Using Fiber-
Optic Strain Sensors (FOSS) Technology 117
V.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
V.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
V.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
V.1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
V.2 Load-Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 120
V.3 Sensitivity Analyses on Analytical Moment Calculations for Calibration
Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
V.3.1 Numerical Results using Exact and Approximate Methods . . . . . 126
V.3.2 E↵ects of Uncertainty on the Moment Calculations . . . . . . . . . 129
V.4 Finite-Element (FEA) Analyses . . . . . . . . . . . . . . . . . . . . . . . . 131
V.4.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
V.4.2 Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
V.5 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
V.5.1 Calibration Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
V.5.2 Estimation Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
V.6 Design of The Experimental Test-bed Structure . . . . . . . . . . . . . . . 138
V.6.1 Description of the Experimental Test-bed . . . . . . . . . . . . . . 138
V.6.2 Description of the Computational Model . . . . . . . . . . . . . . . 144
V.6.3 Instrumentation and Test Apparatus . . . . . . . . . . . . . . . . . 145
V.7 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
V.8 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 152
V.9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
VIConclusions 155
BIBLIOGRAPHY 158
Appendices 168
vii
A Generalization of the Subsystem Modeling Approach on a Nonlinear
Finite-Element Model of a 3-Span Reinforced Concrete (RC) Bridge 169
A.1 Finite-Element Model Specification . . . . . . . . . . . . . . . . . . . . . . 169
A.2 Time-History Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.3 Application of the Subsystem Approach for Response Prediction . . . . . 172
B Alternative Analyses on the Yokohama-Bay Bridge (YBB) Datasets 177
B.1 Processed Time Histories from the Main-Shock (EQ1) dataset . . . . . . . 177
B.2 Statistical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2.1 Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2.2 Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . 188
C Sensitivity Analyses of the Modal Approach 197
C.1 FBG Sensor Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
C.2 Number of FBG Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
C.3 E↵ects of Measurement Noise . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.4 Type of Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
C.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
C.5.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
C.5.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
C.6 Conclussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
viii
List of Figures
II.1 Photograph of the large scale Soil-Foundation-Superstructure test model setup. 13
II.2 Distribution of the sensors on the model. . . . . . . . . . . . . . . . . . . . . 14
II.3 Response of sample channel A7 of the pile foundation model, due to excitation
EL1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II.4 Architecture of the neural network in the global approach. . . . . . . . . . . . 21
II.5 Flowchart of the computational model using the global approach. The integra-
tion of a single trained neural network with RK45 and the updating procedure
at each time-step is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
II.6 Flowchart of the computational model using the subsystem approach. The in-
tegration of multiple trained neural networks with RK45 and the updating pro-
cedure at each time-step is shown. . . . . . . . . . . . . . . . . . . . . . . . 25
II.7 The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channels A2 through A7 on
the SBSI system.The network was trained using level-1 EL-Centro data and the
base excitation from the same earthquake was fed back to the created model. . 28
II.8 The displacement and velocity predictions from the computational model super-
posed on the original corresponding measurements from channels A2 through
A7 on the SBSI system.The network was trained using level-1 EL-Centro data
and the base excitation from a di↵erent earthquake level-1 Kobe was fed back
to the created model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
II.9 The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channels A2 through A7 on
the SPSI system.The network was trained using level-6 Kobe data and the base
excitation from the same earthquake was fed back to the created model. . . . 30
II.10 The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channelsA2 throughA7 on the
SBSI system.The networks were trained using level-1 EL-Centro data and the
base excitation from the same earthquake was fed back to the created model. . 32
II.11 The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channels A2 through A7 on
the SBSI system.The networks were trained using level-1 EL-Centro data and
the base excitation from a di↵erent earthquake level-1 Kobe was fed back to the
created model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
ix
II.12 The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channelsA2 throughA7 on the
SPSI system.The networks were trained using level-6 Kobe data and the base
excitation from the same earthquake was fed back to the created model. . . . 35
II.13 The displacement and velocity predictions from the computational model su-
perposed on the original corresponding signals from channels A2 through A7
on the SPSI system.The networks were trained using level-6 Kobe data and the
baseexcitationfromadi↵erentearthquake level-6 El-Centro wasfedbacktothe
created model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
II.14 The flowchart of the identification-detection framework. The network is trained
first during the identification step (left-part). New data is fed to the trained
network during the detection step (right-part). . . . . . . . . . . . . . . . . . 37
II.15 The detected changes in the SFSI systems quantified by the average normalized
error measure. Each subfigure corresponds to a di↵erent SFSI system. For each
system, results from three di↵erent networks are shown. . . . . . . . . . . . 39
II.16 The observed vertical cracks on the piles after all the tests. . . . . . . . . . . 43
II.17 Comparison of the normalized change results from the neural network detection
and the reported natural frequencies in the pile foundation model. . . . . . . . 44
III.1 View of the Yokohama-Bay Bridge. . . . . . . . . . . . . . . . . . . . . . 55
III.2 Location of the Yokohama-Bay Bridge (Source: Google). . . . . . . . . . 55
III.3 Sensor map on Yokohama Brige. . . . . . . . . . . . . . . . . . . . . . . . . 56
III.4 Sample time histories and corresponding FFT plots corresponding to EQ1. . . 61
III.5 In the top figure, the measured channel S5R - lateral (Y) acceleration is shown.
Belowthisisthelinearmodelestimateusingleast-squaremethod. Thethirdfig-
ure shows the non-linear residual (i.e., the di↵erence of the previous two signals.
The main shock (EQ1) dataset is used.. . . . . . . . . . . . . . . . . . . . . 70
III.6 In the top figure, the measured channel S5R - lateral (Y) acceleration is shown.
Belowthisisthelinearmodelestimateusingleast-squaremethod. Thethirdfig-
ure shows the non-linear residual (i.e., the di↵erence of the previous two signals.
The first aftershock (EQ2) dataset is used. . . . . . . . . . . . . . . . . . . . 71
III.7 Identifiedfundamentalnaturalfrequenciesanddampingratiosfromall10datasets. 72
III.8 Architecture of the neural network used in this study. . . . . . . . . . . . . . 75
III.9 The neural network estimates of the nonlinear residual. The first row represents
the entire record of the nonlinear residual from channel S5R(Y) superposed on
the corresponding neural network estimate. The second row shows the errore(t)
plot. The third row shows a segment of 5 sec. from the same record. . . . . . 76
III.10 Flowchart of the developed nonlinear computational model. . . . . . . . . . . 78
x
III.11 Displacementtimehistoryestimatesfromthecomputationalmodel(dottedline)
superposed on the corresponding displacements from the measured data (solid
line). The rows correspond to channels S4(Y) and S7(Y), respectively. The
computational model is created using EQ1 and is tested using EQ1. A segment
of 100 sec. is shown from the strong motion part of the records. . . . . . . . . 80
III.12 Displacementtimehistoryestimatesfromthecomputationalmodel(dottedline)
superposed on the corresponding displacements from the measured data (solid
line). TherowscorrespondtochannelsS4(Y)andS7(Y),respectively. Thecom-
putational model is created using EQ1 and is validated using EQ2. A segment
of 100 sec. is shown from the strong motion part of the records. . . . . . . . . 82
III.13 A segment of the base excitation used for the identification of the damping ratio. 83
III.14 A segment of the displacement time history estimate corresponding to channel
S5R(Y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
IV.1 Photosofthesweptplateexperimentwiththreelinesoffiberopticstrainsensors
placed on the top surface. (Property of NASA Dryden Flight Research Center) 96
IV.2 Description of the three loading cases. The loading values are in (kg). . . . . . 99
IV.3 The flowchart of the FOSS approach. . . . . . . . . . . . . . . . . . . . . . . 102
IV.4 The flowchart of the modal approach. . . . . . . . . . . . . . . . . . . . . . 104
IV.5 The optimized locations of the sensors along with the vibration mode shapes. . 105
IV.6 The estimated (FOSS) displacements along the two fiber lines for all three load
cases, compared to the reference photogrammetry results. Each row represents
a di↵erent fiber line. Each column represents a di↵erent loading case. For each
subfigure, the parameter RMS is calculated using Eq. IV.10. . . . . . . . . . . 108
IV.7 The measured strains at two of the top surface fiber lines superposed on the
strains obtained from the FEA model. Each row corresponds to a distinct fiber
line on the top surface of the plate. Each column corresponds to a di↵erent
loading condition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
IV.8 The estimated (Modal) displacements along the two fiber lines for all three load
cases, compared to the reference photogrammetry results. Each row represents
a di↵erent fiber line. Each column represents a di↵erent loading case. For each
subfigure, the parameter RMS is calculated using Eq. IV.10. . . . . . . . . . . 110
IV.9 The first row shows the displacements at the top-middle fiber based on the FEA
model, modal estimate, photogrammetry, and the FOSS approach, respectively.
The second row shows the corresponding estimation errors for the two approaches.112
V.1 Flowchartoftheload-estimationalgorithmforflexure-dominatedstructure.122
V.2 The sample single-story frame under investigation. . . . . . . . . . . . . 123
V.3 The first subfigure shows the moments M
ab
and M
ba
for di↵erent values
ofEI
G
/EI
C
. The exact and the the approximate results are superposed.
The second subfigure shows the normalized absolute percent error be-
tween the exact and the approximate solutions for M
ab
. . . . . . . . . . 127
xi
V.4 The first subfigure shows the moments M
ab
and M
ba
for di↵erent values
of H/L when I
G
=3I
C
. The exact and the the approximate results are
superposed. The second subfigure shows the normalized absolute percent
error between the exact and the approximate solutions for M
ab
. . . . . . 128
V.5 The sample single-story frame with indeterministic parameters. . . . . . 129
V.6 Normalized probability distribution functions pdfs of the calculated mo-
ments M
ab
, M
ba
, and M
cb
,where I
1
=I
2
=I
3
= 1. . . . . . . . . . . . . 132
V.7 Normalized probability distribution functions pdfs of the calculated mo-
ments M
ab
, M
ba
, and M
cb
,where I
1
=I
3
= 3 and I
2
= 1. . . . . . . . . . 133
V.8 A sketch of the four-story building-model created in Femap
r
. . . . . . . 134
V.9 Description of the five loading cases. . . . . . . . . . . . . . . . . . . . . 136
V.10 Strains and moments from the calibration load case, and the estimated
section properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
V.11 Estimated moments (dotted line) from various loading cases superposed
on the exact analytical moments (solid line) from the FEA. . . . . . . . 139
V.12 A photo of the four-story experimental test-bed structure. . . . . . . . . 140
V.13 Drawing of a typical building floor cross-section with corresponding di-
mensions in inches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
V.14 Photos of some of the details in the test-bed structure. . . . . . . . . . . 142
V.15 Details of the base angle-brackets. . . . . . . . . . . . . . . . . . . . . . 143
V.16 3D view of the computational model designed in Femap
r
. . . . . . . . . 144
V.17 The classical mode shapes and the corresponding natural frequencies. . . 145
V.18 The torsional mode shapes and the corresponding natural frequencies. . 146
V.19 Location of the fibers on the columns of the test-bed. . . . . . . . . . . . 147
V.20 Photos of the installed fibers on the columns of the test-bed structure. . 148
V.21 Solidworks
r
model of the test-bed structure and the test apparatus. . . 149
V.22 The test-bed and the pulley structure. . . . . . . . . . . . . . . . . . . . 150
V.23 Four point-loads applied on the test-bed, using the pulley structure. . . 151
V.24 Comparisonbetweenexperimentalandcomputationalstrainmeasurements.152
A.1 An undeformed side-view (x-z plane) of the bridge. . . . . . . . . . . . . . . 170
A.2 Location of the introduced 4 frame hinges on the columns of the bridge, for
nonlinear time-history analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.3 Comparison of theundeformed stateof thebridgewith thedeformed state, after
the time-history analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
A.4 The plastic rotations from hinge 26H1(located at the bottom of the shorter
column), after the nonlinear time-history analysis. . . . . . . . . . . . . . . . 174
A.5 The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from 8 nodes (sensors) on the bridge
deck.The networks were trained using EL-Centro data and the base excitation
from the same earthquake was fed back to the created model. . . . . . . . . . 176
B.1 Acceleration Plots from dataset EQ1. . . . . . . . . . . . . . . . . . . . 178
xii
B.2 Acceleration Plots from dataset EQ1. . . . . . . . . . . . . . . . . . . . 179
B.3 Acceleration Plots from dataset EQ1. . . . . . . . . . . . . . . . . . . . 180
B.4 Acceleration Plots from dataset EQ1. . . . . . . . . . . . . . . . . . . . 181
B.5 Acceleration Plots from dataset EQ1. . . . . . . . . . . . . . . . . . . . 182
B.6 Acceleration Plots from dataset EQ1. . . . . . . . . . . . . . . . . . . . 183
B.7 Covariance matrix corresponding to dataset EQ1. . . . . . . . . . . . . . . . 185
B.8 Acceleration variances for 10 datasets. . . . . . . . . . . . . . . . . . . . . . 186
B.9 Subplotsoftheacceleration variancesfor10datasets. Notethat amplitudescale
is not the same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.10 Acceleration variances from EQ1 dataset, subdivided into 12 segments (frames). 188
B.11 Mean acceleration variances for the nine aftershocks. The combined histograms
depict these variance parameters: mean, (mean + standard deviation), (mean
+2⇥ standard deviation), minimum and maximum values for the aftershocks. . 189
B.12 POV values of the acceleration covariance matrix from dataset EQ1. . . . . . 190
B.13 Undamped classical mode shapes obtained using dataset EQ1. . . . . . . . . . 192
B.14 POM mode shapes obtained using dataset EQ1. . . . . . . . . . . . . . . . . 192
B.15 Undamped classical mode shapes obtained using dataset EQ5. . . . . . . . . . 193
B.16 POM mode shapes obtained using dataset EQ5. . . . . . . . . . . . . . . . . 193
B.17 Undamped classical mode shapes obtained using dataset EQ10. . . . . . . . . 194
B.18 POM mode shapes obtained using dataset EQ10. . . . . . . . . . . . . . . . 194
B.19 MAC values from the undamped classical mode shapes and the POM modes
corresponding to dataset EQ1. . . . . . . . . . . . . . . . . . . . . . . . . . 196
C.1 The vibration mode shapes and the optimum FBG sensor locations. . . 198
C.2 Thee↵ectsofthenumberofsensorsontheestimationofthedisplacement
shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
C.3 The e↵ects of measurement noise on the estimation of the displacement
shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
C.4 The e↵ects of di↵erent loading conditions on the estimation of the dis-
placement shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
xiii
Abstract
The recent advancements in computational capabilities and sensing technologies pro-
vide an excellent opportunity to develop, test, and validate data-driven mathematical
models for system identification, condition assessment, and health monitoring of struc-
tural systems that may be vibrating in linear and/or nonlinear ranges. In this study,
measurements from various large-scale, complex, experimental systems, as well as full-
scale real-life multi-input-multi-output (MIMO) structures are used to develop robust
mathematical frameworks for response prediction, change detection, nonlinear damping
estimation, in addition to displacement-field and operating-load estimation. The systems
underconsiderationaretheYokohamaBayBridgewhichwassubjectedtothe2011Great
East Japan Earthquake; large-scale experimental soil-foundation-superstructure interac-
tion systems subjected to various earthquake excitations with systematically increasing
levels of intensity; swept wing-like experimental aluminum plates developed at the NASA
Dryden Flight Research Center and instrumented with state-of-the-art fiber-optic sen-
sors; and a four-story experimental test-bed designed, developed and fabricated at the
University of Southern California. The vibration signatures from these systems are used
toassesstheviabilityofexistingparametricandnonparametricidentificationapproaches,
and to propose new hybrid data-driven computational modeling methods that can accu-
rately capture the correct physics of the underlying complex systems. This dissertation
is a collection of analytical, computational, and experimental studies that capitalizes on
the availability of large datasets to develop tools that can interpret these datasets, and
to establish robust frameworks that can extract physically meaningful information, for an
informed decision-making.
xiv
Chapter I
Introduction
I.1 Background and Motivation
S
TRUCTURAL health monitoring has been an important research topic in various
engineering fields for more than 3 decades. Several methodologies have been devel-
oped, tested, and validated, particularly in the area of system identification and damage-
detection. With the advancements in sensor-based technologies such as data acquisition
systems, inexpensive sensors, etc., as well as the improvements in the computational tools
such as processing speed, storage, etc., data-driven health monitoring approaches have
been receiving increased attention. The availability of large amounts of data (Big Data)
provides important additional information about the systems under investigation, and an
opportunity for an in-depth analysis of the system characteristics. On the other hand,
dealing with large amounts of data poses challenges in interpreting the results (Data An-
alytics) and extracting meaningful conclusions from the massive number of observations.
Consequently, appropriate frameworks need to be developed and evaluated.
While several data-driven techniques are available within the identification-detection
framework, there is a paucity of studies that concentrate on data-driven approaches for
1
computational modeling and response prediction. Having reduced-order computational
modelsisveryimportantwhendealingwithcomplexmulti-componentnonlinearsystems,
(e.g., soil-foundation-superstructure interaction systems subjected to strong excitation).
A conventional way of modeling such systems is through using finite-element or finite-
di↵erence packages, where critical assumptions need to be made about the system bound-
ary conditions. In multi-component systems, with varying soil-layers interacting with the
foundation and the superstructure, it is a daunting task to develop a model that accu-
rately represents the full-scale physical system. A robust data-driven modeling approach
is an alternative, where no assumptions are needed (i.e., “model-free”) and the developed
reduced-order models can be used for dynamic response prediction in complex nonlinear
systems.
In general, methods involving health monitoring techniques can be categorized into
two broad classes: parametric and nonparametric. The selection of a method is a func-
tion of several considerations such as the complexity of the system, the materials used
for construction, the spatial resolution of the sensors, etc. Parametric techniques can
provide important information about the physical characteristics of the system, such as
damping, sti↵ness, natural frequencies, etc. However, most of these methods are subject
to one or more limitations such as assuming linear systems, stationary excitations, chain-
like physical topology, etc. Nonparametric techniques (such as support vector machines,
neural networks, Chebyshev Polynomials, etc.), which do not requireapriori information
about the physical nature of the structural system, have a significant advantage when
dealing with complicated structures, where accurate parametric models are not available.
On the other hand, the physical interpretation of the results from some nonparametric
methods might be challenging, given their “black-box” nature (such as neural networks).
2
While the literature has plenty of studies exploring certain parametric and nonparametric
methodsindividually, notmanystudiesareperformedtoexplorethepotentialof“hybrid”
approaches that combine certain parametric and nonparametric methods to capitalize on
the advantages of each one of them. A “hybrid” approach can particularly be useful
when analyzing complex structures (such as full-scale bridges) driven to their nonlinear
response range. Well-known parametric methods can be used to identify the equivalent
linear portion of the response, while the nonparametric method can be incorporated to
model the associated nonlinear forces. The combined results could provide an accurate
representation of the system.
The advancements in the technology relevant to sensors and data-acquisition systems
was associated with parallel development in new methodologies that leverage the bene-
fits of these advanced technologies. This is particularly important for structural health
monitoring of unique structural systems with specific requirements, such as Unmanned
Aerial Vehicles (UAVs). Many aerospace systems, including UAVs, have specific weight
requirements which makes the use of traditional sensors (i.e., strain gauges) not viable.
With the development of fiber-optic strain sensors, which are known for their lightweight,
accuracy, and high spatial resolution, a unique opportunity became available during this
study to develop methodologies for in-flight shape-detection and operational-load estima-
tion of these types of structures. While there are many well-studied techniques for shape
prediction and operational-load estimation for various fields of engineering, it is essential
to assess the viability of the proposed new methodologies by comparing them to these
well-established approaches and identifying the advantages and the limitations.
3
Motivatedbytheprecedingdiscussion,thisdissertationinvestigatesdata-driven,reduced-
order, computational modeling approaches applicable to mechanical, structural, civil in-
frastructure, and aerospace systems. Analytical, computational, and experimental stud-
ies are performed to assess the viability of the approaches. New modeling frameworks
are proposed, tested, and validated using experimental datasets. The proposed models
can be used for system identification, change-detection, and nonlinear response predic-
tion, amongst other applications relevant to structural health monitoring. Discussion
is provided on the advantages and the limitations of the proposed and the investigated
methodologies.
I.2 Scope
With the previous section in mind, Chapter II proposes nonlinear data-driven compu-
tational models for response prediction and change detection. Data is used from three
relatively large-scale experimental soil-foundation-superstrucure-interaction (SFSI) sys-
tems to develop reduced-order computational models for response prediction and change-
detection relevant to structural health monitoring and computational mechanics. The
three systems under consideration consist of identical superstructures with: (a) fixed
base; (b) box foundation; and (c) pile foundation. The three SFSI systems were devel-
oped and experimentally tested at Tongji University. In the first part of the study, a
computational time-marching prediction framework is proposed by incorporating trained
neural network(s) within an ODE solver, and predicting the dynamic response (i.e., dis-
placement and velocity) of the SFSI systems to various earthquake excitations. Two
4
approaches are investigated; a global approach and a subsystem approach. Both ap-
proaches are tested and validated with linear and nonlinear systems, and their respective
pros and cons are discussed. In the second part of the study, the trained neural networks
from the global approach are further used for change-detection in the SFSI systems. The
detected changes in the systems are then quantified through a measure of a normalized
error index. Challenges related to the physical interpretation of the quantified changes in
the SFSI systems are addressed and discussed. It is shown that the general procedures
adopted in this study provide a robust nonlinear model that is reliable for computational
studies, as well as furnishing a robust tool for detecting and quantifying inherent change
in the system.
Structural health monitoring of large multi-span flexible bridges is particularly impor-
tant because of their important role in civil infrastructure and transportation systems. In
Chapter III, the response of the Yokohama-Bay Bridge (YBB), a three-span cable-stayed
bridge, to the 2011 Great East Japan Earthquake is used to perform multi-input-multi-
output (MIMO) system identification studies. The extensive multi-component measure-
ments are also used to develop and validate data-driven nonlinear mathematical models
that can predict the response of YBB to various earthquake records, and can accurately
estimateitsdampingcharacteristicswhenthesystemisdrivenintothenonlinearresponse
range. A combination of least-square (parametric) and neural network (nonparametric)
approaches are used to develop “hybrid” mathematical models, along with time-marching
techniques, for dynamic response calculations. It is shown that the nonlinear mathemati-
cal models perform better than the equivalent linear models, both for response prediction
and damping estimation. The importance of having an accurate approach for quantifying
the damping due to the variety of nonlinear features in the YBB response is shown. This
5
study demonstrates the significance of constructing robust mathematical models that can
capture the correct physics of the underlying system, and that can be used for compu-
tational purposes to augment experimental studies. Given the lack of suitable data sets
for full-scale structures under extreme loads, the availability of the long-duration mea-
surements from the 2011 Great East Japan Earthquake and its many strong aftershocks
provides an excellent opportunity to perform the analyses presented in this study.
In Chapter IV, a new deformation-shape sensing methodology is investigated for the
purposes of real-time condition assessment, control, and health monitoring of flexible
lightweight aerospace structures. The Fiber Optic Strain Sensing (FOSS) technology was
recently proposed by the NASA Dryden Flight Research Center (DFRC). The method-
ology implements the use of fiber optic sensors to obtain strain measurements from the
target structure, and to estimate the corresponding displacement field. In this chap-
ter, the methodology is investigated through an experimental aluminum wing-like swept
plate model. The proposed algorithm is implemented for three distinct loading cases and
compared to a well-established modal-based shape estimation algorithm. The estimation
resultsfrombothmethodsarealsocomparedtoreferencedisplacementsfromphotogram-
metry and computational analyses. The estimation error for each method is quantified
using the root-mean-square (RMS) measure, and the range of validity of the approach for
damage-detection is established. Furthermore, the disadvantages and the advantages of
each method are discussed, demonstrating the additional benefits of using the proposed
FOSS methodology to achieve a robust method for monitoring ultra lightweight flying
wings, or next-generation commercial airplanes.
6
Chapter V explores the applicability of the FOSS technology (described in Chapter
IV) along with a load-estimation algorithm developed at DFRC, to estimate operating-
loads on tall building-like structures. Analytical studies are performed to investigate the
sensitivity of the algorithm to uncertainty, as well as its sensitivity to exact vs approxi-
mate moment calculation methods. Computational studies are performed to understand
the di↵erences between the strain measurements from wing-like structures (for which the
algorithm was initially developed) and building-like structures. An experimental building
test-bed structure is designed and developed to assess the algorithm. Discussion is pro-
videdonthedesignconstraints,instrumentation,andthetestapparatus. Adetailedfinite-
element model (FEM) of the test-bed structure is created using the Femap
r
/Nastran
r
packages. The FEM is used to perform various modal analyses to supplement the design
of the test-bed structure. Various challenges and complexities involved in adapting the
load-estimation approach on tall-buildings are acknowledged and discussed.
Chapter VI summarizes the presented methodologies, results, and observations.
7
Chapter II
Nonlinear Data-Driven Computational Models
for Response Prediction and Change Detection
II.1 Introduction
II.1.1 Background
D
ATA-DRIVEN modeling is an important field in computational mechanics with
applications in various engineering disciplines. Developing reduced-order com-
putational models for complex nonlinear systems can be useful for response prediction,
system identification, and structural health monitoring purposes.
Most of the previous studies on this topic concentrate on developing methodologies
for system identification and damage-detection purposes. A recent book by Takewaki
et al. (2012) analyzes various system identification methods for structural health mon-
itoring (SHM). A survey by Kerschen et al. (2006) summarizes the past and present
developments in system identification of nonlinear dynamic systems. A more recent work
by Worden et al. (2008) reviews nonlinear dynamics applications to structural health
monitoring. In general, the methods involving system identification techniques can be
8
categorized into two broad classes: parametric and nonparametric. The selection of an
identificationmethodassuggestedbyMasrietal.(1996), isafunctionofseveralconsider-
ations, such as the complexity of the system, the materials used for construction, the test
setup, etc. Promising parametric techniques such as the least-square, ERA/NExT, and
subspace identification, have been tested and validated in many studies Juang and Pappa
(1985);Masrietal.(1987a);VanOverscheeandDeMoor(1994). Althoughthementioned
parametricmethodshavetheadvantageofestimatingimportantsystemparameters(such
as sti↵ness, damping ratio), most of them make significant assumptions about the system
(such as linearity, stationary excitation, specific physical nature, i.e., chain-like), which
limits their application on other types of systems (such as nonlinear systems with un-
known physical nature). Nonparametric techniques (such as neural networks, machine
learning, etc.), which do not requireapriori information about the physical nature of
the structural system have a significant advantage when dealing with complicated nonlin-
ear structures, where accurate parametric models are not available Chassiakos and Masri
(1996); Hernandez-Garcia et al. (2010); Masri et al. (1993); Ying et al. (2012).
Theneuralnetworkapproachisoneofthemoste↵ectivenonparametricmethodologies
for system identification and damage-detection. Noteworthy works in testing and devel-
oping this approach were performed by Arangio and Beck (2012); Cury and Cr´ emona
(2012); Kao and Loh (2013); Li et al. (2010); Liang (2001); Lin and Jianjun (2010); Masri
et al. (2000); Nakamura et al. (1998); Pei et al. (2004); Suresh et al. (2012); Wang (2011);
Xu et al. (2009), amongst others. The robustness of the approach was tested and vali-
dated (both computationally and experimentally) on various systems, such as mechanical
equipments, fixed-base chain-like building models, etc. However, not many thorough
9
studies were performed to assess the viability of the approach in detecting damage in soil-
foundation-supersturce-interaction(SFSI)systems. Giventhecomplexityofsuchsystems
(especially when driven to their nonlinear response range), it might be a challenging task
to distinguish damage in a certain component (particularly a hidden component such as
piles, soil layers, etc.) from the overall change in the system.
While many studies in the literature concentrate on developing models for system
identification and damage-detection, not many concentrate on developing models for re-
sponse prediction purposes. There are very well-established modeling techniques (i.e.,
finite element, finite di↵erence, etc.) that provide relatively accurate models of various
linear (and even certain nonlinear) structures. However, if an accurate model of an entire
system is desired (i.e., including superstructure, foundation, and soil), major assump-
tions need to be made about the boundary conditions of the soil, the interaction e↵ects
between the soil-foundation and foundation-superstructure, the geometric nonlinearities,
etc. Therefore, it is a very challenging task to model the full system with the existing
tools and accurately represent the real physical conditions. Hence, exploring data-driven
approaches where no assumptions need to be made about the system (i.e., system is un-
known), is a useful modeling alternative for response prediction, especially when dealing
with nonlinear, multi-component interacting systems.
II.1.2 Motivation
Series of systematic dynamic tests were performed on three experimental soil-foundation-
superstructure interaction (SFSI) systems at the Tongji University in China. The avail-
ability of this unique and comprehensive data set generated from vibration tests on rel-
atively large scale (i.e., 1/10) SFSI system models, provides an excellent opportunity to
10
developandassessreduced-ordernonlinearcomputationalmodelsforresponseprediction.
Computational models are created by integrating trained neural networks within an ODE
solver to form a time-marching framework that can dynamically predict the response
of the systems to various excitations. Furthermore, the availability of three di↵erent
physical models (i.e., fixed base, box-foundation, and pile foundation) allows a thorough
assessment of the trained networks as change detection tools by actually comparing the
detection results to visual observations in the soil, foundations, and the superstructures
during and after the tests. Availability of the response datasets from various levels of
excitations on the there SFSI systems allows the quantitative assessment of the proposed
approaches by performing multiple tests and validation of generalization.
II.1.3 Scope
With the preceding in mind, Section 2 describes the shaking table tests including the
models, instrumentation, loading, and the data processing. Section 3 describes the com-
putationalmodelswithspecificsoninput-outputconfigurationoftheneuralnets,theODE
solver, and the testing and validation of both global and subsystem approaches. Section
4 presents the proposed detection approach, the quantified changes, and compares them
to visual inspection results as well as reported changes in the natural frequencies of the
systems. Section 5 discusses the obtained results as well as some of the advantages and
the disadvantages of the proposed approaches. Section 6 presents a conclusive summary
of the study.
11
II.2 Shaking-Table Tests
II.2.1 Description of the Models
Extensive shaking table model tests have been conducted in the State Key Laboratory
at Tongji University to investigate the soil-foundation-superstructure interaction (SFSI)
e↵ects of building-foundation systems. Three relatively large scale (i.e., 1 /10) specimens
were designed and fabricated with di↵erent foundation types, including pile foundation,
box foundation and fixed base. The pile foundation consisted of 9 piles, 1200mm long
each, with a cross-section of (45mm⇥ 45mm). The dimensions of the box foundation
were (650mm⇥ 650mm).
The superstructures were 12-story cast-in-place reinforced concrete (RC) frame mod-
els. Eachsuperstructurehad12floorswithidenticalheightsof300mm. Thecross-sections
of the beams and the columns were (30mm⇥ 60mm) and (50mm⇥ 60mm), respectively.
Shanghai soft soil was used for all the models. The soil was layered from top to bot-
tom into silty clay, powder sand and sand soil, respectively. The total height of the soil
was 1600mm. A flexible container (3000mm in diameter) was used to contain the soil
and the superstructure. Appropriate construction details were employed to simulate the
semi-infinite nature of the soil boundary conditions, and reduce the so-called “box e↵ect”
errors resulting from the wave reflections on the boundaries of the container. Fig. II.1
shows a photograph of the container and one of the superstructure models fixed on the
shaking table. The dimensions of the shaking table were (4.0m⇥ 4.0m) and it had a
maximum excitation capacity of 1.2g. Further details on the experiments, including the
scaling laws used and the soil/structure specifications can be found in Bo et al. (2002); Li
12
et al. (2004); Lu et al. (2005, 2004). For the purposes of this study, the vibration data
from three test models are investigated (see Table II.1).
Table II.1: The investigated three, 1/10 scale, 12-story RC frame superstructure models.
Shorthand Notations Soil Foundation Dynamic Content
FBS No Soil No Foundation Superstructure
SBSI Layered Soil Box Soil-box foundation-superstructure
SPSI Layered Soil Piles (3⇥ 3) Soil-pile foundation-superstructure
In Table II.1, the test FBS is the fixed-base structure with no soil or foundation
involved, SBSI is the soil-box foundation-superstructure interaction, and SPSI refers to
the soil-pile foundation-superstructure interaction.
Figure II.1: Photograph of the large scale Soil-Foundation-Superstructure test model setup.
13
A1
A2
SD
A6
A4
A3
A5
12X300
A7
250 350
A1
12X300
A7
A6
A5
A4
A3
A2
SD
350 250
b) Box Foundation
c) Pile Foundation
X - Direction
Z - Direction
A1
12X300
A7
A6
A5
A4
A3
A2
a) Fixed Base
Figure II.2: Distribution of the sensors on the model.
II.2.2 Instrumentation and Measurement Locations
The arrangement of the accelerometers on the superstructures is shown in Fig. II.2. As
seen, there are 7 accelerometers on each superstructure. The sensors A1 through A7, as
indicated in Fig. II.2, measure the acceleration response in the horizontal direction. A1
corresponds to the horizontal base acceleration, while A7 corresponds to the horizontal
accelerationatthetopfloor. Theaccelerometerswerelocatedateveryotherfloor,starting
with A2 at the second floor of the superstructure. There is an additional accelerometer
(i.e., channel SD) that is mounted on the surface of the shaking table and represents the
applied excitation. The sampling frequency of the acceleration data was 250 Hz. The
unit of the dimensions shown in the figure is (mm).
14
II.2.3 Loading Schedule and Network/Data ID’s
The sequence of the uni-directional (X-Direction) excitations applied on each of the three
SFSI systems is shown in Table II.2. The three base excitation records used were the
El Centro earthquake record (EL), the Shanghai artificial wave record (SH), and the
Kobe earthquake record (KB). The peak acceleration of the excitations were adjusted to
correspond to the scale of the experimental model. A total of 18 excitations were applied
in the X-Direction to the three models with the level of intensity increasing from test 1
to test 18. The levels of the excitations were categorized into six levels between 1 and 6,
level-1 being the lowest excitation and level-6 being the highest excitation. Furthermore,
thelevelsofexcitationweredeterminedbytheirpeakaccelerationvalues, asseeninTable
II.2.
II.2.4 Data Processing
The recorded acceleration data were band-pass filtered between 0.5 (Hz) and 50 (Hz)
usingzero-phasedistortionfilter. Then,thedatawerenumericallyintegratedwithrespect
to time to obtain the corresponding velocities and displacements. Up to sixth order de-
trendingwasappliedtotheintegratedsignals, toeliminateanypotentialtrendsgenerated
from the integration. Fig. II.3 shows the full record of the acceleration signal (10.748
sec.), along with the computed velocity and displacement records for sample channel A7
(top floor) from dataset I1c.
15
Table II.2: Test schedule and excitation intensity levels.
Test No. Excitation ID Intensity Level / Peak Acceleration (g)
1 EL (I)
Level-1 / (0.093g) 2 SH (II)
3 KB (III)
4 EL (I)
Level-2 / (0.266g) 5 SH (II)
6 KB (III)
7 EL (I)
Level-3 / (0.399g) 8 SH (II)
9 KB (III)
10 EL (I)
Level-4 / (0.532g) 11 SH (II)
12 KB (III)
13 EL (I)
Level-5 / (0.665g) 14 SH (II)
15 KB (III)
16 EL (I)
Level-6 / (0.798g) 17 SH (II)
18 KB (III)
16
0 1 2 3 4 5 6 7 8 9 10
−0.2
−0.1
0
0.1
0.2
Acceleration (g)
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
Velocity (cm/sec)
0 1 2 3 4 5 6 7 8 9 10
−0.5
0
0.5
Displacement (cm)
Time (sec.)
Figure II.3: Response of sample channel A7 of the pile foundation model, due to excitation EL1.
II.3 Computational Models
Two di↵erent approaches are investigated; a global approach, and a subsystem approach.
Both approaches are based on incorporating an ordinary di↵erential equation (ODE)
solver with trained neural network(s) to establish a time-marching framework that can
predict the response of complex linear and nonlinear systems to arbitrary excitations.
With both approaches, reduced-order computational models of the complex SFSI sys-
tems are developed, tested, and validated using the experimental measurements from the
previously described tests.
II.3.1 Input-Output Configuration of the Neural Nets
Consider an n-degree-of-freedom system that is governed by the following equation:
M¨ y(t)+g(y(t), ˙y(t)) =Mu¨ s(t) (II.1)
17
where ¨ s(t)isthebaseaccelerationvector,y(t)istherelativedisplacementsmatrixwithre-
spect to the base, g(y(t), ˙y(t)) is the matrix of restoring forces,M =diag(m
1
,m
2
,...,m
n
)
is the diagonal mass matrix, and u=(1,1,...,1)
T
is n⇥ 1 unit column vector, where n is
thenumberofdegrees-of-freedomofthesystem. Itisassumedthatthesystemparameters
are not available for this analysis, i.e., the mass matrixM and the restoring forces matrix
g(y(t), ˙y(t)) are unknown. In order to transform the above equation from a continuous to
a discrete time domain formulation, the following expressions are introduced: ¨ s
k
=¨ s(t
k
),
¨y
k
= ¨y(t
k
), ˙y
k
= ˙y(t
k
), and y
k
=y(t
k
), where the subscript k is the time index.
In this study, the experimental measurements for ¨ s(t) and ¨x(t) are available, where
¨x(t) is the matrix of absolute (recorded) accelerations. However, the needed relative
accelerations ¨y(t) can be calculated easily using ¨y(t)= ¨x(t)¨ s(t). Furthermore, ˙y(t)
and y(t) can be obtained by conventional numerical integration. Equation II.1 can be
rewritten as follows:
¨y(t)=M
1
(g(y(t), ˙y(t))Mu¨ s(t)) (II.2)
Note that Eq. II.2 is a general expression and can be used for linear and nonlinear sys-
tems. Theinputsoftheneuralnetworkswereselectedtobe(y
k
, ˙y
k
,¨ s
k
)
T
whichessentially
represent the right-hand-side of Eq. II.2, and the outputs were selected to be ¨y
k
,which
represent the left-hand-side of Eq. II.2. The choice of this specific input-output con-
figuration allows the integration of the trained networks within the ODE solver to form
the time-marching framework of the reduced-order computational models. The developed
models would only require the initial displacements and velocities (i.e., initial conditions)
ofthesysteminordertodynamicallypredicttheresponseofthesystemtoanyexcitation.
18
The neural networks with the same input-output configuration can be further utilized for
change detection purposes in the systems under investigation.
II.3.2 Ordinary Di↵erential Equation (ODE) Solver
The equation of motion defined in Eq. II.1 can be transformed into system of ordinary
di↵erential equations (ODEs) and can be rewritten as follows:
˙y(t)= ˙y(t)
¨y(t)=M
1
(g(y(t), ˙y(t))Mu¨ s(t))
(II.3)
Let’s definez=[y, ˙y]
T
and ˙z=[ ˙y, ¨y]
T
. As a result, the system can be characterized with
the equation ˙z(t)= f(z,t) subjected to the initial conditions z(t
0
)= z
0
. The Fourth-
Order Runge-Kutta Method was used to solve the initial value problem. Based on the
method, the value of the function at time-step k+1 can be estimated as follows:
z
k+1
=z
k
+(1/6)(k
1
+2k
2
+2k
3
+k
4
) (II.4)
where k
1
, k
2
, k
3
, and k
4
are defined as follows:
k
1
= t
f(t
k
,z
k
)
k
2
= t
f(t
k+1/2
,z
k
+k
1
/2)
k
3
= t
f(t
k+1/2
,z
k
+k
2
/2)
k
4
= t
f(t
k+1
,z
k
+k
3
)
(II.5)
19
The integration step size t
is an important parameter in the solver. It has to be small
enough to produce the necessary accuracy, yet it should not be too small as it e↵ects
the computational eciency. One of the most ecient ways of implementing the Runge-
Kutta Method is using a variable step size (implemented in this study). In this approach,
the function f is evaluated at 5 di↵erent points and a 5
th
order estimate is established.
Then, a 4
th
order estimate is made using 4 of the 5 evaluated points. The two estimates
are then compared and if they are relatively close, the step size is assumed to be small
enough. If the two estimates are not close, a smaller integration step size is chosen. This
approach is often referred to as the Runge-Kutta 45 (RK45) Method. Further details
on the Runge-Kutta methods can be found in Dormand and Prince (1980), Bogacki and
Shampine (1989), Shampine and Reichelt (1997), amongst others.
II.3.3 Global Approach
In the global approach, one neural network is trained to represent the entire system by
providing inputs (displacements, velocities, and base acceleration) from all the available
degrees-of-freedom (DOFs) of the superstructure. The corresponding outputs are the
accelerations from all DOFs of the superstructure. The approach assumes the system
is unknown. It does not assume any specific physical configuration (i.e., chain-like),
neither it assumes anything about the system characteristic (i.e., linear vs. nonlinear).
Therefore, the approach is general and applicable on variety of structural systems with
di↵erent configurations and characteristics.
Network Architecture and Training
The architecture of the proposed network is shown in Fig. II.4. The network has 13
20
Hidden Layer 1 Hidden Layer 2
Input Layer Output Layer
1
2
3
4
10
2
3
4
10
W
b
Sigmoid
Transfer
Function
1
W
b
Linear
Transfer
Function
y
k
˙y
k
¨ s
k
ˆ
¨y
k
Figure II.4: Architecture of the neural network in the global approach.
input vectors (6 displacements, 6, velocities, 1 base acceleration). The network has two
hidden layers and an output layer that contains 6 output vectors (accelerations). Each
hidden layer has 10 hidden neurons. The two hidden layers have hyperbolic tangent
sigmoid transfer functions. The hyperbolic tangent transfer function is nonlinear, easily
di↵erentiable,hasanoutputrangeof( 11),andisknownforitscomputationaleciency.
The output layer has a linear transfer function. These neural transfer functions calculate
the layers’ output from their input. As seen from the figure, the weighted inputs are
summed with the bias before going in the transfer function. Both the weights and the
biases are initialized using a random scheme. It should be noted that decreasing the
number of hidden layers and neurons might lead to insucient training, while increasing
them too much might lead to overtraining (over-fit), consequently the network might not
generalize very well. Keeping this in mind, the number of hidden layers and neurons in
21
this approach werechosen to achieve arelatively high level of accuracy given the available
computational resources.
Severalneuralnetworktrainingalgorithmsweredevelopedalongtheyears. Theadap-
tive random search method (ARS) proposed by Masri and Bekey (1980) promises an im-
proved convergence rate of the minimizer of a given cost function. An improved optimiza-
tionprocedurebasedontheadaptiverandomsearchalgorithmwasalsoproposedbyMasri
et al. (1999). Furthermore, Zhang et al. (2010) presented the Levenberg-Marquardt back-
propagation algorithm and described it as the harmonization between the Gauss-Newton
method and the steepest descent method. Lyn Dee et al. (2010) studied and compared
the performance of several neural network training algorithms and suggested that the
Levenberg-Marquardt backpropagation algorithm provides the best results for the pur-
poses of data-driven modeling in structural systems. In this study, the networks were
trained using the batch mode of the Levenberg-Marquardt backpropagation algorithm.
For this type of training, the weights and the biases are updated once in each epoch,
after all the inputs are presented. The mean squared normalized error (mse) is used as
the performance function during the training. In this analysis, the networks satisfied the
specified (mse) performance criteria after an average of 50 epochs.
Flowchart of the Global Approach
Figure II.5 shows the flowchart of the proposed framework. For a given ground motion ¨ s
k
theaccelerationattime-stepk,
ˆ
¨y
k
, isestimatedusingonetrainednetworkthatrepresents
the system. Then, the estimated acceleration
ˆ
¨y
k
is provided to the ODE solver and the
displacement and the velocity at the next time-step (i.e., t
k+1
) are obtained. Then, the
net’sinputsareupdatedwiththenewestimatesof ˆy and
ˆ
˙y, alongwiththegroundmotion
22
x
n 1
k
RK45
ˆ
¨ y
n
k
Neural
Network
¨ s
k
ODE Solver
y
1
k
y
2
k
y
n
k
Initial
Conditions
ˆ
¨ y
1
k
ˆ
¨ y
2
k
ˆ
¨ y
k
ˆ y
k+1
ˆ
˙ y
k+1
ˆy
k
= ˆy
k+1
ˆ
˙y
k
=
ˆ
˙y
k+1
¨ s
k
=¨ s
k+1
Update
net
2
RK45
ˆ
¨ y
nk
net
1
net
n
Neural Networks
¨ s
k
¨ s
k
¨ s
k
ODE Solver
y
1
k
y
2
k
y
n
k
˙ y
n
k
˙ y
2
k
˙ y
1
k
Initial
Conditions
ˆ
¨ y
1
k
ˆ
¨ y
2
k
ˆ
¨ y
k
ˆ y
k+1
ˆ
˙ y
k+1
ˆy
k
= ˆy
k+1
ˆ
˙y
k
=
ˆ
˙y
k+1
¨ s
k
=¨ s
k+1
Update
y
3
k
˙ y
3
k
¨ s
k
˙ x
n 1k
¨ s
k
˙ y
1
k
˙ y
2
k
˙ y
n
k
Figure II.5: Flowchart of the computational model using the global approach. The integration
of a single trained neural network with RK45 and the updating procedure at each time-step is
shown.
value at the next time-step, and the procedure is repeated. In the performed tests, the
systems were initially at rest (which can be assumed to be the case in most practical
applications). Therefore, the initial conditions of the systems were zero.
It should be noted that having one global network representing a complex unknown
systemisextremelyuseful. However, fordensermonitoringsystems(i.e., moredegrees-of-
freedom/sensors), it might become a daunting task to train the network with a relatively
good performance, which in turn can lead to increased estimation errors. The estimation
errors when combined with the truncation error of the ODE solver can lead to instability
problems within the computational model. Furthermore, for large systems with more
23
degrees-of-freedom (i.e., bridges or dams), the computational cost and e↵ort associated
with proposed model might become significant.
II.3.4 Subsystem Approach
In this approach, the system is represented by a combination of multiple neural networks.
Foreachdegree-of-freedom(DOF),asingleneuralnetworkistrainedwithinputsfromthe
DOF itself and the neighboring DOFs. For DOF 1, the network is trained using inputs
from DOFs 1 and 2. For DOF n, the network is trained using inputs from DOFs n and
n1. For any other DOF i, the network is trained using inputs from DOFs i1, i,
and i+1. As a result, the first and last networks (i.e., net 1 and net n)have5inputs
and 1 output. All other networks have 7 inputs and 1 output. Figure II.6 shows the
flowchart of the approach. As seen in the figure, the flowchart is very similar to the global
approach. However, instead of integrating 1 network within the ODE solver (as it was
the case in the previous approach), n = 6 networks are integrated within the ODE solver.
It should be noted that the network architecture used in in this approach is similar to
the one explained earlier and shown in Fig. II.4. However, given the smaller amount of
inputs in these networks, only 5 neurons were used in each hidden layer (as opposed to 10
neurons in the previous approach). Furthermore, the training algorithm discussed earlier
is implemented in this approach as well.
The proposed subsystem approach can be particularly advantageous when dealing
with densely instrumented complex systems (many degrees-of-freedom). Decomposing
the system by training multiple neural networks can help achieving a higher estimation
accuracy, as opposed to using large amount of inputs to train one neural network. Fur-
thermore, given the relatively bounded size of the subsystem networks, the training can
24
Hidden Layer 1 Hidden Layer 2
Input Layer Output Layer
1
2
3
4
10
2
3
4
10
W
b
Sigmoid
Transfer
Function
1
W
b
Linear
Transfer
Function
y
k
˙y
k
¨ s
k
ˆ
¨y
k
y
n 1
k
net
i
RK45
ˆ
¨ y
nk
net
1
net
n
Neural Nets
¨ s
k
¨ s
k
¨ s
k
ODE Solver
y
1k
y
2k
y
nk
˙ y
nk
˙ y
2k
˙ y
1k
Initial
Conditions
ˆ
¨ y
1k
ˆ
¨ y
ik
ˆ
¨ y
k
ˆ y
k+1
ˆ
˙ y
k+1
ˆy
k
= ˆy
k+1
ˆ
˙y
k
=
ˆ
˙y
k+1
¨ s
k
=¨ s
k+1
Update
˙ y
n 1k
¨ s
k
¨ s
k
¨ s
k
¨ s
k
˙ y
i
k
˙ y
i 1k
˙ y
i+1k
y
i 1
k
y
i+1
k
y
i
k
Figure II.6: Flowchart of the computational model using the subsystem approach. The integra-
tion of multiple trained neural networks with RK45 and the updating procedure at each time-step
is shown.
be implemented faster with negligible computational cost and e↵ort. The tradeo↵ on the
other hand, is that when integrated with the ODE solver, multiple networks need to be
called at each time-step (which might e↵ect the speed of the time-marching framework),
as opposed to one network in the global approach.
25
II.3.5 Testing and Validation
Theresultsfromfour di↵erent computationalmodelsarepresented. Twoofthepresented
models were created using the global approach, and the other two were created using the
subsystem approach. For each approach, a computational model was created using the
lowest intensity data (i.e., level-1 excitation) and another model was created using the
highest intensity data (i.e., level-6 excitation). During level-1 excitation, the response
of the systems was mostly within the linear range. On the other hand, during level-6
excitation, the systems were driven into the nonlinear response range.
Global Approach
Figure II.7 shows the response (i.e., relative velocities and displacements) predicted form
the computational model (solid line) superposed on top of the measured response (dotted
line). The right column represents the velocities (cm/sec) and the left column repre-
sents the displacements (cm). Each row represents the response from a di↵erent channel.
The top row for example, represents channel A7 which is located at the top floor. The
computational model was developed using measurements from the box foundation (SBSI)
system. The network was trained using vibration data corresponding to level-1 El-Centro
excitation. In order to test the computational model, level-1 El-Centro base excitation
(same excitation used in training) was fed back to the model. The results are shown
in Fig. II.7. As seen in the figure, the predictions from the computational model are
excellent and match the measured signals for all the channels.
In order to validate the computational model under discussion, a di↵erent base exci-
tation was fed to the model (i.e., level-1 Kobe excitation). The results are shown in Fig.
II.8. As in the previous figure, the predicted relative velocities and displacements (solid
26
line) are superposed on top of the corresponding original measurements (dotted line). It
is seen that most of the predicted main oscillations match the original signals very well.
However, the displacement predictions slightly diverge from the original displacements
after about 5 seconds for most of the channels. This is mainly due to propagation of the
neural network estimation error and the truncation error from the ODE solver. Given
the time-marching nature of the proposed models, errors accumulate and propagate with
each time-step, which might cause the divergence of the estimates or instability of the
models. This further emphasizes that with the dynamic time-marching nature of the
proposed framework, it is much more challenging to achieve robust models, compared to
the traditional neural networks that are used to provide static fit.
The results shown so far were from one computational model trained using level-1
El-Centro data (i.e., linear response from the system). Another model was developed
using level-6 Kobe data (i.e., using nonlinear response from the systems). The results
from the pile foundation (SPSI) system are shown in Fig. II.9. It should be noted that
the scale of the response (y-axis) in Fig. II.9 is much greater than Fig.’s II.7 and II.8
given the intensity-level of the data used. The results in Fig. II.9 are the outcome of
feeding level-6 Kobe base excitation to the computational model that was created using
level-6 Kobe data (i.e., the base excitation used in training of the network was fed back
to the computational model). As seen, the velocity predictions from all channels are
fairly acceptable (especially for the main oscillations), but are not as good as the previous
prediction of the linear case in Fig. II.7. It is also seen that the displacement predictions
fromalmostallofthechannelsdivergeafterfewseconds. Themodelwasfurthervalidated
by feeding a base excitation di↵erent than the one used in training, and similar behavior
was observed. The results show that developing a viable computational model using
27
0 2 4 6 8 10
−0.5
0
0.5
Channel A7
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A6
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A5
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A4
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A3
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10
−5
0
5
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A2
Vel. (cm/sec)
Time (sec)
FigureII.7: Thedisplacementandvelocitypredictionsfromthecomputationalmodelsuperposed
on the original corresponding signals from channels A2 through A7 on the SBSI system.The net-
work was trained using level-1 EL-Centro data and the base excitation from the same earthquake
was fed back to the created model.
28
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A7
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A6
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A5
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A4
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A3
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A2
Vel. (cm/sec)
Time (sec)
Figure II.8: The displacement and velocity predictions from the computational model super-
posed on the original corresponding measurements from channels A2 through A7 on the SBSI
system.The network was trained using level-1 EL-Centro data and the base excitation from a
di↵erent earthquake level-1 Kobe was fed back to the created model.
29
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A7
Disp. (cm)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A6
Disp. (cm)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A5
Disp. (cm)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A4
Disp. (cm)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A3
Disp. (cm)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10 12
−10
0
10
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A2
Vel. (cm/sec)
Time (sec)
FigureII.9: Thedisplacementandvelocitypredictionsfromthecomputationalmodelsuperposed
on the original corresponding signals from channels A2 through A7 on the SPSI system.The
network was trained using level-6 Kobe data and the base excitation from the same earthquake
was fed back to the created model.
nonlinear response data from the systems is more challenging than the case where linear
response was used (i.e., Fig.’s II.7 and II.8).
Subsystem Approach
As mentioned earlier, the systems in this approach are modeled using a combination
of several trained neural networks integrated within an ODE solver. Data from level-
1 El-Centro test case applied on the SBSI system was used to train 6 neural networks
30
which were then connected to the ODE solver. Then, the base excitation from the same
earthquake was fed back to the model and the results are shown in Fig. II.10. The
predictions (solid line) are superposed on top of the original measurements (dotted line).
As seen, the match between the predictions and the measurements is excellent for both
the velocities and the displacements from all the channels.
In order to validate the same model and assess how well it generalizes with di↵erent
excitations, level-1 Kobe base excitation was used. The results are shown in Fig. II.11.
It is seen, that there is a very good match between the predictions and the original mea-
surements throughout the entire record from each channel (i.e., no divergence is observed
as in the validation of the previous approach). Satisfactory results were observed with
other base excitations as well.
While the proposed subsystem modeling approach promises to be robust with linear
systems, it is very important to assess its viability when dealing with nonlinear systems.
Therefore, a computational model was created using the highest-intensity data corre-
sponding to the SPSI system driven to its nonlinear response range using level-6 Kobe
earthquake. As explained previously, 6 neural networks were trained and connected with
the ODE solver to form the proposed model. The model was first tested by feeding the
same base excitation used in training. Very promising results were obtained and are
shown in Fig. II.12. It is seen that the predictions match the original signals very well for
most of the channels. For channels, A5, A6, and A7 some computational (non-physical)
oscillations were observed towards the end of the records. This might be due to propa-
gation of the computational errors. But for most of the channels, the main oscillations
are predicted fairly accurately. It is worth noting the scale di↵erence (y-axis) between
31
0 2 4 6 8 10
−0.5
0
0.5
Channel A7
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A6
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A5
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A4
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A3
Disp. (cm)
0 2 4 6 8 10
−0.5
0
0.5
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10
−5
0
5
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10
−5
0
5
Channel A2
Vel. (cm/sec)
Time (sec)
Figure II.10: The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channelsA2 throughA7 on the SBSI system.The
networks were trained using level-1 EL-Centro data and the base excitation from the same earth-
quake was fed back to the created model.
32
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A7
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A6
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A5
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A4
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A3
Disp. (cm)
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10 12
−4
−2
0
2
4
Channel A2
Vel. (cm/sec)
Time (sec)
Figure II.11: The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channelsA2 throughA7 on the SBSI system.The
networksweretrainedusinglevel-1 EL-Centro dataandthebaseexcitationfromadi↵erentearth-
quake level-1 Kobe was fed back to the created model.
33
Fig.’s II.10 and II.12, to better appreciate the intensity level of the data used from the
nonlinear response of the SPSI system (i.e., Fig. II.12).
In order to further validate the model, a di↵erent base excitation (i.e., level-6 El-
Centro) was used. The prediction results are shown in Fig. II.13. Despite of the highly
nonlinear nature of the data used in training of the model, as well as the new (di↵erent)
base excitation fed into the model, it is seen that the prediction results are very good.
No traces of divergence or instability are observed. Similar results were obtained from
feeding other base excitations to the developed model. As a result, it is seen that the
subsystem approach has a promising potential to be used as a reliable modeling tool for
complex linear and nonlinear systems, and to develop accurate and robust computational
models for response prediction of such systems under strong earthquake excitations
II.4 Change-Detection in the SFSI Systems
II.4.1 Methodology
Thetrainednetworksincorporatedwithinthecomputationalmodelsintheearliersections
can be further utilized (with the same input-output configuration) in an identification-
detection framework in order to detect changes in the SFSI systems. The availability of
systematic measurements from 6 intensity levels of three di↵erent earthquakes applied on
three large-scale SFSI systems provides an excellent opportunity to assess the viability
of the proposed general detection scheme with the linear and the nonlinear states of the
SFSI systems.
As shown in Fig. II.14, the method consists of two main steps. During the first step
(identification step), the neural network is trained using data from a certain (reference)
34
0 2 4 6 8 10 12
−1
0
1
Channel A7
Disp. (cm)
0 2 4 6 8 10 12
−1
0
1
Channel A6
Disp. (cm)
0 2 4 6 8 10 12
−1
0
1
Channel A5
Disp. (cm)
0 2 4 6 8 10 12
−1
0
1
Channel A4
Disp. (cm)
0 2 4 6 8 10 12
−1
0
1
Channel A3
Disp. (cm)
0 2 4 6 8 10 12
−1
0
1
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10 12
−10
0
10
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10 12
−10
0
10
Channel A2
Vel. (cm/sec)
Time (sec)
Figure II.12: The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channelsA2 throughA7 on the SPSI system.The
networks were trained using level-6 Kobe data and the base excitation from the same earthquake
was fed back to the created model.
35
0 2 4 6 8 10
−2
0
2
Channel A7
Disp. (cm)
0 2 4 6 8 10
−2
0
2
Channel A6
Disp. (cm)
0 2 4 6 8 10
−2
0
2
Channel A5
Disp. (cm)
0 2 4 6 8 10
−2
0
2
Channel A4
Disp. (cm)
0 2 4 6 8 10
−2
0
2
Channel A3
Disp. (cm)
0 2 4 6 8 10
−2
0
2
Channel A2
Disp. (cm)
Time (sec)
0 2 4 6 8 10
−10
0
10
Channel A7
Vel. (cm/sec)
0 2 4 6 8 10
−10
0
10
Channel A6
Vel. (cm/sec)
0 2 4 6 8 10
−10
0
10
Channel A5
Vel. (cm/sec)
0 2 4 6 8 10
−10
0
10
Channel A4
Vel. (cm/sec)
0 2 4 6 8 10
−10
0
10
Channel A3
Vel. (cm/sec)
0 2 4 6 8 10
−10
0
10
Channel A2
Vel. (cm/sec)
Time (sec)
Figure II.13: The displacement and velocity predictions from the computational model super-
posed on the original corresponding signals from channelsA2 throughA7 on the SPSI system.The
networks were trained using level-6 Kobe data and the base excitation from a di↵erent earthquake
level-6 El-Centro was fed back to the created model.
36
A1
A2
SD
A6
A4
A3
A5
12X300
A7
250 350
A1
12X300
A7
A6
A5
A4
A3
A2
SD
350 250
b) Box Foundation
c) Pile Foundation
X - Direction
Z - Direction
A1
12X300
A7
A6
A5
A4
A3
A2
a) Fixed Base
Create NN Model
1) Identification Step
Inputs
Targets
A1
A2
SD
A6
A4
A3
A5
12X300
A7
250 350
A1
12X300
A7
A6
A5
A4
A3
A2
SD
350 250
b) Box Foundation
c) Pile Foundation
X - Direction
Z - Direction
A1
12X300
A7
A6
A5
A4
A3
A2
a) Fixed Base
NN Model
2) Detection Step
X
- +
Change
Inputs
Targets
NN Outputs
Figure II.14: The flowchart of the identification-detection framework. The network is trained
first during the identification step (left-part). New data is fed to the trained network during the
detection step (right-part).
earthquake. Duringthesecondstep(detectionstep),thetrainednetworkisfedinputdata
recorded from the same SFSI system when it is subjected to a di↵erent earthquake (not
used in training). If the output of the neural network matches the target response from
the SFSI system, then there is no observable change in the characteristics of the system.
If the network output and the target response from the SFSI system do not match, then
there is a change in the system, and the di↵erence between the output and the target is
the measure which quantifies the change.
The networks were trained following the global approach (i.e., one network repre-
sents the entire system). As a result, the network characteristics of the global approach
explained earlier (i.e., input-output configuration, architecture, training, etc.), are imple-
mented in this section as well. Given the global approach, the proposed identification-
detection framework is general and does not assume a specific physical configuration or a
certain system characteristic; hence, it is applicable on wide range of structural systems.
37
II.4.2 Results
In order to get the most out of the available datasets and extract as much information
about the condition of the three SFSI systems, 9 neural networks were trained for the
analyses in this section. The trained neural networks, simulation data, and the corre-
sponding three SFSI systems were organized by adopting a simple naming convention.
For the three earthquakes, EL, KB, and SH, the following IDs were assigned, I, II, and
III,respectively. Thethreeboundaryconditions(SFSIsystems)wereassignedthefollow-
ing codes; (a), (b), and (c), representing fixed base, box foundation, and pile foundation,
respectively. Each excitation intensity level was assigned a numerical value ranging from
1 for the lowest intensity level, to 6 for the highest intensity level. As a result, each record
can be distinguished by a 3-letter code, (i.e., I1a represents El-Centro, level-1, applied on
the fixed base system). Furthermore, each trained network can be distinguished by the
adopted 3-letter code (i.e., Net-III6c represents the neural network trained using Level-6
Kobe record, applied on the pile foundation). For each SFSI system, 3 networks were
trained using the lowest levels (level-1) of each of the three earthquakes. Each trained
network was then simulated 6-times, using the training data and the data from all other
intensitylevelsofthesameearthquake(i.e., Net-I1awastrainedwithI1a, thensimulated
with datasets I1a through I6a).
In order to detect the changes in the system after each intensity level, the average
normalized error was calculated as follows:
ˆ e =
n
X
k=1
h
i
e
i

n
X
k=1
h
i
(II.6)
38
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
1.2
Intesity level of simulated data
Average normalized error
Fixed Base (a)

Net−I1a
Net−II1a
Net−III1a
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
1.2
Intesity level of simulated data
Box Foundation (b)

Net−I1b
Net−II1b
Net−III1b
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
1.2
Intesity level of simulated data
Pile Foundation (c)

Net−I1c
Net−II1c
Net−III1c
Figure II.15: The detected changes in the SFSI systems quantified by the average normalized
error measure. Each subfigure corresponds to a di↵erent SFSI system. For each system, results
from three di↵erent networks are shown.
where ˆ e is the average normalized error, n is the number of the channels (n = 6), h
i
is the
weight applied to the error from each channel i (h
i
= 1), and e
i
is the normalized error
at channel i. The normalized error (e) at each channel is defined by Eq. C.2:
e =
kx
r
x
nn
k
kx
r
k
=
kx
e
k
kx
r
k
(II.7)
whereeisthenormalizederror,x
r
istherecorded(target)signal,x
nn
istheneuralnetwork
output, x
e
is the di↵erence between the recorded signal x
r
and the neural network output
x
nn
(i.e., x
e
=x
r
x
nn
), andkx
e
k is the Euclidean Norm of vector x
e
.
Figure II.15 shows the results of the simulations for all three SFSI systems using all
9 trained networks. Each subfigure in Fig. II.15 represents a distinct SFSI system. The
x-axis in each subfigure corresponds to the intensity level of the simulated data, and
the y-axis corresponds to the dimensionless average normalized error. The intention in
training three networks for each of the three available earthquake records, was to see
39
how well the detection approach could depict the changes in the system by comparing
the results from three uncorrelated networks, each trained and simulated with a di↵erent
earthquake record, but representing the same system. Looking at Fig. II.15, it can be
seen that for each of the SFSI systems, the simulation results from each of the three
trained networks are relatively similar. Since all of the networks are trained with level-1
earthquakes, simulating the trained networks with the training data yields to minimal
errors. Therefore, the simulation errors at level-1 are minimal. The errors propagate as
the intensity level of the simulation data increases to reach a maximum at the highest
intensity level. As expected, the results from all of the trained networks follow a similar
pattern for each system. Comparing the average normalized errors for the three SFSI
systems in Fig.II.15, it can be seen that there is a significant error increase from level-1 to
level-6 in the fixed base and the pile foundation systems. On the other hand, the increase
in the error in the box foundation is not as drastic, meaning that there is less change in
the box foundation system.
II.4.3 Visual Observations During and After the Tests
A significant challenge of the neural network approach is the physical interpretation of
the detected changes. Given the large-scale experimental nature of the current study it
was possible to compare the results with observed physical changes in the three SFSI
systems after the experiments. Post-experiment physical inspection of the SFSI systems
shows changes in various parts of the three systems.
In the case of the fixed-base model, there was a systematic increase in the cracks
along the superstructure as the intensity-level of the excitation increased. After level-2
excitations, verticalcrackswereobservedonthebeamsoffloors1through4. Afterlevel-3
40
excitations, the width of the observed initial cracks increased. After level-4 excitations,
cracks were observed on the columns of the first floor, in addition to increased cracks
on the beams. Furthermore, part of a concrete column on the 3
rd
floor crushed. After
level-6 excitations, more parts of the concrete crushed on the columns of the 2
nd
and 3
rd
floors. After all the tests, the width of the vertical cracks ranged from 0.1mm to 1.0mm.
It is seen, that the detected changes in the fixed-foundation model shown in Fig. II.15
agree with the systematic increase in the observed cracks and damages throughout the
experiment.
In the case of the box-foundation model, no major cracks were observed on the su-
perstructure. The range of the crack widths was from 0.05mm to 0.08mm on the first 5
floors. Furthermore, theboxfoundationwasinspectedaftertheexperimentandnocracks
were found. On the other hand, the system started settling after level-3 excitations and
throughout the remaining tests. The final settlement was about 6.5cm. Based on the
physical observations, the amount of the cracks on the superstructure and their corre-
sponding sizes were significantly less than the fixed-base model. The detection results in
Fig. II.15 agree with the visual inspection. As seen, the change from level-1 to level-6 is
not very drastic, with an average normalized error of less than 0.4. Since there was no
damage in the box-foundation, and very little damage in the superstructure, the detected
change might be due to the system settlement and changes in soil layers as well as the
interaction e↵ects between the soil, foundation, and the superstructure.
Inthecaseofthepile-foundationmodel, nocrackswereseenonthesuperstructurebe-
fore level-5 excitations. After level-5 and level-6 excitations, cracks ranging from 0.02mm
to 0.5mm wide have been observed on the columns. It is interesting to see that while
no cracks were observed on the superstructure during the first 4 levels of excitation, the
41
detection results in Fig. II.15 show significant changes in the SPSI system during the first
4 levels of excitation. To further investigate the other elements of the SPSI system, the
piles were dug out after the tests and dense cracks were observed at the top of the piles
with less cracks at the tip of the piles, see Fig. II.16. Of course, it is not very easy to
conclude whether these cracks formed systematically starting with the level-1 tests and
propagated throughout the other tests, or it happened only during the higher intensity
excitations. But, they might have contributed to the detected changes during the fist 4
levels of excitation shown in Fig. II.15. It was also seen, that the system started set-
tling throughout the tests and had an uneven final settlement of 10.7cm in one corner
and 7.6cm in the opposite corner. The changes in the soil characteristics combined with
the interaction e↵ects between the soil-piles-supersturcutre might be another contribut-
ing factor to the changes detected (especially during the first 4 excitation levels) from the
neural nets.
As a result, it is seen that it is relatively straightforward to interpret the detection
results from the globally trained neural networks for the fixed-base model. It should also
be noted that most of the relevant neural network identification-detection studies in the
literature concentrate on fixed-base synthetic or experimental models. However, it is seen
in this study that as the models get more complex (especially the pile-foundation model),
it is much more challenging to interpret the obtained changes and attribute them to dam-
age in certain components of the systems. This is due to the large number of components
in each system (i.e., foundations, soil, superstructure), and the interaction e↵ects of all
these elements with each other. Nevertheless, even for these complex interacting systems,
the proposed global detection scheme provides a quantified measure of an overall change
in the system which can be useful for various control and monitoring purposes. The next
42
Figure II.16: The observed vertical cracks on the piles after all the tests.
section further investigates the change detection results by comparing it to the changes
in the reported natural frequencies of the SPSI system after each excitation level.
II.4.4 Comparison of the Detection Results to Changes in Natural Fre-
quencies in the Pile Foundation (SPSI) System
According to Li et al. (2004); Lu et al. (2004), the natural frequencies of the SPSI system
was measured before performing the shakings for each intensity level. The system fre-
quencieswereobtainedbyapplyingwhite-noiseexcitationbeforeeachexcitationleveland
analyzingtheobtainedresponse. Furtherdetailsonhowthefrequencieswereobtainedcan
be found in the mentioned publications. The reported natural frequencies were (2.643,
2.139, 1.636, 1.384, 1.258, 1.133) Hz, measured after each intensity level starting from
level-1 through level-6, respectively (i.e., the reported frequency after level-1 excitations
43
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
1.2
Intesity level of simulated data
Average normalized error

Net−I1c
Net−II1c
Net−III1c
Δω
c
0 0.2 0.4 0.6 0.8 1 1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Detected normalized changes using neural networks
Normalized frequency changes Δω
c

Net−I1c
Net−II1c
Net−III1c
(a) Reported frequency changes compared
with detected neural network errors.
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
1.2
Intesity level of simulated data
Average normalized error

Net−I1c
Net−II1c
Net−III1c
Δω
c
0 0.2 0.4 0.6 0.8 1 1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Detected normalized changes using neural networks
Normalized frequency changes Δω
c

Net−I1c
Net−II1c
Net−III1c
(b) Correlation of the reported frequency
changes with the detected changes.
Figure II.17: Comparison of the normalized change results from the neural network detection
and the reported natural frequencies in the pile foundation model.
was 2.643 Hz). It is seen, that the frequency values decrease as the number of tests in-
crease, indicating changes (as reflected by the lowering of the dominant frequency) in the
system, as a result of the high intensity excitations.
A normalized index of the changes in the reported fundamental natural frequencies is
provided by the following equation:
!
c
=
!
i
!
ref
!
ref
(II.8)
where !
c
is the normalized change in the fundamental frequency, !
i
is the reported
frequency before each testing stage, and !
ref
is the reference frequency. In this study,
!
ref
is chosen to be the frequency obtained at the initial (unchanged) stage of the system
(i.e., !
ref
=2.643 was chosen the SPSI system under investigation.)
The normalized quantities from Eq. II.8 are dimensionless representations of the
change in the system obtained after every level of ground excitation, starting with the
44
initial (reference) case. The reported normalized frequency changes were compared to
the dimensionless normalized changes in the system obtained using the neural network
change-detection approach. The comparison results are plotted in Fig. II.17(a). The
horizontal axis in Fig. II.17(a) represents the excitation level, while the vertical axis
represents the percent change in the system. It is seen that the results from the neural
networks are analogous to the frequency-based results for the first 4 levels of excitations.
For levels 5 and 6, the neural network results indicate a greater change in the system than
the frequency-based results. To further investigate the correlation between the detected
and the reported changes, the neural network detection errors were plotted against the
normalized frequency changes and the results are shown in Fig. II.17(b). As seen in the
figure, the relation between the two is not linear. In fact, the plot shows that the neural
network detection approach is much more sensitive to changes in the SPSI system than a
simple frequency-based analysis.
II.5 Discussion
In the first part of this study, a time-marching framework is proposed for the dynamic
response prediction of the large-scale experimental SFSI systems. Two di↵erent model-
ing approaches are investigated; a global approach and a subsystem approach. For both
approaches, the physical configuration (i.e., chain-like, bridge-like, etc.) and the system
characteristics (i.e., linear vs. nonlinear) are assumed to be unknown. The input-output
configuration of the neural networks was chosen to enable the integration with the ODE
solver for prediction purposes, as well as to use them for change-detection purposes. For
the global approach, it is seen that the computational models perform relatively well with
45
linear systems (with slight instabilities during the validation tests). However, the models
su↵er from severe instabilities with nonlinear systems. In the global approach, the entire
system is represented by one network which makes the training and the integration with
the ODE solver more challenging from the computational aspect. The errors propagat-
ing from the network estimation and the truncation error of the RK45 contribute to the
observed instabilities. These instabilities might become more severe when dealing with
densely instrumented systems (i.e., more sensors/DOFs) where it becomes even a bigger
challenge to represent the system with one network. It is worth mentioning that mod-
eling a nonlinear SFSI system is more challenging than a linear system, which in turn
contributes to the increased level of divergences observed with the nonlinear models.
The computational models developed using the subsystem approach on the other
hand, prove to be more reliable and robust. Both the testing and the validation results
presented in this study show that excellent prediction accuracy was achieved for both
linear and nonlinear systems. An important feature of the subsystem approach is that
it uses a combination of several networks (as opposed to a single network in the global
approach) to represent the system. As a result, it is much easier to train networks with
significantly fewer inputs and outputs and achieve netter prediction results. Furthermore,
the complexity of the networks do not change with increased DOFs in system. It should
be noted that both the global and the subsystem approaches are general (i.e., applicable
without restrictions on topology). In fact, the generality of the subsystem approach
was validated on a 3-span finite-element model of a bridge structure and the results are
presented in Appendix A.
For the second part of the study, the networks trained with the global approach were
further used for change-detection in the SFSI systems. For each SFSI system, six neural
46
networks were trained: the first three networks were trained with level-1 excitations from
three di↵erent earthquakes, and the rest were trained with level-6 excitations. It is seen
that for each excitation level (i.e., level-1 and level-6), the results from three uncorrelated
networksarerelativelysimilar. Itisalsoseenthatthedetectednormalizedchangesmatch
the normalized changes in the system natural frequency of the SBSI and SPSI systems.
For the fixed-base model, it is relatively straightforward to interpret the detected changes
and correlate them with the physical damages (changes) observed on the superstructure
after each test. However, for the more complex systems (i.e., involving soil, foundation,
andsuperstructure)itisnotverytrivialtolinkthedetectedchangeswithaspecificchange
inthephysicalsystem. Inthisstudy, thiswasbestillustratedwiththeSPSIsystemwhere
no changes were observed on the superstructure after the first 4 levels of excitation, but
significant changes were detected (during the first 4 levels) in the overall system using
the proposed approach. While this demonstrates the e↵ectiveness and the sensitivity of
the approach in capturing the overall change in complex SFSI systems (including changes
in hidden components), it emphasizes the danger of potential misinterpretation of the
results for these type of systems. Therefore for SFSI systems, one should be extremely
careful in attributing detected changes to physical damage on the superstructure, as the
they might be due to changes in soil, foundation, superstructure, or the interaction e↵ects
between all the components.
47
II.6 Conclusion
In this study, data-driven reduced-order nonlinear computational models are developed
for response prediction in soil-foundation-superstructure systems based on data from rel-
atively large-scale experiments performed on three SFSI systems. Two modeling ap-
proaches are proposed; a global approach and a subsystem approach. Both approaches
are based on incorporating trained neural network(s) within ODE solvers to achieve a dy-
namic time-marching prediction framework. Both approaches are general and applicable
on wide range of structural systems. The approaches are tested and validated with exper-
imental data from three di↵erent SFSI systems. The advantages and the disadvantages
of the approaches are discussed and it is shown that the models created with the subsys-
tem approach are more robust and reliable for both linear and nonlinear systems. The
trained networks from the global approach are further used for change-detection in the
three SFSI systems. It is shown that while it is relatively easy to physically interpret the
detected changes in the fixed-base system, it is a challenging task to correlate the quan-
tified detection results with visual observations in the pile-foundation system due to the
complexity of the system and the interaction e↵ects between various components of the
system. Thedangerofpotentialmisinterpretationoftheneuralnetworkchange-detection
results in multi-component interacting systems is emphasized. Analyzing the three sys-
tems modeled at Tongji University, along with the various earthquake records, allowed a
quantitative assessment of the proposed methodologies and a thorough understanding of
the investigated approaches through several checks of validation and generalization.
48
II.7 Acknowledgment
This work was partially supported by the project (Grant No. 51178349) of the National
Natural Science Foundation of China, and the Kwang-Hua Fund for the College of Civil
Engineering, Tongji University.
49
Chapter III
Development and Validation of Nonlinear
Computational Models of Dispersed Structures
Under Strong Earthquake Excitation
III.1 INTRODUCTION
III.1.1 Background
S
TRUCTURAL health monitoring and system identification of multi-span bridges
is particularly important because of the vital role they play in civil infrastructure
systems. System identification of a complex multi-input-multi-output (MIMO) system
during strong earthquakes allows the identification of dominant dynamic features of the
system from structural measurements. The development of data-driven reduced-order
mathematical models of such systems is important for several control and health moni-
toring applications. Furthermore, constructing robust mathematical models that capture
the correct physics of the underlying system are important for computational purposes
to augment experimental studies. While there have been several studies in developing
50
reduced-order linear mathematical models, not much work has been performed in de-
veloping nonlinear reduced-order mathematical models based on vibration measurement
analysis of multi-span bridges. Many studies use ambient vibration data to perform
system identification of various structural systems. One of the advantages of ambient
measurements is that they can be recorded for a long time (hours or even days), resulting
in a very accurate equivalent linear estimates of the system parameters. On the other
hand, one disadvantage of earthquake records is their duration, which is usually relatively
short(2030seconds). Asaresult, thereisalackofsuitabledatasetsforfull-scalestruc-
tures under extreme loads, that are crucial for the proper characterization of nonlinear
behavior of such structures.
Several works have been performed in the identification of bridges. Figueiredo et al.
(2013) provided a background on the condition assessment of bridges by providing a sur-
vey on its past, present, and future, Smyth et al. (2003) performed system identification
of the Vincent Thomas suspension bridge using earthquake records, Siringoringo and
Fujino (2006) studied the observed dynamic performance of the Yokohama-Bay Bridge
from system identification using seismic records, Siringoringo and Fujino (2007) studied
the dynamic characteristics of a curved cable-stayed bridge identified from strong motion
records, Siringoringo and Fujino (2008a) applied system identification techniques to long-
span cable-supported bridges using seismic records, Siringoringo and Fujino (2008b) ana-
lyzedthecharacteristicsofsuspensionbridgeusingambientvibrationresponse,Siringoringo
et al. (2011) studied the dynamic characteristics of an overpass bridge in a full-scale
destructive test, Hu et al. (2012) investigated the continuous dynamic monitoring of a
lively footbridge for serviceability assessment and damage detection. Siringoringo and
51
Fujino (2012); Siringoringo et al. (2013) performed the response analysis of Yokohama-
Bay Bridge after the 2011 Great East Japan Earthquake. Other relevant works include
Abdel-Gha↵ar and Scanlan (1985a,b), Nagayama et al. (2005), Ceravolo and Tondini
(2012), Zhou et al. (2012), Koo et al. (2013), amongst others. There is a small number
of publications that treat the response of such nonlinear distributed-parameter systems
as multi-input-multi-output. Smyth et al. (2003) incorporated the MIMO approach in
the analysis of the Vincent Thomas bridge. The lack of such studies is not only due to
the complexity of the analysis when adopting the approach, but also due to lack of the
necessary sensor resolution on most of the bridges (not enough sensors) to perform such
analysis. Another important dynamic feature of a multi-span cable-stayed bridge is its
damping characteristics. An accurate damping ratio estimate is particularly important
from the design standpoint, as such an estimate cannot be easily obtained from engineer-
ing drawings. While equivalent linear models can provide relatively accurate estimates
of the damping ratio when modeled using ambient vibration data, it may not accurately
reflect the inherent damping encountered during strong earthquakes.
III.1.2 Motivation
This study analyzes the response of a relatively large (three-span) cable-stayed bridge,
Yokohama-Bay Bridge, to the 2011 Tohoku earthquake (the Great East Japan earth-
quake). A total of 10 datasets (1 main shock, and 9 strong aftershocks) are used in the
analysis. Response from 66 channels located at various parts of the bridge is analyzed.
System identification is preformed using well-known state-of-the-art methodologies. A
unique aspect of this study is the availability of relatively long strong earthquake records
(the main shock is 600 seconds), which significantly helps to improve the accuracy of the
52
vibration-signature based analysis and thus obtain an accurate nonlinear model of the
system. Another important feature of the study is the availability of extensive multi-
component measurements from a relatively dense sensor array (66 channels) that allows
a meaningful analysis of the system. The available measurements are used to develop
a reduced-order nonlinear mathematical model of the bridge. The mathematical model
is used to predict important system dynamic characteristics such as natural frequency,
damping ratio, and is also used to predict the response of the system to other earthquake
records. The robustness of the proposed model is assessed by comparing it to an equiva-
lent linear mathematical model, as well as by comparing the obtained system character-
istics to results from previous analyses from ambient monitoring of the Yokohama-Bay
Bridge. Given the lack of recorded response of large flexible bridges driven into nonlin-
ear response range, this study provides an excellent opportunity to assess the viability
of the developed MIMO nonlinear mathematical models in accurately characterizing the
Yokohama-Bay Bridge.
III.1.3 Scope
Withtheprecedinginmind, Section2describestheYokohama-BayBridge(YBB)includ-
ing its general characteristics, the sensor network, and the dynamic characteristics based
on ambient measurements. Section 3 describes the 2011 East Japan Earthquake includ-
ing the earthquake response of the Yokohama-Bay Bridge and an overview of the earth-
quake response data sets. In Section 4, system identification studies are presented. The
identification methods are discussed and the results based on equivalent-linear systems
are reported. Also, the nonlinearities in the YBB response are detected and quantified.
Section 5 presents the development of the reduced-order nonlinear mathematical models.
53
Architecture of neural networks used to identify the nonlinear forces are discussed, as well
as the training results from the neural networks. In Section 6, validation of the reduced-
order nonlinear mathematical models is performed. Sections 7 and 8 are discussion and
conclusions, respectively.
III.2 Yokohama-Bay Bridge (YBB)
III.2.1 Characteristics
The Yokohama-Bay Bridge was completed in 1988 and is located at the entrance of
Yokohama Harbor, see Fig. III.1. The left subfigure in Fig. III.2 shows the map of Japan
and the locations of Yokohama-Bay Bridge (denoted as A) and the epicenter of the 2011
East Japan Earthquake (denoted as B). The right subfigure in Fig. III.2 shows a zoomed
version of the map where YBB is denoted as A. The bridge is at a distance of about 42
km south of Tokyo and is a crucial part the Yokohama-Tokyo bay-shore expressway. It
is a continuous three-span cable-stayed bridge. The main girder consists of a double-deck
steel truss-box. The mid-span is 460 m and each side-span is 200 m. The bridge has two
H-shaped towers of 172 m height and 29.25 m width. The upper deck has 6 lanes and is
part of the Yokohama expressway bay-shore Route, while the lower deck has 2 lanes and
is part of the National Route.
III.2.2 Sensor Network
There are 85 acceleration sensors (66 of which are available and used for the analysis in
this study) installed at 36 locations throughout the bridge. The sensors on the middle
girder are located at a spacing of 115 m. Figure III.3 shows the location of the sensors
54
Figure III.1: View of the Yokohama-Bay Bridge.
Epicenter of 2011 East Japan
Earthquake
Yokohama-Bay Bridge
Tokyo
10 km
200 km
Figure III.2: Location of the Yokohama-Bay Bridge (Source: Google).
55
S9
P1 P2
P3 P4
T1
T3 T4
T2
S5R,L S4 S6
S7
S3
S2 S8
S9
S1
T5 T6
T8
T7
K4
K2
K1
K3
K7
K6
K5
K8
B3 B4
G1
460m 200m 200m
Z+
X+
S5R
S5L
S6R
S6L
S8L
S8R
S4 S2
S1 S3 S7
Y+
X+
To Homoku
To Oguro
Yokohama
Harbour
Pacific
Ocean
Triaxial Accelerometer (SA-355CT)
Biaxial (Y,Z) Accelerometer (SA-255CT)
Biaxial (X,Y) Accelerometer (SA-255CT)
Legend
Silt Layer
Hard-Soil Layer
T7L
T3L
T1
T3R
T5
S3
T7R
K4
K3
P2
Y+
X+
T8L
T4L
T2
T4R
T6
S7
T8R
K6
K5
P3
Yokohama
Harbour
Pacific
Ocean
B3
S1
K1
P1
K2 Y+
X+
B4
S9
K8
K7
P4

Figure III.3: Sensor map on Yokohama Brige.
installedontheYokohamaBridge. ThemeasurementsareinX(longitudinal),Y(lateral),
and Z (vertical) directions. Each sensor has a frequency measurement range from 0.05 to
35 Hz, and an accuracy of 15⇥ 10
3
amp/cm/s
2
. The sampling frequency of the records
used in this study is 100 Hz. As seen from Fig. III.3, sensors are attached on the piles of
the bridge which makes it a great opportunity to assess multi-input modeling approaches.
On the other hand, it is seen that there is lack of sensors on the bridge cables.
56
Table III.1: Identified natural frequencies from ambient measurements and FEA.
Mode Frequency (Hz)
Lateral-Vertical Ambient FEA
1
st
symmetric bending 0.27 0.28
(tower-girder same phase)
1
st
asymmetric bending 0.38 0.42
(tower anti phase)
1
st
asymmetric bending 0.68 0.70
(tower-girder anti phase)
2
nd
symmetric bending n/a 1.08
III.2.3 Natural Frequencies Based on Ambient Measurements
The natural frequencies of the bridge corresponding to the first few modes were identified
fromambientmeasurements. TheresultswerepublishedbytheMetropolitanExpressway
Public Corporation in 1991, and were later reported and compared to finite-element-
analysis (FEA) results by Siringoringo and Fujino (2006). The results are summarized in
Table III.1. As seen in the table, the frequencies are identified using sensors in the lateral
and vertical directions only. There is a good match between the FEA and the ambient
measurements results. The fundamental frequency corresponding to the 1
st
symmetric
bending was identified to be 0.27 Hz.
57
III.3 Great East Japan 2011 Earthquake
III.3.1 Overview
OnMarch11,2011,at14:46JapanStandardTime(JST),NorthEasternJapanwasstruck
by the Great East Japan Earthquake with a moment magnitude of 9.0. It was the most
powerful earthquake to have hit Japan, and the fifth most powerful earthquake in the
world since the modern recording began in early 1900s.
III.3.2 Earthquake Response of Yokohama-Bay Bridge
Yokohama-Bay Bridge is the second longest-span cable-stayed bridge in the Eastern part
of Japan. Japan Meteorological Agency (JMA) seismic intensity of more than 5 (out of
maximumscaleof7)wasrecordedatthebridgelocationduringthe2011GreatEastJapan
Earthquake. The epicenter of the earthquake was about 398 km away from the bridge
(see Fig. III.2), with focal depth of 24 km. The response from the bridge superstructure
was represented by large vibration in the transverse direction. The maximum girder
displacement and acceleration were recorded in the transverse direction at the center of
the mid-span of the girder (see sensor S5 in Fig. III.3). The values of the maximum
displacement and the maximum acceleration were 60 cm and 299.2 cm/s
2
,respectively.
The maximum transverse displacement in the towers was 54.6 cm (at location T1) and
the maximum transverse acceleration was 656.9 cm/s
2
(at location T2). See Fig. III.3
for the location of the sensors.
58
Table III.2: List of the earthquake records.
Earthquake ID Mw JMA Description Data Length (sec) Max Input Acc (cm/s
2
)
EQ1 9.0 5 Main shock 600 83.32
EQ2 7.7 4 Aftershock 1 480 35.27
EQ3 7.5 3 Aftershock 2 240 6.06
EQ4 6.5 2 Aftershock 3 60 3.84
EQ5 6.1 3 Aftershock 4 150 7.16
EQ6 6.7 3 Aftershock 5 120 3.11
EQ7 6.4 3 Aftershock 6 120 3.82
EQ8 6.2 4 Aftershock 7 60 5.31
EQ9 6.4 4 Aftershock 8 120 15.65
EQ10 6.1 3 Aftershock 9 120 5.62
III.3.3 Earthquake Response Data Sets
There are a total of 10 data sets available for the analyses in this study. Table III.2
presents a brief description of each available data set. As seen, there is 1 main shock and
9 aftershocks. It should be noted that the length of the response data corresponding to
the main shock is 600 sec. Furthermore, looking at the intensity of the aftershocks, it is
seen that some of the aftershocks are strong earthquakes in their own right.
Figure III.4 shows sample records in the transverse (Y) and the vertical (Z) directions
with the corresponding Fast-Fourier-Transform (FFT) plots. The sensors used in the
figure are sensor G1 (located in the ground - free field measurement) and sensor S5R
(located) at the center of the mid-span of the bridge-deck (girder). The strength of the
main shock (EQ1) and its length can be seen in Fig. III.4. As mentioned earlier, the
59
response of the girder in the lateral (transverse) direction at the center of the mid-span
is just below 300 cm/s
2
.
III.4 System Identification Studies
III.4.1 Overview of Some System Identification Methods
Three state-of-the-art system identification techniques are used to identify the modal pa-
rameters of the bridge. The methods are: (1) MNExT-ERA, (2) SRIM, and (3) LSID.
It should be noted that these methods are developed based on linear system theory and
the parameters obtained from them represent an equivalent-linear system. Given the
linearity assumption, only linear viscous damping is considered in the damping estima-
tions. Anyothersourceofinherentenergydissipationmechanismsuchas, cables, friction,
and connections, is modeled as part of the equivalent-linear damping. As a result, the
modal parameters obtained from these techniques are associated with both estimation
and modeling errors.
Abriefsummaryofeachmethodisprovidedinthissection. Themeasuredacceleration
data was sampled at 100 Hz with a corresponding Nyquist frequency of 50 Hz. The
frequencies of interest in this study are much less than 35 Hz, therefore the measured
acceleration data was band-pass filtered between 0.05 and 30 Hz. The results presented
in the next section are obtained using the main shock (EQ1) dataset.
60
0 200 400 600
−200
−100
0
100
200
Channel G1 − lateral (Y)
time (sec.)
Acceleration (cm/s
2
)
0 200 400 600
−200
−100
0
100
200
Channel S5R − lateral (Y)
time (sec.)
Acceleration (cm/s
2
)
0 200 400 600
−200
−100
0
100
200
Channel G1 − vertical (Z)
time (sec.)
Acceleration (cm/s
2
)
0 200 400 600
−200
−100
0
100
200
Channel S5R − vertical (Z)
time (sec.)
Acceleration (cm/s
2
)
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
FFT − Channel G1 − lateral (Y)
Frequency (Hz)
|Y(f)|
0 0.5 1 1.5 2
0
10
20
30
FFT − Channel S5R − lateral (Y)
Frequency (Hz)
|Y(f)|
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
FFT − Channel G1 − vertical (Z)
Frequency (Hz)
|Z(f)|
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
FFT − Channel S5R − vertical (Z)
Frequency (Hz)
|Z(f)|
Figure III.4: Sample time histories and corresponding FFT plots corresponding to EQ1.
61
Multiple-Reference Natural Excitation Technique Combined with Eigensys-
tem Realization Algorithm (MNExT-ERA)
The natural excitation technique is based on obtaining the cross-correlation function be-
tween the acceleration responses of two degrees of freedoms in the structure that is sub-
jected to a broadband excitation. The obtained cross-correlation function has the same
analytical form as the free-vibration of the structure James et al. (1993). The Eigensys-
temRealizationAlgorithm(ERA)canthenbeusedtoextractthemodalparametersfrom
the cross-correlation function Juang and Pappa (1985). While in the case of building-like
structures it is much easier to choose a single reference degree-of-freedom to calculate the
cross-correlation functions, it is much more challenging in bridges to choose a single refer-
ence DOF that assures the identification of all the dominant modes. Therefore, multiple
references (vector of references) were chosen in this study He et al. (2009). Six reference
channels were chosen for this study (S5R(X,Y,Z) and S6R(X,Y,Z)). Accelerations were
used from 49 channels on the superstructure in all three directions (i.e., X, Y, and Z).
Theresponsecross-correlationfunctionswereestimatedbycalculatingtheinverseFourier
transform of the corresponding cross-spectral density (CSD). CSD was estimated using
Welch’s averaged modified periodogram method of spectral estimation, using Hanning
windows with 50% overlap.
System Realization Using Information Matrix (SRIM)
In this approach, a state-space-based realization is used to identify the system observabil-
ity matrix from the information matrix consisting of input-ouput data correlation Juang
(1997). The system matrices are then identified, from which the modal parameters are
estimated. It should be noted that this is an input-output method, as opposed to the
62
Table III.3: Summary of the methodologies used for system identification.
Method Input/Output
MNExT-ERA Output-Only
SRIM Input-Output
LSID Input-Output
NExT-ERA method which is an output-only method. In this method, acceleration re-
sponsefrom49(output)channelsonthesuperstructurewereused,alongwithacceleration
response from 9 (input) channels (i.e., K1(X,Y,Z), K2(X,Y,Z), and K3(X,Y,Z)).
Time-Domain Least-Square Identification (LSID)
The time-domain least-square identification approach is an input-output method that
estimates the system matrices, appearing in the vector equation of motion, by reducing
the estimation error in a least-square sense, see Masri et al. (1987a,b). Further details on
the approach are presented in the upcoming sections. As in the SRIM method, 49 output
channels and 9 input channels are used for the identification purposes.
The methods are briefly summarized in Table III.3.
III.4.2 System Identification Results Based on Equivalent-Linear Sys-
tem
The results of the identification are summarized in Table III.4. The table shows the
frequencies and the damping ratios identified from the first 10 modes using the three
above-listed methods. It is seen that the frequency estimates from all three methods
63
are fairly consistent and within the same range. However, the identified damping ratios
vary quite a bit from one method to another. This is mainly due to the fact that all
three methods are based on linear system theory and the identified damping ratio is an
equivalent viscous damping and does not take into account other major sources of energy
dissipation such as materials, connections, cables, etc. Depending on the method used,
theerrorcanbeduetoothercontributingfactorsaswell. InthecaseofMNExT-ERA,one
major assumption is that the excitation is broadband (white noise) signal. This is clearly
not the case with the main shock (EQ1) dataset, therefore it is a violation of one of the
assumptions in the method, and might be a contributing factor in the estimation error.
Nevertheless, with all three methods, consistent frequency estimations are obtained. It
shouldalsobenotedthatsincetheanalysesareperformedusingdatainallthreedirections
(X,Y,andZ),mostoftheidentifiedmodalparametersarecoupled, anddonotnecessarily
represent one distinct direction.
III.4.3 Detection and Quantification of Nonlinearities in Response
This least-square time-domain system identification (LSID) approach depends on two
stages. The first stage is the identification of the reduced-order equivalent-linear model.
During the second stage, instead of treating the unmodeled response as an error, it is
assumed to be the non-linear dynamics yet to be modeled. The equation of motion can
be written as follows:
M
1
11
C
11
˙x
1
(t)+M
1
11
K
11
x
1
(t)+M
1
11
M
10
¨x
0
(t)+M
1
11
C
10
˙x
0
(t)+M
1
11
K
10
x
0
(t)=I¨ x
1
(t)
(III.1)
64
Table III.4: System identification results.
Mode MNExT-ERA SRIM LSID
! (Hz) ⇣ % ! (Hz) ⇣ % ! (Hz) ⇣ %
1 0.319 4.81 0.316 1.80 0.320 1.76
2 0.422 4.85 0.384 3.20 0.335 1.63
3 0.467 5.69 0.481 2.59 0.417 1.58
4 0.570 3.18 0.521 7.09 0.478 1.96
5 0.780 4.24 0.728 2.61 0.557 3.77
6 0.885 6.48 1.006 3.33 0.667 6.63
7 1.006 3.57 1.141 2.89 0.755 5.49
8 1.117 1.89 1.254 1.85 0.827 2.63
9 1.336 5.64 1.615 3.55 0.931 1.81
10 1.759 1.98 1.774 1.38 1.010 2.88
65
Table III.5: The abbreviated notation for the identified system matrices.
Matrix Product M
1
11
C
11
M
1
11
K
11
M
1
11
M
10
M
1
11
C
10
M
1
11
K
10
Abbreviation
2
A
3
A
4
A
5
A
6
A
Dimension n
1
⇥ n
1
n
1
⇥ n
1
n
1
⇥ n
0
n
1
⇥ n
0
n
1
⇥ n
0
wherex
1
(t)=[x
11
(t),...,x
1n
1
(t)]
T
, ˙x
1
(t)=[˙ x
11
(t),..., ˙ x
1n
1
(t)]
T
,and ¨x
1
(t)=[¨ x
11
(t),...,¨ x
1n
1
(t)]
T
represent the measured displacement, velocity, and acceleration, respectively. While,
x
0
(t)=[x
01
(t),...,x
0n
0
(t)]
T
, ˙x
1
(t)=[˙ x
11
(t),..., ˙ x
1n
1
(t)]
T
,and ¨x
1
(t)=[¨ x
11
(t),...,¨ x
1n
1
(t)]
T
represent the measured base (ground) displacement, velocity, and acceleration, respec-
tively. The symbol n
1
represents the number of degrees-of-freedom of the system and the
symbol n
0
represents the number of base excitations. The matrices M
11
,C
11
, and K
11
are the discretized system matrices corresponding to the mass, damping and the sti↵ness
matrices. The system matrices can be represented using a abbreviated notation as in
Table III.5:
Applying Eq. III.1 to discrete time-steps, t=[t
1
,...,t
N
], where N is the time step
index, yields parallel matrix equations as follows:
M
1
11
C
11
˙x
1
(t
1
)+M
1
11
K
11
x
1
(t
1
)+M
1
11
M
10
¨x
0
(t
1
)+M
1
11
C
10
˙x
0
(t
1
)+M
1
11
K
10
x
0
(t
1
)=I¨ x
1
(t
1
)
.
.
.
M
1
11
C
11
˙x
1
(t
N
)+M
1
11
K
11
x
1
(t
N
)+M
1
11
M
10
¨x
0
(t
N
)+M
1
11
C
10
˙x
0
(t
N
)+M
1
11
K
10
x
0
(t
N
)=I¨ x
1
(t
N
)
(III.2)
66
The components of
j
A matrices mentioned above (i.e., the components of the mass,
damping and sti↵ness matrices) are unknowns and need to be identified. The above
equation can be written in the following form:
ˆ
Rˆ ↵ =
ˆ
b (III.3)
ˆ
R =
2
6
6
6
6
6
6
6
6
6
6
6
6
4
R 00 ··· 0
0 R 0 ··· 0
00 R 00
.
.
.
.
.
. ···
.
.
.
.
.
.
00 ··· ··· R
3
7
7
7
7
7
7
7
7
7
7
7
7
5
(III.4)
ˆ ↵ =

↵ 1
↵ 2
···↵ n1

T
(III.5)
ˆ
b =

b
1
b
2
···b
n1

T
(III.6)
where ˆ ↵ contains all the unknown parameters to be identified,
ˆ
R is equivalent to a
coecient matrix assembled from data measurements, and
ˆ
b is the right-hand-side of
Eq. III.1. The coecient can be calculated using ˆ ↵ =
ˆ
R
†
ˆ
b,where † represents the
psuedoinverse of a matrix. The identified equivalent linear system matrix coecients can
be inserted back into equation III.3 to obtain an estimate of the acceleration vector
ˆ
b,
which can be denoted as b
est
.Then,thedi↵erence(
ˆ
bb
est
) is treated as the nonlinear
residual F
nl
(instead of treating it as error). The residual can be modeled using several
techniques. Some of these techniques include non-parametric methods such as neural
67
networks, basis function fitting, amongst others. Neural networks are used in this study
to model the nonlinear residual.
Itispossibletoobtainthemodalpropertiesofthestructureusingtheidentifiedsystem
matrices M
1
11
C
11
and M
1
11
K
11
. It should be noted that in practical base-excitation
problemsthemassmatrixM
11
isunknown. Themodalfrequencies, dampingcoecients,
and mode shapes can be derived by solving a classical eigenvalue problem as follows:
Az = z (III.7)
A =
2
6
4
0I
M
1
11
K
11
M
1
11
C
11
3
7
5 (III.8)
Sincetheabovesystemmatricesareidentified,thematrixAisknownandhasadimension
of 2n
1
⇥ 2n
1
. The eigenvalues k
and the eigenvectors z
k
may be complex numbers and
come in complex conjugate pairs. The physical modal frequencies, !
i
, and the modal
damping coecients, ⇣ i
, are related to the obtained eigenvalues as follows:
!
i
=
p
<( 2i1
)
2
+=( 2i1
)
2
=
p
<( 2i
)
2
+=( 2i
)
2
,i=1,···,n
1
(III.9)
⇣ i
=
<( 2i1
)
p
<( 2i1
)
2
+=( 2i1
)
2
=
<( 2i
)
p
<( 2i
)
2
+=( 2i
)
2
,i=1,···,n
1
(III.10)
where <(.) denotes the real part of a complex number and =(.) denotes the imaginary
part of a complex number.
The time-domain identification approach was implemented using data in the longitu-
dinal, lateral, and vertical directions. A total of 58 channels were used for the analysis.
68
A total of 49 channels from the superstructure (i.e., n
1
= 49), and a total of 9 channels
were used from the base and the foundation of the bridge (i.e., n
0
= 9).
Figures III.5 and III.6 show the system identification results from a representative
channel (S5RY). Figure III.5 corresponds to the results from dataset (EQ1), and Figure
III.6correspondstodataset(EQ2). Asseeninbothfigures, theequivalentlinearestimate
captures the main oscillations of the measured acceleration very well. The di↵erence
between the estimate and the measured acceleration is treated as the nonlinear residual
and is plotted on the third row of each figure.
Figure III.7 shows the identified fundamental natural frequencies (!) and the cor-
responding damping ratios (⇣ ) for all available datasets. The left sub-figure shows the
identified natural frequencies for each dataset. It is seen that the highest fundamental
naturalfrequencyisobservedduringthemainshock(datasetEQ1). Thisisnotsurprising,
as during the main shock (the strongest ground motion), the system will undergo most
of the nonlinearities which might cause increase in the sti↵ness (induced partly by the
bridge cables) and correspondingly increase in the natural frequency. It should be noted
that the intensity of the ground motion does not decrease gradually from aftershock EQ2
through EQ10. Some aftershocks are stronger than the other, the epicenter is closer to
the bridge, the strong motion part of the record is longer in some records. Because of all
thementionedfactors, itisdiculttoobserveatrendintheidentifiednaturalfrequencies
corresponding to aftershocks EQ2 to EQ10. However, it is seen in the left sub-figure that
all the identified frequencies during the aftershocks are less than the frequency identified
during the main shock, which is consistent with the expectations discussed above. Is it
seen that the average natural frequency identified during the aftershocks is about 0.27
69
100 110 120 130 140 150 160 170 180 190 200
−200
−100
0
100
200
Time (sec.)
Acceleration (cm/s
2
)
Channel S5R − lateral (Y)

Measured Acceleration
100 110 120 130 140 150 160 170 180 190 200
−200
−100
0
100
200
Time (sec.)
Acceleration (cm/s
2
)

Equivalent Linear Estimate
100 110 120 130 140 150 160 170 180 190 200
−200
−100
0
100
200
Time (sec.)
Acceleration (cm/s
2
)

Nonlinear Residual
fig_leastSquareID_y2013_m2_d21_h12_m32_s29.pdf: corresponding to: EQ1
Figure III.5: In the top figure, the measured channel S5R - lateral (Y) acceleration is shown.
Below this is the linear model estimate using least-square method. The third figure shows the
non-linear residual (i.e., the di↵erence of the previous two signals. The main shock (EQ1) dataset
is used.
70
100 110 120 130 140 150 160 170 180 190 200
−200
−100
0
100
200
Time (sec.)
Acceleration (cm/s
2
)
Channel S5R − lateral (Y)

Measured Acceleration
100 110 120 130 140 150 160 170 180 190 200
−200
−100
0
100
200
Time (sec.)
Acceleration (cm/s
2
)

Equivalent Linear Estimate
100 110 120 130 140 150 160 170 180 190 200
−200
−100
0
100
200
Time (sec.)
Acceleration (cm/s
2
)

Nonlinear Residual
fig_leastSquareID_y2013_m2_d21_h12_m38_s5.pdf: corresponding to: EQ2
Figure III.6: In the top figure, the measured channel S5R - lateral (Y) acceleration is shown.
Below this is the linear model estimate using least-square method. The third figure shows the
non-linear residual (i.e., the di↵erence of the previous two signals. The first aftershock (EQ2)
dataset is used.
71
EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 EQ8 EQ9 EQ10
0.25
0.26
0.27
0.28
0.29
0.3
0.31
0.32
0.33
Earthquake ID
Fundamental Frequency (Hz)
EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 EQ8 EQ9 EQ10
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Earthquake ID
Damping Coefficient (%)
Figure III.7: Identified fundamental natural frequencies and damping ratios from all 10 datasets.
Hz, which matches the identified fundamental frequency obtained from ambient vibration
tests reported in the literature, Siringoringo and Fujino (2008a).
The right sub-figure in Fig. III.7 shows the identified damping ratios from all 10
datasets. It is seen that the damping ratio corresponding to the main shock is ⇣ =
1.5%. Considering the extent of the inherent nonlinearities during the main shock (EQ1),
the equivalent linear estimate of the damping ratio is much lower than expected. The
estimates using the aftershock datasets (where the contribution of nonlinearities are less
than the main shock) seem to be reasonable. In the upcoming sections, the accuracy of
the obtained damping results from EQ1 will be validated using a linear computational
model and the significance of including the nonlinear terms will be investigated using the
nonlinear computational model.
72
III.5 DevelopmentOfReduced-OrderNonlinearComputa-
tional Models
There is a paucity of publications dealing with measurements from large-scale real struc-
turalsystemsfordevelopingdata-drivenreduced-ordernonlinearmodelsofcomplexbridge
systems. In this study, such reduced-order nonlinear models were developed by using a
combination of parametric and nonparametric approaches, and employing both physi-
cal and computational tools. Two models are developed and investigated in this study:
an equivalent linear model and a nonlinear model. In both cases, the system matrices
identified using the least-square approach are used.
III.5.1 Equivalent Linear Model
In the equivalent linear model, the Fourth-Order Runge-Kutta (RK4) method is used to
solve the di↵erential equation governing the motion of the bridge given by:
I¨ x
1
(t)=
2
A˙ x
1
(t)
3
Ax
1
(t)
4
A¨ x
0
(t)
5
A˙ x
0
(t)
6
Ax
0
(t) (III.11)
where the system matrices are identified using the time-domain identification approach
explained earlier, the base motion (x
0
(t), ˙x
0
(t), ¨x
0
(t)) are earthquake-specific measure-
ments, and the bridge motion (x
1
(t), ˙x
1
(t)) is calculated using the ordinary di↵erential
equation (ODE) solver, by providing their respective initial conditions (x
1
(0), ˙x
1
(0)).
73
III.5.2 Noninear Model
In the nonlinear model, the Fourth-Order Runge-Kutta method is also used to solve the
di↵erential equation governing the bridge motion, given by:
I¨ x
1
(t)=
2
A˙ x
1
(t)
3
Ax
1
(t)
4
A¨ x
0
(t)
5
A˙ x
0
(t)
6
Ax
0
(t)F
nl
(t) (III.12)
The additional term in the right hand side of the equation (F
nl
(t)) is the nonlinear
residual defined as the di↵erence between the measured acceleration and the equivalent
linear estimate obtained by the least-square approach. In order for this nonlinear (new)
termtobeusedinatime-marchingmodulesuchastheRunge-Kutta,ithastobemodeled
in a robust way so that it is sensitive to various earthquake records. In this study, the
neural network approach is used to model the nonlinear part of the response.
Identification of Nonlinear Forces Using Neural Networks
A typical neural network architecture used in this study is shown in Fig. III.8. The
neural network has an input layer, one hidden layer, and an output layer. The hidden
layer has 15 hidden neurons. The hidden layer has hyperbolic tangent sigmoid transfer
functions, while the output layer has a linear transfer function. Initially, the inputs of
the neural network were chosen to be the displacements (y
k
), velocities ( ˙y
k
), and base
accelerations (¨s
k
) at time t
k
. The outputs are the nonlinear residuals (
ˆ
Fnl
k
). However,
giventhecomplexdynamicsystemsubjectedtoseverenon-stationaryexcitations,delayed
inputswereusedtoincorporatetheresponsefromthesystemmemory(i.e., fromprevious
time steps), hence to enhance the training of the neural network and better capture the
nonlinear characteristics of the system. As a result, the nonlinear residual at delayed
74
time step (
ˆ
Fnl
k1
) is provided as an additional input. The components of the input and
output layers can be seen in Fig. III.8.
net
2
RK4
G
n
G
1
G
2
Error
m
1
m
2
net
1
net
n
Neural Network
f
1
f
1
f
1
ODE Solver
y
1
y
2
x
n
˙ x
n
˙ y
2
˙ y
1
Initial
Conditions
Update
Hidden Layer
Input Layer
Output Layer
2
3
4
15
W
b
Sigmoid
Transfer
Function
1
W
b
Linear
Transfer
Function
Displacement
Base
Acceleration
Velocity
Nonlinear
Force
Hidden Layer
Input Layer
Output Layer
2
3
4
15
W
b
Sigmoid
Transfer
Function
1
W
b
Linear
Transfer
Function
ˆ
Fnl
k
Fnl
k 1
y
k
˙ y
k
¨ s
k
Figure III.8: Architecture of the neural network used in this study.
The neural network was trained using the Levenberg-Marquardt backpropagation al-
gorithm (see Section II.3.3). A sample neural network output is shown in Fig. III.9. The
firstrowshowstheentirerecordofthenonlinearresidualfromchannelS5R(Y)superposed
on the corresponding neural network estimate. The second row shows the time-history
of the error (i.e., the di↵erence between the nonlinear residual and the neural network
estimate). The third row depicts a 5 sec. segment from the plot in the first row. As seen
in the figure, the neural network output and the target nonlinear residual force match
very well.
75
0 100 200 300 400 500 600
−100
−50
0
50
100
Nonlinear Residual (cm/sec
2
)
Time (sec.)
CH S5R(Y) RMS = 0.39646

Nonlinear residual
NN output
0 100 200 300 400 500 600
−100
−50
0
50
100
Time (sec.)
Error e(t) (cm/sec
2
)
50 50.5 51 51.5 52 52.5 53 53.5 54 54.5 55
−20
−10
0
10
20
Time (sec.)
Nonlinear Residual (cm/sec
2
)

Nonlinear residual
NN output
target_output_NN_y2013_m2_d21_h12_m53_s22.pdf: corresponding to: FNL
Figure III.9: The neural network estimates of the nonlinear residual. The first row represents
the entire record of the nonlinear residual from channel S5R(Y) superposed on the corresponding
neuralnetworkestimate. Thesecondrowshowstheerrore(t)plot. Thethirdrowshowsasegment
of 5 sec. from the same record.
76
III.6 ValidationOfReduced-OrderNonlinearComputational
Models
III.6.1 Overview of Time-Marching Approaches for Dynamic Response
Calculations
Oneofthemostcommonandwell-establishedtime-marchingapproachestosolveordinary
di↵erential equations (ODEs) is the fourth-order Runge-Kutta Method. In this approach,
the value at the next time-step is determined based on the value at the present time-
step and the weighted average of four increments, where each increment is a function of
the integration step-size and the estimated slope specified by the right-hand side of the
di↵erential equation to be solved. In this study, the integration step-size was chosen to
be 1/10
th
of the sampling time-step. The fourth-order Runge-Kutta Method was used
to solve Eq.III.11 (for the linear mathematical model) and EQ. III.12 (for the nonlinear
mathematical model).
Inthecaseofthenonlinearmathematicalmodel, theimplementationoftheprocedure
is particularly challenging because of the two sources of error involved in the procedure:
the neural network estimation error, and the truncation error associated with the ODE
solver at each time-step. As mentioned earlier, the neural network performance was
enhanced by providing the nonlinear residual estimate at time k,(
ˆ
Fnl
k
), as an additional
input to the network at time k+1, along with ˆ y
k+1
,
ˆ
˙ y
k+1
, and ¨ s
k+1
. While the truncation
error of the ODE solver was decreased by using a relatively smaller integration step-size
(i.e., t=0.001). A detailed flowchart of the nonlinear computational model is shown in
Fig. III.10. AsseeninFig. III.10, ateachtime-stepthenonlinearforceisestimatedusing
the neural network and added to the right-hand side of Eq. III.12 along with the velocity,
77
ˆ y
k
ˆ
˙ y
k
¨ s
k
ˆ
Fnl
k
Neural
Network
net
2
RK4
Hidden Layer 1 Hidden Layer 2
Input Layer
Output Layer
G
n
G
1
G
2
Error
m
1
m
2
net
1
1
2
3
2
3
net
n
Gn
˙ y
n
y
n
f
1
Neural Network
W
b
Sigmoid
Transfer
Function
1
W
b
Linear
Transfer
Function
f
1
f
1
f
1
ODE Solver
y
1
y
2
x
n
˙ x
n
˙ y
2
˙ y
1
Initial
Conditions
Update
ˆ
Fnl
k 1
ˆ
Fnl
k 1
=
ˆ
Fnl
k
ODE
Solver
ˆ y
k+1
ˆ
˙ y
k+1
ˆ y
k
=ˆ y
k+1
ˆ
˙ y
k
=
ˆ
˙ y
k+1
¨ s
k
=¨ s
k+1
Update
Figure III.10: Flowchart of the developed nonlinear computational model.
displacement, and base motion, then the ODE solver is used to solve the equation and
estimate the response at the next time-step.
III.6.2 Assessing the Accuracy of Linear vs Nonlinear Dynamic Re-
sponse Calculations
Inordertoassesstheviabilityofthedevelopedlinearandnonlinearcomputationalmodels,
testing and validation is performed using various available earthquake records. During
the testing stage, the same earthquake data used in creating the models is fed back to
the models, and the displacement response of the system is compared to the available
measurements. During the validation stage, di↵erent earthquake data is used (i.e., ones
that have not been used in modeling the equivalent-linear part or the nonlinear residual
part) to see how well these computational models generalize when unknown datasets are
78
introduced. Just like the testing stage, the displacement responses obtained from the
computational models are compared to the corresponding measurements.
Testing
Figures III.11(a) and III.11(b) show the results from the linear and the nonlinear models,
respectively. Both figures show the results from two representative channels; S4(Y) and
S7(Y), respectively. In each subfigure, 100 sec. segment of the displacement response
estimate is superposed on the corresponding response from the measured data. It is seen
from Fig. III.11(a) that the results from the linear model show an excellent match with
the target displacement for about 70 sec. However, after 70 seconds some instabilities are
observed in both channels. Similar results were obtained from the rest of the channels
as well. Figure III.11(b) shows the results from the nonlinear model. It is seen that the
estimates match the target displacements fairly well. Furthermore, as opposed to the
linear model, no instabilities are observed throughout the record. Similar results were
observed in the other channels.
Validation and Generalization
Figures III.12(a) and III.12(b) show the validation results, where new earthquake data
(unknowntothemodels)arefedtothecomputationalmodels. Theresultsshownarefrom
dataset (EQ2). Similar trends are observed as in the testing stage. The estimates from
the linear model are good for about 20 seconds, then instabilities are observed in most
of the channels. While the results from the nonlinear model are stable throughout the
entire record, and match the main oscillations of the target displacement records fairly
79
100 110 120 130 140 150 160 170 180 190 200
−40
−20
0
20
40
Channel S1(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−40
−20
0
20
40
Channel S4(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−40
−20
0
20
40
Channel S7(Y)
Displacement (cm)
Time (sec.)
fig_NN_RK_FNL_y2013_m1_d27_h21_m58_s32.pdf: corresponding to: EQ1EQ1
(a) Linear model.
100 110 120 130 140 150 160 170 180 190 200
−40
−20
0
20
40
Channel S1(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−40
−20
0
20
40
Channel S4(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−40
−20
0
20
40
Channel S7(Y)
Displacement (cm)
Time (sec.)
fig_NN_RK_Linear_y2013_m1_d27_h21_m58_s30.pdf: corresponding to: EQ1EQ1
(b) Nonlinear model.
Figure III.11: Displacement time history estimates from the computational model (dotted line)
superposed on the corresponding displacements from the measured data (solid line). The rows
correspond to channels S4(Y) and S7(Y), respectively. The computational model is created using
EQ1 and is tested using EQ1. A segment of 100 sec. is shown from the strong motion part of the
records.
80
well. Similar trends were observed when feeding other earthquake data (EQ3 through
EQ10) to the computational models.
III.6.3 Dependance of Bridge Damping Parameters on Nature of Com-
putational Model
In order to identify the fundamental natural frequency (!
1
) and the corresponding damp-
ing ratio (⇣ 1
) using the developed linear and nonlinear computational models, new syn-
thetic base excitations were generated. The first 50 seconds of the new records are iden-
tical to the ones recorded during the main shock (EQ1), then for the rest of the record,
the excitation is artificially set to be zero. A sample excitation is shown in Fig. III.13.
Tracking the free-vibration of the superstructure during zero-excitation period (i.e., af-
ter the initial 50 seconds), one can observe the fundamental natural frequency and can
compute the damping ratio using the logarithmic decrement approach using =lnx
1
/x
2
and ⇣ =/
p
(2⇡ )
2
+ 2
,where x
1
and x
2
are two consecutive peaks in the free-vibration
response, and ⇣ is the desired damping ratio.
The displacement response estimates from channel S5R(Y) (located at the mid-span
of the bridge-deck) are shown in figures III.14(a) and III.14(b). Fig. III.14(a) shows
the displacement estimate from the linear model, and Fig. III.14(b) shows the estimate
from the nonlinear model. The time between two consecutive peaks ( t
i
) is calculated
for four cycles during the free-vibration part of the record (i.e., after t = 50sec.). Then,
the inverse of the average is calculated to obtain an estimate of the fundamental natural
frequency
ˆ
f=4/( t
1
+ t
2
+ t
3
+ t
4
). The natural frequency results obtained from
the linear and the nonlinear models are consistent with each other and are
ˆ
f
ln
=0.328
(Hz) and
ˆ
f
nln
=0.326 (Hz), where
ˆ
f
ln
represents the result from the linear model and
81
100 110 120 130 140 150 160 170 180 190 200
−20
−10
0
10
20
Channel S1(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−20
−10
0
10
20
Channel S4(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−20
−10
0
10
20
Channel S7(Y)
Displacement (cm)
Time (sec.)
fig_NN_RK_Linear_y2013_m1_d27_h21_m53_s2.pdf: corresponding to: EQ1EQ2
(a) Linear model. 100 110 120 130 140 150 160 170 180 190 200
−20
−10
0
10
20
Channel S1(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−20
−10
0
10
20
Channel S4(Y)
Displacement (cm)
100 110 120 130 140 150 160 170 180 190 200
−20
−10
0
10
20
Channel S7(Y)
Displacement (cm)
Time (sec.)
fig_NN_RK_FNL_y2013_m1_d27_h21_m53_s4.pdf: corresponding to: EQ1EQ2
(b) Nonlinear model.
Figure III.12: Displacement time history estimates from the computational model (dotted line)
superposed on the corresponding displacements from the measured data (solid line). The rows
correspond to channels S4(Y) and S7(Y), respectively. The computational model is created using
EQ1 and is validated using EQ2. A segment of 100 sec. is shown from the strong motion part of
the records.
82
10 20 30 40 50 60 70 80
−4
−3
−2
−1
0
1
2
3
4
5
Channel K1(Y)
Acceleration (cm/s
2
)
Time (sec.)
Figure III.13: A segment of the base excitation used for the identification of the damping ratio.
ˆ
f
nln
represents the result from the nonlinear model. These results also match previously
reported results that are calculated using di↵erent system identification techniques.
The damping ratio is calculated for four consecutive cycles, then the average is cal-
culated. The damping ratio obtained for the linear model is
ˆ
⇣ ln
=1.63% and for the
nonlinear model is
ˆ
⇣ nln
=4.26%. This di↵erence in the damping ratio between the linear
andthenonlinearmodelcanalsobeobservedbylookingatfiguresIII.14(a)andIII.14(b).
Thedecay in theresponsefromthelinear model (Fig. III.14(a)) is not clearly visible(i.e.,
ˆ
⇣ ln
=1.63%), while the decay in the response from the nonlinear model (Fig. III.14(b))
can be seen clearly (i.e.,
ˆ
⇣ nln
=4.26%).
Althoughtheresultsfromthelinearcomputationalmodelagreeswiththeleast-square
identification result (i.e., ⇣ =1.5%), both values only represent the equivalent linear part
ofthesystemanddonottakeintoaccountthenonlinearpartoftheresponse. Theaddition
ofthenonlinearresidualinthenonlinearcomputationalmodelisreflectedintheobtained
corresponding damping ratio (i.e.,
ˆ
⇣ nln
=4.26%). Although both computational models
provideanaccurateestimateofthenaturalfrequency, thenonlinearcomputationalmodel
is a better representative of the complex nonlinear system under investigation. When the
bridgeundergoessignificantnonlinearforces,anequivalentlinearmodelmightnotcapture
83
40 45 50 55 60 65 70
−2
−1
0
1
2
Channel S1(Y)
Displacement (cm)
40 45 50 55 60 65 70
−2
−1
0
1
2
Channel S5R(Y)
Displacement (cm)
40 45 50 55 60 65 70
−2
−1
0
1
2
Channel S7(Y)
Displacement (cm)
Time (sec.)
fig_free_vib_linear_y2013_m1_d27_h21_m43_s56.pdf: corresponding to: EQ1EQ1
(a) Linear model.
40 45 50 55 60 65 70
−2
−1
0
1
2
Channel S1(Y)
Displacement (cm)
40 45 50 55 60 65 70
−2
−1
0
1
2
Channel S5R(Y)
Displacement (cm)
40 45 50 55 60 65 70
−2
−1
0
1
2
Channel S7(Y)
Displacement (cm)
Time (sec.)
fig_free_vib_nonlinear_y2013_m1_d27_h21_m43_s57.pdf: corresponding to: EQ1EQ1
(b) Nonlinear model.
Figure III.14: A segment of the displacement time history estimate corresponding to channel
S5R(Y).
the full system dynamic characteristics (especially, damping ratio). The inclusion of the
nonlinear residual term can provide a more accurate estimate of the damping ratio. The
results further emphasize the importance of having an accurate approach for quantifying
the damping due to variety of nonlinear features in the YBB response. Furthermore,
the demonstrated approach is useful as a general methodology for other structures (not
necessarily bridges).
III.7 Discussion
Inthefirstpartofthisstudy,measurementsfrom10earthquakesandthecorresponding66
channels were organized in an ecient dataset to be used for various types of analyses in
the context of structural health monitoring, system identification, and damage detection.
During this process, the acceleration measurements were first band-pass filtered, then
integrated to obtain the corresponding velocity and displacement signals.
84
The initial analysis involved using state-of-the-art system identification methodolo-
gies to identify the modal parameters of the system including the natural frequencies and
equivalent viscous damping ratios. Both input-output and output-only methods were
used. The three methods were MNExT-ERA, SRIM, and LSID. All three methods are
based on linear system theory and are therefore associated with both modeling and es-
timation errors. It was found that the identified natural frequencies are comparable to
each other and are in a range consistent with previously reported results in the literature.
However, the identified damping ratios from the three methods were not consistent. This
is mainly due to the fact that the identified damping ratios represent the linear viscous
damping and do not take into account other sources of energy dissipation. In a complex
system, such as the Yokohama-Bay Bridge, there are multiple sources of energy dissi-
pation, such as local geometric nonlinearities, tower-to-deck connections, nonlinearities
associated with the cables, etc. Therefore, it is seen that modeling the bridge as a linear
system, is not an adequate approach for the purposes of damping ratio estimation.
Asaresult,therewasaneedtodevelopareduced-ordercomputationalmodelthatnot
only accurately represents the bridge, but also takes into account the contribution of the
associated severe nonlinear forces that occur during major earthquakes, such as the ones
beinganalyzedinthisstudy. Ahybridmodelingapproachisproposedinthisstudy,where
the equivalent linear part is modeled using a well-known parametric least-square identifi-
cationapproach,whiletheidentifiednonlinearforcesaremodeledusingthenonparametric
neural network approach. The combined contributions are then integrated into an ODE
solver to dynamically predict the response of the system to various non-stationary exci-
tations. It is shown that the proposed nonlinear mathematical models are able to predict
natural frequencies as well as damping ratios that take into account not only the linear
85
viscous damping, but also the contribution from the other energy dissipation sources. For
example, the damping ratio associated with mode 1 from EQ1 was estimated for both lin-
ear and nonlinear computational models and are given as
ˆ
⇣ ln
=1.63% and
ˆ
⇣ nln
=4.26%,
respectively. Further analyses were performed to show the importance of including the
nonlinear forces in these models. The nonlinear mathematical models prove to be robust
and stable when used for response prediction, while the linear models accurately esti-
mate the system response for short duration but su↵er from numerical instabilities after
a certain time. It is shown throughout the analysis, that the nonlinear forces are a very
important part of such complex systems and should not be ignored. Estimating accurate
damping ratios is crucial from the engineering design perspective, and in this study it is
shownthatrelyingonlinearmodelsonlyisnotsucientforaccuratedampingestimation.
While the results presented in this study show the importance of including the non-
linear e↵ects in the proposed mathematical models, the authors are aware that these
results can be further improved. This can be done by performing sensitivity analysis on
the parameters of the neural networks as well as trying di↵erent nonparametric nonlinear
modeling approaches such as function fitting, machine learning, amongst others. These
approaches are currently under investigation by the authors.
III.8 Summary and Conclusions
System identification approaches have a significant role in extracting dominant features
from structural measurements. There is a need to construct robust mathematical models
for computational purposes to augment experimental studies. Furthermore, it is impor-
tant to have models that capture the correct physics of the underlying system and allow
86
the proper characterization of the system’s nonlinear behavior. Given the lack of suit-
able data sets for full-scale structures under extreme loads, the strong shaking of the
2011 Great East Japan earthquake and its several strong aftershocks provide a unique
opportunity to develop, test, and validate such mathematical models. In this study, the
recorded response of Yokohama-Bay Bridge (a large flexible bridge), driven into nonlinear
response range, was used to develop mathematical models for computational purposes.
The extensive multi-component measurements from relatively dense sensor array were
also analyzed for system identification purposes. In the first part of the study, three well-
established linear system identification methods (i.e., MNExT-ERA, SRIM, and LSID)
were used to identify the modal characteristics of the system (i.e., natural frequencies and
damping ratios). It was seen that the natural frequency estimates matched the expecta-
tions, while the damping ratio estimates were not consistent. The inconsistent damping
estimates were mainly due to the fact that all three identification methods were based
on linear system theory that takes into account only the linear viscous damping, while
the available measurements were mostly from the strong shaking of the bridge driven to
its nonlinear range. The analysis emphasized the importance of having mathematical
models that accurately takes into account the nonlinear characteristics of the system that
can be used for response prediction as well as damping estimation. In the second part
of the study both linear and nonlinear mathematical models were developed, validated,
and compared. The nonlinear mathematical model was developed using a combination
of parametric (least-square) and non-parametric (neural network) approaches along with
time-marching(Runge-Kutta)techniquesfordynamicresponsecalculations. Itwasshown
that the nonlinear mathematical models are more accurate and reliable from the linear
models both in response prediction as well as damping estimation. The demonstrated
87
approach is a general methodology and can be used for various other structural systems
(not necessarily bridges). The findings in this study were reported in Derkevorkian et al.
(2013).
III.9 Acknowledgment
Yokohama-Bay Bridge (YBB) has the densest bridge seismic monitoring system in the
world. The records from the past 25 years have been very valuable to understand the
performance of the bridge under various seismic excitations. The authors acknowledge
the Tokyo Metropolitan Expressway Company Limited who generously granted the per-
mission to use the valuable datasets from the YBB. This study was also supported in
part by a contract from the California Department of Transportation (Caltrans). The
assistance of Dr Charles Sikorsky of Caltrans is appreciated.
88
Chapter IV
Evaluation of a Strain-Based Deformation Shape
Estimation Algorithm for Control and
Monitoring Applications
IV.1 Introduction
IV.1.1 Motivation
Displacement-field prediction is an important research area in structural health monitor-
ing (SHM), structural control, and condition assessment of aerospace structures. Algo-
rithmsthatcanestimatethedeflectionshapeofawingcanbeusedforin-flightmonitoring
of various lightweight structures, flexible wings, as well as various commercial airplanes.
Recent advancements in the sensing technology allows the use of fiber optic sensors on
variousaerospacestructures, toobtainstrainmeasurementsthatcanbeusedintheshape
detection algorithms. These optical sensors, such as the Fiber Bragg Gratings (FBGs),
areknownfortheirlightweightandaccuracy, whichmakesthemapplicabletowiderange
of structures.
89
A promising shape sensing methodology has recently been developed at NASA Dry-
den Flight Research Center (DFRC). This technique allows the use of fiber optic strain
sensor (FOSS) technology to obtain surface strain measurements and to estimate the de-
flection shapes through a series of displacement transfer functions (DTF). The obtained
displacements can be used for various control and monitoring applications. In this paper,
the methodology is presented and analyzed through comparison to a classical modal-
based shape prediction approach. The analyses are performed using experimental data
obtained from the various loading tests performed at DFRC, as well as computational
data obtained from the finite-element-analysis (FEA). The estimation error for both ap-
proaches are quantified using the RMS measure, and a range of validity is proposed for
each approach for damage detection purposes.
IV.1.2 Background
Structural health monitoring of aerospace structures is an extremely important problem
both in the defense and the commercial aviation industries. Real-time condition assess-
ment and damage detection of flying aerospace structures have been studied by many
researchers. Some of the noteworthy works on this topic include papers by, Zhang and
Li (2005), Cusano et al. (2006), Sekine et al. (2006), Su and Cesnik (2006), Oliver et al.
(2007), Wang and Inman (2007), Moncayo et al. (2010), Dixit and Hanagud (2011),
Mieloszyk et al. (2011), amongst others.
The strain-based deflection shape estimation of flying structures, which is the topic of
thispaper,isacrucialparttowardsachievingareliablehealthmonitoringmethodologyfor
such structures. Several studies have been performed to estimate the displacements using
90
strain measurements. An inverse finite element method (IFEM) shape sensing method-
ology was introduced by Tessler and Spangler (2003) and Tessler and Spangler (2005)
which depends on a least-square variational principle to estimate the deformation field of
3D plate and shell elements using strain measurements. Application of the approach for
structural health monitoring was discussed in Tessler (2007) and Vazquez et al. (2005).
Gherloneetal.(2011)andGherloneetal.(2012)furtherinvestigatedtheapproachandits
application on 3D frame structures. Many other studies rely on a modal-based algorithm
thatusesthemodeshapesofthestructuretotransformthemeasuredstrainsintodisplace-
ments. Foss and Haugse (1995) used modal test results to develop strain-to-displacement
transformations, Kirby et al. (1995) investigated the optimum sensor layout to perform
such transformation, Davis et al. (1996) discussed about incorporating fiber optic sensors
for shape and vibration mode sensing, Li and Ulsoy (1999) presented a strain-gauge-
based method for the high-precision vibration measurement of a beam, Chopra (2002)
reviewed the state-of-the-art of smart structures and integrated systems including sensors
and shape estimation, Kim and Cho (2004) estimated the deflection of a simple beam
using classical beam theory, Kang et al. (2007) studied dynamic structural displacements
usingdisplacement-strainrelationshipandmeasuredstraindata, Rappetal.(2007)inves-
tigated dynamic shape estimation by the modal approach, Rapp et al. (2009) investigated
the displacement field estimation for a two-dimensional structure using its modal proper-
ties, Kim et al. (2011) presented shape estimation for rotating structures using the modal
approach, and Derkevorkian et al. (2012) performed sensitivity analyses on the modal
algorithm.
The advancement in the shape prediction algorithms was associated with parallel de-
velopment in the sensing technology. Fiber-optic strain sensors such as the FBGs showed
91
a significant potential to be used in complex aerospace structures. Their light weight,
accuracy, and high spatial resolution distinguish them from the traditional strain gauges,
and make them applicable to a variety of aeroelastic systems such as UAVs and flexi-
ble flying wings that have strict weight requirements. Several studies have investigated
the viability of such sensors for monitoring and control applications. Wood et al. (2000)
studied the development of fiber optic sensors for health monitoring of future aircraft,
Kuang et al. (2001) investigated the performance of FBGs embedded in advanced com-
posite materials, Lee et al. (2002) presented an experimental study on the durability of
fiber optic sensors in aerospace grade composite laminate, Zhou and Sim (2002) reviewed
the state-of-the-art in damage detection in composite structures using fiber optic sensors,
Stewartetal.(2003)investigatedthermalfiberopticsensorsfornondestructiveevaluation,
Richards (2004) studied the fiber optic sensors for the development of FOSS technology
for SHM applications, Fan and Kahrizi (2005) reported the experimental results involving
orthogonal FBG array embedded in graphite/epoxy composite structures, Stewart et al.
(2005) investigated a health monitoring technique for composite materials utilizing em-
bedded thermal fiber optic sensors, Emmons et al. (2009) characterized the performance
of embedded optical FBGs used as strain sensors, and Emmons et al. (2010) investigated
the influence of strain state distribution on the accuracy of embedded optical FBGs used
as strain sensors.
The in-flight break-up of NASA’s Helios wing on June 2003, further emphasized the
importance of having a robust shape detection methodology that can be implemented on
light weight flexible unmanned vehicles such as the Helios. The investigation report, Noll
etal.(2004), suggeststhatpartofthecollapsemighthavebeenduetotheexcessivedefor-
mationsthewinghadsu↵eredduringtheflight. Theavailabilityofamonitoringtechnique
92
is crucial in long-span flexible wings such as the Helios, that had a wingspan of 75.29 m.
Driven from the incident and the need for a viable shape prediction methodology, and
benefiting from the advanced fiber optic sensors, a new methodology was proposed by the
Dryden Flight Research Center (DFRC), (see Ko et al. (2007), Ko and Richards (2009),
Bakalyar and Jutte (2012)). The proposed Fiber Optic Strain Sensing (FOSS) method-
ology was developed such that it combines a sophisticated strain sensing technology that
acquires data from the target structure in real-time, and a robust and computationally
ecient algorithm that accurately estimates the real-time deformation field of the target
structure using the measured data. In this paper, the proposed new method is analyzed
and compared to the modal-based approach. The range of validity is quantified through
the measure of root-mean-square (RMS). Experimental results are presented to show the
e↵ectiveness of the FOSS method in estimating the deformation shape of a cantilever
wing-like swept plate. It is shown that the proposed FOSS approach is a viable deflection
shape prediction methodology with a potential of application for control and monitoring
purposes.
IV.1.3 Scope
This paper investigates the FOSS shape estimation methodology proposed by DFRC
and compares it, through the RMS parameter, to the classical modal-based estimation
approach, using both experimental and computational (FEA) models. Furthermore, the
potential application of the approach as a damage detection tool is examined through
identifying its range of validity. The experimental model (swept plate) was designed,
fabricated, and tested at the NASA Dryden Center. An equivalent computational (FEA)
model was also created using FEMAP
R
with an embedded NX NASTRAN
R
solver.
93
The section on experimental models presents the test apparatus and the test procedure.
The section on computational models describes the FEA model used in the analysis along
with the loading cases used. The section on shape detection algorithms explains both the
FOSS algorithm and the modal algorithm providing flowcharts of the procedure for both
algorithms. The section on results compares the results from the FOSS and the modal
approaches along with the photogrammetry and the FEA reference results. The RMS
parameter is used to measure the estimation error.
IV.2 Experimental Models
IV.2.1 Test Apparatus
The test plate under investigation was designed, fabricated, and instrumented at NASA’s
Dryden Flight Research Center. Fiber-Optic strain sensors were placed at the top and
the bottom surfaces of the plate and strain data were collected from the plate while being
subjected to various loading conditions. The plate was made of 6061T6 aluminum with
a Young’s Modulus of E = 68.94 GPa and a mass density of 2712.63 kg/m
3
. It was 305
mm wide, 4.83 mm thick, and had a span of approximately 1.30 m. Three fibers (with
100 sensors each) were located at the top and the bottom surfaces, respectively. The fiber
lines were placed 12.7 mm from the leading and the trailing edges, while the third fiber
line was located in the middle of the plate. The plate was swept horizontally by a 45
degrees angle from the backstop test frame. Fig. IV.1(a) shows the test plate mounted
on a backstop test frame forming a cantilever fixture. The three lines of fibers on the
top surface of the plate can be seen in the figure. The white circular dots along each
fiber line correspond to the locations where photogrammetry images were taken. Fig.
94
IV.1(b) shows the swept plate subjected to uniformly distributed loads applied at the
bottom surface of the plate. The uniform e↵ect was created by applying equally spaced
concentrated point loads along the two edges of the plate. The swept nature of the plate
along with its fixed end, provide a meaningful simplification of a realistic wing in various
aerospace structures. Furthermore, it helps detecting and observing realistic challenges
that might not be possible to observe in traditional rectangular test plates that have been
studied extensively in the literature.
IV.2.2 Test Procedure
Threeloadingcasesareinvestigatedinthisstudy; thetrailing-edgeloadcase,theuniform-
load case, and the point-load case. For the trailing-edge load case, 6 gravity point loads
were applied at the leading edge of the plate with no loads applied at the opposite edge.
For the uniform-load case, 12 equally spaced point loads were applied downwards (gravity
direction), 6 loads on each edge of the plate, see Fig. IV.1(b). For the point-load case, a
single point load was applied in the gravity direction at the tip of the plate. See Table
II.1 for a summary of the load cases. In the case of the trailing-edge loads, the two point
loads closer to the fixed end were 2.72 kg while the rest of the point loads were 1.36 kg.
In the case of the uniform-load, all the applied loads were 0.91 kg each. In the case of
the point-load, the single applied load is 4.98 kg. Sensors were located every 12.7 mm
along the length of each fiber, and corresponding strain values were collected from each
sensor. Each fiber line along the span of the plate was around 1.27 m, resulting in 100
strain measurements from each fiber.
As mentioned earlier, photogrammetry images were collected from each fiber as a
reference measure of deflection. Images were collected from 11 locations along the span
95
(a) Photo of the unloaded swept plate.
(b) Photo of the uniformly-loaded swept plate.
Figure IV.1: Photos of the swept plate experiment with three lines of fiber optic strain sensors
placed on the top surface. (Property of NASA Dryden Flight Research Center)
96
Table IV.1: The investigated load cases for the swept plate experiment - All the loads are in the
gravity direction.
Load Name Test Code Number of Point Loads Loading Values Location
Trainling-edge Load LE 6 two 2.72 kg, four 1.36 kg Trailing edge
Uniform-Load UL 12 0.91 kg Both edges
Point-Load SP 1 4.98 kg Tip
of each fiber (see the circular white dots in Fig. IV.1). Then, the images were used to
obtain the deflections at those locations. Interpolation was used to obtain the deflections
between two consecutive photogrammetry measuring points.
IV.3 Computational Model
IV.3.1 Description of the Model
A finite element model was created to represent the swept-plate experimental plate. The
computational package FEMAP
R
with an embedded NX NASTRAN
R
solver was used
to create and analyze the finite element model. The FEA model was created such that
its physical properties and geometrical constraints match the actual experimental model
(described earlier) as closely as possible. The FEA model had 10533 nodes and 10236
elements. Eachelementwasmodeledasa4-nodedCQUAD4plateelement,whichtendsto
be less sti↵ than other element types. Each plate element contained 5 degrees-of-freedom
(DOF) on each node. The 5 DOF’s are the translation in X,Y,Z directions (T
x
,T
y
,T
z
)
and the rotation in X and Y directions (R
x
,R
y
). The material property of each element
was assigned based on the specifications of 6061-T6 Aluminum described earlier. Fixed
97
constraints were applied at the root of the plate. Three lines of nodes were re-numbered
along the span of the FEA model to represent the three fiber lines on the top surface of
the experimental plate. The three lines were located at the two edges and the middle of
the plate surface.
IV.3.2 Loading of the Models
The loading cases for the computational model were created to match the loading config-
uration of the experiment. As seen in Fig. IV.2 the three loading configurations are the
edge-load, the uniform-load, and the point-load cases. The values and the orientations of
the loads match the ones used in the experiment and are summarized in Table II.1.
EachsubfigureinFig. IV.2showsthefixed-endoftheplate,thedeformedshapeofthe
plate, and the loading values. The maximum deflections for the edge-load, uniform-load,
and the point-load cases are 11.58, 19.67, and 23.90 cm, respectively.
IV.4 Shape Detection Algorithms
In this study, the Fiber-Optic Strain Sensing (FOSS) algorithm is evaluated through
experimental and computational models. The viability of the algorithm in predicting dis-
placement fields is assessed through comparing the estimates to photogrammetry results
and results from other state-of-the-art shape prediction algorithms i.e., the modal algo-
rithm. Both the FOSS and the modal algorithms rely on obtaining strain measurements
using Fiber Bragg Gratings (FBGs) to estimate the corresponding displacement shapes.
98
(a) Edge-load case.
(b) Uniform-load case.
(c) Point-load case.
Figure IV.2: Description of the three loading cases. The loading values are in (kg).
99
By measuring the Bragg wavelength B
, the strain information at a specific location can
be calculated using the following equation:
✏ =
1
(1p
e
)
✓
B
B
◆
(IV.1)
where ✏ is the measured strain, p
e
is the strain-optic coecient, B
is the Bragg reflected
wavelength, and B
is the change in the Bragg wavelength. Many studies have been
conducted to experimentally verify the viability of using fiber optic sensors as strain
measuring tools and to compare it to traditional strain sensors. Promising results were
obtained in many works such as the ones by Lee et al. (2002), Zhou and Sim (2002),
Richards (2004), amongst others.
IV.4.1 Fiber-Optic Strain Sensing (FOSS) Algorithm
AdeflectionshapepredictionalgorithmwasdevelopedatNASA’sDrydenFlightResearch
Center using strain measurements from optical Fiber Bragg Gratings. The advancement
in the sensing technology allows the use of FBGs on lightweight aerospace structures to
measure real-time strain data. The FOSS approach depends on the classical beam theory
to develop displacement transformation functions (DTF) by descretizing the surface of
thestructureintromultiplesegments. Eachsegmentcanberepresentedbyalinearbeam-
curvature equation. The DTF equations were initially given by Ko and Richards (2009);
Ko et al. (2007):
ˆ y
i
=
( l)
2
6c
i1
✓✓
3
c
i
c
i1
◆
✏
i1
+✏
i
◆
+ˆ y
i1
+ ltan✓ i1
(IV.2)
100
where ˆ y
i
is the estimated deflection at location i, c
i
is the distance from the neutral axis
at location i, l is the axial distance between two adjacent sensor locations, ✏
i
is the
strain measured at location i, and tan✓ i
is the slope at location i and is defined by the
following equation:
tan✓ i
=
( l)
2c
i1
✓✓
2
c
i
c
i1
◆
✏
i1
+✏
i
◆
+tan✓ i1
(IV.3)
In most realistic conditions, the distance from the surface to the neutral axis (c
i
) varies
alongthespanofthewing. Inordertoaccountforthechangeinthec
i
value,thefollowing
equation was suggested by Ko and Richards (2009):
c
i
=
✏
it
✏
it
✏
ib
h
i
(IV.4)
where ✏
it
is the measured strain from the top surface at location i, ✏
ib
is the measured
strain from the bottom surface at location i, and h
i
is the full-depth at location i.It
was soon realized that Eq. IV.4 causes numerical stability when ✏
ib
= ✏
it
. As a result, a
modified approach was suggested by DFRC, which used the distance separating the top
and bottom fibers (i.e., the full-beam depth, h), instead of the distance from the surface
to the neutral axis (c). Furthermore, the di↵erence between top and bottom strains were
used (✏
b
✏
t
), as opposed to the individual surface strains (✏
b
or ✏
t
). The modified DTF
equations were reported by Bakalyar and Jutte (2012) as follows:
ˆ y
i
=
( l)
2
6h
i1
✓✓
3
h
i
h
i1
◆
(✏
b
✏
t
)
i1
+(✏
b
✏
t
)
i
◆
+ˆ y
i1
+ ltan✓ i1
(IV.5)
101
FEA model
Displacement mode shapes:
Strain mode shapes:
DST
matrix
[ T ]
Measured
strains, ✏
Modal
algorithm
Experimental
model
Estimated eflection
shape, ˆ y
Measured
strains, ✏
FOSS
algorithm
Experimental
model
Estimated deflection
shape, ˆ y
FEA deflection,
y
X
-
+
Estimation
error
Photogrammetry Reference deflection,
y
X -
+
Estimation
error
Wing-depth,
h
i
Figure IV.3: The flowchart of the FOSS approach.
h
i
is the depth of the beam (wing) at location i, ✏
b
is the strain measured at the bottom
surface, ✏
t
is the strain measured at the top surface, and tan✓ i
is the slope at location i
and is defined by the following equation:
tan✓ i
=
( l)
2h
i1
✓✓
2
h
i
h
i1
◆
(✏
b
✏
t
)
i1
+(✏
b
✏
t
)
i
◆
+tan✓ i1
(IV.6)
It is worth mentioning that at the fixed-end of the plate, it is assumed that both the
deflection and the slope are zero, i.e., y
0
= tan✓ 0
= 0. Further details on the derivation
of the FOSS algorithm can be found in Ko et al. (2007), Ko and Richards (2009), and
Bakalyar and Jutte (2012).
Figure IV.3 shows the flowchart of the FOSS algorithm. As seen, the strains (✏) are
measured from the experimental model and provided to the algorithm along with depth
information (h) of the test plate (wing). Then, the displacements are estimated (ˆ y) from
the algorithm and compared to the ones obtained through photogrammetry.
102
IV.4.2 Modal Approach
As mentioned earlier, the modal approach has been studied extensively in the literature
and has been applied to various types of structures for the purposes of shape estimation.
In this study, the modal approach is presented and compared to the FOSS estimation
method. As in the FOSS approach, it uses FBGs to obtain strain measurements from the
structure to predict the deformation shape based on the following equation Kang et al.
(2007); Rapp et al. (2007):
{ˆ y}
N⇥ 1
=[T]
N⇥ M
{"}
M⇥ 1
(IV.7)
where ˆ y is the estimated displacement vector at N locations, ✏ is the measured strain vec-
toratM locations,and[T]isanN⇥ M matrixcalledDisplacement-Strain-Transformation
(DST) matrix.
The DST matrix [T] can be calculated using Eq. IV.8:
[T]
N⇥ M
=[]
N⇥ n
·([ ]
T
n⇥ M
·[ ]
M⇥ n
)
1
·[]
T
n⇥ M
. (IV.8)
where [] is the displacement mode shape matrix, [ ] is the strain mode shape matrix,
and n is the desired number of modes to be included in the calculations. The number of
modes n is determined based on the desired accuracy, the higher the number of modes
the more accurate the estimations are. Note that the number of measured strains M
can be much less than the number of estimated displacements N, i.e., with few strain
measurements, the displacement-field of a relatively large area can be predicted.
Fig. IV.4 shows the flowchart of the modal approach. As seen in the figure, the
displacement mode shapes [] and the strain mode shapes [ ] are extracted from the
103
FEA model
Displacement mode shapes:
Strain mode shapes:
DST
matrix
[ T ]
Measured
strains, ✏
Modal
algorithm
Experimental
model
Estimated deflection
shape, ˆ y
Measured
strains, ✏
FOSS
algorithm
Experimental
model
Estimated deflection
shape, ˆ y
FEA deflection,
y
X
-
+
Estimation
error
Photogrammetry Reference deflection,
y
X -
+
Estim
ation
error
Wing-depth,
h
i
Figure IV.4: The flowchart of the modal approach.
FEA model to form the DST matrix. The strains (✏) are measured from the experimental
model using the fiber optic sensors. Then, the DST matrix along with the measured
strains are provided to the modal algorithm and an estimate is obtained for the deflection
shape (ˆ y). The estimated displacements are then compared to the real displacements
from the FEA and the estimation error is computed.
According to Foss and Haugse (1995), Kirby et al. (1995), Davis et al. (1996), Kang
et al. (2007), and Kim et al. (2011), the estimation quality of the modal approach can
be improved by determining the placement of the FBG sensors that corresponds to the
lowest condition number ( ) of the DST matrix ([T]). The condition number of the DST
matrix is given by the following equation:
 =k[T]k·k[T]
1
k (IV.9)
wherek·k is the norm of the matrix. In general,  is greater or equal than 1, (i.e.,  = 1).
The case when  = 1 means the matrix is well-conditioned and most of the information is
preserved. In this study, only the first four modes of the structure (i.e., n = 4) were used,
104
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Normalized fiber length
Normalized vibration mode shapes

Vibration mode 1
Vibration mode 2
Vibration mode 3
Vibration mode 4
Sensor location
Plate top−surface
Plate bottom−surface
Figure IV.5: The optimized locations of the sensors along with the vibration mode shapes.
along with the data from four FBG sensors (i.e., M =n = 4). It is desired to obtain the
locations of the FBG sensors along the span of the fiber so that the condition number
of the DST matrix is minimum. The number of possible placement configurations of the
four FBG sensors depends on the mesh size of the computational model from which the
modal data (i.e., and matrices) are extracted. A denser mesh results in more nodes
along the span of each fiber, consequently, more locations for each FBG sensor to be
placed on. The model used for this study has 50 nodes (sensors) along each fiber. The
Matlab
R
function nchoosek.m is used to determine the possible location combinations
for the four sensors and the condition number ( ) is calculated for each combination. The
combination with the lowest condition number is then picked as the optimum location for
the four sensors.
105
Figure C.1 shows the first four normal mode shapes of the plate along with the ob-
tained optimum locations of the four sensors to be used in the modal algorithm. The cor-
responding four frequencies of the plotted four mode shapes were: (1) 2.342, (2) 14.677,
(3) 41.096, and (4) 80.530, respectively. The reported frequency values are in (Hz). The
x-axis in Fig. C.1 represents the normalized fiber length, and the y-axis represents the
normalized magnitudes of the mode shapes. The sensor locations shown in the figure
correspond to the condition number  =1.424, which was the lowest obtained value. The
optimum locations of the sensors were at nodes 4, 10, 18, and 34, assuming the fixed-end
being node 1 and the free-end being node 50 along a given fiber line.
IV.5 Results
In this section, the results from the FOSS and the modal estimation approaches are
presented and compared to the photogrammetry reference results using the root-mean-
square (RMS) measure. Using the RMS results, the range of validity of the methods are
compared.
IV.5.1 FOSS Results
The FOSS method was implemented using the strain data collected from the swept plate
experiment. The photogrammetry imaging results were the reference displacements and
were used to compute the RMS parameter based on the following equation:
RMS =
v
u
u
t
N
X
i=1
(y
i
ˆ y
i
)
2
,
N
X
i=1
y
2
i
(IV.10)
106
wherey
i
isthereferencephotogrammetrysignalatlocationi,ˆ y
i
istheestimateatlocation
i,andN isthenumberofsensors(nodes). Itshouldbenotedthatphotogrammetryimages
weretakenonlyat11locationsalongeachfiberline. Therefore, thephotogrammetrydata
wereinterpolatedsoitcanbeusedtocomputetheRMS.FigureIV.6showstheestimation
resultsobtainedbyFOSS.Eachcolumnrepresentsadi↵erentloadingcase. Columnoneis
the trailing-edge load case, column two is the uniform-load case, and column three is the
point-load case. Each row represents a di↵erent fiber line. Row one represents the middle
fiber, and the second row represents the leading-edge fiber. The x-axis is the normalized
fiber length, and the y-axis is the displacement (cm). The RMS value for each subfigure
was calculated using Eq. IV.10 and printed on top of each subfigure. The superposed
curves in each subfigure indicate a very good match. As seen, the obtained RMS error
values are below 2% for all the presented cases. The visual fit of all the curves along with
the obtained RMS values indicate an excellent estimation by the FOSS approach. The
quantified measure of the errors shows a big potential for the approach to be applied as a
damage-detection tool. Based on the presented results, if there was more than 2% change
in the structure in the vicinity of the fiber lines, it would have been detectable by the
FOSS method, i.e., the range of validity of the approach as a damage detection tool is for
changes (damages) above 2%.
IV.5.2 Modal Results
As mentioned earlier, the modal approach requires knowledge of the vibration and the
strain mode shapes of the structure. In this study, the FEA model was used to extract
theneededmodeshapes. Oneofthechallengesinmodelingusingfinite-elementpackages,
is the ability to represent the real (experimental) model as accurately as possible. Several
107
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Middle fiber
Trailing−edge load: RMS =0.0049

Photogrammetry
FOSS estimate
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Uniform−load: RMS =0.0159
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Point−load: RMS =0.0192
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Leading−edge fiber
RMS =0.0185
Normalized fiber length
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
RMS =0.0122
Normalized fiber length
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
RMS =0.0101
Normalized fiber length
Figure IV.6: The estimated (FOSS) displacements along the two fiber lines for all three load
cases, compared to the reference photogrammetry results. Each row represents a di↵erent fiber
line. Each column represents a di↵erent loading case. For each subfigure, the parameter RMS is
calculated using Eq. IV.10.
factors a↵ect the accuracy of the modeling, including boundary conditions, mesh size,
material properties, amongst others. Figure IV.7 shows a sample of strains obtained
from the FEA model superposed over the measured strains from the experiment. The
columns correspond to the uniform and the point-load cases, and the rows correspond
to the middle and the leading-edge fiber lines. The plot x-axis is the normalized fiber
length and the y-axis is the strain values. As seen from the figure, the FEA strains match
the experimental strains indicating a relatively accurate computational representation
of the actual experimental plate. The strain output from the FEA was compared to the
experimentalstrainandquantifiedusingtheroot-mean-square(RMS)andthecorrelation
(R) coecients. The RMS coecient is defined in Eq. IV.10, while the R coecient can
be calculated as follows:
R =
P
N
i=1
(y
i
¯ y)(ˆ y
i

¯
ˆ y)
q
P
N
i=1
(y
i
¯ y)
2
P
N
i=1
(ˆ y
i

¯
ˆ y)
2
(IV.11)
108
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x 10
−3
Uniform−load: RMS =0.12, R =0.98
Middle fiber
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x 10
−3
RMS =0.11, R =0.99
Normalized fiber length
Leading−edge fiber
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x 10
−3
Point−load: RMS =0.11, R =0.97
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x 10
−3
RMS =0.086, R =0.98
Normalized fiber length
Figure IV.7: The measured strains at two of the top surface fiber lines superposed on the strains
obtained from the FEA model. Each row corresponds to a distinct fiber line on the top surface of
the plate. Each column corresponds to a di↵erent loading condition.
where y
i
is the reference measurement at location i,ˆ y
i
is the FEA output (estimate) at
location i,¯ y is the mean of the reference measurement y,
¯
ˆ y is the mean of the estimate
ˆ y, and N is the number of sensors. A perfect correlation corresponds to R = 1.
As seen in Fig. IV.7, the RMS and R values are reported at the top of each subfigure.
TheaveragenormalizedRMSerrorbetweentheFEAstrainoutputsandtheexperimental
strainswas10%. Whiletheaveragecorrelationcoecient(R)betweenthetwostrainswas
0.98. The 10% average RMS error is attributed to measurement noise, modeling error,
uncertainties in the fabrication of the test plate, and many other factors. Both the RMS
and R values, along with the plots in Fig. IV.7, show a very good match between the
FEA and the experimental measurements. It is worth mentioning that for more realistic
test articles, such as a wing with a complex geometry and composite materials, achieving
a good correlation between the FEA model and the experimental model might not be
a trivial task. Given the numerous parameters involved in such complex models, more
109
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Trailing−edge load: RMS =0.0364
Middle fiber

Photogrammetry
Modal estimate
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Uniform−load: RMS =0.031
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Point−load: RMS =0.0202
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Trailing−edge load: RMS =0.035
Leading−edge fiber
Normalized fiber length

Photogrammetry
Modal estimate
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Uniform−load: RMS =0.0368
Normalized fiber length
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Point−load: RMS =0.0276
Normalized fiber length
Figure IV.8: The estimated (Modal) displacements along the two fiber lines for all three load
cases, compared to the reference photogrammetry results. Each row represents a di↵erent fiber
line. Each column represents a di↵erent loading case. For each subfigure, the parameter RMS is
calculated using Eq. IV.10.
sophisticated approaches might be needed, such as finite element updating, to achieve an
acceptable degree of match between the experiment and the FEA.
Once the FEA model was completed, the mode shapes were obtained and the modal
estimation algorithm was implemented. The estimated displacements are shown in Fig.
IV.8. As in the previous case, the modal estimates are superposed on the reference pho-
togrammety displacements. The RMS error values for the di↵erent cases range between
2% to 3.7%. Therefore, the range of validity of the approach for change detection in this
specific experiment is above 3.7%. Meaning, any change in the structure below 3.7%,
cannot be distinguished from the estimation error. Changes above the obtained limit
however, can be attributed to structural change (damage). The slightly narrower range
of validity of the approach compared to the FOSS, can partly be attributed to the mod-
eling error involved in the estimation. As reported in Fig. IV.7, the average RMS error
between the strain curves from the FEA and the reference experimental strains is about
110
10%. Consequently, the corresponding displacement estimation is negatively a↵ected by
the existing strain errors.
Tofurtherassesstheexperimentalresultsobtainedbythetwoapproaches,theabsolute
errors were calculated as follows:
e=ˆ yy (IV.12)
where e is the absolute error (cm), ˆ y is the estimated displacement, and y is the reference
displacement. Figure IV.9 shows the obtained errors for the middle fiber line. The first
row of the figure plots the displacements from FEA, photogrammetry, FOSS estimate,
and the modal estimate. The shown displacements correspond to the middle fiber line,
while each column corresponds to a di↵erent loading case. As seen, there is a very good
agreement between all four superposed plots in each subfigure. The second row of Fig.
IV.9 shows the calculated absolute errors using Eq. IV.12. It is seen that the errors from
both approaches have similar trends, with the modal estimate showing slightly higher
errors for all three loading cases. The maximum obtained error from all the three cases is
about 0.57 cm at the tip of the fiber. Similar results were obtained from the other fiber
lines.
IV.6 Discussion
It is seen from the above results, that the FOSS methodology shows a significant poten-
tial as a reliable shape-sensing and damage-detection technique. The estimation results
obtained by the FOSS technique are comparable to the finite-element results, and the
experimental photogrammetry results. Furthermore, the FOSS approach shows a wider
range of validity as a damage-detection tool with estimation errors bounded below 2%,
111
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Displacement (cm)
Trailing−edge load

FEA
Modal estimate
Photogrammetry
FOSS estimate
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Uniorm−load
0 0.2 0.4 0.6 0.8 1
−20
−15
−10
−5
0
Point−load
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
Normalized fiber length
Error (cm)

Modal error
FOSS error
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
Normalized fiber length
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
Normalized fiber length
Figure IV.9: The first row shows the displacements at the top-middle fiber based on the FEA
model, modal estimate, photogrammetry, and the FOSS approach, respectively. The second row
shows the corresponding estimation errors for the two approaches.
using the RMS measure, while the estimation errors from the modal approach are below
4%.
The estimation results obtained from the modal technique can be improved by in-
creasing the number of modes (4 modes were used in this study), increasing the number
of sensors (4 sensors were used in this study), and reducing the modeling error by having
a more accurate finite element model. It should be noted that the test plate in this exper-
iment has a monolithic nature with uniform material properties and constant thickness,
which makes the modeling in the computational package relatively simple. However, for
a realistic complex composite wing, achieving a robust computational model that closely
matches the real structure might be a daunting task, and might significantly increase the
modelingerror. Thechoiceofthenumberofsensorsandthemodesusedinthisstudywas
based on a sensitivity analysis that was performed and reported in Derkevorkian et al.
(2012) and shown in Appendix C.
112
It is worth mentioning that the increase in the number of modes will be associated
with a greater computational e↵ort. A major advantage of the modal approach is that it
can predict a displacement field of a large area with relatively few sensors (i.e., 4 in this
study). This was particularly important when traditional strain gauges were being used
on lightweight structures with very strict weight requirements (i.e., limited number of
strain gauges would be allowed for use). However, with the advancements in the sensing
technology and the development of fiber optic sensors such as the FBGs, it is possible
to record the strains at many more stations without violating the weight requirement of
such lightweight aerospace structures.
As shown in the flowchart of the modal approach in Fig. IV.4, an important aspect
of this approach is its dependence on the displacement transformation matrix which is
formed by the displacement and strain mode shapes of the structure. If the mode shapes
areobtainedfromacorrespondingFEAmodel(asisthecaseinthisstudy), anymodeling
error will be reflected in the estimation error. Furthermore, since the few sensors used
are located at optimized locations that correspond to the lowest condition number of
the DST matrix, any malfunction in one sensor during a real-time flight can leave the
other sensors in non-optimized locations, thus, contributing to an increased estimation
error. Nonetheless, as the obtained results suggest, the modal approach is a classical
methodology that has been studied extensively in the literature, and has been applied to
various types of complex structures, and successful results have been reported.
The recently-developed shape-detection FOSS approach is based on discretizing the
structure into several segments, and describing each segment with a linear classical beam-
curvature function, that are known as DTF functions, and can be written in recursive
format as shown in Eqs. IV.5 and IV.6. As seen in the flowchart of this approach in Fig.
113
IV.3, the method depends on the measured strains from the top and the bottom surfaces
of the plate, the distance between two successive sensors, and the depth of the plate at
each segment.
A major advantage of the FOSS approach is that it does not require additional mod-
eling, which could be substantial for actual full-scale structures. The proposed algorithm
has been developed for high aspect ratio aircraft wings based on one-dimensional beam
theory assumption, which has proven to be valid for the practical applications studied in
BakalyarandJutte(2012)andJutteetal.(2011). Furthermorethealgorithmiscomputa-
tionally very ecient, which makes its implementation practical for real-time monitoring.
As opposed to the modal approach, the FOSS method does not require knowledge of
the modal properties of the structure (i.e., mode shapes, material properties, etc.); conse-
quently, it is not associated with any modeling error. In fact, the only information needed
about the physical nature of the structure is the depth, h
i
.TheFOSSmethodbenefits
from the state-of-the-art in sensing technology by using multiple fiber lines both at the
top and the bottom surfaces of the plate, with sensors located (in this experiment) at
every 12.7 mm along each fiber.
The strain data measured with the FOSS technology has a big potential to be used
for health monitoring purposes. Having a dense distribution of FBG sensors along each
fiber (with FOSS, at a minimum of 0.79 mm spacing) not only allows the detection and
quantificationofsmalldamageatearlystages,butalsolocatesit. Adensearrayofsensors
could also increase the reliability of the measurements, especially if the fibers mounted on
thesurfaceofthestructurearesubjectedtoroughenvironmentalconditions, asaresultof
which some of the sensors could malfunction. Furthermore, embedded fiber-optic sensors
114
(lessvulnerabletoenvironmentalextremeconditions)areexperimentallybeingtestedand
could potentially be used with the FOSS technology.
Aspresentedintheprevioussection, theresultsobtainedfromtheexperimentalFOSS
estimates match the photogrammetry reference displacements with good accuracy. The
RMS parameters for the presented simulation cases are below 2%, indicating a wider
range of validity as a damage detection tool than the model approach where the obtained
RMS parameters are below 4%. The FOSS method for real-time structure shape sensing
was awarded a United States Patent (see, Ko and Richards (2009)), and shows a great
potentialtobeimplementedonlightweightflexiblewings(suchastheHelios)andavariety
of other types of future-generation aircraft and aerospace structures, for the purposes of
shape sensing, damage detection, and structural health monitoring. At the present time,
the FOSS method has also been used under dynamic conditions to measure time-varying
strains with a frequency of up to 25 Hz.
IV.7 Conclussion
This study investigated the FOSS shape detection technology proposed by the NASA
Dryden Flight Research Center (DFRC). This technology employes the state-of-the-art in
sensing technology by using fiber optic sensors to measure strains from an experimental
model, and subsequently estimate the corresponding deformation shape. Several loading
cases were experimentally tested, and promising estimation results were obtained from all
the cases. This study also compared the proposed approach to the widely used classical
modal-based estimation approach, through analyzing the algorithms and quantifying the
results from both methods by using the measure of root-mean-square (RMS) deviation
115
error. The FOSS algorithm does not need to be associated with a corresponding FEA
model,anddoesnotdirectlydependonthephysicalpropertiesofthestructure(i.e.,mode
shapes, material properties, etc.), which makes it more amenable to real-time online ap-
proaches for SHM applications. The proposed FOSS method is less dependent on the
number of sensors than the modal approach. Using more sensors in the modal approach
could lead to extra computational e↵ort when optimizing the sensor locations. Increasing
the number of sensors and the mode shapes will increase the dimensions of the displace-
ment transformation matrix, which in turn could a↵ect the computational eciency. The
findings in this study were reported in Derkevorkian et al. (2013)
It is shown that the FOSS approach has a wide range of validity for damage detection
purposes. The proposed method has the potential of applicability to a wide range of
unmannedvehicleswithspecificweightrequirements,alongwithhighly-flexiblewings,and
various types of other aerospace structures. The presented new technology can provide
a robust tool for real-time shape sensing, condition assessment, and in-flight damage
detection.
IV.8 Acknowledgment
This work was supported by NASA Grant URC NCC NNX08BA44A.
116
Chapter V
Investigation of an Operating-Load Estimation
Algorithm Using Fiber-Optic Strain Sensors
(FOSS) Technology
V.1 Introduction
V.1.1 Background
R
EAL -time operational-load estimation of structural systems is crucial for various
monitoring and control applications. The accurate prediction of applied loads
can be very important in many types of structures such as o↵shore platforms, rotating
wind turbines, flying unmanned aerial vehicles, as well as conventional civil structures
such as multi-span bridges, high-rise buildings, amongst others. Some of the noteworthy
works in the area are the following: White et al. (2009) investigated potential methodolo-
gies to estimate the wind turbine blade operational loading and deflections with inertial
measurements. White et al. (2010) proposed a model updating method to evaluate the
operational monitoring method for wind turbines. Ahmari and Yang (2013) suggested an
inverse analysis method for load identification in plates, considering bounded uncertain
117
measurements. Arsenault et al. (2013) reported the development of FBG strain sensor
system for structural health monitoring in wind turbines.
Realizing the importance of load-estimation algorithms for the monitoring and con-
trol of aerospace structures, researchers at the NASA Dryden Flight Research Center
performed several key studies. Lizotte and Lokos (2005) proposed a deflection-based
aircraft structural load estimation algorithm and experimentally tested it on the active
aeroelastic wing F/A-18 aircraft. Richards and Ko (2010) obtained a patent on a process
for using surface strain measurements to obtain operational loads for complex structures.
With the development of the Fiber-Optic Strain Sensing (FOSS) technology at NASA
Dryden Center Emmons et al. (2010, 2009); Ko and Richards (2009); Ko et al. (2007);
Lee et al. (2002); Richards (2004); Stewart et al. (2003, 2005), Bakalyar and Jutte (2012)
experimentallyvalidatedthealgorithmproposedbyRichardsandKo(2010)usingvarious
plateelements. Nicolasetal.(2013)usedtheproposedalgorithmtoestimateout-of-plane
loads of a large-scale carbon-composite wing.
Estimating the lateral pressure loads (i.e., wind loads) on tall high-rise buildings is
important for design, control, and monitoring applications. There is a paucity of method-
ologiesthatcanaccuratelypredictthelateralpressureloadsonbuildingsurfaces. Mostof
such methodologies are not practical to be implemented on full-scale high-rise buildings.
Many studies in the literature address the limitations of current techniques that are used
to estimate static equivalent wind loads on tall buildings. Zhou et al. (1999) examines
two code methods for equivalent static wind load estimation and demonstrates that they
mayleadtosomeundesirableloade↵ects. ChenandKareem(2001)addressesthecurrent
design practice for wind load estimation on bridges and shows that the estimated load
distributions may not be a physically accurate description of the real applied loads. Zhou
118
et al. (2002) points out the scatter among the wind loads predicted by various interna-
tionalcodesandstandards. Tamuraetal.(2008)presentsaguidefornumericalprediction
of wind loads on buildings. Lou et al. (2012) shows an experimental and zonal modeling
for wind pressures on double-skin facades of a tall building.
V.1.2 Motivation
The recent advancements in the sensing technology allows the use of sophisticated, high-
resolution, light-weight Fiber Brag Grating (FBG) strain sensors. These sensors are
known for their accuracy and their durability in rough environmental conditions. The
Fiber-OpticStrainSensing(FOSS)technologydevelopedattheNASADrydenFlightRe-
searchCenterutilizesFBGstrainsensorswithinlongfiberstoachievearobustmonitoring
system that can be used for various real-time data sensing applications. The availability
of the FOSS technology along with the development of the load-estimation algorithms
at the Dryden Center, combined with the need for a robust data-driven methodology to
estimate lateral loads on tall buildings, provide a great opportunity to investigate the
viability of adopting the approach to monitor lateral loads (i.e., wind loads) acting on
buildings.
V.1.3 Scope
Motivatedfromthepreceding,thisstudyevaluatestheapplicabilityoftheload-estimation
algorithms developed at the NASA Dryden Flight Research Center combined with the
state-of-the-artFOSSstrain-sensingtechnologiestoestimatelateralloadsontallbuilding-
like structures. This study includes the analytical investigation of exact and approximate
119
methodologies for moment calculations in typical frames encountered in buildings. Sensi-
tivity analyses are performed to demonstrate the e↵ects of the relative sti↵ness between
the horizontal and the vertical members, as well as the hight-to-width ratio of the frame,
on the calculated moments. Furthermore, analyses are performed to show the e↵ects of
uncertainty in section properties on the moment calculations. A finite-element model of a
building structure is created and analyzed using finite-element software (Nastran
r
). The
results of the load-estimation algorithm are presented from various loading cases and the
challengesarediscussed. Furthermore, anexperimentaltest-bedstructureisdesignedand
a test procedure is proposed for further analyses.
V.2 Load-Estimation Algorithm
The load-estimation algorithm depends on obtaining strain measurements using Fiber
Bragg Gratings (FBGs). The strain information at a specific location can be calculated
by measuring the Bragg wavelength B
, as shown in Eq. V.1.
✏ =
1
(1p
e
)
✓
B
B
◆
(V.1)
where ✏ is the measured strain, p
e
is the strain-optic coecient, B
is the Bragg reflected
wavelength, and B
is the change in the Bragg wavelength. Important works that show
the validity of the FBG sensors include Lee et al. (2002), Zhou and Sim (2002), Richards
(2004), amongst others.
As seen in Fig. V.1, the load-estimation algorithm developed by Richards and Ko
(2010)consistsoftwophases; calibrationphaseandestimationphase. Duringthecalibra-
tionphase, aknownpoint-loadisappliedatthetipofthestructureandthecorresponding
120
strain at each sensor location i is measured (i.e., ✏
i(clb)
). For a uniform cantilever beam-
like structure, the moment M
i
at each location i can be computed by multiplying the
known point-load at the tip by it’s distance from the sensor location i (i.e., M
i
=P⇥ L
i
,
where L
i
is the distance from the tip of the structure to the location of sensor i, and P
is the point-load applied at the tip (free-end)). The equivalent section property

EI
c

i
at each location i can then be estimated by dividing the computed moments M
i
by the
measured strains ✏
i(clb)
, as shown in Eq. V.2:
M
i
✏
i(clb)
=
✓
EI
c
◆
i
(V.2)
whereE istheYoung’smodulus,I isthemomentofinertia, andcisthedistancefromthe
neutralaxis. Itshouldbenotedthatthealgorithmdependsontheflexuralcharacteristics
of the structure and it assumes the strain is due to bending-only (not shear). It should
also be noted that for a more complex structure (i.e., a building consisting of interacting
beams and columns with varying sti↵ness characteristics), the relative sti↵ness values of
the members might be needed to calculate the exact moments needed for the calibration.
Further discussion will be provided on this in the upcoming sections.
During the estimation phase, the section properties determined from the calibration
phaseareusedalongwiththenewstrainmeasurements✏
i(new)
fromnew(di↵erent)applied
loads, to estimate the corresponding moment
ˆ
M
i
, as shown in Eq. V.3.
ˆ
M
i
= ✏
i(new)
⇥ ✓
EI
c
◆
i
(V.3)
121
Measured
strains,
✏
DTF
equation
Experimental
model
Estimated deflection
shape, ˆ y
Wing-depth, c
i
Estimation
Phase
Calibration
Phase
M
i
✏
i(clb)
=
✓
EI
c
◆
i
ˆ
M
i
= ✏
i(new)
✓
EI
c
◆
i
Figure V.1: Flowchart of the load-estimation algorithm for flexure-dominated structure.
Ifneeded,onecanthenusethemomentinformationtocalculatetheequivalentdistributed
load along the fiber line.
V.3 Sensitivity Analyses on Analytical Moment Calcula-
tions for Calibration Phase
As seen in the previous section, an accurate calculation of the moments during the cal-
ibration phase is essential for the proposed load-estimation algorithm. In most of the
previous studies related to this approach, the testing and the validation was performed
on relatively simple structures (i.e., cantilever, uniform and homogeneous beam or plate
element). In such structures, the analytical calculation of the moment is relatively simple
andcanbeperformedbymultiplyingthecalibrationpoint-loadatthefree-endofthebeam
with the distance from the sensor station. No information is needed about the physical
characteristics of the beam or the plate (i.e., moment of inertia, sti↵ness). The objective
of this study, as mentioned earlier, is to extend the technique and assess its viability when
122
used with more complex structures, such as buildings with various interacting elements
(i.e., beams and columns with varying sti↵ness characteristics). In the classical theory
of structures literature, there are well-established techniques to calculate the moments
of statically indeterminate rigid frames. Such techniques include matrix force method,
moment distribution method, slope-deflection method, amongst others Hsieh and Mau
(1995). In this study the slope-deflection method will be demonstrated symbolically for
a one-bay frame. Then, numerical results from the exact solution will be compared to
corresponding approximate solutions. The sensitivity of the approximate approach to
relative sti↵ness between horizontal and vertical members, as well as to frame height and
width will be analyzed.
Let us consider the frame shown in Fig. V.2. As seen, the frame consists of one bay.
The columns have moment of inertia of I
C
and the girder has a moment of inertia of I
G
.
The width of the bay is denoted by L and the height of the frame is denoted by H.A
Measured
strains, ✏
DTF
equation
Experimental
model
Estimated deflection
shape, ˆ y
Wing-depth, c
i
Estimation
Phase
Calibration
Phase
M
i
✏
i(clb)
=
✓
EI
c
◆
i
ˆ
M
i
= ✏
i(new)
✓
EI
c
◆
i
I
C
I
G
I
C
P
a
b c
d
H
L
Figure V.2: The sample single-story frame under investigation.
123
single point-load P is applied at one side of the frame. The frame has four joints denoted
by a,b,c, and d. It is assumed that the Young’s Modulus of the material for the columns
and the girder is E.
Based on the slope-deflection method the moment at each joint (i.e., joint a, M
ab
)
comprises of the moment due to the end rotation ✓ a
while the other end b is fixed, the
moment due to the end rotation ✓ b
while the other end a is fixed, the moment due to
the relative deflection ab
between the ends of the member ab, and the moment due to
potential loads along the span of the member. In case of the frame shown in Fig. V.2,
since the bottom supports are fixed, ✓ a
= ✓ d
= 0. Also given the symmetric nature of the
frame, ab
= dc
=. Since there are no loads applied on the span of any member, the
moment due to loads on span is M
F
= 0. Therefore the frame-specific moment equations
can be written as follows:
M
ab
=2EI
C
/H⇥ (✓ b
3 /H)
M
ba
=2EI
C
/H⇥ (2✓ b
3 /H)
M
bc
=2EI
G
/L⇥ (2✓ b
+✓ c
)
M
cb
=2EI
G
/L⇥ (2✓ c
+✓ b
)
M
cd
=2EI
C
/H⇥ (2✓ c
3 /H)
M
dc
=2EI
C
/H⇥ (✓ c
3 /H)
(V.4)
The three unknowns (i.e., ✓ b
, ✓ c
, and) in the above equations can be solved using
the following equations:
124
M
ba
+M
bc
=0
M
cb
+M
cd
=0
PV
ab
V
dc
=0
(V.5)
where V
ab
=(M
ab
+M
ba
)/H and V
dc
=(M
dc
+M
cd
)/H. The above expressions can be
organized in a matrix from as follows:
K✓ =F (V.6)
where
K =
2
6
6
6
6
6
4
4E
⇣
I
C
H
+
I
G
L
⌘
2E
⇣
I
G
L
⌘
6E
⇣
I
C
H
2
⌘
2E
⇣
I
G
L
⌘
4E
⇣
I
C
H
+
I
G
L
⌘
6E
⇣
I
C
H
2
⌘
6E
⇣
I
C
H
2
⌘
6E
⇣
I
C
H
2
⌘
24E
⇣
I
C
H
3
⌘
3
7
7
7
7
7
5
(V.7)
✓ =
2
6
6
6
6
6
4
✓ b
✓ c
3
7
7
7
7
7
5
(V.8)
F =
2
6
6
6
6
6
4
0
0
P
3
7
7
7
7
7
5
(V.9)
It is seen that the exact moments of the single-story frame are function of I
C
,I
G
,H,L,
and E.
Insomecases,theremightbenoinformationavailableaboutthephysicalpropertiesof
theframe(suchasthemoment-of-inertiaofthebeamsandthecolumns),orperformingthe
125
exact analytical calculations might take time, especially, for bigger multi-bay multi-story
frames with no sophisticated FEA models available. As a result, approximate methods
were developed to estimate the moments of the frame. The main assumption with the
approximate method is that the inflection point of the member is at the midpoint of the
member. Since the inflection point has zero curvature, the bending moment will be zero
as well. As a result, one can use statics equations to compute the moments, without
incorporating any of the physical characteristics of the member into the equation. For
the frame in this example, the approximate moment can be calculated as follows:
ˆ
M
a
=
ˆ
M
b
=
P
2
⇥ H
2
=
P⇥ H
4
(V.10)
V.3.1 Numerical Results using Exact and Approximate Methods
In order to better understand the di↵erence between the exact and the approximate solu-
tion, and to investigate the sensitivity of the exact approach to the member sti↵ness and
the frame geometry, two numerical examples are presented. In Fig. V.3, the sensitivity of
the exact exact solution to the member sti↵ness is examined. It is seen that as the hor-
izontal member (girder) becomes sti↵er, the exact solution approaches the approximate
solution. The second subfigure in Fig. V.3 shows that if the vertical members are sti↵er
thanthehorizontalmember, thenormalizederrorbetweentheexactandtheapproximate
solution is about 31% when I
G
=0.25⇥ I
C
. It is also seen that the normalized error de-
creases to about 4% whenI
G
=5⇥ I
C
. This analysis shows that the approximate method
can be a viable approach to compute the member moments if the horizontal members are
much sti↵er than the vertical members (i.e., I
G
I
C
).
126
1 2 3 4 5
0.5
1
1.5
EI
G
/EI
C
Normalized moments

M
ab
: Exact
M
ba
: Exact
M
ab
= M
ba
: Approx.
1 2 3 4 5
0
10
20
30
40
EI
G
/EI
C
Normalized error (%)
Figure V.3: The first subfigure shows the moments M
ab
and M
ba
for di↵erent values of
EI
G
/EI
C
. Theexactandthetheapproximateresultsaresuperposed. Thesecondsubfig-
ure shows the normalized absolute percent error between the exact and the approximate
solutions for M
ab
.
127
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
H/L
Nomralized moments

M
ab
: Exact
M
ba
: Exact
M
ab
= M
ba
: Approx.
1 2 3 4 5
0
2
4
6
8
10
H/L
Normalized error (%)
Figure V.4: The first subfigure shows the moments M
ab
and M
ba
for di↵erent values of
H/L when I
G
=3I
C
. The exact and the the approximate results are superposed. The
second subfigure shows the normalized absolute percent error between the exact and the
approximate solutions for M
ab
.
Figure V.4 shows the sensitivity of both the exact and the approximate approaches
to the frame geometry. This numerical simulation was performed for constant moment-
of-inertia values for the horizontal and the vertical members, given by I
G
=3I
C
.The
simulationwasperformedforvariousframeheights(withaconstantbaywidth). Thefirst
subfigure shows that both the exact and the approximate solutions are relatively similar
for M
ab
and M
ba
.
When looking at the normalized error in the second subfigure, it is seen that as
the height-to-width ratio increases, the exact and the approximate solutions converge.
Meaning, for tall and slender frames, the approximate solution might provide a good
moment estimate (given, the horizontal members are sti↵er than the columns).
128
P
a
b c
d
I
1
± ✏
1
I
2
± ✏
2
I
3
± ✏
3
L
H
Figure V.5: The sample single-story frame with indeterministic parameters.
V.3.2 E↵ects of Uncertainty on the Moment Calculations
Let’s consider the frame shown in Fig. V.5. This is a similar frame to the one analyzed
earlier with a slight modification in the section properties. As seen in the figure, the
moment-of-inertia for the members ab, bc, and cd are I
1
±✏
1
, I
2
±✏
2
, and I
3
±✏
3
,respec-
tively. The values of I
1
, I
2
, and I
3
are not necessarily equal. On the other hand, ✏
1
, ✏
2
,
and ✏
3
represent the uncertainty in the moment-of-inertia values. In this case, we assume
the width of the bay L and the height of the frame H to be known and deterministic.
Similar to the previous case, a known point-load P is applied on the frame, as shown in
Fig. V.5.
129
Takingintoaccounttheerrors(i.e., uncertainties), thematrixKdefinedearlier, which
is used in the moment calculations, is updated as follows:
K =
2
6
6
6
6
6
4
4E

I
1
±✏
1
H
+
I
2
±✏
2
L

2E

I
2
±✏
2
L

6E
⇣
I
1
±✏
1
(H)
2
⌘
2E

I
2
±✏
2
L

4E

I
3
±✏
3
H
+
I
2
±✏
2
L

6E
⇣
I
3
±✏
3
(H)
2
⌘
6E
⇣
I
1
±✏
1
(H)
2
⌘
6E
⇣
I
3
±✏
3
(H)
2
⌘
12E
⇣
I
1
±✏
1
+I
3
±✏
3
(H)
3
⌘
3
7
7
7
7
7
5
(V.11)
Inordertoassesse↵ectsoftheuncertainparameters,numericalsimulationsareperformed.
The values for ✏
1
, ✏
2
, and ✏
3
are generated by creating a vector of random numbers
that has a zero-mean Gaussian distribution and a standard deviation equal to 10% of
the deterministic value of the corresponding moment-of-inertia. Each vector has 2000
samples. For each simulation, all section properties are assumed to be deterministic (i.e.,
✏ = 0) except one moment-of-inertia, which will have normally distributed random values
(i.e., I ± ✏). This will show the e↵ects of that uncertain parameter on the accuracy
of the calculated moments across the frame. The probability density functions pdf of
the calculated moments are then estimated using kernel density estimator. Assuming a
Gaussian kernel, the bandwidth h of the kernel was chosen to be h=1.06n

1
5
,where is the standard deviation of the data and n is the number sample points.
The results from two cases are shown in Figs V.6 and V.7. Figure V.6 shows the pdfs
from the case where I
1
=I
2
=I
3
= 1. Each subfigure corresponds to a di↵erent moment.
In each subfigure, the pdfs from three simulations are superposed. Each simulation has
a di↵erent random variable indicated in the legend. The normalized deterministic exact
moment is indicated with a dot on the x-axis. The first subfigure represents the distribu-
tions of the moment at pointa at the bottom of the left column in the frame shown in Fig
V.5. As seen, an uncertainty in the section properties of any of the columns, drastically
130
e↵ectsthedistributionofthecalculatedmoment(i.e., M
ab
)atthatlocation. Ontheother
hand, the e↵ects of the uncertainty in the girder’s properties are not as drastic.
Figure V.7 shows simulations from a case where I
1
= I
3
= 3 and I
2
= 1. This case
is similar to strong-column weak-beam condition encountered in most realistic structural
frames. Similar important information can be drawn by analyzing each of the subfigures.
This analysis shows the importance of taking uncertain parameters into account and
their e↵ect on the accuracy of the corresponding moment calculations for the calibration
phase in the load-estimation algorithm.
V.4 Finite-Element (FEA) Analyses
V.4.1 Model Description
A finite-element model of a four-story building was created in Femap
r
and a 3D view
is shown in Fig. V.8. The model consists of 2144 combined horizontal and vertical bar
elements. The vertical bar elements (columns) are modeled to represent a square tube
withdimensionsof(1in⇥ 1in),athicknessof(1/8in),andacross-sectionalareaof(0.4375
in
2
). The horizontal bar elements are modeled to have a cross-sectional area of (0.25 in
2
).
The material used for the elements is 6061-T651 Aluminum.
The width of the frame is (16 in), the height of each floor is (17.5 in), resulting in a
total height of (70 in). Nodal constraints are applied at four nodes at the bottom of the
model. The four nodes are fixed and cannot translate or rotate in any direction (x,y,z).
131
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
0
0.2
0.4
0.6
0.8
1
pdf [−]
M
ab

I
1
± ε
1
I
2
± ε
2
I
3
± ε
3
Exact moment
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
0
0.2
0.4
0.6
0.8
1
pdf [−]
M
ba
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
0
0.2
0.4
0.6
0.8
1
Normalized moment values
pdf [−]
M
cb
FigureV.6: Normalizedprobabilitydistributionfunctions pdfsofthecalculatedmoments
M
ab
, M
ba
, and M
cb
,where I
1
=I
2
=I
3
= 1.
132
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
0
0.2
0.4
0.6
0.8
1
1.2
pdf [−]
M
ab

I
1
± ε
1
I
2
± ε
2
I
3
± ε
3
Exact moment
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
0
0.2
0.4
0.6
0.8
1
1.2
pdf [−]
M
ba
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
0
0.2
0.4
0.6
0.8
1
1.2
Normalized moment values
pdf [−]
M
cb
FigureV.7: Normalizedprobabilitydistributionfunctions pdfsofthecalculatedmoments
M
ab
, M
ba
, and M
cb
,where I
1
=I
3
= 3 and I
2
= 1.
133
Figure V.8: A sketch of the four-story building-model created in Femap
r
.
V.4.2 Load Cases
A total of five load cases are investigated. The load cases under consideration are shown
in Fig. V.9. The first load case shown in Fig. V.9(a) is used for calibration. Two point
loads (100 lbs each) are applied at the tip of the two columns. Both loads are applied
in the same negative X-Direction. The purpose of this load case (calibration case) is to
obtain the strain measurements and estimate the section properties using the calculated
analyticalmoments. Usingtheestimatedsectionpropertiesfromthecalibrationload-case,
the next four load cases are used for testing.
Thesecondloadcase(DistributedPoint-Loads)isshowninFig. V.9(b). Inthiscase,8
point loads (100 lbs each) are applied on two columns (four loads per column). The loads
are all in the same direction. The loads are applied at the intersection of the horizontal
members (girders) with the vertical members (columns) at each floor level. The third
134
Table V.1: The investigated load cases for the four-story building model
Load/Moment Name Test Code Number of Loads Loading Values Location
Two Point-Loads (Calibration) CALIB 2 two 100 lbs Tip (free-end), two columns
Distributed Point-Loads DL 8 Eight 100 lbs Each floor, two columns
Uniform Pressure PL n/a 10 lbs/node Each node, two columns
Single Point-Load SPL 1 100 lbs Tip (free-end), one column
Moment (Torsion) TL 4 10 lbs.in One column, each floor
load case is shown in Fig. V.9(c). Point loads are applied at each node on two columns
to simulate a pressure load on the two columns. Each point load is 10 lbs and there are
a total of 256 node per column. The loads are applied in the same direction.
In order to assess the viability of the algorithm with torsional loads, two load cases
were created. The load case named Single-Point-Load (SPL) is shown in Fig. V.9(d),
where a single point load (100 lbs) is applied at the tip of one column. The other load
casenamedMoment(TL)isshowninFig. V.9(e),wherefourmoments(10lbs.ineach)are
applied(oneperfloor). Themomentsareappliedtorotatethestructurecounterclockwise
about the Z-Axis. The load cases are summarized in Table V.1.
V.5 Computational Results
V.5.1 Calibration Phase
The results from the calibration phase are shown in Fig. V.10. As mentioned earlier,
load case (CALIB) was used on the FEA model to obtain the strain measurements. The
presented strain measurements are obtained from the column labeled (1) in Fig. V.9(a).
TheX-AxisineachofthethreesubfiguresofFig. V.10representsthenormalizedheightof
135
(a) Two point-load (CALIB). (b) Distributed point-loads (DL).
(c) Uniform pressure (PL). (d) Single point-load (SPL).
(e) Concentrated moments (TL).
Figure V.9: Description of the five loading cases.
136
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
0
1
2
x 10
−3
Strain
Calibration Phase
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1000
0
1000
2000
Moment
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
1.1
1.2
x 10
6
Normalized length of the building
Estimated (EI/c)
Figure V.10: Strains and moments from the calibration load case, and the estimated
section properties.
thebuildingstartingfromzerobeingtheground(fixed-end)andendingwithonebeingthe
tip of the building (free-end). The discontinuities seen in the strain and the moment plots
are at the location of intersection between the horizontal and the vertical members. The
third subfigure shows the estimated flexural section properties. The column from which
the strain data is extracted has uniform section properties along the span, therefore,
the estimated EI/c values are identical at every sensor station. Of course, this is a
highly idealized case (proof of concept) where no uncertainty or modeling errors are
involved. More realistic scenarios will be investigated with the FEA, taking into account
uncertain parameters and modeling errors. Furthermore, the approach will be validated
with experimental data using an experimental testbed that is described in the upcoming
sections.
137
V.5.2 Estimation Phase
Usingthesectionpropertiesestimatedinthecalibrationphase,theremainingfourloading
cases are tested. The results are shown in Fig. V.11. As expected, the estimation is
excellent with bending-only loads (i.e., load cases DL and PL), but not as good when
twisting e↵ects are introduced (i.e., load cases SPL and TL). As mentioned earlier, this
is due to the fact that the algorithm takes into account strain from bending-only loads.
It is seen that as the torsional characteristics of the load increase, the estimation quality
decreases (Figs. V.17(c) and V.11(d)).
V.6 Design of The Experimental Test-bed Structure
V.6.1 Description of the Experimental Test-bed
A relatively large-scale experimental building structure was designed and fabricated at
the USC Machine Shop. The photo of the building is shown in Fig. V.12. The total
height of the structure is 72 in and the height of each floor is 17.5 in. As seen in Fig.
V.12, the building has four floors. All components of the structure is made of aluminum.
The columns are made of aluminum tubes. The cross-sectional dimensions of the columns
are (1 in⇥ 1 in), with a thickness of 3/16 in. The plates are also 3/16 in thick and are
made of aluminum. The plates are square-shaped with a dimension of (17 in ⇥ 17 in).
A typical floor cross-section is shown in Fig. V.13. The dimensions shown are in the
unit of inches. It is seen that the plates are connected to the columns via angle-brackets.
Each angle-bracket is connected to the column with two No. 10 bolts arranged diagonally
along one face of the angle. The angle’s other face is connected to the plate with two
No. 10 bolts. No. 10 bolts are also used to connect angle-sti↵eners to the plates. These
138
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1000
0
1000
2000
Moment
Distributed point−loads case
Normalized length of the building
(a) Distributed point-loads (DL).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
0
1
2
3
x 10
4
Moment
Pressure load case
Normalized length of the building
(b) Uniform pressure (PL).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−200
0
200
400
600
Moment
Single point−load case
Normalized length of the building
(c) Single point-load (SPL).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−5
0
5
10
15
Moment
Torsion load case
Normalized length of the building
(d) Concentrated moments (TL).
Figure V.11: Estimated moments (dotted line) from various loading cases superposed on
the exact analytical moments (solid line) from the FEA.
139
Figure V.12: A photo of the four-story experimental test-bed structure.
sti↵eners are designed to be portable so they can easily be added or removed, when
needed. Enlarged photos of both the angle-brackets and the sti↵eners are shown in Figs.
V.14(a) and V.14(b).
The structure is designed to be connected to a heavy base fixture via four large angle-
brackets connected to each column at the bottom of the structure. This can be seen in
Figs. V.15(a) and V.15(b). The large brackets are connected to the columns using M8
bolts. On the other hand, the large brackets will be concected to the base-fixture using
half-inch diameter bolts.
140
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Figure V.13: Drawing of a typical building floor cross-section with corresponding dimen-
sions in inches.
141
(a) A sample sti↵ener attached underneath one of the plates. The angle-
brackets used to connect the columns to the plates can be seen on the
sides.
(b) Enlarged photo of the bolts and the nuts used to
connect the sti↵eners to the plates.
Figure V.14: Photos of some of the details in the test-bed structure.
142
(a) A photo of the base angle-brackets to be connected
to a heave base-fixture.
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(b) A plan drawing of the four base angle-brackets at
the four columns.
Figure V.15: Details of the base angle-brackets.
143
Figure V.16: 3D view of the computational model designed in Femap
r
.
V.6.2 Description of the Computational Model
In order to augment the experimental studies, a finite-element model of the test-bed
structure is created using Femap
r
and is shown in Fig. V.16. The model consists of
45424 elements and 46346 nodes. The element type used to model most of the horizontal
and vertical members are 4-noded quad-shape plate elements. The connecting bolts were
modeledusing2-nodedline-shapebarelements. Thegeometryandthematerialproperties
were designed to match the physical test-bed structure described earlier.
Modal analysis was performed and the corresponding classical and torsional mode
shapes are plotted in Figs V.17 and V.18. The fundamental natural frequency of the
modeled structure is 6.76 Hz. The first torsional frequency is 15.90 Hz. The FOSS
system is currently capable of sampling at a rate of up to 50 Hz. The test-bed structure
was deigned to have two bending modes and two torsional modes below 50 Hz.
144
(a) First classical mode
shape (6.76 Hz).
(b) Second classical mode
shape (30.57 Hz).
(c) Third classical mode
shape (77.18 Hz).
Figure V.17: The classical mode shapes and the corresponding natural frequencies.
V.6.3 Instrumentation and Test Apparatus
Instrumentation
Installing the fibers on the test-bed is an important part of the experimental test proce-
dure. As shown in Section V.5, the load-estimation algorithm currently relies on bending-
only (flexural) strains to estimate the corresponding loads. In the same section, it was
shownthattheestimationqualitywasnotgoodwhendealingwithtorsionalloads. Hence,
there is a need to incorporate additional strain information (i.e., shear strain) in the al-
gorithm. In order to capture a more complete strain-field information, several fiber con-
figurations were considered. It was seen that the middle of the column-face is the best
145
(a) First torsional mode
shape (15.90 Hz).
(b) Second torsional mode
shape (49.33 Hz).
Figure V.18: The torsional mode shapes and the corresponding natural frequencies.
146
(a) Location of the fibers along the
column.
(b) The taping procedure in prepa-
ration for fiber-installation.
Figure V.19: Location of the fibers on the columns of the test-bed.
location for straight fibers to span, because it will capture the most accurate bending-
strains. In order to achieve a rosette-e↵ect and capture the shear strain, fibers were also
placed at 30 degree angles. Initially, a bending radius of 45 degrees was being considered,
butitwasfoundthata45degreebendingwill exceedtheallowablebendingradiusfor the
fiber. Accordingly, the fibers are bent at 30 degree angles and there are 8 locations along
thespanof eachcolumn-facewherethree-componentstraininformationcan bemeasured.
Figure V.19 shows the fiber locations on the columns. As seen, the fibers will be installed
on the outer two faces of each column.
147
(a) Two instrumented columns. (b) A magnified photo of the in-
stalled fibers.
Figure V.20: Photos of the installed fibers on the columns of the test-bed structure.
As seen in Fig. V.19 tapes are placed on the columns to depict the geometry of
the fibers before installing them. While the fibers are extremely thin, the tapes were
designed to provide a
1
4
inch wide pathway to install the fibers. It is worth mentioning
that Fiber Bragg Gratings (FBGs) are placed in the fibers at
1
4
inch distance from each
other, providing very high spatial-resolution sensors. Photos of theinstrumented columns
are shown in Fig. V.20.
148
Figure V.21: Solidworks
r
model of the test-bed structure and the test apparatus.
Test Apparatus
While the main objective of this research is to assess the viability of the approach in
predicting random dynamic pressure loads on buildings, the current study concentrates
on static loads similar to the ones described in Table V.1. In order to apply quasi-static
point-loads at various locations on the test-bed structure simultaneously, two pulley-
structures were designed and are shown in Fig. V.21. Furthermore, a base structure was
designed to fix the instrumented test-bed structure as well as one of the pulley-structures
on it. The dimensions and the spacings of the holes on the base structure were designed
to match the angle-brackets at the bottom of the test-bed structure explained earlier and
shown in Fig. V.15.
149
One of the main goals of the study is to enhance the accuracy of the algorithm when
torsionalloadsareapplied. Asaresult,thesecondpulley-structurewasdesignedtosimul-
taneouslyapplymulti-directionalloadsonthetest-bedandexplorethee↵ectsoftorsional
loads. As seen in Fig. V.21, a 6 feet tall person is incorporated in the Solidworks
r
model
togetaqualitativesenseoftheoveralldimensionsofthemultiplestructuresindiscussion.
V.7 Experimental Results
Preliminary experimental tests were performed to assess the test-bed, the sensors, and
the FEA model. The test-bed and the pulley structure are shown in Fig V.22. Initially, a
Figure V.22: The test-bed and the pulley structure.
free-vibration test was performed by pulling the structure (by hand) and letting it vibrate
freely. The purpose of this test was to identify the fundamental natural frequency of the
test-bedbytrackingthemeasuredstraincycles. Theexperimentallydeterminedfrequency
was6.5Hz,whichisrelativelyclosetothefrequencydeterminedfromtheFEA,whichwas
150
6.76 Hz. It should be noted that during the preliminary tests under discussion, both the
test-bed and the structure were fixed on a temporary base fixture (using supplementary
sandbags).
Then, four 25 lbs point-loads were applied on the test-bed, using sandbags (as shown
inFigV.23). Thestrainmeasruemtnsfromthecolumnswerethencomparedtoequivalent
Figure V.23: Four point-loads applied on the test-bed, using the pulley structure.
strains from the FEA. A sample result is shown in Fig V.24. As seen in Fig V.24, there
is a very good match between the superposed strain plots from the experimental and the
computational models. This indicates, along with the relatively good frequency match,
thatthefiniteelementmodelisarelativelygoodrepresentativemodeloftheexperimental
151
0 50 100 150 200 250 300
−100
0
100
200
300
400
500
Sensor index
Strain (micro)

FEA
Experiment
Figure V.24: Comparison between experimental and computational strain measurements.
test-bed structure and can be used for validation purposes throughout the next phases of
the research.
V.8 Discussion and Conclusions
The load-estimation algorithm under investigation was developed at the NASA Dryden
Flight Research Center providing a robust data-driven model-free approach to estimate
operating loads on flexible wing-like aerospace structures. The objective of this study is
to assess the viability of adapting the approach and applying it on civil structures, specif-
ically, tall building structures. Another objective of this study is to identify the various
complexities and challenges that might arise when using the approach with building-like
structures, and to propose potential ideas to address such challenges.
The first part of this study concentrated on performing analytical studies on the mo-
ment calculations that constitute a major part of the algorithm. It was shown that the
knowledge of the interacting horizontal and vertical members in typical frames, along
152
with their relative section properties, has a big impact on the accuracy of the moment
calculations. This is particularly important, since it shows thatapriori information is
needed for the approach to yield viable estimation results. Approximate moment cal-
culation methods were investigated where no knowledge is required about the physical
characteristics of the frame members, however, it was shown that the results converge
to the exact solution only if the horizontal members are much sti↵er than the vertical
members. In current design practice, most lateral resisting moment frames are geared
toward strong-column weak-beam concept, which further emphasizes that the exact mo-
ment calculation methods might be needed for our purposes. Furthermore, sensitivity
analysis was performed on the e↵ects of uncertainty on the moment calculations and it
was shown that an uncertain parameter in a certain frame member might significantly
impact the accuracy of the calculated moment in other frame members.
This study also pinpoints the discontinuous nature of the strain measurements from
building-like structures due to the floor-slabs along their span (or horizontal members
such as girders). In this study, it was shown that such discontinuities do not a↵ect the
accuracy of the load estimation algorithm if the calibration phase is performed properly.
However,itiscrucialtovalidatethefindingsfromthecomputationalmodelbyperforming
experimental analyses.
Asmentionedintheprevioussections,theload-estimationapproachiscurrentlybased
on bending-only strains for flexural behavior of wing-like structures. A common phe-
nomena in tall buildings, as well as many other types of structures, is the existence of
loads with twisting (torsional) e↵ects. The computational analysis in this study using a
preliminary finite-element model further emphasizes the importance of having a robust
153
approach that takes into account a more complete strain-field information, as opposed to
the bending-strains only.
With the above challenges in mind, the final part of this study concentrates on the
design of an experimental test-bed structure with the corresponding sophisticated finite-
element model. Details on the instrumentation, sensor location, sampling frequency,
the mode shapes of the test-bed, and the test apparatus for the load application were
discussed. Preliminary tests were performed and sample strain readings were reported.
Havingarelativelylarge-scaleexperimentaltest-bedstructurealongwithadetailedfinite-
element computational model is an excellent opportunity to perform various tests that
might address several of the challenges mentioned in this study, as well as explore other
potential complications that might arise. The load-estimation approach under investiga-
tion along with the FOSS sensing technology provide an excellent opportunity to develop
arobustframeworkthatcanestimatereal-timeoperatingloadsontallbuildings, andcon-
sequently, have a great impact on several design, control, and monitoring applications.
V.9 Acknowledgements
The assistance of Dr. Lance Richards, Francisco Pena, and the technicians at the NASA
Dryden Flight Research Center and the AERO Institute is highly appreciated.
154
Chapter VI
Conclusions
T
HIS dissertation is a collection of studies focusing on the extension and evaluation
of data-driven methodologies for applications ranging from developing mathemat-
ical models for response prediction and change detection in complex multi-component
systems, to nonlinear system identification, nonlinear damping estimation, deformation-
shape prediction, and operational-load estimation in various structural systems. The
proposed techniques along with the presented analyses, rely on evaluating and building
onexistingmethodologies,aswellasdevelopingnewtoolsandframeworksforinterpreting
largedatasetswiththepurposeofextractingmeaningfulinformationrelevanttocondition
assessment, health monitoring, and structural control.
With the constant advancements in the state-of-the-art of sensing technologies, it is
possible to obtain long-duration measurements (i.e., on the order of days or weeks) from
target structural systems. This provides an excellent opportunity to analyze, investigate,
and diagnose the performance of such systems. The availability of large sets of data
from full-scale structures allows the assessment of various techniques that have previously
been tested using mainly synthetic data generated from numerical models. Despite its
many advantages, the constant stream of measurements can create challenges in storing,
155
organizing,andinterpretingtheresults. Thefindingsinthisstudydemonstratethedanger
of misinterpreting the results from experimental datasets when used with certain existing
methodologies, henceit emphasizesthe importance of capturingthecorrect physics of the
system to be identified or monitored.
Thestudyevaluatestherobustnessandtherangeofvalidityofseveralexistingmethod-
ologies when used with experimental measurements, indicating their advantages and lim-
itations. The study also presents a convenient way of organizing and processing large sets
of data (i.e., earthquake records) to be used with the methods under investigation. This
study also focuses on developing mathematical models of nonlinear structural systems.
Due to the availability of long-duration multiple earthquake records, it is shown that in-
cluding the nonlinear e↵ects in the modeling procedure is crucial for accurate damping
estimation. Quantifying the nonlinear damping, in turn, is very important for structural
design, control, and monitoring purposes.
The testing and the validation of the investigated frameworks and approaches were
done using data from full-scale structures and large-scale experimental systems. The var-
ious investigated datasets include measurements from 85 sensors on the Yokohama Bay
Bridge that was subjected to the 2011 Great East Japan Earthquake and its many af-
tershocks; three di↵erent large-scale experimental soil-foundation-superstructure systems
subjected to 3 earthquakes with 6 intensity levels each; and data from 6 experiential
plate articles developed at the NASA Dryden Flight Research Center and instrumented
using state-of-the-art fiber-optic sensors. Furthermore, various analytical, numerical, and
computational (finite-element) studies were performed to complement the experimental
studies.
156
This work identifies and analyzes the numerous complications and diculties that
might potentially arise from combining high spatial-resolution sensing networks, signal
processing algorithms, damage detection approaches, and mathematical modeling frame-
works, to achieve a robust way of identifying, assessing and monitoring complex nonlinear
structural systems.
157
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167
Appendices
168
Appendix A
Generalization of the Subsystem Modeling
Approach on a Nonlinear Finite-Element Model
of a 3-Span Reinforced Concrete (RC) Bridge
A.1 Finite-Element Model Specification
A finite-element model of a 3-span reinforced concrete (RC) bridge was created using
SAP2000 v14.1.0. The material used was 4000Psi concrete. The cross-section of the
two columns was 4 ft wide and 6 ft deep. The rebar material for both longitudinal and
confinement bars was A615 Grade 60. For the longitudinal bars 3 #9 bars were used
in each direction. While for the confinement bars (ties), 3 #4 bars were used in each
direction. The bridge-deck was modeled as a rectangular box (6 ft deep and 40 ft wide)
with flange thickness of 1ft and web thickness of 2 ft. The longer column was 35 ft and
the shorter column was 25 ft. Each column was modeled using 3 elements, while each
span was modeled using 3 elements totaling in 9 elements for the deck (i.e., 10 nodes
(senseors)). A side-view of the bridge is shown in Fig.A.1. The mid-span is 150 ft long
and the 2 side-spans are 110 ft long. The boundary conditions were defined as rollers for
169
Figure A.1: An undeformed side-view (x-z plane) of the bridge.
the beginning and the end of the bridge deck, while the boundary conditions for the two
columns were fixed and pinned, respectively.
A.2 Time-History Analyses
Time-historyanalyseswereperformedonthebridge. Inordertoperformnonlinearanaly-
sis, 4 deformation-controlled frame hinges were introduced to the model. Also, a hysteric
behavioroftypeTakeda (availableinSAP2000)wasdefinedbeforethenonlinearanalysis.
The locations of the hinges on the columns are shown in Fig. A.2. As seen, the hinges
(denoted by FH1) are located at the top and the bottom of both columns. For the time-
history analysis, El-Centro record was applied to the bridge in the lateral (y-direction).
170
FigureA.2: Locationoftheintroduced4framehingesonthecolumnsofthebridge, fornonlinear
time-history analysis.
171
For the nonlinear time-history analysis, the direct integration method was used. Figure
A.3 shows the deformed (from linear analysis) and the undeformed states of the bridge.
Figure A.4 shows the plastic rotations from hinge 26H1 located at the bottom of
the shorter column. The x-axis represents the plastic-rotation (radians), and the y-axis
represents the moment (Kip-ft). While this is a slightly nonlinear system (based on the
properties of the hinge defined in SAP2000), one can increase the nonlinearity (the level
of plastic rotations) as desired.
A.3 Application of the Subsystem Approach for Response
Prediction
As mentioned earlier, the subsystem approach relies on training multiple neural networks
and integrating them with RK45 to achieve a dynamic prediction scheme. The approach
wasimplementedonthebridgebyusingtheresponsefrom8nodes(channels). Theresults
are shown in Fig. A.5. The displacement and velocity predictions are superposed on the
original signals from SAP2000. All presented results are in the lateral (y-direction). The
first and last rows correspond to the nodes closer to the end points of the superstructure
(deck), therefore, the magnitudes of vibrations are not very high. While, the response
from the mid-span is shown in rows 4 and 5, therefore, the response magnitudes are
greater. It is seen from the figure that there is an excellent match between the predicted
and the original response. This demonstrates that the subsystem approach for developing
computational models has a big potential to be applied on various structural systems,
regardless of system topology. Further studies can and should be performed by increasing
172
(a) Undeformed state of the bridge before the analysis.
(b) Deformed state of the bridge after the time-history analysis.
Figure A.3: Comparison of the undeformed state of the bridge with the deformed state, after the
time-history analysis.
173
Figure A.4: The plastic rotations from hinge 26H1(located at the bottom of the shorter column),
after the nonlinear time-history analysis.
174
thelevelofnonlinearityofthebridge,introducingnoisetothesyntheticresponseobtained
from the finite element package, using other time-history recordings, etc.
175
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−0.1
0
0.1
Disp. (ft)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
0 1 2 3 4 5 6
−1
0
1
Vel. (ft/sec)
Time (sec)
FigureA.5: Thedisplacementandvelocitypredictionsfromthecomputationalmodelsuperposed
on the original corresponding signals from 8 nodes (sensors) on the bridge deck.The networks were
trained using EL-Centro data and the base excitation from the same earthquake was fed back to
the created model.
176
Appendix B
Alternative Analyses on the Yokohama-Bay
Bridge (YBB) Datasets
B.1 Processed Time Histories from the Main-Shock (EQ1)
dataset
Figures B.1, B.2, B.3, B.4, B.5, and B.6, show the processed time histories from the
main-shock (EQ1) dataset. All the sub-figures have unified plotting scales (i.e., X-axis
and Y-axis limits). Acceleration records are plotted from 66 channels that are used in the
analysis.
B.2 Statistical Approaches
B.2.1 Covariance Matrix
One way of representing and analyzing large data sets is calculating the associated accel-
eration covariance matrix (ACM). The formulation of the ACM is given as follows:
177
0 100 200 300 400 500 600
−500
0
500
P1 S1(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 S3(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/2)S5R(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/2)S5L(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 S7(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T1(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T3R(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T3L(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T5(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T7R(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T8R(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T8L(X)
Time (sec.)
Accel. (cm/sec.
2
)
Figure B.1: Acceleration Plots from dataset EQ1.
178
0 100 200 300 400 500 600
−500
0
500
P3 T2(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T4R(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T4L(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T6(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 K2(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 K1(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 K4(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 K3(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 K5(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 K6(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 G1(X)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 S1(Y)
Time (sec.)
Accel. (cm/sec.
2
)
Figure B.2: Acceleration Plots from dataset EQ1.
179
0 100 200 300 400 500 600
−500
0
500
P1−P2 S2(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 S3(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/4) S4(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/2)S5R(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/2)S5L(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/4) S6R(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/4) S6L(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 S7(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T1(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T3R(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T3L(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 T5(Y)
Time (sec.)
Accel. (cm/sec.
2
)
Figure B.3: Acceleration Plots from dataset EQ1.
180
0 100 200 300 400 500 600
−500
0
500
P2 T7L(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T8R(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T8L(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T2(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T4R(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T4L(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 T6(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 K2(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 K1(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 K4(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 K3(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 K5(Y)
Time (sec.)
Accel. (cm/sec.
2
)
Figure B.4: Acceleration Plots from dataset EQ1.
181
0 100 200 300 400 500 600
−500
0
500
P3 K6(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 G1(Y)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 S1(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1−P2 S2(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 S3(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/4) S4(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/2)S5R(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/2)S5L(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/4) S6R(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
(L/4) S6L(Z)H19
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 S7(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P1 K2(Z)
Time (sec.)
Accel. (cm/sec.
2
)
Figure B.5: Acceleration Plots from dataset EQ1.
182
0 100 200 300 400 500 600
−500
0
500
P1 K1(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 K4(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P2 K3(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 K5(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 K6(Z)
Time (sec.)
Accel. (cm/sec.
2
)
0 100 200 300 400 500 600
−500
0
500
P3 G1(Z)
Time (sec.)
Accel. (cm/sec.
2
)
Figure B.6: Acceleration Plots from dataset EQ1.
183
C =
1
m
A
T
A (B.1)
where m is the number of measurement channels (in this study, m = 66), C is the ACM
matrix with dimensions of [m⇥ m]. Each column of matrix A contains acceleration data
for each channel, and is defined as:
A =
2
6
6
6
6
6
4
↵ (t
1
,1) ↵ (t
1
,2) ··· ↵ (t
1
,66)
.
.
.
.
.
.
.
.
.
.
.
.
↵ (t
k
,1) ↵ (t
k
,2) ··· ↵ (t
k
,66)
3
7
7
7
7
7
5
(B.2)
The covariance matrices are symmetrical by definition, and the order of the matrix
corresponds to the number of channels. The diagonal elements represent the acceleration
variancesforeachchannel, whiletheo↵-diagonalelementsrepresentsthecross-correlation
between channels. A sample covariance matrix is plotted in Fig.B.7.
FigureB.8showsthediagonalvariancesextractedfromtheACMmatricescorrespond-
ing to 10 di↵erent earthquakes. It is seen that datasets EQ1 and EQ2 have large variance
amplitudes, while it is dicult to observe the aptitude changes in the other aftershocks.
As a result, a di↵erent interpretation of the same figure is presented in Fig. B.9.
It is seen from Fig. B.9 that the variance amplitudes have similar behavior for all
datasets, but are di↵erent in their magnitudes. It can also be seen that channels 33 and
40consistentlyhavelargeramplitudescomparedtotheotherchannels. Itshouldbenoted
that channel 33 corresponds to sensor “T1” in the lateral (Y) direction, and channel 40
corresponds to sensor “T2” in the lateral (Y) direction. These sensors are located on the
top of the bridge tower, one on each side.
184
Figure B.7: Covariance matrix corresponding to dataset EQ1.
185
10
20
30
40
50
60
1
2
3
4
5
6
7
8
9
10
0
2
4
6
x 10
6
Channel number
Earthquake ID
Acc
2
, (cm sec
−2
)
2
Figure B.8: Acceleration variances for 10 datasets.
AnotherwayofutilizingtheACMmatricesisbysubdividingeachrecordintomultiple
segments (frames), and then track the change in the variance amplitudes. Dataset EQ1
was divided into twelve 50-second frames, and the ACMs were calculated for each frame.
Figure B.10 shows the amplitude variation along the twelve frames. Non-surprisingly,
the highest amplitudes are observed during the strongest parts of the ground motion.
The nine aftershocks are represented by their minimum, maximum, mean, and stan-
dard deviation values in Fig. B.11. It is seen that the maximum aftershock variance
amplitudes can be estimated by adding the mean variances to 2⇥ the standard deviation
values.
186
0 10 20 30 40 50 60
0
2
4
6
x 10
6
Earthquake1
Acc
2
0 10 20 30 40 50 60
0
5
10
15
x 10
5
Earthquake2
0 10 20 30 40 50 60
0
5000
10000
Earthquake3
Acc
2
0 10 20 30 40 50 60
0
1000
2000
3000
4000
Earthquake4
0 10 20 30 40 50 60
0
1
2
3
4
x 10
4
Earthquake5
Acc
2
0 10 20 30 40 50 60
0
2000
4000
6000
Earthquake6
0 10 20 30 40 50 60
0
0.5
1
1.5
2
x 10
4
Earthquake7
Acc
2
0 10 20 30 40 50 60
0
5000
10000
15000
Earthquake8
0 10 20 30 40 50 60
0
2
4
6
8
x 10
4
Earthquake9
Channel ID
Acc
2
0 10 20 30 40 50 60
0
2000
4000
6000
Earthquake10
Channel ID
Figure B.9: Subplots of the acceleration variances for 10 datasets. Note that amplitude scale is
not the same.
187
10
20
30
40
50
60
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
x 10
6
Channel number
Frame number
Acc
2
, (cm sec
−2
)
2
Figure B.10: Acceleration variances from EQ1 dataset, subdivided into 12 segments (frames).
B.2.2 Proper Orthogonal Decomposition
The Karhunen-Loeve transform Ghanem and Spanos (1991) is a linear technique that
transforms a data set to a new coordinate system. Based on this transformation, the
largest variance by any projection of the dataset lies on the first axis, the second largest
variance on the second axis, and so on. This technique is also referred as principal
component analysis or proper orthogonal decomposition Kerschen and Golinval (2002a).
The proper orthogonal decomposition of the acceleration covariance matrix can be
expressed as follows:
[C][e]= [e] (B.3)
where [e] is the matrix of the proper orthogonal modes (POM), and is the vector of the
corresponding proper orthogonal values (POV). The POVs determine the contribution
188
0 10 20 30 40 50 60 70
0
2
4
6
8
10
12
14
x 10
5
Channel number
Acc
2
, (cm sec
−2
)
2

Mean + 2*st. dev
Mean + st. dev
Mean
Max
Min
Figure B.11: Mean acceleration variances for the nine aftershocks. The combined histograms
depict these variance parameters: mean, (mean + standard deviation), (mean +2⇥ standard
deviation), minimum and maximum values for the aftershocks.
189
0 10 20 30 40 50 60 70
0
1
2
3
4
5
6
7
8
9
10
x 10
6
Mode number
POV values
Figure B.12: POV values of the acceleration covariance matrix from dataset EQ1.
of each mode to the bridge’s vibrational energy. As seen in Fig. B.12 the first several
orthogonal modes have significant contribution, while the amplitudes associated with
higher modes are almost negligible. In fact, the first 6 modes correspond to the 90%
range of the eigenvalue range.
Several studies were performed on the interpretation of the proper orthogonal modes
exploringthepotentialcorrelationwiththeclassicalundampedmodeshapes. Someofthe
relevant works are by Kerschen and Golinval (2002b), Feeny and Liang (2003), Kerschen
et al. (2005), amongst others. In the mentioned works, it is shown that while proper
orthogonal modes (POMs) is a promising alternative way to interpret the vibrational
characteristics of a structure, it is not a trivial task to correlate the POMs to classical
190
mode shapes, especially for highly-damped, nonlinear, non-symmetric structures, with
unknown mass distribution.
GiventhattheYokohama-Baybridgeisarelativelysymmetricstructure,andassuming
that it is only slightly damped, the proper orthogonal modes were calculated using the
acceleration records from the bridge-deck. A total of 23 channels in all three directions
were used (i.e., sensors with S label on the diagram in Fig. III.3). The obtained proper
orthogonal modes (POMs) are plotted and compared to the undamped classical mode
shapes from the least-square method. The same channels were used in calculating the
modes from both approaches. The results for the first 4 mode shapes from 3 earthquake
datasets (EQ1, EQ5, and EQ10, respectively) are plotted in Figures B.13, B.14, B.15,
B.16, B.17, and B.18.
Comparingclassicalmode1(Fig. B.13)toPOM1(Fig. B.14), obtainedusingdataset
EQ1,itisseenthattheyhavesimilardeformationshape. Ontheotherhand,thepotential
relationship between the other plotted mode shapes is not very easy to detect visually.
No clear similarities are observed in the results from dataset EQ5. Looking at the results
from dataset EQ10, it is seen that classical mode 2 (Fig.B.17) matches POM 2 (Fig.
B.18).
In order to further assess the correlation between the classical and energy modes, the
modal assurance criterion (MAC) values associated with all the identified modes from
dataset EQ1 are calculated and plotted in Fig. B.19.
Themodalassurancecriterion(MAC)iscalculatedtoquantifythecorrelationbetween
mode shapes. The MAC value between two mode shapes
ˆ
i
and
ˆ
j
is defined as:
MAC
i,j
=
|
ˆ
H
i
ˆ
j
|
(
ˆ
H
i
ˆ
i
)(
ˆ
H
j
ˆ
j
)
(B.4)
191
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode1fr
1
= 0.31989
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode2fr
2
= 0.33582
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode3fr
3
= 0.39166
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode4fr
4
= 0.50857
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode5fr
5
= 0.514
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode6fr
6
= 0.55912
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode7fr
7
= 0.58692
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode8fr
8
= 0.68613
Figure B.13: Undamped classical mode shapes obtained using dataset EQ1.
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode1
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode2
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode3
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode4
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode5
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode6
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode7
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode8
Figure B.14: POM mode shapes obtained using dataset EQ1.
192
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode1fr
1
= 0.26427
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode2fr
2
= 0.31562
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode3fr
3
= 0.34107
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode4fr
4
= 0.36456
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode5fr
5
= 0.49832
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode6fr
6
= 0.55494
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode7fr
7
= 0.58923
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode8fr
8
= 0.67772
Figure B.15: Undamped classical mode shapes obtained using dataset EQ5.
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode1
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode2
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode3
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode4
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode5
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode6
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode7
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode8
Figure B.16: POM mode shapes obtained using dataset EQ5.
193
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode1fr
1
= 0.26793
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode2fr
2
= 0.28228
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode3fr
3
= 0.36202
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode4fr
4
= 0.37294
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode5fr
5
= 0.39736
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode6fr
6
= 0.51276
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode7fr
7
= 0.56313
0
100
200
300
400
500
600
700
800
−50
0
50
−40
−20
0
20
40
Mode8fr
8
= 0.58727
Figure B.17: Undamped classical mode shapes obtained using dataset EQ10.
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode1
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode2
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode3
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode4
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode5
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode6
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode7
0
100
200
300
400
500
600
700
800
−50
0
50
−50
0
50
Mode8
Figure B.18: POM mode shapes obtained using dataset EQ10.
194
wherethesubscriptH,denotestheHermitianofamatrix. TheMACvaluerangesbetween
0 and 1; 0 for orthogonal, and 1 for identical mode shapes.
In Fig. B.19, it is seen that some of the modes are highly correlated with each other
resulting in MAC values close to 1 (i.e., perfect correlation). For example, the MAC
number between energy mode 1 and classical mode 1 is 0.96. This is in accordance with
the relatively similar mode 1 shapes observed in Figures B.13 and B.14. It is also seen
that, for higher energy modes, the MAC values are very low. This is also in accordance
with Fig. B.12, where it is shown that only the first few POD values (i.e., energy modes)
have significant contribution.
It is seen that the proper orthogonal mode shapes (POMs) are a useful way of inter-
preting the vibrational energy characteristics associated with each mode. As seen from
the previous figures, and the MAC values, it is very dicult to define a specific correla-
tion pattern between the classical undamped mode shapes and the POMs. However, the
lower-ordermodesconsistentlyhavehigherMACvalues, showingthatthereisarelatively
high correlation between the first several mode shapes. As shown through the POD val-
ues, only the first 4-5 modes have significant contribution, while the higher modes can be
disregarded.
195
2
4
6
8
10
12
14
16
0
5
10
15
20
25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Classical modes
Energy modes
MAC
Figure B.19: MAC values from the undamped classical mode shapes and the POM modes cor-
responding to dataset EQ1.
196
Appendix C
Sensitivity Analyses of the Modal Approach
C.1 FBG Sensor Placement
According to Foss and Haugse (1995), Kirby et al. (1995), Davis et al. (1996), Kang
et al. (2007), and Kim et al. (2011), the estimation quality of the modal approach can
be improved by determining the placement of the FBG sensors that corresponds to the
lowest condition number ( ) of the DST matrix ([T]). The condition number of the DST
matrix is estimated by the following equation:
 =k[T]k·k[T]
1
k (C.1)
wherek·kisthenormofthematrix. Ingeneral,a isgreaterorequalthan1,(i.e.,  = 1).
The case when  = 1 means the matrix is well-conditioned and most of the information is
preserved. In this study, we choose the first four modes of the structure (i.e., n = 4) and
we use data from four FBG sensors (i.e., M =n = 4). It is desired to obtain the locations
of the FBG sensors along the span of the fiber so that the condition number of the DST
matrix is a minimum. The number of possible placement configurations of the four FBG
sensors depends on the mesh size of the computational model from which the modal data
197
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Normalized fiber length
Normalized vibration mode shapes

Vibration mode 1
Vibration mode 2
Vibration mode 3
Vibration mode 4
Sensor location
Figure C.1: The vibration mode shapes and the optimum FBG sensor locations.
(i.e., and matrices) are extracted. A denser mesh results in more nodes along the
spanofeachfiber; consequently, morelocationsforeachFBGsensortobeplacedon. The
model used for this study has 35 nodes (sensors) along each fiber. The Matlab
R
function
nchoosek.m is used to determine the possible location combinations for the four sensors,
and the condition number ( ) is calculated for each combination. The combination with
the lowest condition number is then picked as the optimum location for the four sensors.
Using the modal data from the top-middle fiber, the optimum locations for the four
FBGsensorsonthenormalizedunit-lengthfiberareasfollows: 0.0857,0.3429,0.6000,and
0.8857. These locations correspond to the lowest condition number  =1.3579. Figure
C.1showsthenormalizedvibrationmodeshapes,alongwiththeobtainedoptimumsensor
198
0 0.2 0.4 0.6 0.8 1
−16
−14
−12
−10
−8
−6
−4
−2
0
2
Normalized fiber length
Displacement (in)
Estimated displacements

actual
2 sensors
3 sensors
4 sensors
0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
Normalized fiber length
Normalized error (%)
Estimation errors

2 sensors
3 sensors
4 sensors
Figure C.2: The e↵ects of the number of sensors on the estimation of the displacement
shape.
locations aligned on top. The x-axis represents the normalized fiber length [01], while
the y-axis represents the normalized mode shapes [11].
C.2 Number of FBG Sensors
The number of sensors to be used in the system-identification process highly depends
on the required estimation accuracy; e.g., in very light aerospace structures, such as
unmanned flying wings, relatively high level of precision is usually expected. In order
to assess the e↵ects of the number of sensors on the modal approach, three study cases
are created using 2, 3, and 4 sensors, respectively. For each case, the displacements are
estimated and the corresponding normalized estimation errors are calculated using the
following equation:
e =
|yˆ y|
|y|
⇥ 100% (C.2)
199
where e is the normalized error, y is the real displacement, and ˆ y is the estimated dis-
placement. The results of the case studies are shown in Fig. C.2. The left sub-figure
shows the estimated displacements for the cases of 2, 3, and 4 sensors, along with the
actual displacement, plotted in solid line. The horizontal axis represents the fiber length
normalized between 0 and 1 (the actual length of the top-middle fiber is 51 in), and the
vertical axis represents the displacement in inches. The sub-figure on the right side shows
the estimation errors obtained using Eq. C.2. As seen in both figures, in the case of
using 2 sensors, the estimated displacements have an average error of about 50%, but
they still roughly follow the trend of the actual displacement. When using 3 sensors, the
estimation results drastically improve with an average normalized error of about 15%.
For the case of 4 sensors, the estimated displacements are essentially on top of the actual
displacements, with an average normalized error of about 0.5%. In all three cases, the
number of modes (n) included in the calculations is equal to the number of sensors (M),
(i.e., M = n). As expected, increasing the number of sensors and the number of modes
used in the algorithm highly improves the accuracy of the estimation. Depending on the
type of application, it is up to the user to decide the number of sensors and the modes to
be included.
C.3 E↵ects of Measurement Noise
Measurementnoiseisoftenencounteredinreal-lifesituationssuchasexperimentaltesting.
Arelativelylargeamountofnoisemightsignificantlyimpacttheviabilityofthealgorithm
and negatively a↵ect the quality of the corresponding estimation. In order to assess the
e↵ects of noise on the modal estimation approach, synthetic noise is created and added to
200
0 0.2 0.4 0.6 0.8 1
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
Normalized fiber length
Displacement (in)
Estimated displacements

actual
0% noise
1% noise
5% noise
10% noise
20% noise
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
16
18
20
Normalized fiber length
Normalized error (%)
Estimation errors

0% noise
1% noise
5% noise
10% noise
20% noise
FigureC.3: Thee↵ectsofmeasurementnoiseontheestimationofthedisplacementshape.
the strain “measurements”, and several case studies are considered. The synthetic noise
is created by calculating the root-mean-square (rms) of the measurement signal, then
multiplying the desired percentage of the rms by a vector of normally-distributed random
numbers. The resulting noise vector is then added to the original measurement signal.
The Matlab
R
function randn.m can be used to generate a vector of normally distributed
pseudorandom numbers.
Four study cases are considered using data with 0%, 1%, 5%, 10%, and 20% noise,
respectively. For all the cases, four sensors and four mode shape are used, (i.e., M =
n = 4). The obtained results are presented in Fig. C.3. As in the previous section,
the horizontal axis represents the normalized length of the fiber (the top-middle fiber in
this case), the vertical axis in the left sub-figure represents the displacements (in), while
the vertical axis in the right sub-figure represents the normalized errors (%). There are
6 displacement plots in the left sub-figure, the “real” displacement and the estimates
corresponding to the 5 study cases. In all cases, the estimates follow the trend of the real
201
0 0.2 0.4 0.6 0.8 1
−8
−7
−6
−5
−4
−3
−2
−1
0
Normalized fiber length
Displacement (in)
Estimated displacements

Uniform
Single point
Leading edge
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
Normalized fiber length
Normalized error (%)
Estimation errors

Uniform
Single point
Leading edge
Figure C.4: The e↵ects of di↵erent loading conditions on the estimation of the displace-
ment shape.
displacements, but as the noise level increases, the accuracy of the estimates decreases.
Looking at the error plot, it is seen that the estimates corresponding to 0% and 1% noise
have error values less than 1% for more than 75% of the length of the fiber. The errors
are higher towards the fixed end of the plate (i.e., beginning of the fiber), and seem to
significantly decrease towards the free-end of the plate (i.e., end of the fiber). It is noted
that the errors increase along the entire length of the fiber, as the noise level increases.
For a significantly high level of noise (i.e., 20%), the estimation errors stay in an average
value of around 15%. In general, it is observed that for low levels of noise, the estimation
accuracy is relatively high, especially towards the middle and the free-end of the fixed
swept plate.
202
C.4 Type of Loading
The sensitivity of the modal algorithm to various loading conditions is assessed in this
section. The analysis shown in the previous sections were based on the uniform- load
case, as described in Table II.1. In this section, the results of the uniform-load case is
compared to the single-point and the leading-edge load cases (see Table II.1). The latter
two loading cases induce torsional and twisting e↵ects that can not be captured in the
caseofauniformlydistributedload. Suche↵ectsareusuallyencounteredinrealisticflight
conditions, and it is essential to investigate the ability of the algorithm to estimate the
displacement shapes in the presence of these type of loads. In all three loading cases, data
from the top-middle fiber is used, with no artificially induced noise.
The results of the simulations are presented in Fig. C.4. As in the previous sections,
the left sub-figure shows the estimated displacements compared to the actual displace-
ments (solid line), and the right sub-figure presents the corresponding normalized errors.
It is seen that in all three loading cases, the estimation error drastically decreases towards
the free-end of the plate. Furthermore, there is an embedded error near the fixed-end of
the plate for all the cases. It is noted that the leading-edge loading case contains the
strongest torsional e↵ects; consequently, it has the highest normalized error at the fixed-
endoftheplate(around30%). Ontheotherhand, theuniform-loadcasedoesnotcontain
any torsional e↵ects, and its normalized error at the fixed-end is relatively low (around
1%). In the case of the single-point load case, the normalized error at the fixed-end is
around 14%. It is also noted that the type of loading a↵ects the location of the fiber
along which the error is minimum (i.e., 0%). In the case of the uniform-load case, the er-
ror converges to its minimum value relatively quickly, and a viable estimation is achieved
203
alongmore90%ofthefiberlength. While, intheothertwoloadingcases, theerrorsdon’t
converge as quickly (as a result of the higher initial value), and an optimum estimation
is achieved along around 55% of the fiber length.
C.5 Discussion
C.5.1 Advantages
As seen from the previous analysis, the real-time displacement-shape prediction of a
lightweight aerospace structure using its modal properties promises to be an e↵ective
approach for the purposes of structural health monitoring and control. With the modal
approach, it is possible to estimate a relatively large displacement field using only few
fiber-optic strain sensors. As shown earlier, the modal technique under investigation pro-
vides satisfactory results under various loading conditions (including twisting loads) and
in the presence of measurement noise. It is possible to decrease the estimation error by
increasing the number of the fiber-optic sensors used for measurment, and increasing the
number of higher modes of vibration incorporated in the calculations. Furthermore, the
algorithm for the modal approach is relatively simple, and does not require a significant
computational e↵ort. As a result, its implementation for online monitoring can be more
ecient and practical, compared to other algorithms that have higher computational
demands.
204
C.5.2 Limitations
Itisseenthatthemodalapproachprovidesacceptableestimationsalongmorethanhalfof
thefiberlength. However, theestimationisnotasaccurateasonegetsclosertothefixed-
end of the plate. Also, in the presence of torsional loads, the estimation error converges
to its minimum value at a much a slower rate compared to the translation-only load cases
(i.e., theuniform-loadcase). Thefactthatthemodalapproachdependsonmeasurements
fromonlyafewfiber-opticsensors(3-4inthecaseofthisstudy),makestheaccuracyofthe
measurements and the functionality of the sensors a high priority, especially during real-
time flights. During real flights, it is not uncommon for some sensors to malfunction. In
thecaseofthemodalapproach,suchmalfunctionscansignificantlya↵ecttheperformance
of the algorithm, as the remaining sensors will be in non-optimum locations (i.e., higher
condition number and non-accurate estimation). Finally, the modal approach depends on
the displacement and strain mode shapes of the structure to be monitored and controlled.
These mode shapes are usually calculated through state-space and FEA models based on
a priori knowledge of the physical properties of the structure. During flights, any change
in these properties (due to damage) might a↵ect the recorded strain data, and lead to
inaccurate estimations when it is combined with the original (undamaged) mode shapes.
C.6 Conclussions
Through the sensitivity analysis performed in this study, it has been shown that an
estimation with a relatively high level of accuracy can be achieved using only few FBG
sensors. Furthermore, promising estimation results have been obtained in the presence of
various factors, such as measurement noise and torsional loads. Based on the obtained
205
results, it is seen that the modal algorithm is a promising methodology for real-time
deformation-shape prediction. When combined with the recent developments in fiber-
optic FBG sensors, the proposed approach has a big potential to be used in lightweight
unmanned aerospace vehicles and next-generation commercial airplanes for the purposes
of structural health monitoring and condition assessment.
206

Abstract (if available)

Abstract The recent advancements in computational capabilities and sensing technologies provide an excellent opportunity to develop, test, and validate data‐driven mathematical models for system identification, condition assessment, and health monitoring of structural systems that may be vibrating in linear and/or nonlinear ranges. In this study, measurements from various large‐scale, complex, experimental systems, as well as full‐scale real‐life multi‐input‐multi‐output (MIMO) structures are used to develop robust mathematical frameworks for response prediction, change detection, nonlinear damping estimation, in addition to displacement‐field and operating‐load estimation. The systems under consideration are the Yokohama Bay Bridge which was subjected to the 2011 Great East Japan Earthquake

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Asset Metadata

Creator Derkevorkian, Armen (author)

Core Title Studies into data-driven approaches for nonlinear system identification, condition assessment, and health monitoring

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 04/02/2014

Defense Date 03/02/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag condition assessment,data‐driven approaches,large‐scale experiments,mathematical modeling,nonlinear systems,OAI-PMH Harvest,sensing technologies

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Masri, Sami F. (committee chair), Ioannou, Petros (committee member), Wellford, L. Carter (committee member)

Creator Email derkevor@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-372814

Unique identifier UC11297364

Identifier etd-Derkevorki-2314.pdf (filename),usctheses-c3-372814 (legacy record id)

Legacy Identifier etd-Derkevorki-2314.pdf

Dmrecord 372814

Document Type Dissertation

Format application/pdf (imt)

Rights Derkevorkian, Armen

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA