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Studies into vibration-signature-based methods for system identification, damage detection and health monitoring of civil infrastructures

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Studies into vibration-signature-based methods for system identification, damage detection and health monitoring of civil infrastructures

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Content Studies Into Vibration-Signature-Based Methods for System
Identification, Damage Detection and Health Monitoring of Civil
Infrastructures
by
Reza Jafarkhani
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(Civil Engineering)
May 2013
Copyright 2013 Reza Jafarkhani
Dedicated to my beloved family back in Iran and to my late grandma, who passed away
just before my defense.
ii
Acknowledgements
I would first and foremost like to express my deep and sincere gratitude to my advisor, Professor
Sami F. Masri, for his invaluable guidance, support, and encouragement during the course of my
long graduate studies and preparation of this work. His breath of knowledge, mentorship, and ex-
pertise have been invaluable resources for me in choosing the direction of my work and fulfilling
my desire to study in a multi-disciplinary setting. I would also like to thank the other members of
my Ph.D committee, Professor L. Carter Wellford, Professor James C. Anderson, Professor Erik
A. Johnson, and Professor Roger G. Ghanem. I am particularly thankful to Professor Ghanem for
his cooperation of providing the exclusive access to the CPU clusters on HPCC. The assistance of
Professor M. Saiid Saiidi of the University of Nevada, Reno and Dr. Nathan Johnson, his former
PhD student, is highly appreciated. I am also thankful to Professor Bin Xu, Dr. Jia He and Dr.
Ren Zhou of the Hunan University, China for providing access to the experimental data set.
I would like to thank my colleagues in our research group for their friendship and hours of useful
discussions. They are Dr. Farzad Tasbihgoo, Dr. Reza D. Nayeri, Dr. Mohammadreza Jahan-
shahi, Dr. Hae-Bum (Andrew) Yun, Ali Bolourchi, Miguel G. Hernandez, Vahid Keshavarzzadeh
and Armen Derkevorkian.
Last but not least, I must thank my beloved family back in Iran, especially my parents for con-
stantly encouraging and inspiring me throughout my studies since the first day I went to school
28 years ago, and my brothers and sister for their love and support.
Finally, I gratefully acknowledge the partial support by the National Science Foundation (NSF).
iii
Table of Contents
Abstract xii
I INTRODUCTION 1
I.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I.3 Motivation and Technical Challenges . . . . . . . . . . . . . . . . . . . . . . . . 4
I.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
II COMPARISON OF DIFFERENT GLOBAL OPTIMIZATION TECHNIQUES FOR
HIGH-DIMENSIONAL PROBLEMS 7
II.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
II.1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II.2 Optimization Methods under Discussion . . . . . . . . . . . . . . . . . . . . . . 8
II.2.1 CMA-ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II.2.2 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
II.3 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
II.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II.4.1 Problem-order effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II.4.2 Population size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
II.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
III FINITE ELEMENT MODEL UPDATING USING EVOLUTIONARY STRATEGIES 21
III.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
III.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
III.1.2 Motivation and Technical Challenges . . . . . . . . . . . . . . . . . . . 23
III.1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
III.2 Overview of Identification Approach . . . . . . . . . . . . . . . . . . . . . . . . 25
III.2.1 Subspace Method for System Identification . . . . . . . . . . . . . . . . 25
III.2.2 Formulation of Cost Function . . . . . . . . . . . . . . . . . . . . . . . 27
III.3 Experimental Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
III.3.1 Description of Test Bridge Structure . . . . . . . . . . . . . . . . . . . . 28
III.3.2 Destructive Shaking Procedure . . . . . . . . . . . . . . . . . . . . . . . 29
III.3.3 Instrumentation, Data Acquisition and Filtering . . . . . . . . . . . . . . 30
iv
III.3.4 NASTRAN
R
Finite Element Model . . . . . . . . . . . . . . . . . . . 33
III.3.5 Choice of Initial Parameters and Weights . . . . . . . . . . . . . . . . . 36
III.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
III.4.1 Preliminary Damage Tracking . . . . . . . . . . . . . . . . . . . . . . . 37
III.4.2 Model Updating Results . . . . . . . . . . . . . . . . . . . . . . . . . . 41
III.4.3 Validation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
III.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
IV EV ALUATION AND APPLICATION OF SOME DATA-DRIVEN APPROACHES
FOR THE DEVELOPMENT OF EQUIV ALENT LINEAR SYSTEM FOR NON-
LINEAR STRUCTURES 59
IV .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
IV .1.1 Vibration-Signature-Based Nonlinear System Identification Methods: A
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
IV .1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
IV .2 Overview of System Identification Methods Under Discussion . . . . . . . . . . 66
IV .2.1 Linear System Using Least-Squares Method . . . . . . . . . . . . . . . . 66
IV .2.2 Symmetric Linear System Using Least-Squares Method . . . . . . . . . 69
IV .2.3 Restoring Force Surface (RFS) Method for Chain-Like Systems . . . . . 71
IV .2.4 Model Updating Method . . . . . . . . . . . . . . . . . . . . . . . . . . 74
IV .2.5 Sub-Space Identification Method . . . . . . . . . . . . . . . . . . . . . . 75
IV .2.6 Iterative Prediction-Error Minimization Method . . . . . . . . . . . . . . 78
IV .3 Experimental Case Study: Hunan University Building Model Structure . . . . . . 79
IV .3.1 Description of Test Building Model . . . . . . . . . . . . . . . . . . . . 79
IV .3.2 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
IV .4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
IV .4.1 Identified Mass, Damping and Stiffness Matrices . . . . . . . . . . . . . 84
IV .4.2 Identified Classical Frequencies and Damping Values . . . . . . . . . . . 87
IV .4.3 Restoring Forces in Chain-Like System . . . . . . . . . . . . . . . . . . 91
IV .5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
V IDENTIFICATION OF NONLINEAR STRUCTURAL MODELS USING ARTIFI-
CIAL NEURAL NETWORKS 97
V .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
V .1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
V .1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
V .2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
V .2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
V .2.2 Neural Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . 103
V .2.3 Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
V .3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
V .3.1 Performance in Detection of Change . . . . . . . . . . . . . . . . . . . . 107
V .3.2 General Neural Network Model Combined with ODE Solvers . . . . . . 110
v
V .3.3 Estimation of Structural Mass for Chain-Like Systems . . . . . . . . . . 114
V .4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
VI Conclusion 119
VI.1 Overview of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
VI.1.1 Chapter II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
VI.1.2 Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
VI.1.3 Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
VI.1.4 Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
BIBLIOGRAPHY 125
AppendixA 136
vi
List of Figures
II.1 Concept behind the covariance matrix adaptation for generation evolution. Two phe-
nomena are observed as the generations develop: 1- The center of the new generation is
shifted toward global minimum due to the weighting of the population for reproduction
process. 2- The distribution shape adapts to an ellipsoidal or ridge-like landscape along
the principle axis, which is the eigenvector corresponding to the largest eigenvalue of
the covariance matrix. (figure adapted from http://en.wikipedia.org/wiki/CMA-ES). . . 10
II.2 Plot of 2-D Rosenbrock function. . . . . . . . . . . . . . . . . . . . . . . . . . 13
II.3 Plot of 2-D Schwefel function. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
II.4 Plot of 2-D Ackley function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II.5 Plot of 2-D Rastrigin function. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II.6 Plot of 2-D Griewank function. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II.7 Effects of problem-order on the performance of evolutionary optimization meth-
ods (GA and CMA-ES) for Ackley function. Population size in each generation
is set as 100. The results are based on 100 simulations for each case. In each
plot, the solid line shows the average of normalized error in the final solution
vector while the dashed line indicates the average number of function evaluation
to reach the solution. The statistical distribution for normalized error of the fi-
nal solution and average number of function evolutions are also shown in small
subplots. Note that the relatively small mean value and standard deviation of
the normalized error (with respect to the dimension range = 32.768) shows the
robustness and fidelity of these methods in solving high-order problems. . . . . 16
II.8 Effects of population size on the performance of evolutionary optimization meth-
ods (GA and CMA-ES) for Rastrigin function of ordern = 5. The population
size of Pop = 5; 10; 25; 50; 100; 250; 500; 1000 is implemented. The results
are based on 100 simulations for each case. In each plot, the solid line shows
the final function value while the dashed line indicates the average number of
function evaluation to reach the solution. The statistical distribution for the fi-
nal function value and number of function evolutions are also shown in small
subplots. Note that the success rate to reach better results (i.e., lower function
value) strongly improves for larger population sizes, at the expense of higher
computational effort (i.e., higher number of function evolutions). . . . . . . . . 18
III.1 Flowchart of finite element model updating process. . . . . . . . . . . . . . . . . . 28
III.2 Rendering of the bridge structure (adapted from Johnson et al. (2006)). . . . . . . . . 29
vii
III.3 NEES@Reno combined time-history record for the sequence of the excitations applied
to the bridge model in the transverse direction. Windows indicated by WN-i represent
the i
th
white noise excitation test, and windows designated by T-n denote earthquake-
like test number n. Note the significant difference in test levels covering the 390-sec
record. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
III.4 Top View (a) and Elevation View (b) of the NEES@Reno bridge, together with selected
sensor locations for this study. S1 to S5 denote the location of accelerometers on the
bridge deck. (adapted from Johnson et al. (2006)). . . . . . . . . . . . . . . . . . . 31
III.5 First three identified transverse mode shapes of the tested structure in the undamaged
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
III.6 Finite element model updating results for the calibration study. Plots (a) and (b) show
the variation of the modification factors with function evaluation through the optimiza-
tion procedure for Scenario 1 while plots (c) and (d) corresponds to Scenario 2. Each
function evaluation consists of a finite element analysis to find the modal properties
of the analytical model for the given set of input parameters and then computation of
the cost function. The dark thick vertical line on the Right-Hand-Side (RHS) indicates
the end of optimization process. In each plot, the RHS small panel provides a high-
resolution plot of the modification factors when the identification procedure converged.
Numbers on the RHS indicate the index of the system parameter being identified and the
straight lines point to the corresponding curve. The correspondence of these parameters
to the physical properties of the tested structure is available in Table III.7. . . . . . . . 34
III.7 Probability density function (pdf) of the parameters estimation error. A total of 100 sim-
ulations for scenario 2, with maximum of 5000 iterations for each test, were conducted
for this study using CMA-ES. The cost function to be optimized consisted a parameter
vector of order 13. X(1) to X(12) represent the stiffness on the top and bottom of the
columns in the transverse direction and X(13) denotes the longitudinal stiffness of the
bents. The correspondence of these parameters to the physical properties of the tested
structure is available in Table III.7. Initial parameters set was selected randomly for
each simulation to evaluate the robustness of the algorithm. In each plot panel, a thin
line indicates the outline of the histogram of the parameter estimation error, and the
solid line represents the estimated Gaussian pdf having a matching mean () and stan-
dard deviation () to the corresponding histogram. Note that the abscissa and ordinate
ranges are not identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
III.8 Preliminary damage detection through tracking of the standard deviation of the moving
average parameter () in the ARMA model. Four typical values of the forgetting factor
( = 0.90, 0.95, 0.98 and 0.995) and different window sizes for system identification
were considered in this study. The first row illustrates the synchronized plot of the
excitation to the structure in the transverse direction. . . . . . . . . . . . . . . . . . 40
viii
III.9 Finite element model updating results for Scenario 1, using CMA-ES. Plots (a) to (d)
show the variation of the modification factors with function evaluation through the op-
timization procedure for white-noise excitation windows WN-2 to WN-5, respectively.
Based on system identification results using recorded response measurements by 5 sen-
sors, a 4-dimensional cost function was optimized to detect and quantify the overall
damage in each bent. Parameters 1-3 represent the remaining stiffness of the bents in
the transverse direction and parameter 4 is the corresponding value for the longitudinal
stiffness of all bents (Please also see the caption of Figure III.6 for further details). . . 44
III.10 Finite element model updating results for Scenario 2, using CMA-ES. Plots (a) to (d)
show the variation of the modification factors with function evaluation through the op-
timization procedure for white-noise excitation windows WN-2 to WN-5, respectively.
Based on system identification results using recorded response measurements by 17
sensors, a 13-dimensional cost function was optimized to detect and quantify the over-
all and localized damage for each column. Parameters 1-12 represent the remaining
stiffness in the transverse direction on the bottom and top of the columns and parameter
13 is the corresponding value for the longitudinal stiffness of all bents (Please also see
the caption of Figure III.6 for further details). . . . . . . . . . . . . . . . . . . . . 45
III.11 Finite element model updating results for Scenario 1, using GA. Plots (a) to (d) show the
variation of the modification factors with function evaluation through the optimization
procedure for white-noise excitation windows WN-2 to WN-5, respectively. Based on
system identification results using recorded response measurements by 5 sensors, a 4-
dimensional cost function was optimized to detect and quantify the overall damage in
each bent. Parameters 1-3 represent the remaining stiffness of the bents in the transverse
direction and parameter 4 is the corresponding value for the longitudinal stiffness of all
bents (Please also see the caption of Figure III.6 for further details). . . . . . . . . . . 46
III.12 Finite element model updating results for Scenario 2, using GA. Plots (a) to (d) show the
variation of the modification factors with function evaluation through the optimization
procedure for white-noise excitation windows WN-2 to WN-5, respectively. Based
on system identification results using recorded response measurements by 17 sensors,
a 13-dimensional cost function was optimized to detect and quantify the overall and
localized damage for each column. Parameters 1-12 represent the remaining stiffness
in the transverse direction on the bottom and top of the columns and parameter 13 is
the corresponding value for the longitudinal stiffness of all bents (Please also see the
caption of Figure III.6 for further details). . . . . . . . . . . . . . . . . . . . . . . 47
III.13 Finite element model updating results using output-only data for test window WN-2.
Plots (a) and (b) show the variation of the modification factors with function evaluation
through the optimization procedure for Scenario 1 while plots (c) and (d) corresponds
to Scenario 2. The correspondence of these parameters to the physical properties of
the tested structure is available in Table III.11. For comparison purposes, the reader
is referred to the corresponding figures using input-output data for damage detection
in test window WN-2 which are illustrated in Figures III.9(a), III.10(a), III.11(a), and
III.12(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix
III.14 Linear regression plot relating the damage index and the quantified damage. The ab-
scissa shows the quantified damage, identified through the model updating procedure in
the first scenario (column #4 and column #5 in Table III.7 to Table III.10), and the ordi-
nate is the damage index introduced by Park and Ang (1985) (column #2 in Table III.7
to Table III.10), which is a practical measure of damage based on dissipated hysteretic
energy and ductility demand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
IV .1 Model of a MDOF chain-like system. . . . . . . . . . . . . . . . . . . . . . . . . 71
IV .2 Experimental case study building model before and after the installation of the MR
damper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
IV .3 (a) Top view and (b) Elevation view of the tested building structure . . . . . . . . . . 80
IV .4 (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and
(d) Displacement at each floor for Case 0 (No MR damper) . . . . . . . . . . . . . . 81
IV .5 (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and
(d) Displacement at each floor for Case 1 (MR damper with 0.00A input current) . . . 82
IV .6 (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and
(d) Displacement at each floor for Case 2 (MR damper with 0.05A input current) . . . 82
IV .7 (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and
(d) Displacement at each floor for Case 3 (MR damper with 0.10A input current) . . . 82
IV .8 (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and
(d) Displacement at each floor for Case 4 (MR damper with 0.15A input current) . . . 83
IV .9 Four identified frequencies and corresponding mode-shapes of the tested building struc-
ture in the excitation direction for Case 0. . . . . . . . . . . . . . . . . . . . . . . 89
V .1 Schematic diagram of damage detection using neural networks. (Nakamura et al. (1998)) 102
V .2 Diagram of the neural network in MATLAB
R
(Adapted from the software manual). . 103
V .3 Neural Network (NN) with Input of Measured Displacements, Velocities, and Excita-
tions and Output of System Accelerations. . . . . . . . . . . . . . . . . . . . . . . 104
V .4 Neural Network (NN) with Input of Measured Relative Displacements, and Relative
Velocities, and Output of Restoring Forces on All Stories . . . . . . . . . . . . . . . 104
V .5 Neural Network (NN) with Input of Measured Relative Displacements, and Relative
Velocities, and Output of Restoring Forces on Each Story . . . . . . . . . . . . . . . 105
V .6 Performance of different neural networks models for detection of change in the system. 108
V .7 General neural network model combined with an ODE solver to predict the response of
the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
V .8 Measured vs. predicted displacement of the structure at each floor for Case 0 using RK4.5111
V .9 Measured vs. predicted displacement of the structure at each floor for Case 1 using RK4.5112
V .10 Measured vs. predicted displacement of the structure at each floor for Case 2 using RK4.5112
V .11 Measured vs. predicted displacement of the structure at each floor for Case 3 using RK4.5113
V .12 Measured vs. predicted displacement of the structure at each floor for Case 4 using RK4.5113
V .13 Model of a MDOF chain-like system. . . . . . . . . . . . . . . . . . . . . . . . . 114
V .14 Flowchart of structural mass estimation for chain-like systems using optimization meth-
ods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
x
V .15 Estimation of mass of the floors using CMA-ES for Case 0 and Case 3. Initial popula-
tion is randomly selected from on a uniform distribution in the range of 0 25kg and
equal weights for all floors are considered in the definition of the cost function. . . . . 116
V .16 Estimated mass of the floors using CMA-ES for Cases 0 to 4. . . . . . . . . . . . . . 117
xi
Abstract
Civil infrastructures play a vital role in human societies. Recent catastrophic events due to the de-
ficiency, failure or malfunction of these systems, claiming many lives and resulting in substantial
economic loss, have attracted extensive attention focused on reviewing and amending the design
and maintenance procedures of civil infrastructures. In addition to the possible failure of struc-
tural components, long-term forms of damage due to deterioration or fatigue may also necessitate
regular monitoring of civil structures. Therefore, depending on the importance, use and risk, the
structure of interest needs to be equipped with inspection, monitoring and maintenance systems.
Structural Health Monitoring (SHM) is generally associated with any engineering methodology
whose aim is to detect, locate and quantify the damage in the target system. Vibration-based tech-
niques, as the most conventional SHM approaches, acquire and analyze the structural response
using a variety of sensors mounted at different locations on the structure. The main goal of the
study reported herein is to investigate and evaluate different vibration-signature-based methods for
system identification, damage detection and health monitoring of civil structures. Various well-
known techniques such as finite element model updating approach and damage detection methods
based on artificial neural networks are studied and evaluated. Experimental data from two case
studies, a quarter-scale two-span bridge system, tested at the University of Nevada, Reno, and a
1/20 scale 4-story building equipped with smart devices of magneto-rheological (MR) damper,
are used for investigation and validation purposes. Guidelines are established for the optimum
selection of the dominant control parameters involved in the application of some of the robust
SHM approaches for achieving reliable SHM results under realistic conditions.
xii
Chapter I
INTRODUCTION
I.1 Background
C
IVIL infrastructures such as high-rise buildings, dams, bridges and lifelines play a vital
role in human societies. The investment of the United States in civil infrastructure is es-
timated to be $20 trillion. Recent catastrophic events due to the deficiency, failure or malfunction
of these systems, claiming many lives and resulting in substantial economic loss, have attracted
extensive attention focused on reviewing and amending the design and maintenance procedures
of civil infrastructures.
One of the challenging issues in designing civil infrastructures is the high initial cost of these as-
sets. Performance-based design, as a trade-off approach, may allow the structure to undergo some
ductile inelastic deformation with an acceptable level of damage. Such damage is component-
based rather than system-based, meaning that the structure as a whole will still perform to a
desired level, despite the failure in one or more isolated elements. This design philosophy re-
sults in a cost-effective design and higher system reliability, at the expense of possible localized
damage. Therefore, depending on its importance, use and risk, the structure needs to be equipped
1
with inspection, monitoring and maintenance systems. Furthermore, long-term forms of damage
due to deterioration or fatigue may also necessitate regular monitoring of civil infrastructures.
Consequently, the ability to continuously monitor the integrity of civil infrastructures provides
the opportunity to reduce maintenance costs, while increasing the safety to the public.
The topic of Structural Health Monitoring (SHM) is generally associated with any engineer-
ing methodology whose aim is to detect, locate and quantify the damage in the target system.
Vibration-based SHM techniques acquire and analyze the structural response using a variety of
sensors mounted at different locations on the structure. Broadly speaking, SHM may be inter-
preted as a “continuous system identification of a physical or parametric model of the structure
using time-dependent data” (Brownjohn (2007)).
Vibration-based methods are the most conventional approaches for SHM, but other methodolo-
gies such as optical or acoustic emission techniques have also been developed and employed in
this field. While SHM is usually associated with on-line global damage detection in the structure,
it is often followed by off-line local damage detection strategies (e.g., Non-Destructive Evalua-
tion (NDE)) if any damage is detected. Meanwhile, it should be noted that damage is not the
only source of variations in the measured dynamic response of the system. Hence, understanding
and detecting (quantifying) the effects of other influencing parameters such as system nonlinear-
ity, temperature variations, unobserved excitations, soil-structure interactions, and measurement
noise, should also be carefully investigated and considered (Nayeri et al. (2008)).
Application areas of SHM may include, but are not to be limited to the following categorizes
(Brownjohn (2007)):
2
Structures subject to long-term movement or degradation of materials
Feedback loop to improve future design
Assessment of post-earthquake structural integrity
Fatigue assessment
Complementary to performance-based design philosophy
Modifications to an existing structure
Monitoring of structures affected by external works or during demolition
I.2 Literature Review
Qualitative and non-continuous methods of SHM have long been used to evaluate structures and
machineries for their capacity to serve the intended purpose. For instance, since the beginning of
the 19
th
century, the sound of a hammer striking on the train wheel was a useful tool to detect if
damage was present. In the last half a century, the development of quantifiable SHM approaches
has been closely coupled with the evolution of digital computing hardware and inexpensive, fast
networking technologies. Significant developments in the field have originated from major con-
struction projects, such as offshore gas/oil production installations, large dams and highways
bridges. During the 1970s and 1980s, the oil industry received the greatest attention and research
effort to develop damage identification methods for offshore platforms. Applications of vibration-
based methods of SHM in the aerospace community started during the late 1970s and early 1980s
in conjunction with the development of the space shuttle. The development of a composite fuel
tank for a reusable launch vehicle in mid-1990s motivated studies of damage identification for
3
composite materials. Since the early 1980s, the vibration-based damage assessment of bridge
structures and buildings has been of interest to the civil engineering community (Farrar and Wor-
den (2007)).
Some representative publications that provide a comprehensive overview of the broad interdis-
ciplinary field of SHM, the main technical challenges, as well as promising proposed approaches
that have the potential of being useful tools for damage detection purposes in different classes of
civil infrastructure systems, are available in the works of Doebling et al. (1996), Hou et al. (2000),
Wang et al. (2001), Chang et al. (2003), Staszewski et al. (2004), Hera and Hou (2004), Ko and
Ni (2005), Kim et al. (2007), Farrar and Worden (2007), Brownjohn (2007), Wang (2007) and
Chandrashekhar and Ganguli (2009).
I.3 Motivation and Technical Challenges
Generally speaking, any SHM process may be defined in terms of the following four-step statisti-
cal pattern recognition paradigm (Farrar and Worden (2007)):
(i). Operational evaluation
(ii). Data acquisition, normalization and cleansing
(iii). Feature selection and information condensation
(iv). Statistical model development for feature discrimination
Each step is involved with many technical challenges to the adaptation of SHM that are common
to all applications of this technology. These challenges include (Farrar and Worden (2007)):
How to optimally define the number and location of the sensors?
4
How to identify the appropriate features that are sensitive to small damage levels?
How to discriminate the changes due to damage from those caused by other parameters
such as changing environmental and/or test conditions?
What are the appropriate statistical methods to discriminate features from undamaged and
damaged structures?
Every civil infrastructure is usually unique and may require developing a special and customized
SHM strategy. It should be noted that the chosen SHM strategy might be a trade-off solution
to different aspects of the problem. For instance, one of the basic current technical challenges
in SHM is that, with a minimum of optimally located sensors, any damage to be detected must
have significant effects on the underlying structural dynamic properties (e.g., stiffness, mass or
damping) of the system, which, in turn results in a measurable change in the observed dynamic
response. Unfortunately, this is not usually the case in real operational structures, since typi-
cal localized damage will not significantly influence the dominant lower-frequency modes of the
monitored structure. With increasing widespread availability of sensor networks and data acquisi-
tion and communication capabilities, one may think of a dense, fine-grained sensor architecture as
an intuitive solution, but any increase in the size of the problem will introduce new impediments
in the form of processing huge amounts of multi-faceted data sets with embedded mathematical
and computational challenges.
I.4 Scope
The main goal of the study reported herein is to investigate and evaluate different vibration-
signature-based methods for system identification, damage detection and health monitoring of
5
civil structures. The following chapter reports the performance of two stochastic methods of
global optimization for a subset of well-known benchmark functions. The application of these
methods in finite element model updating approaches for damage detection purposes is investi-
gated in Chapter 3. The case study is a quarter-scale, two-span bridge system, experimentally
tested at the University of Nevada, Reno. Chapter 4 reports the performance of different sys-
tem identification approaches for experimentally recorded data of a 1/20 scale 4-story building
equipped with smart devices of magneto-rheological (MR) damper. This case study is also stud-
ied in Chapter 5 to investigate the application of artificial neural networks for identification of
nonlinear structural models. The last chapter provides a brief overview of the whole dissertation
and highlights the concluding remarks.
6
Chapter II
COMPARISON OF DIFFERENT GLOBAL
OPTIMIZATION TECHNIQUES FOR
HIGH-DIMENSIONAL PROBLEMS
II.1 Introduction
S
TRUCTURAL health monitoring through the use of finite element model updating tech-
niques for dispersed civil infrastructures usually deals with minimizing a complex, non-
linear, non-convex, high-dimensional cost function with several local minima. Hence, global op-
timization algorithms with promising performance have received considerable attention for finite
element model updating purposes. In recent years, different types of optimization algorithms have
been designed and applied to solve real-life optimization problems, especially with engineering
applications. Some of the most well-known approaches include classical deterministic methods
such as quasi-Newton method as well as stochastic approaches such as genetic algorithms, evo-
lutionary strategies, particle swarm optimization, hybrid evolutionary-classical methods, other
non-evolutionary methods such as simulated annealing (SA), tabu search (TS) and others.
7
Global optimization methods can be categorized as follows:
(i). Deterministic methods, in which the computation is completely determined by previously
sampled values . This group of optimization approaches are not generally successful on
non-convex high-dimensional functions. Some examples are algorithms based on real al-
gebraic geometry, interval optimization methods and branch and bound methods.
(ii). Stochastic methods, that incorporate probabilistic elements in the optimization procedure.
Genetic algorithms, evolutionary strategies and particle swarm optimization belong to this
group of optimization methods.
II.1.1 Scope
In this chapter, we empirically investigate the global search performance of some well-known op-
timization packages on a subset of standard test problems. The remainder is organized as follows:
A brief explanation of each optimization method, definition of the benchmark functions, common
termination criteria, size of problems and initialization scheme are presented. The results of sta-
tistical study based on an ensemble of 100 simulations for each case are provided and discussed.
In particular, the effects of function order as well as population size on the performance of these
methods are investigated, followed by the summary and concluding remarks.
II.2 Optimization Methods under Discussion
II.2.1 CMA-ES
Evolutionary Strategy based on Covariance Matrix Adaptation, abbreviated as CMA-ES, was pro-
posed for the first time in 1994 (Ostermeier et al. (1994)) and has been considerably developed
8
since then. In this method, any new population is generated based on the multivariate normal
mutation distribution with adapted covariance matrix. The adaptation is based on increasing the
likelihood of previously realized successful mutation steps, as well as exploiting the evolution
path of the distribution mean of the strategy.
Theoretical concept behind CMA-ES can be summarized as follows (Akimoto et al. (2012)).
If the probability density function of the multivariate normal distribution with mean vector m
and covariance matrix
2
C is represented asN(x;m;
2
C), the CMA-ES starts with the initial
parameters ofm
0
,
0
,C
0
,p
0

= 0 andp
0
C
= 0 and repeats the following steps:
(i). Generate independent sample pointsx
1
;x
2
;:::;x

fromN(x;m;
2
C).
(ii). Evaluate the function values at sample pointsf(x
1
);f(x
2
);:::;f(x

):
(iii). Update the parameters of the algorithm as follows.
Mean vector:
m
t+1
=

X
i=1
W
R
i
X
i
(II.1)
whereR
i
is the ranking off(x
i
).W
R
i
represents the weight for theR
i
th
highest point and
has the following properties:
0W
i
W
j
18i>j &

X
i=1
W
i
= 1 (II.2)
Global step-size:

t+1
=
t
exp

c

d

p
t+1

d

d

(II.3)
wherec

andd

denote learning rate and the damping parameter, respectively.
d
is the
9
Figure II.1: Concept behind the covariance matrix adaptation for generation evolution. Two phenomena
are observed as the generations develop: 1- The center of the new generation is shifted toward global
minimum due to the weighting of the population for reproduction process. 2- The distribution shape adapts
to an ellipsoidal or ridge-like landscape along the principle axis, which is the eigenvector corresponding to
the largest eigenvalue of the covariance matrix. (figure adapted from http://en.wikipedia.org/wiki/CMA-
ES).
expectation of the chi distribution with d degrees of freedom andp

is an evolution path
being updated as
p
t+1

= (1c

)p
t

+
s
c

(2c

)
P

i=1
W
2
i
(C
t
)

1
2
(m
t+1
m
t
)

t
(II.4)
Covariance matrix:
C
t+1
= (1c
1
c

)C
t
+c
1
p
t+1
C
(p
t+1
C
)
T
+c

X
i=1
W
R
i
x
i
m
t

t

x
i
m
t

t

T
(II.5)
wherec
1
andc

are learning rate parameters andp
C
is an evolution path, being updated as
p
t+1
C
= (1c
c
)p
t
C
+
s
c
c
(2c
c
)
P

i=1
W
2
i
(m
t+1
m
t
)

t
(II.6)
c
c
represents the learning rate for the evolution path update.
Figure II.1 schematically illustrates the main concept behind the covariance matrix adaptation for
a 2-D optimization problem. As shown, the search direction is modified such that the candidate
solutions in the new generation are more likely to be sampled along the principle axis. Further
10
details concerning this approach are available in Ostermeier et al. (1994). A module, written in
MATLAB
R
by the developers of this optimization method, is used for this study.
II.2.2 Genetic algorithm
Genetic algorithms are considered as a computational analogy of adaptive systems. They are
modeled based on the principles of the evolution of generations via natural selection, mutation
and crossover. Evaluation of the individuals is performed using a fitness (cost) function. General
paradigm of genetic algorithm methods involves:
(i). Initialization: Randomly generation of an initial population.
(ii). Selection: Selection a proportion of the existing population based on fitness to breed a new
generation.
(iii). Reproduction: Production of the new generation population through genetic operators such
as crossover and mutation.
(iv). Termination: Repeating step 2 and 3 until satisfying solution is obtained.
For this study, the ga module in the Optimization Toolbox of MATLAB
R
is exploited.
II.3 Test functions
In the field of global optimization methods, it is common to compare different algorithms using
a large test set. Five well-known benchmark function are considered in this section to evaluate
and compare the performance of global optimization methods under discussion. The benchmark
functions include:
(i). Rosenbrock function: Function with a deep valley with the shape of a parabola.
11
(ii). Schwefel function: Function composed of a great number of peaks and valleys.
(iii). Ackley function: Function with an exponential term that covers its surface with numerous
local minima.
(iv). Rastrigin function: Function made up of a large number of local minima whose value
increases with the distance to the global minimum.
(v). Griewank function: Function with a product term that introduces interdependence among
the variables.
Rosnebrock function is symmetric, uni-modal and non-separable with the global minimum at
x
e
= 1:0. The rest of the test problems have a high number of local optima (multi-modal), and are
scalable in the problem dimension. The Rastrigin and Schwefel functions are additively separable,
while Ackley function is separable, in that the global optimum can be located by optimizing each
variable independently. Griewank is considered a partially separable function. The known global
minimum is located at x
e
= 0:0 for all functions, except for the Schwefel function, where the
global minimum within [500; 500]
n
resides at x
e
= 420:9687. All the test problems have a
global minimal function value of 0. Figure II.2 to II.6 show 2-D plots of the functions under
investigation. The definition of each function as well as the range of interest and the global
minimum are also provided in these figures. The termination criteria in the optimization process
are defined as follows:
f 10
4
f 10
10
x 10
10
Max
eval
=1
12
−2
−1
0
1
2
−2
0
2
0
500
1000
1500
2000
X
1
X
2
f(x
1
, x2)
Full range plot
Figure II.2: Plot of 2-D Rosenbrock function.
f =
P
n1
i=1

100(x
i+1
x
2
i
)
2
+ (1 x
i
)
2

2:048 x
i
2:048
Min @ x
e
= 1:0
−500
0
500
−500
0
500
0
500
1000
1500
2000
X
1
X
2
f(x
1
, x2)
Full range plot
Figure II.3: Plot of 2-D Schwefel function.
f =
P
n
i=1

x
i
sin

p
jx
i
j

+ 418:9829

500 x
i
500
Min @ x
e
= 420:9687
13
−20
0
20
−20
0
20
0
5
10
15
20
25
X
1
X
2
f(x
1
, x2)
(a) Full range plot
−2
−1
0
1
2
−2
0
2
0
2
4
6
8
X
1
X
2
f(x
1
, x2)
(b) Focus around global minimum
Figure II.4: Plot of 2-D Ackley function.
f =ae
b
r
P
n
i=0
x
2
i
n
e
P
n
i=0
c:cos(x
i
)
n
+a +e
32:768 x
i
32:768 a = 20 b = 0:2 c = 2
Min @ x
e
= 0:0
−5
0
5
−5
0
5
0
20
40
60
80
100
X
1
X
2
f(x
1
, x2)
(a) Full range plot
−1
−0.5
0
0.5
1
−1
0
1
0
10
20
30
40
50
X
1
X
2
f(x
1
, x2)
(b) Focus around global minimum
Figure II.5: Plot of 2-D Rastrigin function.
f = 10n +
P
n
i=1

x
2
i
10 cos(2x
i
)

5:12 x
i
5:12
Min @ x
e
= 0:0
14
−600
−400
−200
0
200
400
600
−500
0
500
0
50
100
150
200
X
1
X
2
f(x
1
, x2)
(a) Full range plot
−6
−4
−2
0
2
4
6
−5
0
5
0
0.5
1
1.5
2
2.5
X
1
X
2
f(x
1
, x2)
(b) Focus around global minimum
Figure II.6: Plot of 2-D Griewank function.
f =
P
n
i=1
x
2
i
4000

Q
n
i=1
cos

x
i
p
i

+ 1
600 x
i
600
Min @ x
e
= 0:0
II.4 Results and discussion
Preliminary investigation of optimization methods under discussion shows that the success rate to
reachf
stop
strongly depends on the population size and problem dimension with other parameters
of less of influence. Therefore, it is decided to study the effects of these two parameters on the
performance of global optimization methods. A statistical study based on an ensemble of 100
simulations for each case is conducted. The starting point x
0
is sampled uniformly within the
initialization intervals. All the data history of each optimization process is recorded in a MAT file
with double digit precision.
II.4.1 Problem-order effects
The dimensionality of the search space is an important factor in the complexity of the problem.
In order to establish different degrees of difficulty in the problems, we have chosen a search space
of dimensionality ofn = 5; 10; 25; 50; 100 for each test functions. For both CMA-ES and GA
15
10
1
10
2
0
0.06
0.12
0.18
0.24
0.3
0.36
0.42
0.48
Function Dimension (n)
ǫ =
| x- xmin| √
n
Ackley function: Population = 100

10
1
10
2
0
0.6
1.2
1.8
2.4
3
3.6
4.2
4.8
x 10
5
NumberofFunctionEvaluations(N)
ε
N
0 0.1 0.2
0
10
20
30
40
50
ǫ
μ = 1e−05
σ = 6.8e−06
n = 5
0 0.1 0.2
μ = 0.00097
σ = 0.0033
n = 10
0 0.1 0.2
μ = 0.0079
σ = 0.0073
n = 25
0 0.1 0.2
μ = 0.038
σ = 0.016
n = 50
0 0.1 0.2
μ = 0.14
σ = 0.026
n = 100
0 1 2
x 10
4
0
10
20
30
40
50
N
μ = 1.4e+04
σ = 1.9e+03
0 2 4
x 10
4
μ = 2.5e+04
σ = 2.8e+03
0 1 2
x 10
5
μ = 7.3e+04
σ = 1.5e+04
0 1 2
x 10
5
μ = 1.1e+05
σ = 3e+04
0 1 2
x 10
5
μ = 1.1e+05
σ = 3.3e+04
(a) GA
10
1
10
2
2
2.95
3.9
4.85
5.8
6.75
7.7
8.65
9.6
x 10
−7
Function Dimension (n)
ǫ =
| x- xmin| √
n
Ackley function: Population = 100

10
1
10
2
0
1.2
2.4
3.6
4.8
6
7.2
8.4
9.6
x 10
4
NumberofFunctionEvaluations(N)
ε
N
0 0.5 1
x 10
−6
0
10
20
30
40
50
ǫ
μ = 2.5e−07
σ = 1.1e−07
n = 5
0 0.5 1
x 10
−6
μ = 2.2e−07
σ = 5.9e−08
n = 10
0 0.5 1
x 10
−6
μ = 2.5e−07
σ = 4.1e−08
n = 25
0 0.5 1
x 10
−6
μ = 2.8e−07
σ = 3e−08
n = 50
0 0.5 1
x 10
−6
μ = 3.1e−07
σ = 2.2e−08
n = 100
0 2000 4000
0
10
20
30
40
50
N
μ = 4.1e+03
σ = 1.8e+02
0 5000 10000
μ = 9.5e+03
σ = 3.7e+02
0 1 2
x 10
4
μ = 1.8e+04
σ = 5.8e+02
0 1 2
x 10
4
μ = 1.8e+04
σ = 3.6e+02
0 2
x 10
4
μ = 3.1e+04
σ = 6.1e+02
(b) CMA-ES
Figure II.7: Effects of problem-order on the performance of evolutionary optimization methods
(GA and CMA-ES) for Ackley function. Population size in each generation is set as 100. The
results are based on 100 simulations for each case. In each plot, the solid line shows the average
of normalized error in the final solution vector while the dashed line indicates the average number
of function evaluation to reach the solution. The statistical distribution for normalized error of the
final solution and average number of function evolutions are also shown in small subplots. Note
that the relatively small mean value and standard deviation of the normalized error (with respect
to the dimension range = 32.768) shows the robustness and fidelity of these methods in solving
high-order problems.
16
algorithms, all 100 runs in each case are performed with the default strategy parameter defined in
the packages except for the population size, which is selected asPop = 100.
The performance of optimization methods is highly influenced by the problem dimension. As
shown in Figure II.7, where exemplary results are illustrated for Ackley function, normalized er-
ror of the solution vector slightly increases for higher function dimensions. As expected, both op-
timization methods require significantly more function evolutions to reach the solution for higher
problem orders. The statistical distribution for normalized error of the final solution and average
number of function evolutions are also plotted in this figure. For ease of comparison between
different cases, the error is defined as the deviation of the final solution vector from exact global
minimum, normalized with respect to the problem-order. Based on this definition, the error can
be represented as:
=
jx
e
x
e
min
j
p
n
(II.7)
where is the normalized error,x
e
is the final solution found by the optimization method,x
e
min
is the exact global minimum, and n is the function dimension. Note that the relatively small
mean value and standard deviation of the normalized error shows the robustness and fidelity of
these methods in solving high order problems. Comparing Figure II.7(a) and Figure II.7(b) also
shows that the average number of required function evaluations in CMA-ES to reach the solution
increases almost linearly with the problem-order. On the other hand, the corresponding plot for
GA illustrates superlinear behavior. The Ackley function represents the typical picture for the
effects of problem-order on the performance of global optimization methods.
17
10
0
10
1
10
2
10
3
0
10
20
30
40
50
60
70
Population Size
f(x)
Rastrigin function: n = 5

10
0
10
1
10
2
10
3
0
2.5
5
7.5
10
12.5
15
17.5
x 10
4
NumberofFunctionEvaluations(N)
f(x)
N
0 50
0
50
100
f(x)
μ = 25
σ = 13
Pop. = 5
0 20
μ = 14
σ = 7
Pop. = 10
0 10 20
μ = 7.8
σ = 4
Pop. = 25
0 5 10
μ = 5
σ = 2.7
Pop. = 50
0 5 10
μ = 3.1
σ = 2.1
Pop. = 100
0 2 4
μ = 1.3
σ = 0.97
Pop. = 250
0 2
μ = 0.44
σ = 0.68
Pop. = 500
0 0.5 1
μ = 0.05
σ = 0.22
Pop. = 1000
0 1e4 2e4
0
10
20
30
40
50
N
μ = 3.1e+03
σ = 2e+03
0 5e3
μ = 2.1e+03
σ = 7e+02
0 2e3 4e3
μ = 2.9e+03
σ = 4e+02
0 5e3 1e4
μ = 5.1e+03
σ = 7e+02
0 1e4 2e4
μ = 1e+04
σ = 1.5e+03
0 2e4 4e4
μ = 2.3e+04
σ = 4.7e+03
0 5e4 1e5
μ = 4.1e+04
σ = 9.2e+03
0 1e5 2e5
μ = 7e+04
σ = 1.4e+04
(a) GA
10
0
10
1
10
2
10
3
0
6
12
18
24
30
36
42
Population Size
f(x)
Rastrigin function: n = 5

10
0
10
1
10
2
10
3
0
1.5
3
4.5
6
7.5
9
10.5
x 10
4
NumberofFunctionEvaluations(N)
f(x)
N
0 50
0
50
100
f(x)
μ = 15
σ = 11
Pop. = 5
0 10
μ = 5.6
σ = 3.3
Pop. = 10
0 5
μ = 2.3
σ = 1.5
Pop. = 25
0 2 4
μ = 1.1
σ = 0.95
Pop. = 50
0 2
μ = 0.48
σ = 0.67
Pop. = 100
0 0.5 1
μ = 0.05
σ = 0.22
Pop. = 250
0 5
x 10
−5
μ = 4.7e−06
σ = 7e−06
Pop. = 500
0 1
x 10
−5
μ = 3e−06
σ = 2.9e−06
Pop. = 1000
0 1e3 2e3 3e3
0
10
20
30
40
50
N
μ = 1.5e+03
σ = 2.6e+02
0 2e3 4e3
μ = 1.8e+03
σ = 2.3e+02
0 5e3 1e4
μ = 3.1e+03
σ = 5.2e+02
0 5e3 1e4
μ = 4.7e+03
σ = 1.1e+03
0 1e4 2e4
μ = 6.2e+03
σ = 1.8e+03
0 1e4 2e4
μ = 9.9e+03
σ = 2.2e+03
0 1e4 2e4
μ = 1.9e+04
σ = 1.9e+03
0 5e4
μ = 3.8e+04
σ = 2.6e+03
(b) CMA-ES
Figure II.8: Effects of population size on the performance of evolutionary optimization meth-
ods (GA and CMA-ES) for Rastrigin function of order n = 5. The population size of
Pop = 5; 10; 25; 50; 100; 250; 500; 1000 is implemented. The results are based on 100 simu-
lations for each case. In each plot, the solid line shows the final function value while the dashed
line indicates the average number of function evaluation to reach the solution. The statistical dis-
tribution for the final function value and number of function evolutions are also shown in small
subplots. Note that the success rate to reach better results (i.e., lower function value) strongly
improves for larger population sizes, at the expense of higher computational effort (i.e., higher
number of function evolutions).
18
II.4.2 Population size effects
The effects of the population size on the performance of evolutionary methods are also investi-
gated in this study. To do so, the population sizes ofPop = 5; 10; 25; 50; 100; 250; 500; 1000 are
implemented for a given problem size for each test function. Figure II.8 illustrates the perfor-
mance versus population size for Rastrigin function of ordern = 5, based on 100 simulations for
each case. The statistical distribution for the final function value and average number of function
evolutions are also shown in small subplots. As shown in this figure, the success rate to reach
better results (i.e., lower function value) improves for larger population sizes, at the expense of
higher computational effort (i.e., higher number of function evolutions).
II.5 Concluding Remarks
In this study, the performance of two global optimization methods are empirically investigated on
a subset of well-known test functions. The global optimization algorithms under discussion were
Genetic Algorithm (ga modules in MATLAB
R
) and an evolutionary strategy called CMA-ES. A
suit of five standard test functions with a search space of dimensionalityn = 5; 10; 25; 50; 100
was considered to study the effects of the problem-order on the performance of the optimization
methods. In addition, the effects of population size on the performance of evolutionary methods
are investigated for a subset of population sizes ofPop = 5; 10; 25; 50; 100; 250; 500; 1000. For
each case, an ensemble of 100 simulations was generated to reach a reliable statistical data set.
Based on the comparison of these results, the following conclusions can be made:
Evolutionary stochastic optimization methods are generally successful in solving high-
dimensional problems.
19
As expected, both optimization methods require significantly more function evolutions to
reach the solution for higher problem orders.
Increasing the population size remarkably improves the performance of these methods at
the expense of higher number of function evaluations. The study shows that the optimal
population size takes a wide range of values depending on the cost function. For a given
objective function, the optimum population size may be tuned through calibration process
with the help of a statistical analysis.
For multi-modal functions, CMA-ES shows better performance than GA in the sense that
it returns smaller final function value with less average number of required function evalu-
ations to reach the solution. For instance, while CMA-ES outperform GA on Ackley and
Rastrigin functions (as shown in Figures II.7 and II.8), it significantly falls behind GA on
Rosenbrock function. Noting that Rosenbrock is the only uni-modal non-separable test
function of this study, this indicates that the performance of these optimization packages
varies with the topography of the functions. This conclusion also agrees with the findings
of the developers of CMA-ES, reported in Hansen and Kern (2004).
20
Chapter III
FINITE ELEMENT MODEL UPDATING USING
EVOLUTIONARY STRATEGIES
III.1 Introduction
I
N this chapter, the performance of global optimization methods which were discussed in the
previous chapter is investigated for damage detection purposes, through the finite element
model updating approach. The case study is a quarter-scale, two-span, reinforced concrete bridge
system, which was investigated experimentally at the University of Nevada, Reno. The damage
sequence in the structure was induced by a range of progressively-increasing excitations in the
transverse direction of the specimen. Intermediate nondestructive white noise excitations and
response measurements were used for system identification and damage detection purposes. It is
shown that, when evaluated together with the strain gauge measurements and visual inspection
results, the applied finite element model updating algorithm of this study could accurately detect,
localize, and quantify the damage in the tested bridge columns throughout different phases of the
experiment.
21
III.1.1 Literature Review
The finite element model updating method has been studied for many years as an important subject
in the mechanical and aerospace engineering fields. It has also developed into a major research
area within the field of SHM, responding to an increasing demand for evaluating the integrity of
civil infrastructures. Many research projects have been conducted to develop a successful tool in
structural health monitoring through finite element model updating methods. Most of these tech-
niques are based on searching for an admissible set of structural parameters to minimize an error
function involving the analytical and measured dynamic response. The success of these method-
ologies strongly depends on having a suitable definition for the cost function, an appropriate
analytical model, an accurate system identification approach, and an effective robust optimization
algorithm for global minimization.
Some representative publications that provide a comprehensive overview of the broad interdis-
ciplinary field of finite element model updating for SHM, the main technical challenges, as well
as promising proposed approaches that have the potential of being useful tools for damage de-
tection purposes in different classes of civil infrastructure systems, are available in the works
of Farhat and Hemez (1993), Adeli and Cheng (1994a), Zimmerman and Kaouk (1994), Adeli
and Cheng (1994b), Friswell and Mottershead (1995), Doebling et al. (1996), Levin and Lieven
(1998), Atalla and Inman (1998), Fritzen et al. (1998), Doebling et al. (1998), Hemez and Doe-
bling (2001), Teughels et al. (2002), Jaishi and Ren (2005), Chu et al. (2008), Ni et al. (2008),
and Cheung and Beck (2009).
22
III.1.2 Motivation and Technical Challenges
One of the basic current technical challenges in SHM is that, with a minimum of optimally lo-
cated sensors, any damage to be detected must have significant effects on the underlying struc-
tural dynamic properties (e.g., stiffness, mass or damping) of the system, which, in turn results
in a measurable change in the observed dynamic response. Unfortunately, this is not usually the
case in real operational structures, since typical localized damage of interest in practical SHM
applications may not induce significant influence on the dominant lower frequencies and the cor-
responding mode shapes of the monitored structure (unless it happens to occur at locations of high
strain energy).
With increasing widespread availability of sensor networks and data acquisition and commu-
nication capabilities, one may think of a dense, fine-grained sensor architecture as an intuitive
solution, but any increase in the size of the problem will introduce new impediments in the form
of processing huge amounts of multi-faceted data sets with embedded mathematical and com-
putational challenges. Of particular relevance to potential applications of finite element model
updating approaches for SHM in conjunction with dispersed systems are issues dealing with min-
imization of a complex, non-linear, non-convex, high-dimensional cost function with several local
minima. The more complicated the structure is with greater number of variables, the less likely
the optimal solution is found by means of conventional deterministic optimization methods.
The major strides that have been achieved in the recent past with regard to the development
of numerical optimization techniques for engineering applications, coupled with the tremendous
increase of computational power, bring SHM through model updating approaches for large-scale
23
structures within the realm of practicality. Specifically, the fact that stochastic optimization tech-
niques, such as evolutionary algorithms, simulated annealing, and other random search methods
have shown promising performance in solving global optimization problems is one of the main
drivers for the growing interest in the investigation and implementation of these methods for
large-scale finite element model updating approaches.
III.1.3 Scope
As a component of a collaborative multi-university, multi-disciplinary project utilizing the Net-
work for Earthquake Engineering Simulation (NEES), a comprehensive series of experimental
studies have been recently conducted at the University of Nevada, Reno (NEES@Reno) on large
bridge systems. Recorded from densely-instrumented test specimens with a very large number
of accelerometers, displacement transducers, and strain gauges at several locations and in differ-
ent orientations, this collection of data provides a unique opportunity for applications in various
fields of earthquake engineering, including the development and evaluation of structural health
monitoring methodologies and damage detection techniques.
The main goal of the study reported here is to investigate the performance of two global opti-
mization methods in finite element model updating approaches for damage detection purposes.
The developed damage detection method was implemented on the NEES@Reno recorded data
from the shake-table experiments conducted on a quarter-scale, two-span bridge system. The
specimen was gradually damaged due to a sequence of low (Peak Ground Acceleration = 0.075g)
to high (PGA = 2.11g) amplitude progressive excitations in the transverse direction. Intermediate
nondestructive white noise excitations were also applied to the structure for system identification
and damage detection purposes.
24
The remainder of this chapter is organized as follows. Section III.2 provides an overview of the
system identification approach and finite element model updating technique used in this study;
section III.3 describes the test bridge, data collection, and computational model; and section III.4
reports and discusses the damage-detection results and investigates the effectiveness of the pro-
posed approach. Section III.5 highlights the concluding remarks.
III.2 Overview of Identification Approach
III.2.1 Subspace Method for System Identification
The subspace state-space system identification algorithm (n4sid module in the System Identifi-
cation Toolbox of MATLAB
R
) was employed for this study. Subspace algorithms have been
shown to be computationally very efficient and robust, specially for large data sets and large-scale
systems. The two main steps in the subspace system identification methods can be summarized
as follows (De Cock and De Moor (2003)):
(i). Estimating the Kalman filter state sequence of the dynamical system without any prior
knowledge of the mathematical model, through an orthogonal or oblique projections of row
spaces of data block Hankel matrices, and then determining the order, the observability
matrix and/or the state sequence, by applying a singular value decomposition.
(ii). State space model realization through the solution of a linear least-squares problem.
Various linear algebra algorithms such as QR and singular value decomposition may be imple-
mented in different stages of this procedure. The main principles of subspace identification meth-
ods with related mathematical derivations are available in De Cock and De Moor (2003).
25
Thek
th
time step in the discrete-time state-space representation of a Linear, Time-Invariant (LTI)
model can be expressed as:
x
k+1
= Ax
k
+ Bu
k
+w
k
y
k
= Cx
k
+ Du
k
+v
k
(III.1)
wherex,y andu represent the state, output and input vectors, respectively. Process noise (w) and
measurement noise (v) are assumed to be zero-mean, stationary, white-noise vector sequences.
The state space realization to estimate matrices A (dynamical system matrix), B (input matrix),
C (output matrix) and D (feedthrough matrix) can be achieved by means of the subspace system
identification algorithm (Ljung (1986)). It is impossible to measure the input termu in the case
of ambient vibration; however, it can be modeled as white noise in the following form:
x
k+1
= Ax
k
+w
k
y
k
= Cx
k
+v
k
(III.2)
This simplified model is suitable as long as the input does not contain some dominant frequency
components in addition to white noise; otherwise, those frequency components cannot be sepa-
rated from the eigenfrequencies of the system (Skolnik et al. (2006)). If eigendecomposition of
the state matrix (A) is represented in the form of:
A =
1
(III.3)
where and are eigenvector and eigenvalue matrices respectively, the modal properties of a
continuous-time structural system can be subsequently derived from (Nayeri (2007)):
26
= diag(
i
j

i
) (III.4)

i
j

i
=
ln(
i
j

i
)
t
(III.5)
!
i
=
q

2
i
+

2
i
(III.6)

i
= cos

tan
1
(

i

i
)

(III.7)

e
i
= C
e
i
(III.8)
where!
i
,
i
and
e
i
represent the natural frequency, damping ratio, and the mode shape of the
i
th
mode of the system, respectively. Through the use of stability diagrams, the physical modal
properties of the structure can be accurately identified and distinguished from the spurious ones
usually generated due to the sensor noise and measurement errors.
III.2.2 Formulation of Cost Function
The cost function describes a Potential Energy Surface (PES) in the parameter space, and its
global minimum optimizes the desired objective. One of the most common feature-extraction
methods in finite element model updating is based on correlating the measured system response,
in the frequency or time domain, with the corresponding quantities in the analytical model. For
the study reported herein, the cost function to be minimized in the model updating process is
calculated by cumulatively summing over the first n dominant modes of the structure. Each
term, corresponding to one of the structural modes, is the summation of two weighted, normal-
ized values which quantify the deviation of the analytical frequency and mode shape from the
corresponding measured ones. From this definition, the cost function can be written as follows:
J(
e
) =
n
X
i=1
2
6
4W
f
i
j
f
(e)
i
f
(a)
i
f
(e)
i
j
2
+W

e
i
0
B
@1
j
e
(e)
i
T

e
(a)
i
j
2
(
e
(e)
i
T

e
(e)
i
)(
e
(a)
i
T

e
(a)
i
)
1
C
A
3
7
5 (III.9)
27
White noise excitation and structure response
Identify experimental f i and Ф i using N4SID
Select initial values for parametrs
Update finite element model
Identify analytical f i and Ф i of FEM
Calculate the cost function (J( α)) between
analytical and experimental modal data
Optimization Method:
Termination criteria
reached?
NO
YES
STOP
Figure III.1: Flowchart of finite element model updating process.
where
e
is the set of input parameters to be identified,f
i
and
e
i
represent the natural frequency
and the mode shape of thei
th
mode, andW
i
denotes the corresponding weight. Superscripts (e)
and (a) stand for experimental and analytical results, respectively. The flowchart in Figure III.1
illustrates the steps in finite element model updating process.
III.3 Experimental Case Study
III.3.1 Description of Test Bridge Structure
The case study to illustrate the application of the method under discussion is a two-span reinforced
concrete bridge tested experimentally at the University of Nevada, Reno. The quarter-scale speci-
men has equal span-length of 30 ft (9.14 m) and three double-column bents with variable column
heights of 6 ft (1.83 m), 8 ft (2.44 m), and 5 ft (1.52 m) respectively. The column section is a
28
Figure III.2: Rendering of the bridge structure (adapted from Johnson et al. (2006)).
1 ft diameter circular reinforced concrete with 1.56% longitudinal steel ratio. The deck of the
bridge is a precast slab with post-tensioned reinforcement in both the longitudinal and transverse
directions. It consists of six superstructure beams (three on each span) resting on the ledges at
the top of the columns. An amount of 280 kip (1245 kN) of additional mass is also provided on
the slab to consider scaling effects. More detailed information about this structure and the related
experimental study are available in Johnson et al. (2006). Figure III.2 shows a rendering of the
bridge structure.
III.3.2 Destructive Shaking Procedure
The bridge was tested under simulated excitations based on records from the Century City Coun-
try Club (1994 Northridge earthquake). Different levels of shaking amplitude, covering the range
of PGA = 0.075g 2.11g, were conducted on the structure. Intermediate, white-noise excitations
were also applied for system identification purposes (Johnson et al. (2006)). Although the spec-
imen underwent these excitations at different times, a combined 390-sec time-history for each
record was used in this paper. The corresponding combined time-history record for the sequence
of excitations in the transverse direction is shown in Figure III.3, and its specifications are pre-
sented in Table III.1.
29
Table III.1: NEES@Reno information of the combined time-history record for the excitations in the
transverse direction as shown in Figure III.3.
Test Ground Motion Tstart (sec) T
end
(sec) PGA(g)
WN-1 White Noise 0 60 0.07
Test-13 Low Earthquake 60 75 0.18
Test-14 Moderate Earthquake 75 90 0.31
WN-2 White Noise 90 150 0.07
Test-15 High Earthquake 150 165 0.68
Test-17 High Earthquake 165 180 1.26
WN-3 White Noise 180 240 0.07
Test-18 Severe Earthquake 240 255 1.65
WN-4 White Noise 255 315 0.07
Test-19 Extreme Earthquake 315 330 2.11
WN-5 White Noise 330 390 0.07
0 50 100 150 200 250 300 350 400
−2
−1
0
1
2
Time (sec)
Amplitude (g)
WN−1
T−13
T−14
WN−2
T−15
T−17
WN−3
T−18
WN−4
T−19
WN−5
Figure III.3: NEES@Reno combined time-history record for the sequence of the excitations applied to
the bridge model in the transverse direction. Windows indicated by WN-i represent the i
th
white noise
excitation test, and windows designated by T-n denote earthquake-like test number n. Note the significant
difference in test levels covering the 390-sec record.
III.3.3 Instrumentation, Data Acquisition and Filtering
Extensive instrumentation consisting of 298 channels (25 for slab displacements, 3 for footing
slip, 68 for column curvatures, 15 for column shear, 14 for slab accelerations, 1 for support frame
acceleration, 104 for longitudinal reinforcement strain, 56 for transverse reinforcement strain, and
12 for shake table response) were used to record the data at the frequency rate of 100 Hz (Johnson
et al. (2006)). From a practical point of view, real structures outside of the lab environment do not
generally enjoy such a comprehensive level of instrumentation. Consequently, it was decided for
the purposes of the study reported here, to also investigate a case under the assumption that only
30
Top View
Elevation View
Bent 1 Bent 2 Bent 3
N S
E
W
AL5 (N-S)
AT5 (E-W) AT1 (E-W)
AL1 (N-S)
AL3 (N-S)
AT3 (E-W)
AL4 (N-S)
AT4 (E-W) AT2 (E-W)
AL2 (N-S)
S1 S2 S3 S4 S5
360in [9.14m]
114in [2.9m]
360in [9.14m]
14in [0.36m]
72in [1.83m]
96in [2.44m]
60in [1.52m]
Safety Frame
Curvature Transducers
12in [0.3m]
Accelerometer
(a) Top View
Top View
Elevation View
Bent 1 Bent 2 Bent 3
N S
E
W
AL5 (N-S)
AT5 (E-W) AT1 (E-W)
AL1 (N-S)
AL3 (N-S)
AT3 (E-W)
AL4 (N-S)
AT4 (E-W) AT2 (E-W)
AL2 (N-S)
S1 S2 S3 S4 S5
360in [9.14m]
114in [2.9m]
360in [9.14m]
14in [0.36m]
72in [1.83m]
96in [2.44m]
60in [1.52m]
Safety Frame
Curvature Transducers
12in [0.3m]
Accelerometer
(b) Elevation View
Figure III.4: Top View (a) and Elevation View (b) of the NEES@Reno bridge, together with selected
sensor locations for this study. S1 to S5 denote the location of accelerometers on the bridge deck. (adapted
from Johnson et al. (2006)).
1 Transverse (f = 3.09 Hz)
st
2 Transverse (f = 4.15 Hz)
nd
3 Transverse (f = 12.90 Hz)
rd
(a) 1
st
Transverse mode shape (f = 3.09 Hz)
1 Transverse (f = 3.09 Hz)
st
3 Transverse (f = 12.90 Hz)
rd
(b) 2
nd
Transverse mode shape (f = 4.15 Hz)
1 Transverse (f = 3.09 Hz)
st
2 Transverse (f = 4.15 Hz)
nd
3 Transverse (f = 12.90 Hz)
rd
(c) 3
rd
Transverse mode shape (f = 12.90 Hz)
Figure III.5: First three identified transverse mode shapes of the tested structure in the undamaged state.
31
a limited number of sensors is available. The following two scenarios were considered:
(i). Only time-history records in both the longitudinal and transverse directions of the bridge
obtained from the five accelerometers on the deck are available.
(ii). Time-history records of the accelerometers on the deck as well as column curvature trans-
ducers are available.
Figure III.4 illustrates the location of the selected instrumentation for this study, deployed by the
NEES@Reno team on the tested bridge structure.
The first three transverse modes, in addition to the first longitudinal mode, were considered in
the calculation of the cost function. Note that, since the deck of the bridge is rigid, considering
the physics of the problem, it is impossible to evaluate the stiffness of the structure and detect
damage in the column level by just using the data from the sensors on the superstructure. There-
fore the second scenario requires more data from the curvature transducers on the top and the
bottom of the columns. In the definition of mode shapes for the cost function in the second sce-
nario, the measurements from the curvature transducers were used to determine the rotational
degrees of freedom about the longitudinal axis of the structure at the distance of 8.5 in (21.6 cm)
from the end of the columns. As illustrated in Figure III.5, based on the data processing conducted
for this study, the first transverse mode was identified as a dominant deck translation with slight
in-plane rotation (f = 3.09 Hz); the second mode as a dominant deck in-plane rotation with slight
translation (f = 4.15 Hz), and the third one as a deck bending (f = 12.90 Hz). The first longitudinal
mode was in translation and dominated the longitudinal response (f = 2.99 Hz). System identifi-
cation results for the aforementioned modes with low-pass filtered data at 25 Hz or at any higher
range were found to be identical. Consequently, a low-pass filter of 25 Hz was applied to all the
32
processed records.
III.3.4 NASTRAN
R
Finite Element Model
With the assumption that the behavior of the tested bridge would stay in the linear region when it
underwent low amplitude white-noise excitation, a NASTRAN
R
computer model was developed
using linear beam column elements. The NASTRAN
R
model emulated the SAP2000 model pro-
vided by the NEES@Reno research team who conducted the bridge test. Gross section properties
were used for all of the elements, but the stiffness of the reinforced columns were calibrated and
updated to represent the equivalent cracked moment of inertia.
As mentioned earlier, two scenarios were considered for investigating the model updating pro-
cedure:
Scenario 1: Modal properties were identified based on the records from the 5 accelerome-
ters on the deck. Based on system identification results from white-noise excitation records
after each destructive shaking test, a 4-dimensional cost function (3 parameters for the stiff-
ness of the bents in the transverse direction + 1 parameter for the longitudinal stiffness of
all bents) was optimized to quantify the overall damage in the bents.
Scenario 2: Modal properties were identified based on the records from 17 sensors (5 ac-
celerometers on the deck + 12 curvature transducers on the top and bottom of the columns).
A 13-dimensional cost function (12 parameters for the stiffness on the top and bottom of
the columns in the transverse direction + 1 parameter for the longitudinal stiffness of all
columns) was optimized to detect and quantify the overall and localized damage for each
column, after each destructive shaking test.
33
0 100 200 300 400 500 600 700 800
10
−8
10
−6
10
−4
10
−2
10
0
10
2
f=0.00855404757959426 (0.00855404757959426)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 100 200 300 400 500 600 700
0
0.2
0.4
0.6
0.8
1
2
1
3
4
Object Variables (4D)
Function evaluation
Modification factor
0 100 200 300 400 500 600 700 800
10
−4
10
−3
10
−2
10
−1
10
0
1
3
4
2
Standard Deviations of All Variables
function evaluations
0 100 200 300 400 500 600 700 800
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(a) Scenario 1 - CMA-ES
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (4D)
4
1
3
2
(b) Scenario 1 - GA
0 1000 2000 3000 4000 5000 6000 7000
10
−6
10
−4
10
−2
10
0
10
2
f=0.0770622933459472 (0.0770554599006103)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 1000 2000 3000 4000 5000 6000
0
0.2
0.4
0.6
0.8
1
1
10
6
2
5
11
8
12
13
7
3
9
4
Object Variables (13D)
Function evaluation
Modification factor
0 1000 2000 3000 4000 5000 6000 7000
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
13
3
1
4
2
12
9
11
10
5
8
6
7
Standard Deviations of All Variables
function evaluations
0 1000 2000 3000 4000 5000 6000 7000
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(c) Scenario 2 - CMA-ES
0 0.5 1 1.5 2 2.5
x 10
4
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (13D)
12
8
11
3
4
7
5
1
10
6
13
2
9
(d) Scenario 2 - GA
Figure III.6: Finite element model updating results for the calibration study. Plots (a) and (b) show
the variation of the modification factors with function evaluation through the optimization procedure for
Scenario 1 while plots (c) and (d) corresponds to Scenario 2. Each function evaluation consists of a finite
element analysis to find the modal properties of the analytical model for the given set of input parameters
and then computation of the cost function. The dark thick vertical line on the Right-Hand-Side (RHS)
indicates the end of optimization process. In each plot, the RHS small panel provides a high-resolution plot
of the modification factors when the identification procedure converged. Numbers on the RHS indicate the
index of the system parameter being identified and the straight lines point to the corresponding curve. The
correspondence of these parameters to the physical properties of the tested structure is available in Table
III.7.
34
Table III.2: Calibration results for analytical finite element model based on the identified experimental
modal properties of the undamaged state of the bridge using subspace method. S1 to S5 represent the
location of the accelerometers on the deck as shown in Figure III.4.
Mode Method Freq. (Hz) MAC
Mode Shape Values
S1 S2 S3 S4 S5
1
st
Transverse
Experimental 3.09
0.995
0.67 0.57 0.38 0.26 0.08
Analytical 3.10 0.63 0.55 0.43 0.30 0.15
2
nd
Transverse
Experimental 4.15
0.995
0.42 0.16 -0.23 -0.50 -0.70
Analytical 4.22 0.38 0.07 -0.23 -0.51 -0.73
3
rd
Transverse
Experimental 12.90
0.996
0.42 -0.42 -0.61 -0.36 0.39
Analytical 13.83 0.41 -0.35 -0.64 -0.34 0.42
1
st
Longitudinal
Experimental 2.99
1.00
0.44 0.44 0.45 0.45 0.46
Analytical 2.94 0.45 0.45 0.45 0.45 0.45
For calibration purposes, as shown in Figure III.6, finite element model updating for both sce-
narios was conducted to reproduce the response of the bridge specimen in the undamaged state.
The identified parameters through the model updating process were modification factors to be
applied to the gross section properties of the columns in the finite element model, eventually rep-
resenting the equivalent cracked reinforced concrete. For Scenario 1, shown in Figure III.6(a) and
Figure III.6(b), parameters 1-3 represent the modification factors for the stiffness of the bents in
the transverse direction and parameter 4 for the longitudinal stiffness of all bents. For Scenario
2, illustrated in Figure III.6(c) and Figure III.6(d), parameters 1-12 correspond to the stiffness in
the transverse direction on the bottom and top of the columns and parameter 13 to the longitu-
dinal stiffness of all bents. The correspondence of these parameters to the physical properties of
the tested structure is available in Table III.7. The resulting set of modification factors agreed
fairly well with the factor (33% of the gross moment of inertia) calculated by the NEES@Reno
researchers (Johnson et al. (2006)) using the slope of the elastic region in the elasto-plastic ide-
alized moment-curvature relationship obtained from an analysis performed through a computer
program called RCMC (Moment-Curvature analysis of confined and unconfined Reinforced Con-
crete sections).
35
Calibration results including obtained analytical and experimental frequencies as well as Modal
Assurance Criterion (MAC) values for the corresponding mode shapes are listed in Table III.2.
MAC values greater than 99.5% with frequency differences less than 2% (except the third trans-
verse mode which is of less importance) clearly indicate that there is a satisfactory agreement
between the modeled and the observed modal properties.
III.3.5 Choice of Initial Parameters and Weights
Parameter selection is a key issue in model updating procedures. The confidence in different
measured test values and initial parameter estimations can be expressed through the weights in
the cost function. Proper weighting factors can improve the optimization results significantly;
however, this requires a good deal of knowledge about the assumptions made in the finite element
modeling of the system, as well as the possible error sources in the analysis.
For typical structures, the system response is generally dominated by the low frequency modes.
In general, the higher the number of parameters is to be identified, the more modes are required to
be included in the cost function. On the other hand, using higher modes may not be reliable due to
not only the measurement noise but also the discretization effect in analytical finite element mod-
eling. Consequently, a plausible choice leads to setting higher weighting on the dominant modes,
while avoiding too little weighting factors for higher modes to preserve their embedded informa-
tion in the cost function (e.g., the inverse of the natural frequency). Nevertheless, model updating
results cannot be considered unique since they ultimately depend on user-defined weights and
constraints, as well as incorporating all the well-known limitations associated with inverse prob-
lems.
36
Upon implementing all of the aforementioned considerations, different sets of weighting factors
in the model updating algorithm used in this study were evaluated for a synthetic damage detec-
tion scenario in the structure. To this end, a randomly damaged computer finite element model
was generated for each simulation and the original undamaged model was updated to capture the
modal properties of the damaged state through the optimization of the cost function. Scenario 2
(availability of data from 17 sensors) with a cost function involving a parameter vector of order 13
was considered for the simulation. Statistical data based on an ensemble of 100 simulations, with
a maximum of 5000 iterations for each test (needed due to the stochastic nature of the algorithm),
were investigated to determine the proper selection of the identification parameters. Despite the
fact that even the accurate computational simulation of realistic damage phenomena is a chal-
lenging problem in its own right, random initial parameter choice was employed for each run
in order to evaluate the robustness of the algorithm under discussion. Figure III.7 illustrates the
probability density function of the random error in each parameter estimation for the weighting
set ofW
f
e
= W

e
=f4 2 1 1g (4, 2 and 1 for the first three transverse modes, and 1 for the first
longitudinal mode). For the weighting set off4 2 1 1g, the model updating algorithm was found
to be capable of detecting, localizing, and quantifying the damage in all sections of the structure,
with a high level of confidence (95% level of confidence with an error margin of less than 4% for
all parameters). Therefore, this weighting set was adopted in the rest of the study.
III.4 Results and discussion
III.4.1 Preliminary Damage Tracking
Finite element model updating is inherently a time-consuming process which makes it impracti-
cal for on-line prediction of system parameters. On the other hand, change-detection in modal
37
−4 −2 0 2 4
0
0.5
1
pdf
X(1)
μ = −0.21%
σ = 1.25%
−4 −2 0 2 4
0
0.5
1
X(2)
μ = −0.32%
σ = 1.50%
−1 0 1 2
0
0.5
1
1.5
2
X(3)
μ = 0.22%
σ = 0.58%
−5 0 5
0
0.2
0.4
0.6
0.8
pdf
X(4)
μ = 0.37%
σ = 1.82%
−4 −2 0 2 4
0
0.5
1
1.5
X(5)
μ = −0.11%
σ = 1.17%
−0.5 0 0.5
0
2
4
6
X(6)
μ = −0.08%
σ = 0.22%
−1 −0.5 0 0.5
0
1
2
3
4
X(7)
μ = −0.14%
σ = 0.32%
−2 0 2
0
0.5
1
1.5
2
2.5
X(8)
μ = 0.08%
σ = 0.69%
−2 0 2
0
0.5
1
1.5
Error (%)
pdf
X(9)
μ = −0.27%
σ = 0.99%
−2 0 2
0
0.5
1
1.5
Error (%)
X(10)
μ = 0.28%
σ = 0.93%
−1 0 1
0
1
2
3
4
Error (%)
X(11)
μ = −0.29%
σ = 0.48%
−4 −2 0 2 4
0
0.2
0.4
0.6
0.8
Error (%)
X(12)
μ = −0.25%
σ = 1.27%
−0.2 0 0.2 0.4
0
2
4
6
8
10
Error (%)
X(13)
μ = 0.05%
σ = 0.13%
Figure III.7: Probability density function (pdf) of the parameters estimation error. A total of 100 simu-
lations for scenario 2, with maximum of 5000 iterations for each test, were conducted for this study using
CMA-ES. The cost function to be optimized consisted a parameter vector of order 13. X(1) to X(12) rep-
resent the stiffness on the top and bottom of the columns in the transverse direction and X(13) denotes the
longitudinal stiffness of the bents. The correspondence of these parameters to the physical properties of the
tested structure is available in Table III.7. Initial parameters set was selected randomly for each simulation
to evaluate the robustness of the algorithm. In each plot panel, a thin line indicates the outline of the his-
togram of the parameter estimation error, and the solid line represents the estimated Gaussian pdf having a
matching mean () and standard deviation () to the corresponding histogram. Note that the abscissa and
ordinate ranges are not identical.
38
properties (i.e., frequencies, mode shapes, and system damping) is quite well known as an easily
quantifiable structural damage index in the field of SHM (Masri et al. (2008)). Hence, a prelimi-
nary damage tracking technique that relies on detection of any shift in these quantities to trigger
the model updating procedure may drastically increase its efficacy. Notice that, actual damage
in real structures manifests itself in different complex forms (Masri et al. (2008)) and therefore,
one of the practical challenges in conjunction with damage detection in physical systems is the
monitoring indicators. For example, it is well recognized in the structural dynamics community
that, solely detecting frequency shifts is a poor precursor of damage in realistic systems, due to its
relative insensitivity to small changes (Nayeri et al. (2008), Peeters and De Roeck (2001), Sohn
et al. (1999), Cornwell et al. (1999)). Therefore, a reliable change-detection scheme should pro-
vide a meaningful correlation between the estimated system parameters and physical measures of
the structural dynamic properties of the monitored system.
Considering all the above mentioned aspects, a preliminary damage tracking technique was de-
veloped for this study using a recursive autoregressive moving average model (Recursive ARMA)
to produce a near-real-time (on-line) monitoring method to detect any significant change in the
system response. The first order single-output recursive ARMA structure to predict the value of
the cost function is defined as:
(1 +L)J(t) = (1 +L)(t) (III.10)
whereJ is the real-time cost function value (defined by Equation III.9) based on extracted modal
properties from a given window size of the past response record. and are Autoregressive
(AR) and Moving Average (MA) parameters, represents the white noise disturbance value and
39
0 50 100 150 200 250 300 350 400
−2
0
2
Amplitude (g)
0 50 100 150 200 250 300 350 400
0
0.5
1
λ =0.9
σ
θ
0 50 100 150 200 250 300 350 400
0
0.5
1
λ =0.95
σ
θ
0 50 100 150 200 250 300 350 400
0
0.5
1
λ =0.98
σ
θ
0 50 100 150 200 250 300 350 400
0
0.5
1
λ =0.995
σ
θ
Time (sec)

= 5~15*T
1
= 10~30*T
1
= 20~60*T
1
Figure III.8: Preliminary damage detection through tracking of the standard deviation of the moving
average parameter () in the ARMA model. Four typical values of the forgetting factor ( = 0.90, 0.95,
0.98 and 0.995) and different window sizes for system identification were considered in this study. The
first row illustrates the synchronized plot of the excitation to the structure in the transverse direction.
L denotes the lag operator in time series analysis. Any change in the dynamic system throughout
the experiment, resulting in the fluctuation of the value of the cost function, would be presumably
reflected in the estimated parameters by the ARMA model. This procedure was implemented
in MATLAB
R
(using the rarmax module) for the response of the structure to the combined
excitation in the transverse direction.
Successive values of the standard deviation of the MA parameter () in the recursive ARMA
model over a 5-second window size is shown in Figure III.8. The plot illustrates the effects of
different lengths of the response record used for system identification, as well as the forgetting
factor variant,, which weighs past information exponentially less as time goes on. As expected,
using a short-length of response (5T
1
15T
1
, where T
1
is the fundamental period of the lin-
40
earized system) for system identification would result in fast damage detection at the expense of
a possible false-positive damage indication (indication of damage when none is present), while
employing a too-long window size (20T
1
60T
1
) would increase the possibility of false-negative
damage indication (no indication of damage when damage is present). Both types of these errors
are clearly undesirable, since the former would cause unnecessary downtime and consequent eco-
nomic disruption, while the latter would bring safety issues to the system.
Figure III.8 also compares the effects of four typical values for the forgetting factor ( = 0.90,
0.95, 0.98 and 0.995). Similar to the system identification window size, the forgetting factor value
showed a significant effect on the reaction time to changing system parameters, and an inverse
effect on ignoring noise. The smaller forgetting factor was more robust in detection of instanta-
neous damage to the system, but also more vulnerable to noise. A window size of (10T
1
30T
1
)
for system identification, and a forgetting factor of = 0.95 were found to be suitable for re-
liable instantaneous damage tracking of the investigated structure and therefore implemented in
this study. To simulate the real-time continuous monitoring of the structure, this proposed pre-
liminary damage tracking method was used to trigger the model updating procedure whenever
the monitored quantity settled down to a steady-state near-zero value after exceeding a predefined
threshold.
III.4.2 Model Updating Results
Damage detection using input-output data
The finite element model updating procedure was employed for both scenarios at different lev-
els of damage to the structure. Table III.3 to Table III.6 display the identified frequencies and
41
the corresponding mode shapes (measured at the location of the accelerometers on the deck) of
the structure in the various time windows depicted in Figure III.3 (WN-2 to WN-5) for the first
longitudinal and the first three transverse modes. These identified quantities were used in the
calculation of the cost function for the finite element model updating procedures illustrated in
Figures III.9 to III.12.
Figure III.9 and Figure III.10 show the finite element model updating results using CMA-ES for
scenario 1 and 2, respectively. Similar results using GA optimization algorithm in the finite ele-
ment updating procedure are illustrated in Figure III.11 and Figure III.12. As shown in Figure III.9
and Figure III.11, based on system identification results using recorded response measurements
by 5 sensors, a 4-dimensional cost function was optimized to detect and quantify the overall dam-
age in each bent. Parameters 1-3 represent the remaining stiffness of the bents in the transverse
direction and parameter 4 is the corresponding value for the longitudinal stiffness of all bents. As
illustrated in Figure III.10 and Figure III.12 for scenario 2, based on system identification results
using recorded response measurements by 17 sensors, a 13-dimensional cost function was opti-
mized to detect and quantify the overall and localized damage for each column. Parameters 1-12
indicate the remaining stiffness in the transverse direction on the bottom and top of the columns
and parameter 13 represents the corresponding value for the longitudinal stiffness of all bents.
The correspondence of these parameters to the physical properties of the tested structure is avail-
able in Table III.7. In each figure, plots (a) to (d) show the variation of the modification factors
with function evaluation through the optimization procedure for white-noise excitation windows
WN-2 to WN-5, respectively. Diminishing fluctuation of the parameters being identified through
the optimization process indicates the convergence of these quantities to their final values. The
dark thick vertical line denotes the end of optimization process. In each plot, the RHS small panel
42
Table III.3: System identification results for test window WN-2. S1 to S5 represent the location of the
accelerometers on the deck as shown in Figure III.4. These values were used for the finite element model
updating procedures illustrated in Figures III.9(a), III.10(a), III.11(a), and III.12(a).
Mode Freq. (Hz)
Mode Shape Values
S1 S2 S3 S4 S5
1
st
Transverse 2.46 0.68 0.54 0.40 0.26 0.08
2
nd
Transverse 3.44 0.34 -0.04 -0.25 -0.51 -0.75
3
rd
Transverse 12.33 0.40 -0.41 -0.62 -0.37 0.38
1
st
Longitudinal 2.82 0.44 0.44 0.44 0.45 0.45
Table III.4: System identification results for test window WN-3. These values were used for the finite
element model updating procedures illustrated in Figures III.9(b), III.10(b), III.11(b), and III.12(b).
Mode Freq. (Hz)
Mode Shape Values
S1 S2 S3 S4 S5
1
st
Transverse 1.53 0.69 0.54 0.40 0.26 0.10
2
nd
Transverse 1.82 0.31 0.01 -0.23 -0.50 -0.78
3
rd
Transverse 11.95 0.39 -0.43 -0.62 -0.38 0.37
1
st
Longitudinal 2.02 0.44 0.44 0.45 0.45 0.45
Table III.5: System identification results for test window WN-4. These values were used for the finite
element model updating procedures illustrated in Figures III.9(c), III.10(c), III.11(c), and III.12(c).
Mode Freq. (Hz)
Mode Shape Values
S1 S2 S3 S4 S5
1
st
Transverse 1.39 0.55 0.48 0.45 0.39 0.34
2
nd
Transverse 1.57 0.39 0.11 -0.18 -0.47 -0.77
3
rd
Transverse 11.94 0.42 -0.41 -0.62 -0.39 0.34
1
st
Longitudinal 1.82 0.44 0.45 0.45 0.44 0.45
Table III.6: System identification results for test window WN-5. These values were used for the finite
element model updating procedures illustrated in Figures III.9(d), III.10(d), III.11(d), and III.12(d).
Mode Freq. (Hz)
Mode Shape Values
S1 S2 S3 S4 S5
1
st
Transverse 1.34 0.61 0.53 0.42 0.33 0.22
2
nd
Transverse 1.56 0.45 0.12 -0.16 -0.44 -0.75
3
rd
Transverse 11.81 0.40 -0.41 -0.62 -0.40 0.35
1
st
Longitudinal 2.05 0.45 0.44 0.45 0.44 0.45
43
0 100 200 300 400 500 600 700
10
−6
10
−4
10
−2
10
0
10
2
10
4
f=0.022278146303274 (0.022278146303274)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 100 200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
1
3
4
2
Object Variables (4D)
Function evaluation
Modification factor
0 100 200 300 400 500 600 700
10
−3
10
−2
10
−1
10
0
1
3
4
2
Standard Deviations of All Variables
function evaluations
0 100 200 300 400 500 600 700
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(a) WN-2
0 100 200 300 400 500 600 700
10
−6
10
−4
10
−2
10
0
10
2
10
4
f=0.0920226048981052 (0.0920226048981052)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 100 200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
3
1
4
2
Object Variables (4D)
Function evaluation
Modification factor
0 100 200 300 400 500 600 700
10
−4
10
−3
10
−2
10
−1
10
0
1
3
4
2
Standard Deviations of All Variables
function evaluations
0 100 200 300 400 500 600 700
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(b) WN-3
0 200 400 600 800 1000 1200
10
−6
10
−4
10
−2
10
0
10
2
f=0.0112279604806573 (0.0112279604806573)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 200 400 600 800 1000
0
0.2
0.4
0.6
0.8
1
3
1
4
2
Object Variables (4D)
Function evaluation
Modification factor
0 200 400 600 800 1000 1200
10
−4
10
−3
10
−2
10
−1
10
0
3
1
4
2
Standard Deviations of All Variables
function evaluations
0 200 400 600 800 1000 1200
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(c) WN-4
0 100 200 300 400 500 600 700 800 900
10
−6
10
−4
10
−2
10
0
10
2
10
4
f=0.0542586123177907 (0.0542586123177907)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
3
1
2
4
Object Variables (4D)
Function evaluation
Modification factor
0 100 200 300 400 500 600 700 800 900 1000
10
−4
10
−3
10
−2
10
−1
10
0
1
3
2
4
Standard Deviations of All Variables
function evaluations
0 100 200 300 400 500 600 700 800 900
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(d) WN-5
Figure III.9: Finite element model updating results for Scenario 1, using CMA-ES. Plots (a) to (d) show
the variation of the modification factors with function evaluation through the optimization procedure for
white-noise excitation windows WN-2 to WN-5, respectively. Based on system identification results using
recorded response measurements by 5 sensors, a 4-dimensional cost function was optimized to detect and
quantify the overall damage in each bent. Parameters 1-3 represent the remaining stiffness of the bents
in the transverse direction and parameter 4 is the corresponding value for the longitudinal stiffness of all
bents (Please also see the caption of Figure III.6 for further details).
44
0 500 1000 1500 2000 2500 3000 3500
10
−6
10
−4
10
−2
10
0
10
2
f=0.573210901416043 (0.573197285074013)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 500 1000 1500 2000 2500 3000
0
0.2
0.4
0.6
0.8
1
2
1
4
3
9
10
11
12
6
13
7
8
5
Object Variables (13D)
Function evaluation
Modification factor
0 500 1000 1500 2000 2500 3000 3500
10
−4
10
−3
10
−2
10
−1
10
0
13
1
3
9
11
4
2
7
10
6
5
12
8
Standard Deviations of All Variables
function evaluations
0 500 1000 1500 2000 2500 3000 3500
10
−0.8
10
−0.6
10
−0.4
10
−0.2
10
0
Scaling (All Main Axes)
function evaluations
(a) WN-2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
10
−6
10
−4
10
−2
10
0
10
2
f=0.0255993276075954 (0.0255993276075954)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 1000 2000 3000 4000
0
0.2
0.4
0.6
0.8
1
11
12
10
9
2
3
1
4
13
6
5
7
8
Object Variables (13D)
Function evaluation
Modification factor
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
10
−4
10
−3
10
−2
10
−1
10
0
13
9
10
11
1
4
2
12
3
8
6
5
7
Standard Deviations of All Variables
function evaluations
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
Function evaluation
Modification factor
(b) WN-3
0 1000 2000 3000 4000 5000 6000 7000
10
−6
10
−4
10
−2
10
0
10
2
f=0.0168001319943607 (0.0167567532999125)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 1000 2000 3000 4000 5000 6000
0
0.2
0.4
0.6
0.8
1
11
12
9
10
2
3
1
4
13
6
5
7
8
Object Variables (13D)
Function evaluation
Modification factor
0 1000 2000 3000 4000 5000 6000 7000
10
−4
10
−3
10
−2
10
−1
10
0
13
12
11
3
9
10
4
2
1
6
7
5
8
Standard Deviations of All Variables
function evaluations
0 1000 2000 3000 4000 5000 6000 7000
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
Function evaluation
Modification factor
(c) WN-4
0 1000 2000 3000 4000 5000 6000
10
−8
10
−6
10
−4
10
−2
10
0
10
2
f=0.0188313752292394 (0.0188313752292394)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 1000 2000 3000 4000 5000
0
0.2
0.4
0.6
0.8
1
10
9
11
12
4
2
1
3
6
7
5
8
13
Object Variables (13D)
Function evaluation
Modification factor
0 1000 2000 3000 4000 5000 6000
10
−4
10
−3
10
−2
10
−1
10
0
11
13
10
9
3
4
12
1
2
7
6
5
8
Standard Deviations of All Variables
function evaluations
0 1000 2000 3000 4000 5000 6000
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(d) WN-5
Figure III.10: Finite element model updating results for Scenario 2, using CMA-ES. Plots (a) to (d)
show the variation of the modification factors with function evaluation through the optimization procedure
for white-noise excitation windows WN-2 to WN-5, respectively. Based on system identification results
using recorded response measurements by 17 sensors, a 13-dimensional cost function was optimized to
detect and quantify the overall and localized damage for each column. Parameters 1-12 represent the
remaining stiffness in the transverse direction on the bottom and top of the columns and parameter 13 is
the corresponding value for the longitudinal stiffness of all bents (Please also see the caption of Figure III.6
for further details).
45
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (4D)
1
3
4
2
(a) WN-2
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (4D)
3
1
4
2
(b) WN-3
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (4D)
3
1
4
2
(c) WN-4
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (4D)
3
1
2
4
(d) WN-5
Figure III.11: Finite element model updating results for Scenario 1, using GA. Plots (a) to (d) show
the variation of the modification factors with function evaluation through the optimization procedure for
white-noise excitation windows WN-2 to WN-5, respectively. Based on system identification results using
recorded response measurements by 5 sensors, a 4-dimensional cost function was optimized to detect and
quantify the overall damage in each bent. Parameters 1-3 represent the remaining stiffness of the bents
in the transverse direction and parameter 4 is the corresponding value for the longitudinal stiffness of all
bents (Please also see the caption of Figure III.6 for further details).
46
0 0.5 1 1.5 2 2.5
x 10
4
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (13D)
1
2
3
12
4
11
9
10
13
6
7
5
8
(a) WN-2
0 0.5 1 1.5 2 2.5
x 10
4
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (13D)
11
12
2
3
4
10
1
9
5
13
6
8
7
(b) WN-3
0 0.5 1 1.5 2 2.5
x 10
4
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (13D)
11
9
2
12
10
1
3
4
13
5
6
7
8
(c) WN-4
0 0.5 1 1.5 2 2.5
x 10
4
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (13D)
4
11
12
10
9
3
2
8
1
7
6
5
13
(d) WN-5
Figure III.12: Finite element model updating results for Scenario 2, using GA. Plots (a) to (d) show
the variation of the modification factors with function evaluation through the optimization procedure for
white-noise excitation windows WN-2 to WN-5, respectively. Based on system identification results using
recorded response measurements by 17 sensors, a 13-dimensional cost function was optimized to detect
and quantify the overall and localized damage for each column. Parameters 1-12 represent the remaining
stiffness in the transverse direction on the bottom and top of the columns and parameter 13 is the corre-
sponding value for the longitudinal stiffness of all bents (Please also see the caption of Figure III.6 for
further details).
47
provides a high-resolution plot of the modification factors when the identification procedure con-
verged. Numbers on the RHS indicate the index of the system parameter being identified and the
straight lines point to the corresponding curve. As shown in these figures, both CMA-ES and GA
converge to pretty close global minimums; however, GA may take more computational effort to
reach the solution, especially for higher order problem.
Note that, as the damage induced in the test structure escalates in response to increasing lev-
els of shaking in the test windows WN-2 through WN-5, the measure of damage provided by
the modification factors assures a diminishing value (since a value of 1.00 for modification factor
indicates an undamaged state, whereas 0.00 represents a completely damaged state). Damage de-
tection results for the test windows WN-2 to WN-5 are also tabulated in Table III.7 to Table III.10,
respectively. In each table, the identified damage quantities ((1.00 - Modification Factor)100%)
through the optimization process for both scenarios as well as their correspondence to the phys-
ical properties of the tested structure are presented. Please note that in the second scenario, the
average of four obtained damage values for each bent (last two columns in Table III.7 to Table
III.10) represents the overall damage to that bent. Comparing this average to the damage value
obtained in the first scenario for each bent (column #4 and column #5 in Table III.7 to Table III.10)
indicates a fair agreement between the results of two scenarios. The damage indices calculated by
the NEES@Reno test team for the bents are also listed for comparison and validation purposes.
48
Table III.7: Damage detection results for test window WN-2. Damage values show the percentage of
the loss of stiffness in the columns with respect to the intact stage. The damage indices calculated by
the NEES@Reno test team for the bents using strain gauge records are also listed. The corresponding
finite element model updating procedures, resulting in these quantities, are illustrated in Figures III.9(a) ,
III.10(a), III.11(a), and III.12(a).
Bent no.
Damage Index Scenario 1 Scenario 2
(NEES@Reno) X(i) CMA-ES GA X(i) Location CMA-ES GA
Bent-1
T
0.28 X(1) 43% 43%
X(1) East Column Bottom 42% 45%
X(2) East Column Top 45% 41%
X(3) West Column Bottom 38% 40%
X(4) West Column Top 39% 38%
Bent-2
T
0.14 X(2) 10% 11%
X(5) East Column Bottom 1% 5%
X(6) East Column Top 21% 15%
X(7) West Column Bottom 12% 12%
X(8) West Column Top 3% 3%
Bent-3
T
0.18 X(3) 35% 35%
X(9) East Column Bottom 37% 36%
X(10) East Column Top 34% 22%
X(11) West Column Bottom 36% 36%
X(12) West Column Top 30% 39%
All Bents
L
- X(4) 16% 16% X(13) All Col. Longitudinal 16% 16%
Table III.8: Damage detection results for test window WN-3. The corresponding finite element model
updating procedures, resulting in these quantities, are illustrated in Figures III.9(b) , III.10(b), III.11(b),
and III.12(b).
Bent no.
Damage Index Scenario 1 Scenario 2
(NEES@Reno) X(i) CMA-ES GA X(i) Location CMA-ES GA
Bent-1
T
1.19 X(1) 80% 79%
X(1) East Column Bottom 79% 75%
X(2) East Column Top 81% 81%
X(3) West Column Bottom 79% 81%
X(4) West Column Top 78% 80%
Bent-2
T
0.61 X(2) 50% 49%
X(5) East Column Bottom 53% 63%
X(6) East Column Top 63% 62%
X(7) West Column Bottom 50% 41%
X(8) West Column Top 50% 44%
Bent-3
T
0.99 X(3) 83% 83%
X(9) East Column Bottom 82% 74%
X(10) East Column Top 82% 79%
X(11) West Column Bottom 84% 90%
X(12) West Column Top 83% 89%
All Bents
L
- X(4) 63% 63% X(13) All Col. Longitudinal 63% 62%
49
Table III.9: Damage detection results for test window WN-4. The corresponding finite element model
updating procedures, resulting in these quantities, are illustrated in Figures III.9(c) , III.10(c), III.11(c),
and III.12(c).
Bent no.
Damage Index Scenario 1 Scenario 2
(NEES@Reno) X(i) CMA-ES GA X(i) Location CMA-ES GA
Bent-1
T
1.63 X(1) 82% 77%
X(1) East Column Bottom 80% 78%
X(2) East Column Top 83% 89%
X(3) West Column Bottom 81% 78%
X(4) West Column Top 80% 77%
Bent-2
T
0.86 X(2) 67% 54%
X(5) East Column Bottom 68% 72%
X(6) East Column Top 70% 70%
X(7) West Column Bottom 65% 60%
X(8) West Column Top 62% 55%
Bent-3
T
1.38 X(3) 88% 84%
X(9) East Column Bottom 87% 89%
X(10) East Column Top 87% 84%
X(11) West Column Bottom 90% 90%
X(12) West Column Top 87% 88%
All Bents
L
- X(4) 71% 67% X(13) All Col. Longitudinal 71% 74%
Table III.10: Damage detection results for test window WN-5. The corresponding finite element model
updating procedures, resulting in these quantities, are illustrated in Figures III.9(d) , III.10(d), III.11(d),
and III.12(d).
Bent no.
Damage Index Scenario 1 Scenario 2
(NEES@Reno) X(i) CMA-ES GA X(i) Location CMA-ES GA
Bent-1
T
2.15 X(1) 84% 84%
X(1) East Column Bottom 83% 79%
X(2) East Column Top 83% 81%
X(3) West Column Bottom 82% 85%
X(4) West Column Top 86% 89%
Bent-2
T
1.15 X(2) 68% 74%
X(5) East Column Bottom 66% 65%
X(6) East Column Top 79% 69%
X(7) West Column Bottom 72% 71%
X(8) West Column Top 66% 80%
Bent-3
T
1.87 X(3) 89% 88%
X(9) East Column Bottom 88% 87%
X(10) East Column Top 89% 88%
X(11) West Column Bottom 88% 89%
X(12) West Column Top 87% 89%
All Bents
L
- X(4) 63% 64% X(13) All Col. Longitudinal 63% 61%
50
The changes in the stiffness of the columns after WN-5 are generally insignificant. Interestingly,
an increase in the stiffness of the structure in transverse direction is also observed after WN-5
(71% damage after WN-4 and 63% after WN-5). Note that such an outcome is not uncommon
when dealing with experimental data. Meanwhile, this may be attributed to some extent to the
self-healing behavior of concrete cracks.
Damage detection using output-only data
In continuous monitoring of a large-scale system, it is usually infeasible to excite the structure
by a measurable artificial source, and even if possible, it will require expensive input devices
such as shakers. Moreover, during real operation, the loading conditions may be substantially
different from the ones used in the modal test. Therefore there is a considerable tendency to-
ward the use of freely available ambient excitation sources for system identification and damage
detection purposes. Various output-only system identification methods for operational modal
analysis have been proposed in the frequency domain (e.g, Peak-Picking method (PP), Complex
Mode Indication Function (CMIF), etc) and time domain (e.g, Instrumental Variable method (IV),
Covariance-Driven Stochastic Subspace Identification (SSI-COV), Data-Driven Stochastic Sub-
space Identification (SSI-DATA), etc) and have successfully been applied to real-life vibration
data (Peeters and Roeck (2001)).
To evaluate the proposed finite element model updating approach for output-only data analysis,
the damage detection procedures for both scenarios are conducted in window test WN-2 and
the results are presented in Figure III.13 and tabulated in Table III.11. Again, subspace method is
applied for modal identification of the structure. As expected, comparison of these results with the
corresponding quantities which are obtained from input-output data analysis (presented in Table
51
0 100 200 300 400 500 600 700 800
10
−8
10
−6
10
−4
10
−2
10
0
10
2
f=0.26706876527626 (0.26706876527626)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 200 400 600
0
0.2
0.4
0.6
0.8
1
1
3
4
2
Object Variables (4D)
Function evaluation
Modification factor
0 100 200 300 400 500 600 700 800
10
−4
10
−3
10
−2
10
−1
10
0
1
3
4
2
Standard Deviations of All Variables
function evaluations
0 100 200 300 400 500 600 700 800
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(a) Scenario 1 - CMA-ES
0 100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (4D)
1
3
4
2
(b) Scenario 1 - GA
0 500 1000 1500 2000 2500 3000 3500 4000
10
−8
10
−6
10
−4
10
−2
10
0
10
2
f=0.013993382609669 (0.0139930842325851)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 1000 2000 3000
0
0.2
0.4
0.6
0.8
1
2
11
3
4
1
9
12
10
6
13
5
8
7
Object Variables (13D)
Function evaluation
Modification factor
0 500 1000 1500 2000 2500 3000 3500 4000
10
−4
10
−3
10
−2
10
−1
10
0
13
2
12
3
4
9
10
1
7
11
5
8
6
Standard Deviations of All Variables
function evaluations
0 500 1000 1500 2000 2500 3000 3500 4000
10
−0.9
10
−0.6
10
−0.3
10
0
Scaling (All Main Axes)
function evaluations
(c) Scenario 2 - CMA-ES
0 0.5 1 1.5 2 2.5
x 10
4
0
0.2
0.4
0.6
0.8
1
Function evaluation
Modification factor
Object Variables (13D)
2
11
12
1
3
4
9
10
8
13
7
6
5
(d) Scenario 2 - GA
Figure III.13: Finite element model updating results using output-only data for test window WN-2. Plots
(a) and (b) show the variation of the modification factors with function evaluation through the optimization
procedure for Scenario 1 while plots (c) and (d) corresponds to Scenario 2. The correspondence of these
parameters to the physical properties of the tested structure is available in Table III.11. For comparison
purposes, the reader is referred to the corresponding figures using input-output data for damage detection
in test window WN-2 which are illustrated in Figures III.9(a), III.10(a), III.11(a), and III.12(a).
52
III.7; see also Figures III.9(a), III.10(a), III.11(a), and III.12(a)) shows an acceptable agreement,
primarily due to the white-noise nature of the excitations. Of course, the performance of the
method for real-life situations will highly depend on the accuracy of the important modal features
of the structure extracted from output-only vibration measurements.
Table III.11: Damage detection results for test window WN-2 using output-only data. The corresponding
finite element model updating procedures, resulting in these quantities, are illustrated in Figure III.13.
Bent no.
Damage Index Scenario 1 Scenario 2
(NEES@Reno) X(i) CMA-ES GA X(i) Location CMA-ES GA
Bent-1
T
0.28 X(1) 41% 41%
X(1) East Column Bottom 38% 46%
X(2) East Column Top 48% 51%
X(3) West Column Bottom 39% 33%
X(4) West Column Top 39% 31%
Bent-2
T
0.14 X(2) 4% 6%
X(5) East Column Bottom 13% 4%
X(6) East Column Top 19% 9%
X(7) West Column Bottom 2% 12%
X(8) West Column Top 12% 16%
Bent-3
T
0.18 X(3) 35% 34%
X(9) East Column Bottom 35% 24%
X(10) East Column Top 29% 19%
X(11) West Column Bottom 42% 49%
X(12) West Column Top 34% 49%
All Bents
L
- X(4) 16% 17% X(13) All Col. Longitudinal 16% 15%
III.4.3 Validation Results
To quantitatively estimate the amount of damage to the structure, the NEES@Reno research team
calculated a damage index for each bent and for each test motion using strain gauge records. This
damage index, developed by Park and Ang (1985) for reinforced concrete, is a practical mea-
sure of damage based on a combination of the amount of dissipated hysteretic energy, and the
maximum displacement demand over ultimate displacement ratio. For validation purposes, these
damage indices are also listed in Table III.7 to Table III.10. Values greater than 1.00 indicate
collapse. However, the probability of collapse at this threshold is approximately 50 percent with
a standard deviation of = 0.54. Damage indices greater than 1.00 represent a higher probability
53
of collapse (Johnson et al. (2006)).
The index represents a mechanistic damage model through the following equation:
DI =

M

u
+

Q
y

u
Z
dE (III.11)
in which
M
,
u
,Q
y
anddE represent maximum deformation under earthquake, ultimate defor-
mation under monotonic loading, calculated yield strength and incremental absorbed hysteretic
energy, respectively. Coefficient is defined as follows:
=

0:447 + 0:073
l
d
+ 0:24n
0
+ 0:314p
t

0:7
!
(III.12)
where
l
d
( 1:7), n
0
( 0:2), p
t
and
!
indicate shear span ratio, normalized axial stress, lon-
gitudinal steel ratio and confinement ratio respectively (Park and Ang (1985)). More detailed
information regarding the calculation of these indices is available in Johnson et al. (2006).
Figure III.14 illustrates the linear regression relating the damage index and the quantified damage
through the model updating procedure. Computed correlation factors (
cmaes
= 0:956;
ga
=
0:946) strongly confirm the accuracy of the detected damage values qualitatively and quantita-
tively. Unfortunately, such a validation is not possible for detected localized damages on the top
and the bottom of the columns; hence, the reliability of these results highly depends upon the ac-
curacy of the identified modal properties of the structure in the different stages of the experiment.
In order to gauge the damage detection results of this study, it is useful to compare them with
54
0 10 20 30 40 50 60 70 80 90 100
10
−1
10
0
Quantified Damage (%)
Damage Index
DI
CMAES
= 0.0802e
0.0349*QD
ρ = 95.6%
DI
GA
= 0.0822e
0.0353*QD
ρ = 94.6%

CMA−ES
GA
Figure III.14: Linear regression plot relating the damage index and the quantified damage. The abscissa
shows the quantified damage, identified through the model updating procedure in the first scenario (column
#4 and column #5 in Table III.7 to Table III.10), and the ordinate is the damage index introduced by Park
and Ang (1985) (column #2 in Table III.7 to Table III.10), which is a practical measure of damage based
on dissipated hysteretic energy and ductility demand.
the observations of the NEES@Reno team throughout the experiment. Table III.12 compares
the finite element model updating results with the visual inspections reported in Johnson et al.
(2006). As explained in the last column of this table, there is a fair agreement between the quali-
tative reported visual inspections and the quantitative identified damage indices through the model
updating approach.
III.5 Concluding Remarks
The underlying objective of this study is to evaluate the performance of two global optimiza-
tion methods in the finite element model updating approaches for damage detection in dispersed
structural systems, which usually deals with minimization of a complex, non-linear, non-convex,
high-dimensional cost function. The case study was a two-span reinforced concrete bridge, exper-
imentally tested at the University of Nevada, Reno. The subspace method for system identification
55
Table III.12: Comparison of finite element model updating results with NEES@Reno observations
Test Window NEES@Reno Observation (Johnson et al. (2006)) Model Updating Results
WN-2
“No damage was observed in the bridge until af-
ter test 13. During test 13, initial hairline flexural
cracks developed in Bent-1.”
The higher detected damage value in Bent-1 (43%
and 43%) agrees with the reported visual inspec-
tion. (Table III.7)
WN-3
“Flexural cracking began in Bent-3 and became sig-
nificant in the columns of both Bents-1 and 3 during
test 15. Also during test 15, initial hairline cracks
began to develop in Bent-2. During test 17, signif-
icant concrete spalling exposed the column lateral
reinforcement in both Bents-1 and 3.”
Significant detected damage values in Bent-1 (80%
and 79%) and Bent-3 (83% and 83%) are in com-
plete agreement with visual observations. Lower
value in Bent-2 (50% and 49%) shows the incipi-
ent stages of damage in the middle of the structure.
(Table III.8)
WN-4
“Significant spalling and exposure of lateral column
reinforcement in Bent-2 became evident during test
18. Also during test 18, the longitudinal reinforce-
ment of Bent-3, the shortest of the bents, became
exposed and initial buckling was observed on the
bottom west side of the west column.”
Very high detected damage value in Bent-3 (88%
and 84%) indicates severe situation in the shortest
bent. Damage values of (82% and 77%) in Bent-
1 and (67% and 54%) in Bent-2 confirm the visual
inspection results. (Table III.9)
WN-5
“Both columns of Bent-3 failed in flexure during test
19. The top and bottom of Bent-3 columns experi-
enced significant plastic hinging and crushing of the
core concrete. Four Bent-3 spirals fractured, and 36
longitudinal bars buckled.”
Complete failure in Bent-3 is clearly reflected in its
significant detected damage value (89% and 88%).
Bent-1 is also in severe condition with more than
84%=84% loss of stiffness. As predicted, since
Bent-2 would be the most unlikely bent to fail dur-
ing the experiment, it has the least damage index of
68%=74%. (Table III.10)
was used to extract the modal parameters (natural frequencies, mode shapes, and modal damping)
of the bridge system. A NASTRAN
R
computer model was developed based on the previous
SAP2000 model provided by the NEES@Reno team and validated with the system identification
results from the measured data. A simple on-line damage detection method, using an ARMA
model, was proposed and employed to trigger the finite element model updating process. Two
scenarios, assuming the availability of limited or large number of sensors were investigated for
the finite element model updating procedure. The feasibility of the proposed finite element model
updating algorithm to accurately detect, localize, and quantify the damage in the columns of the
tested bridge throughout the experiment was investigated and validated by comparison to experi-
mental measurements and visual inspections.
56
Based on the comparison of the results from the application of the finite element model updat-
ing algorithm under discussion with the strain gauge measurements and visual observations, the
following conclusions can be made:
(i). The simple ARMA model proposed for preliminary on-line damage detection can signifi-
cantly increase the efficacy of the model updating process.
(ii). The finite element model updating algorithm presented and applied in this study could
accurately detect and quantify the overall damage in the tested bridge bents throughout the
experiment.
(iii). The proposed method also showed very promising results for damage detection in the sys-
tem using output-only data. This reveals the potential of the technique to provide a useful
tool for SHM purposes in conjunction with promising methods for the identification of
modal properties using available ambient vibration data.
(iv). The finite element model updating algorithm used in this study was shown to be robust and
accurate to detect, localize and quantify the damage in the columns in synthetic simulations;
however, the experimental results could not be completely validated. The reliability of these
results highly depends upon the accuracy of the identified (equivalent) modal properties of
the (damaged, nonlinear) structure in different stages of the experiment.
(v). Detected damage values are highly correlated (
cmaes
= 0:956;
ga
= 0:946) with the
damage index developed by Park and Ang (1985), which is a practical measure of damage
based on dissipated hysteretic energy and ductility demand.
(vi). Both CMA-ES and GA converge to pretty close global minimums; however, GA may take
more computational effort to reach the solution, especially for higher-order problem.
57
Even though the cost function to be optimized in this study was not relatively high-dimensional
(13-D), considering the promising performance of the optimization method under discussion in
solving well-known benchmark problems of global optimization, the general conclusions from
this study are useful in providing guidelines for the application of stochastic optimization methods
to real-world search problems, especially in the implementation of structural health monitoring
for complex, nonlinear distributed systems.
58
Chapter IV
EV ALUATION AND APPLICATION OF SOME
DATA-DRIVEN APPROACHES FOR THE
DEVELOPMENT OF EQUIV ALENT LINEAR
SYSTEM FOR NONLINEAR STRUCTURES
IV .1 Introduction
T
HE performance of six different system identification methods is studied in this section.
The fidelity of the methods under discussion in identifying the structural dynamic prop-
erties of a nonlinear system is evaluated by using the experimental data from a 4-story model
building.
IV .1.1 Vibration-Signature-Based Nonlinear System Identification Methods: A Lit-
erature Review
The large family of system identification methods for estimation of nonlinear models based on
experimental measurements can be classified in several different approaches. The current pre-
59
sentation gives an overview on some essential features in the area which includes a summary
of a comprehensive literature review and classification of methods presented by Kerschen et al.
(2006), variants and developments in each class, and their possible advantages and limitations.
Linearization
Several studies have attempted to find an equivalent linear model that can predict the response of
a nonlinear system. Caughey (1963) generalized the equivalent linearization method of Kryloff
and Bogoliubov to the case of nonlinear dynamic systems with random excitation. The method
operates directly on the equations of motion of a nonlinear oscillator under external Gaussian
excitation and replaces the nonlinear structure by a linear model based on minimizing a statistical
error function between the outputs of two systems. Many developments have been proposed since
to address the drawbacks of the technique such as the requirement of partially knowing the system
or its limitation to specific excitations. More works in this field can be found in Iwan (1973), Wen
(1980), Iwan and Mason (1980), Bruckner and Lin (1987), Roberts and Spanos (1990), Socha and
Soong (1991), Bouc (1994), Rice (1995), Soize and Le Fur (1997), Proppe et al. (2003), Bellizzi
and Defilippi (2003), Belendez et al. (2008), and Socha (2008).
Time-domain methods
Time-domain system identification methods have the advantage of taking direct measurements as
input, resulting in no loss of embedded information about the system in the recorded data, as well
as less time and effort to spend on data acquisition and processing. A fundamental time-domain
approach, called “Restoring Force Surface” (RFS), was proposed by Masri and Caughey (1979)
and Masri et al. (1982) that initiated the analysis of nonlinear structural systems in terms of their
internal RFSs. The original method was extremely appealing for its simplicity of concept; how-
60
ever, it suffered from complicated numerical analysis. It also required all vibration-signatures
(acceleration, velocity, displacement) data at all DOFs. Many researchers tried to improve the
method and overcome these demands in the following years including Masri et al. (1987a), Masri
et al. (1987b), Worden (1990a), Worden (1990b), Mohammad et al. (1992), Duym and Schoukens
(1995), Kerschen et al. (2001), Haroon et al. (2005), Nayeri et al. (2008), and Allen et al. (2008).
Another interesting time-domain technique for nonlinear system identification based on the gen-
eralization of the multivariable ARMAX models for linear systems, called NARMAX (Nonlin-
ear Auto-Regressive Moving Average models with eXogeneous input), has been proposed by
Leontaritis and Billings (1985a) and Leontaritis and Billings (1985b) and received considerable
attention in the structural dynamics community.
Frequency-domain methods
Frequency-domain system identification methods study the data mainly in the form of spectra or
Frequency Response Functions (FRF). Easier computation and more intuitive interpretation are
key advantages of these methods over time-domain approaches. Early frequency-domain methods
for nonlinear system identification were based upon the use of functional series such as V olterra
and Wiener series (Schetzen (1980)). An example of the application of these methods in the field
of structural dynamics was presented by Gifford (1989) in his doctoral dissertation. His pro-
posed method was based on extracting nonlinear parameters by fitting surfaces or hyper-surfaces
to the Higher Order Frequency Response Functions (HOFRF). The method was later improved
and expanded in various works such as Storer and Tomlinson (1993), Khan and Vyas (2001),
and Chatterjee and Vyas (2004). Other frequency-domain approaches have been developed for
identification of nonlinear systems, including methods using nonlinear resonances (e.g., Nayfeh
(1985)), spectral methods based on the reverse path analysis (e.g., Rice and Fitzpatrick (1988));
61
methods using higher-order spectra (e.g., Roberts et al. (1995)); techniques using associated lin-
ear equations (e.g., Feijoo et al. (2004)), or those developed for Hammerstein models (e.g., Bai
(2010)). For a comprehensive overview of this field of research, readers may consult Pintelon and
Schoukens (2001).
Modal methods
Modal analysis represents a dynamic system in the form of its modal parameters. Despite the
popularity and suitability of this traditional method for linear models, its application to highly-
nonlinear systems usually leads to erroneous results. Seminal work of Rosenberg (1962) and
Rosenberg (1966) on the concept of Nonlinear Normal Mode (NNM) has provided an excellent
theoretical foundation for developing nonlinear system identification methods based on modal
analysis; however, new complications will arise due to the amplitude-dependency of NNMs and
their periods. Among many techniques that have been developed based on this methodology is
an early attempt by Szemplinska-Stupnicka (1979) and Szemplinska-Stupnicka (1983) to approx-
imate NNMs using the mode of vibration in resonant conditions.
Investigators have also proposed other modal methods for identification of nonlinear systems.
Masri et al. (1982) introduced a low-order regression analysis in modal space using the classical
RFS method. Bellizzi et al. (2001) proposed an identification method based on comparing exper-
imental coupled nonlinear modes to the predicted ones. Representatives of recent applications of
modal analysis for nonlinear system identification can be found in He et al. (2008) and Platten
et al. (2009).
62
Time-frequency analysis
Time-frequency analysis is based on studying the time-varying nature of the vibrational character-
istics of the nonlinear system by decomposing the signal into a set of simpler components. Spina
et al. (1996) studied the application of time-frequency analysis using Gabor transform on nonlin-
ear oscillations. The Gabor transform identifies a time-variant matrix to decouple the transient
response into a set of quasi-harmonic components. Kitada (1998) proposed a method based on
expanding the response and excitation of the nonlinear system in terms of scaling functions using
wavelet transform. Ghanem and Romeo (2001) presented a wavelet-based approach using orthog-
onal Daubechies scaling functions for model and parameter identification of nonlinear systems.
Kareem and Kijewski (2002) studied the use of the wavelet transform for time-frequency analysis
of wind effects on structures. Pai et al. (2008) proposed a methodology that uses the Hilbert-
Huang transform (HHT) and a Sliding-Window Fitting (SWF) technique to drive time-dependent
dynamic characteristics of nonlinear systems through perturbation analysis.
Black-box modeling
Black-box modeling is a practical data-only-based system identification method that makes no
a priori assumption about the model. When physical insight about the system or the source of
nonlinearity is not available, nonlinear black-box modeling can be considered for system iden-
tification purposes; however, the identified model parameters may not be specifically attributed
to physical information of the structure. Artificial neural network is among the most popular
approaches that have been used for nonlinear mapping between recorded input and output data
of the system. The fundamental paper of Narendra and Parthasarathy (1990) demonstrated that
neural networks can be used effectively for the identification and control of nonlinear dynamical
63
systems. Masri et al. (1992) and Masri et al. (1993) used “dynamic neurons” in a multi-layer
perceptron neural networks structure to represent nonlinear systems. Chen and Billings (1992),
Chassiakos and Masri (1996), Masri et al. (2000), Kosmatopoulos et al. (2001), and Chen (2009)
have also studied using artificial neural networks for black-box nonlinear system identification
purposes. Other methods for non-parametric identification of nonlinear systems include splines
models (e.g. Peifer et al. (2003)), and dynamic fuzzy wavelet neural networks models (e.g. Adeli
and Jiang (2006)).
Structural model updating
Structural model updating techniques compute and update the model parameters through min-
imizing an objective function that measures the deviation of simulated response of the model
from the corresponding real measurement. One of the main applications of these techniques is to
make corrections to the initial finite element model of complex structures which usually suffers
from modeling, parameter, and testing errors. Berman and Flannelly (1971) and Baruch (1978)
were among the first researchers to introduce finite element model updating for linear structures.
Schmidt (1994) used the method of modal state observers to update the parameters of nonlin-
ear dynamic systems. Dippery and Smith (1998) employed the minimum model error estimation
algorithm for updating nonlinear models. Kyprianou et al. (2001) used differential evolution al-
gorithm for minimizing the objective function in the method. Meyer and Link (2003) defined
the objective function based on the difference between the measured and predicted displacement
response in the frequency domain. Yuen and Beck (2003) proposed a model updating method
to quantify the uncertainties in the model parameters. Kerschen and Golinval (2005) proposed a
two-step methodology for decoupling the estimation of the linear and nonlinear parameters of the
finite element model. Muto and Beck (2008) developed an identification method for hysteretic
64
systems using Bayesian updating. For a detailed description of model updating methods, the
readers are referred to Friswell and Mottershead (1995).
IV .1.2 Scope
In this study, the performance of six different system identification methods is investigated, using
experimental data obtained from a 4-story model building. The six methods under discussion are:
(i). Linear System Using Least-Squares Method
(ii). Symmetric Linear System Using Least-Squares Method
(iii). Restoring Force Surface (RFS) Method for Chain-Like Systems
(iv). Model Updating Method
(v). Sub-Space Identification Method
(vi). Iterative Prediction-Error Minimization Method
It is not the intention of this research to compare these methods in detail, but rather to study their
performance in identifying the structural dynamic properties of the system. Comparison of dif-
ferent system identification techniques on operational data is potentially a subjective matter for
several reasons. Firstly, due to the lack of a reliable reference system, the methods can only be
compared relative to one another rather than to the exact solution. Furthermore, the comparison
criterion is also a highly subjective choice depending on the actual application. Some researchers
may suggest that the accuracy of the identified modal properties or the regenerated response is
the critical parameter for comparison of the methods, while others might emphasize the signifi-
cance of other factors such as robustness, computational efficiency, etc (Andersen et al. (1999)).
65
Considering the relatively small size of the structural model under discussion (4 DOF system),
emphasis in this study is placed on studying the identified structural matrices (mass, damping,
and stiffness matrices), and modal properties (frequencies, damping, and mode-shapes) for the
purpose of detecting, localizing and quantifying the nonlinearity in the system throughout the ex-
periment.
The theory behind each method is briefly reviewed in section IV .2. In section IV .3, the test
structure, a 4-story model building which was investigated experimentally at Hunan University in
China, as well as the test procedure are explained. Section IV .4 presents the results and discussion,
and section IV .5 highlights the concluding remarks.
IV .2 Overview of System Identification Methods Under Discussion
IV .2.1 Linear System Using Least-Squares Method
Consider a discrete nonlinear MDOF system that is subjected to directly applied excitation forces
f
1
(t) as well as prescribed support motions x
0
(t). The governing equations of motion for this
multi-input/multi-output nonlinear system can be written as
M
e
11
x
1
(t)+C
e
11
_ x
1
(t)+K
e
11
x
1
(t)+M
e
10
x
0
(t)+C
e
10
_ x
0
(t)+K
e
10
x
0
(t)+f
NL
(t) = f
1
(t) (IV .1)
where the coefficients are defined as follows:
f
1
(t) = column vector of ordern
1
, representing directly applied forces;
x(t) = (x
T
1
(t);x
T
0
(t))
T
= system displacement vector of ordern
1
+n
0
;
x
1
(t) = active DOF displacement vector of ordern
1
;
66
x
0
(t) = prescribed support displacement vector of ordern
0
;
M
e
11
, C
e
11
, K
e
11
= constant matrices that characterize the equivalent inertia, damping, and stiffness
forces associated with the unconstrained DOFs of the system, each of ordern
1
n
1
;
M
e
10
, C
e
10
, K
e
10
= constant matrices that characterize the inertia, damping, and stiffness forces
associated with the support motions, each of ordern
1
n
0
;
f
NL
(t) = ann
1
column vector of nonlinear nonconservative forces.
The governing equation of the linearized version of this system can be presented as
M
e
11
x
1
(t) + C
e
11
_ x
1
(t) + K
e
11
x
1
(t) + M
e
10
x
0
(t) + C
e
10
_ x
0
(t) + K
e
10
x
0
(t) = f
1
(t) (IV .2)
For clarity of presentation, let the six matrices appearing in Equation IV .2 be denoted by
1
A,
2
A,
...,
6
A, respectively. If thei
th
row of a generic matrix
j
A is shown ash
j
A
i
i, the parameter vector

i
, that constitutes all of these elements in six matrices, can be introduced as:

i
= (h
1
A
i
i;h
2
A
i
i;h
3
A
i
i;h
4
A
i
i;h
5
A
i
i;h
6
A
i
i)
T
(IV .3)
Let the response vectorr(t) of order 3(n
1
+n
0
) be defined as
r(t) = ( x
T
1
(t); _ x
T
1
(t); x
T
1
(t); x
T
0
(t); _ x
T
0
(t); x
T
0
(t))
T
(IV .4)
If the excitation and the response of this linear system is measured at time stepst =t
1
;t
2
;:::;t
N
,
then at everyt
k
1
A x
1
(t
k
)+
2
A _ x
1
(t
k
)+
3
Ax
1
(t
k
)+
4
A x
0
(t
k
)+
5
A _ x
0
(t
k
)+
6
Ax
0
(t
k
) = f
1
(t
k
) k = 1; 2;:::;N
(IV .5)
67
With introduction of matrix R as
R =
2
6
6
6
6
6
6
6
6
6
4
r
T
(t
1
)
r
T
(t
2
)
.
.
.
r
T
(t
N
)
3
7
7
7
7
7
7
7
7
7
5
(IV .6)
the grouping of the measurements can be expressed concisely as
^
R^ =
^
b (IV .7)
where
^
R is a block diagonal matrix whose diagonal elements are equal to R, ^ = (^
T
1
; ^
T
2
;:::; ^
T
n
1
)
T
and
^
b is the corresponding vector of excitation measurements. It should be noted that
^
R is of or-
dermn wherem = Nn
1
, andn = 3n
1
(n
1
+n
0
), and therefore, if a sufficient number of
measurements is taken, it will result inm>n. Under these conditions, least-squares procedures
can be used to solve for all the system parameters that constitute the entries in ^ :
^ =
^
R
y
^
b (IV .8)
where
^
R
y
is the pseudo-inverse of
^
R. In the more general case where the measurements asso-
ciated with certain DOFs are more reliable than others and/or measurements accumulated over
certain time periods are to be emphasized differently from the others, a symmetric, nonsingular,
usually diagonal error weighting matrix W can be used with the overdetermined set of equations
in Equation IV .7. Let the deviation error ^ e be
^ e =
^
b
^
R^ (IV .9)
68
Using the weighted least-squares method to minimize the cost functionJ,
J = ^ e
T
W^ e (IV .10)
results in the following approximate solution
^ = (
^
R
T
W
^
R)
1
^
R
T
W
^
b (IV .11)
Considering the diagonal nature of partitioned matrix
^
R, the solution of Equation IV .7 can be
simplified into a set ofn
1
decoupled matrix equations, each of the form:
R^
i
=
^
b
i
i = 1; 2;:::;n
1
(IV .12)
Comparing the orders of
^
R to R shows that the order of R is smaller by a factor ofn
2
1
, making
Equation IV .12 much more computationally efficient. Least-squares techniques can again be used
to find the components of then
1
parameter vectors ^
i
:
^
i
=
^
R
y
^
b
i
(IV .13)
Note that
^
R
y
needs to be computed only once (Masri et al. (1987a)).
IV .2.2 Symmetric Linear System Using Least-Squares Method
When dealing with structural systems, it is a legitimate assumption that structural matrices possess
symmetric properties. For a special case of linear system with symmetric matrices, thei
th
(row)
69
component of then
1
system of equations in Equation IV .2 is presented as
n
1
X
j=i

1
A
ij
x
1
j
+
2
A
ij
_ x
1
j
+
3
A
ij
x
1
j
+
4
A
ij
x
0
j
+
5
A
ij
_ x
0
j
+
6
A
ij
x
0
j

+
i1
X
j=1

1
A
ij
x
1
j
+
2
A
ij
_ x
1
j
+
3
A
ij
x
1
j

= f
1
i
(t) i = 1; 2;:::;n
1
(IV .14)
Leth
j
A
l
i
i denote the elements of a square matrix
j
A that reside in rowi and lie below the diagonal
of the partitioned matrix. For exampleh
j
A
l
i
i = 0 andh
j
A
l
k
i = (
j
A
k;1
;
j
A
k;2
;:::;
j
A
k;k1
).
Similarly, leth
j
A
u
i
i denote the complement ofh
j
A
l
i
i associated with rowi. In other words,h
j
A
u
i
i
corresponds to the diagonal and above the diagonal elements of rowi. For example, in a matrix
j
A of ordern,h
j
A
u
k
i = (
j
A
k;k
;
j
A
k;k+1
;:::;
j
A
k;n
). Using this notation, the following quantities
are defined:

i
= (h
1
A
u
i
i;h
2
A
u
i
i;h
3
A
u
i
i;h
4
A
i
i;h
5
A
i
i;h
6
A
i
i)
T
(IV .15)

i
= (h
1
A
l
i
i;h
2
A
l
i
i;h
3
A
l
i
i)
T
(IV .16)
r
i
(t) = ( x
T
1
i
(t); x
T
1
i+1
(t);:::; x
T
1n
1
(t); _ x
T
1
i
(t); _ x
T
1
i+1
(t);:::; _ x
T
1n
1
(t);
x
T
1
i
(t); x
T
1
i+1
(t);:::; x
T
1n
1
(t); x
T
0
(t); _ x
T
0
(t); x
T
0
(t))
T
(IV .17)
v
i
(t) = ( x
T
1
1
(t); x
T
1
2
(t);:::; x
T
1
i1
(t); _ x
T
1
1
(t); _ x
T
1
2
(t);:::; _ x
T
1
i1
(t);
x
T
1
1
(t); x
T
1
2
(t);:::; x
T
1
i1
(t))
T
(IV .18)
With the definition of
i
,
i
, r
i
(t) and v
i
(t), Equation IV .14 can be expressed as:
r
i
T
(t)
i
+ v
i
T
(t)
i
= f
1
i
(t) (IV .19)
70
which can be rewritten in the form:
r
i
T
(t)
i
= f
i
(t) v
i
T
(t)
i
(IV .20)
Starting from the first row (i = 1), v
i
(t) is a null vector and elements of
1
can be computed
as in general method. For the next rows, system parameters in
i
have been determined in the
previous stage of the calculation and the term on the RHS of Equation IV .20 is known or can be
measured. Thus a system with symmetric matrices can be identified for the given measurements
(Masri et al. (1987a)).
IV .2.3 Restoring Force Surface (RFS) Method for Chain-Like Systems
m
2
1
m
n
m
(1)
G
G
(2)
G
(3)
G
(n)
F (t)
1
2
F (t)
F (t)
n
x (t)
i
Figure IV .1: Model of a MDOF chain-like system.
Consider the MDOF chain-like structure shown in Figure IV .1 which consists ofn elements, each
with a lumped massm
i
, and an arbitrary (unknown) nonlinear restoring functionG
(i)
. The struc-
ture may be subjected to a base excitation x
0
(t), and/or directly applied forces F
i
(t). In the
context of civil structures, this system would be analogous to ann-story building with rigid floor
71
slabs under ambient forces or ground motion excitation. It is assumed that the absolute acceler-
ation at each element, x
i
(t), is directly available from measurement. The other state variables,
_ x
i
(t) and x
i
(t), can be computed through integration of acceleration records. At this stage, we
also need to assume that the applied forceF
i
(t) are measurable. The relative motion between two
consecutive elements can be computed as follows:
z
i
(t) = x
i
(t) x
i1
(t) i = 1; 2;:::;n (IV .21)
A reasonable assumption in the field of structural dynamics is that the restoring force at each
element is only dependent on the relative displacement and velocity across the terminals of that
element:
G
(i)
= G
(i)
(z
i
; _ z
i
) (IV .22)
Therefore the equations of motion for such a system can be written as
m
n
x
n
= F
n
(t) G
(n)
(z
n
; _ z
n
)
m
i
x
i
= F
i
(t) G
(i)
(z
i
; _ z
i
) + G
(i+1)
(z
i+1
; _ z
i+1
) i =n 1;n 2;:::; 1 (IV .23)
Equation IV .23 can be rewritten to express the unknown restoring force functions as
G
(n)
(z
n
; _ z
n
) = F
n
(t)m
n
x
n
G
(i)
(z
i
; _ z
i
) = F
i
(t)m
i
x
i
+ G
(i+1)
(z
i+1
; _ z
i+1
) i =n 1;n 2;:::; 1 (IV .24)
72
which can be presented in the more compact form of
G
(i)
(z
i
; _ z
i
) =
n
X
j=i
(F
j
(t)m
j
x
j
) i = 1; 2;:::;n (IV .25)
Thus, starting from the tip of the chain, one can sequentially determine the time-histories of all
the inter-story restoring forces within the chain. The advantage of this formulation is that the
identification problem of a MDOF system is converted to a set of decoupled SDOF problems. For
the very top element, the restoring force is directly computed by subtracting the inertia force from
the external force measured at the top element. Then, starting from the element right before the
very last, the restoring force can be calculated by subtracting the inertia force from the external
force at that element, plus the restoring force of the previous element which has been computed
in the previous step. It should be noted that since the identification of the restoring force for each
element within the chain is dependent on the restoring force of the previous element, there is an
error accumulation effect, in which the error propagates down the chain and leads to a bigger error
for the lower elements of the chain.
Once the time-history of the restoring functions for all the elements is determined, one can
use suitable basis functions to approximate a nonparametric model for each element. A suit-
able choice of basis functions would be a power series expansion in the doubly indexed series as
follows:
Basis = =
qmax
X
q=0
rmax
X
r=0
z
q
_ z
r
(IV .26)
A third-order expansion is usually sufficient for most of practical applications in structural dy-
73
namics. Thus, forq
max
=r
max
= 3, the set of basis functions becomes
=

1; _ z; _ z
2
; _ z
3
; z; z_ z; z_ z
2
; z_ z
3
; z
2
; z
2
_ z; z
2
_ z
2
; z
2
_ z
3
; z
3
; z
3
_ z; z
3
_ z
2
; z
3
_ z
3

(IV .27)
Standard least-squares methods can then be used to find the individual coefficients associated with
each basis function
G
(i)
=
qmax
X
q=0
rmax
X
r=0
p
(i)
qr
z
q
i
_ z
r
i
G
(i)
=
(i)
p
(i)
(IV .28)
p
(i)
= [
(i)
]
y
G
(i)
where G
(i)
is anN 1 vector whose elements are the time history samples of thei
th
restoring
function; p
(i)
is a ((1+q
max
)(1+r
max
))1 vector of the unknown parameters p
(i)
qr
to be identified
in the process,
(i)
is anN ((1 +q
max
)(1 +r
max
)) matrix of the known time-histories of the
basis functions (Equation IV .26),N is the number of time samples, and the superscripty denotes
the pseudo-inverse. Note that identified damping and stiffness matrices for chain-like systems will
be in symmetric tridiagonal form while the mass matrix is in diagonal form (Masri et al. (1982)).
IV .2.4 Model Updating Method
As discussed earlier, one of the most popular feature-extraction methods in finite element model
updating is based on correlating the measured system response, in the frequency or time domain,
with the corresponding quantities in the analytical model. For the study reported herein, the cost
function to be minimized in the model updating process was similar to the one in the least-squares
method for linear systems. The defined cost function quantifies the deviation of the analytical
74
response from the corresponding measured ones and cumulatively sums them over the N data
points. In other words, for the linear MDOF system governed by Equation IV .2, model updating
procedure was applied for each unconstrained degree of freedom to minimize the following cost
function:
J(^
i
) =kR^
i

^
b
i
k i = 1; 2;:::;n
1
(IV .29)
wherek k measures the Euclidian norm of the vector. The CMA-ES optimization package was
used to minimize the cost function and achieve an optimal set of elements for the structural dy-
namic matrices of the system.
IV .2.5 Sub-Space Identification Method
The last two methods under discussion are based on identifying the state-space representation of
the structural system. In control engineering, a state-space representation is a mathematical model
of a physical system as a set of input, output and state variables related by first-order differential
equations. For an LTI dynamical system, these differential and algebraic equations can be written
in matrix form, which provides a convenient and compact way to model and analyze multi-input,
multi-output (MIMO) systems. Continuous-time state-space representation of an LTI model is
presented as:
_
X(t) = A
c
X(t) + B
c
U(t) +v(t)
Y (t) = C
c
X(t) + D
c
U(t) +w(t) (IV .30)
where X, Y , and U represent the state, output, and input vectors, respectively. Process noise
(v) and measurement noise (w) are assumed to be zero-mean, stationary, white-noise vector se-
quences. The state-space realization will result in estimation of matrices A
c
(dynamical system
75
matrix), B
c
(input matrix), C
c
(output matrix) and D
c
(feedthrough matrix). For the discrete-time
state-space representation of this model, thek
th
time step can be expressed as:
X((k + 1)T ) = AX(kT ) + BU(kT ) +v(kT )
Y (kT ) = CX(kT ) + DU(kT ) +w(kT ) (IV .31)
whereT is the sampling interval. The relationships between the discrete-time state-space matrices
A,B,C, andD and the continuous-time state-space matricesA
c
,B
c
,C
c
, andD
c
are given for
piece-wise-constant input as follows:
A =e
AcT
B =
Z
T
0
e
Ac
B
c
d C = C
c
D = D
c
(IV .32)
To form the state-space representation of the MDOF dynamic system governed by Equation IV .2,
state, output, and input vectors are defined as:
X(t) =
2
6
4
x
1
(t)
_ x
1
(t)
3
7
5 (IV .33)
Y (t) = x
1
(t) (IV .34)
U(t) = f
1
(t) (IV .35)
These definitions will result in the following state-space representation
A
c
=
2
6
4
0 I
M
1
K M
1
C
3
7
5 B
c
=
2
6
4
0
M
1
3
7
5 C
c
=

I 0

D
c
=

0

(IV .36)
76
This state-space representation is not unique in the sense that we get an equivalent equation by
introducing a non-singular transformation matrixT as follows
T
_
X(t) = TA
c
T
1
X(t) + TB
c
U(t) +v(t)
Y (t) = C
c
T
1
X(t) + D
c
U(t) +w(t) (IV .37)
Equations IV .30 and IV .37 are said to be equivalent because they describe the same system and
have the same transform function. On the other hand, care must be taken because two state-space
equations might have the same transfer function without being equivalent.
Inversely, in order to extract the mass, stiffness, and damping matrices of the structure from its
state-space representation, it is necessary to have the identified state-space matrices in the form
of Equation IV .36. Since the identification process usually leads to an estimate of the state-space
matrices up to a similarity transformation T (A
T
= TA
c
T
1
; B
T
= TB
c
; C
T
= C
c
T
1
), the
abstraction of M, C, and K is not straightforward as from [A
c
; B
c
; C
c
]. It can be shown that
there exists a unique similarity transformation P that transforms [A
T
; B
T
; C
T
] to the form of
[A
c
; B
c
; C
c
] in Equation IV .36. A similarity transformationP that satisfies
A
c
= PA
T
P
1
B
c
= PB
T
(IV .38)
C
c
= C
T
P
1
77
is obtained as follows:
P =
2
6
4
C
T
C
T
A
T
3
7
5 (IV .39)
The mass, damping, and stiffness matrices can then be determined as (Jezequel (1997)):
M = (C
T
A
T
B
T
)
1
[K C] = MC
T
A
2
T
2
6
4
C
T
C
T
A
T
3
7
5
1
(IV .40)
The n4sid module in the System Identification Toolbox of MATLAB
R
was employed for this
study.
IV .2.6 Iterative Prediction-Error Minimization Method
The iterative Prediction-Error Minimization algorithm (the pem module in the System Identifi-
cation Toolbox of MATLAB
R
) was used to estimate model parameters of the structure. For the
linear model defined in Equation IV .30, the general symbolic transfer function description is given
by:
Y (t) = GU(t) + He(t) (IV .41)
where G is a transfer function that takes the input U to the outputY . H is a transfer function that
describes the properties of the additive output noise model. PEM uses optimization to minimize
the cost function, defined as follows, to find G and H:
V
N
(G; H) =
t
N
X
t=t
1
e(t)
2
(IV .42)
78
wheree(t) is the difference between the measured output and the predicted output of the model.
For a linear model, this error is defined by the following equation:
e(t) = H
1
(q) [Y (t) G(q)U(t)] (IV .43)
whereq is the lag operator. The subscriptN indicates that the cost is a function of the number of
data samples and becomes more accurate for larger values of N. As with any nonlinear optimiza-
tion algorithm, there is a chance that the model might find a local minimum that is not accurate
for a specific system. When matrices G and H are estimated, one of many available methods can
then be used to derive a minimal state-space realization of the system from the transfer function
matrix.
IV .3 Experimental Case Study: Hunan University Building Model
Structure
IV .3.1 Description of Test Building Model
The case study is a 4-story frame structure which was investigated experimentally at Hunan Uni-
versity, China. The steel-frame building model is 40cm 30cm in plan and 120cm in height
with a total mass of' 52:4kg. All the junctions between the floors (each made of a 10mm thick
steel plate) and the column elements (each made of a 120cm long steel bar with a cross section
of 30mm 5mm) are connected by bolts. In the second phase of the experiment, the structure
is also equipped with a magneto-rheological (MR) damper between the third and the fourth floor
to simulate nonlinear behavior in the system, and the actual nonlinear hysteretic restoring force
is measured by a force transducer. The building model was equipped with one accelerometer at
79
(a) Building without MR damper (b) Building with MR damper
Figure IV .2: Experimental case study building model before and after the installation of the MR damper.
400mm
30mm
300mm
5mm
P 400X300X10 L
P 5X30X1200 L
300mm
300mm
300mm
300mm 300mm
300mm
300mm
300mm
400mm 300mm
X
Y
(a) Top view
30mm
300mm
5mm
P 5X30X1200 L
300mm
300mm
300mm
300mm
400mm 300mm
MR Damper
(b) Elevation view
Figure IV .3: (a) Top view and (b) Elevation view of the tested building structure
80
each floor to record the data at the frequency rate of 1000 Hz throughout the experiment. Figure
IV .2 shows photos of the building model before and after the installation of the MR damper on
the structure. The top view and elevation view of the structure are also illustrated in Figure IV .3.
IV .3.2 Test Cases
For the purpose of nonlinear behavior identification, the experiment was conducted in two stages:
(1) before and (2) after adding the MR damper. The behavior of the structure is assumed to be
linear before it is equipped with the MR damper. In the second stage, four different nonlinearity
scenarios based on the electric current input to the MR damper (0.00A, 0.05A, 0.10A and 0.15A)
were considered. In each of these five cases, the structure was excited at each floor by means
of an impact hammer, and the corresponding accelerations were measured simultaneously by the
accelerometers mounted on each floor, while the velocity and the displacements of the structure
were obtained from the integration of the measured accelerations. Figure IV .4 to Figure IV .8
show the applied force on the structure and the corresponding measurements for Case 0 to Case
4, respectively. The description of these tests are also tabulated in Table IV .1 (Xu et al. (2010)).
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
4
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
3
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
2
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
1
(N)
Time (sec)
(a) Force
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
4
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
3
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
2
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
1
(g)
Time (sec)
(b) Acceleration
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
4
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
3
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
2
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
1
(cm/s)
Time (sec)
(c) Velocity
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
4
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
3
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
2
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
1
(cm)
Time (sec)
(d) Displacement
Figure IV .4: (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and (d)
Displacement at each floor for Case 0 (No MR damper)
81
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
4
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
3
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
2
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
1
(N)
Time (sec)
(a) Force
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
4
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
3
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
2
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
1
(g)
Time (sec)
(b) Acceleration
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
4
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
3
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
2
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
1
(cm/s)
Time (sec)
(c) Velocity
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
4
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
3
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
2
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
1
(cm)
Time (sec)
(d) Displacement
Figure IV .5: (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and (d)
Displacement at each floor for Case 1 (MR damper with 0.00A input current)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
4
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
3
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
2
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
1
(N)
Time (sec)
(a) Force
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
4
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
3
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
2
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
1
(g)
Time (sec)
(b) Acceleration
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
4
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
3
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
2
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
1
(cm/s)
Time (sec)
(c) Velocity
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
4
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
3
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
2
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
1
(cm)
Time (sec)
(d) Displacement
Figure IV .6: (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and (d)
Displacement at each floor for Case 2 (MR damper with 0.05A input current)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
4
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
3
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
2
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
1
(N)
Time (sec)
(a) Force
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
4
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
3
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
2
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
1
(g)
Time (sec)
(b) Acceleration
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
4
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
3
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
2
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
1
(cm/s)
Time (sec)
(c) Velocity
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
4
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
3
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
2
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
1
(cm)
Time (sec)
(d) Displacement
Figure IV .7: (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and (d)
Displacement at each floor for Case 3 (MR damper with 0.10A input current)
82
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
4
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
3
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
2
(N)
0 1 2 3 4 5 6 7 8
0
500
1000
1500
F
1
(N)
Time (sec)
(a) Force
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
4
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
3
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
2
(g)
0 1 2 3 4 5 6 7 8
−10
0
10
Acc
1
(g)
Time (sec)
(b) Acceleration
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
4
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
3
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
2
(cm/s)
0 1 2 3 4 5 6 7 8
−50
0
50
Vel
1
(cm/s)
Time (sec)
(c) Velocity
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
4
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
3
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
2
(cm)
0 1 2 3 4 5 6 7 8
−0.5
0
0.5
Dis
1
(cm)
Time (sec)
(d) Displacement
Figure IV .8: (a) Applied forces and the corresponding recorded (b) Acceleration, (c) Velocity and (d)
Displacement at each floor for Case 4 (MR damper with 0.15A input current)
Table IV .1: Description of the conducted experiments
Case # Input current Impact forces Sequence of excitations
Case 0 No MR damper Single impact force on each floor 1-2-3-4
Case 1 0.00A Two impact forces on each floor 4-3-2-1
Case 2 0.05A Single impact force on the 4th floor and two impact forces on others 4-3-2-1
Case 3 0.10A Two impact forces on each floor 4-3-2-1
Case 4 0.15A Single impact force on the 4th floor and two impact forces on others 4-3-2-1
IV .4 Results
No information about the structure was assumed for the identification of the system, and only the
applied excitations and corresponding system response measurements were used to implement
each of the methods under discussion. Therefore the mass, damping, and stiffness coefficients
for each test case were identified independently with no priori knowledge about the system. The
building model was equipped with one accelerometer at each floor to record the response of the
system. From a practical point of view, real structures outside of the lab environment might not en-
joy such a comprehensive level of instrumentation on all key degrees-of-freedom. Consequently,
it was decided for the purposes of this study, to also investigate a situation under the assumption
that only a limited number of sensors is available. The following two scenarios were considered:
83
(i). Full Instrumentation Recordings: Time-history records obtained from all four accelerom-
eters are available.
(ii). Partial Instrumentation Recordings (Model Order Reduction): Only time-history records
obtained from the accelerometers on the top two floors are available. Considering the se-
quence and timing of the applied impact forces on the structure, this scenario is only con-
templated for cases 1 to 4, using the first 4.0 seconds of the recorded data to avoid the
effects of missing excitations on the system.
The recorded data was processed and high pass filtered with the lowest cut-off frequency of 1 Hz.
IV .4.1 Identified Mass, Damping and Stiffness Matrices
Full Instrumentation Recordings
Tables IV .2 and IV .3 present the obtained mass, damping and stiffness matrices from different
system identification methods for case 0 and 4, respectively. Comprehensive results for all case
studies are tabulated and presented in Appendix A. As expected, other than elements correspond-
ing to the MR damper in the damping matrix (bolded values), no significant change is detected in
other structural dynamic matrices throughout the experiment. The identified values in the damp-
ing matrix corresponding to the location of MR damper (between the third and fourth floors)
monotonically increase as the input current to the MR damper is intensified. Furthermore, there
is a reasonable agreement among the identified matrices obtained from different methods.
84
Table IV .2: Identified mass, damping, and stiffness matrices of the system: Case 0
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
11:10 0:74 0:17 0:71
0:63 11:99 0:27 0:29
0:63 0:49 12:83 0:58
0:88 0:54 0:55 11:67
3
7
7
5
2
6
6
4
42:74 2:85 16:22 4:11
8:20 29:61 9:55 7:29
15:17 7:47 48.23 5:33
0:96 3:69 8:70 12.05
3
7
7
5
2
6
6
4
2:60 1:44 0:02 0:14
1:63 3:27 1:80 0:11
0:06 1:72 3:08 1:45
0:15 0:07 1:58 1:43
3
7
7
5
Sym. Linear System Method
2
6
6
4
11:10 0:74 0:17 0:71
0:74 11:06 0:38 0:30
0:17 0:38 12:13 0:53
0:71 0:30 0:53 10:44
3
7
7
5
2
6
6
4
42:74 2:85 16:22 4:11
2:85 29:60 10:26 6:55
16:22 10:26 48.45 5:80
4:11 6:55 5:80 9.61
3
7
7
5
2
6
6
4
2:60 1:44 0:02 0:14
1:44 2:96 1:64 0:09
0:02 1:64 2:97 1:39
0:14 0:09 1:39 1:25
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
11:34 0 0 0
0 11:15 0 0
0 0 13:06 0
0 0 0 12:55
3
7
7
5
2
6
6
4
39:60 20:83 0 0
20:83 47:78 26:95 0
0 26:95 38.05 11:10
0 0 11:10 11.10
3
7
7
5
2
6
6
4
2:53 1:27 0 0
1:27 2:79 1:52 0
0 1:52 3:07 1:55
0 0 1:55 1:55
3
7
7
5
Model Updating Method
2
6
6
4
11:12 0:72 0:21 0:74
0:60 11:94 0:28 0:30
0:65 0:52 12:83 0:59
0:85 0:51 0:54 11:65
3
7
7
5
2
6
6
4
44:00 2:63 16:23 3:96
7:94 29:87 10:59 7:57
15:03 6:55 47.25 4:83
2:66 2:71 7:56 11.69
3
7
7
5
2
6
6
4
2:60 1:44 0:01 0:14
1:63 3:26 1:80 0:11
0:05 1:71 3:08 1:45
0:15 0:07 1:58 1:43
3
7
7
5
Sub-Space Method
2
6
6
4
14:93 0:13 0:56 0:28
0:01 14:10 0:02 0:17
0:29 0:03 13:81 0:04
0:15 0:10 0:02 12:56
3
7
7
5
2
6
6
4
49:99 2:91 21:66 4:20
10:07 29:36 10:09 7:75
16:39 7:61 50.46 7:16
2:36 3:02 5:95 12.07
3
7
7
5
2
6
6
4
3:60 2:14 0:25 0:06
2:10 3:98 2:13 0:09
0:23 2:05 3:49 1:64
0:01 0:14 1:77 1:60
3
7
7
5
Iterative PEM Method
2
6
6
4
14:50 0:10 0:50 0:18
0:05 14:09 0:18 0:02
0:25 0:10 14:15 0:14
0:11 0:04 0:09 12:78
3
7
7
5
2
6
6
4
46:97 6:92 26:79 1:39
16:36 24:06 5:17 6:10
0:41 8:17 34.47 11:28
5:53 0:99 0:64 6.06
3
7
7
5
2
6
6
4
3:47 2:08 0:26 0:05
2:06 3:92 2:13 0:12
0:24 2:07 3:56 1:69
0:03 0:13 1:78 1:61
3
7
7
5
85
Table IV .3: Identified mass, damping, and stiffness matrices of the system: Case 4
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
13:72 0:53 0:14 0:37
0:55 13:00 0:67 0:12
0:43 1:06 13:17 2:27
0:27 0:16 1:43 10:84
3
7
7
5
2
6
6
4
54:46 9:98 36:75 12:84
32:06 52:49 15:33 46:34
25:80 84:01 326.03 295:32
19:19 55:89 295:02 323.97
3
7
7
5
2
6
6
4
3:17 1:85 0:16 0:08
1:75 3:39 1:70 0:01
0:10 1:47 2:62 1:22
0:03 0:14 1:01 1:12
3
7
7
5
Sym. Linear System Method
2
6
6
4
13:72 0:53 0:14 0:37
0:53 13:26 0:67 0:14
0:14 0:67 13:68 2:53
0:37 0:14 2:53 11:93
3
7
7
5
2
6
6
4
54:46 9:98 36:75 12:84
9:98 47:69 1:55 38:42
36:75 1:55 337.10 342:98
12:84 38:42 342:98 424.73
3
7
7
5
2
6
6
4
3:17 1:85 0:16 0:08
1:85 3:50 1:74 0:02
0:16 1:74 2:93 1:33
0:08 0:02 1:33 1:33
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
13:60 0 0 0
0 12:74 0 0
0 0 14:68 0
0 0 0 13:67
3
7
7
5
2
6
6
4
119:17 59:31 0 0
59:31 80:42 21:11 0
0 21:11 397.79 376:68
0 0 376:68 376.68
3
7
7
5
2
6
6
4
2:73 1:32 0 0
1:32 2:76 1:44 0
0 1:44 2:95 1:51
0 0 1:51 1:51
3
7
7
5
Model Updating Method
2
6
6
4
13:73 0:53 0:13 0:34
0:56 12:99 0:66 0:13
0:43 1:07 13:17 2:28
0:27 0:16 1:44 10:84
3
7
7
5
2
6
6
4
54:92 9:71 34:79 10:80
31:46 51:52 15:28 46:07
25:58 83:75 327.39 295:90
18:87 54:36 293:76 323.73
3
7
7
5
2
6
6
4
3:18 1:85 0:17 0:07
1:75 3:38 1:70 0:01
0:10 1:47 2:62 1:22
0:03 0:14 1:01 1:12
3
7
7
5
Sub-Space Method
2
6
6
4
14:86 0:13 0:27 0:42
0:03 14:29 0:11 0:03
0:06 0:34 14:49 0:17
0:19 0:16 0:43 14:30
3
7
7
5
2
6
6
4
87:93 24:87 86:41 49:44
27:47 45:98 22:36 42:95
0:65 80:53 453.49 429:76
22:58 85:22 394:62 431.57
3
7
7
5
2
6
6
4
3:50 2:11 0:34 0:02
2:08 3:89 2:04 0:07
0:37 1:98 3:40 1:68
0:12 0:08 1:69 1:64
3
7
7
5
Iterative PEM Method
2
6
6
4
14:72 0:29 0:19 0:10
0:13 14:30 0:34 0:00
0:25 0:82 14:96 0:02
0:32 0:11 1:08 14:06
3
7
7
5
2
6
6
4
44:01 7:69 23:05 1:12
15:72 41:06 18:23 11:80
2:17 70:85 395.98 386:26
8:31 50:06 375:62 419.98
3
7
7
5
2
6
6
4
3:45 2:05 0:23 0:06
2:05 3:84 1:98 0:04
0:36 1:94 3:38 1:68
0:15 0:08 1:52 1:51
3
7
7
5
86
Partial Instrumentation Recordings (Model Order Reduction)
For this scenario, it is assumed that only time-history records obtained from the accelerometers
on the top two floors are available. Tables IV .4 and IV .5 present the obtained mass, damping
and stiffness matrices using partial instrumentation recordings for case 1 and 4, respectively.
Comprehensive results for all case studies are tabulated and presented in Appendix A. Similar to
the results from full instrumentation data, other than elements corresponding to the MR damper in
the damping matrix (bolded values), no significant change is detected in other structural dynamic
matrices throughout the experiment. For all methods, the identified values in the damping matrix
corresponding to the location of MR damper (diagonal terms) monotonically increase as the input
current to the MR damper is intensified which is a sign of change in this location. However,
there is much less agreement among the identified matrices from different system identification
methods probably due to short length of recordings.
IV .4.2 Identified Classical Frequencies and Damping Values
To extract the classical frequencies, damping values, and mode-shapes of the structure, the state-
space representation of the system should be constructed using relationships given in Equation
IV .36. The modal properties of the system can then be derived using the Equations III.4 to III.8.
Figure IV .9 schematically illustrates the four identified frequencies and corresponding mode-
shapes of the tested building structure in the excitation direction for case 0.
Tables IV .6 and IV .7 present four extracted classical frequencies and corresponding damping val-
ues of the structure using full instrumentation recordings for cases 0 and 4, respectively. Compre-
hensive results for all case studies are tabulated and presented in Appendix A. As shown in these
87
Table IV .4: Identified mass, damping, and stiffness matrices of the system, using partial instrumentation
recordings: Case 1
Methodn Matrix M C K
(10
5
)
Linear System Method

9:00 1:00
0:85 12:83

243.18 199:32
191:80 236.73

1:10 0:84
1:45 1:40

Sym. Linear System Method

9:00 1:00
1:00 10:77

243.18 199:32
199:32 222.34

1:10 0:84
0:84 0:86

RFS Method for Chain-Like Systems

14:93 0:00
0:00 13:65

296.55 227:11
227:11 227.11

1:64 1:29
1:29 1:29

Model Updating Method

9:02 0:96
0:84 12:88

250.66 194:03
192:14 233.61

1:11 0:85
1:46 1:41

Sub-Space Method

14:77 3:04
0:34 11:33

276.80 274:35
177:99 213.74

2:36 1:91
1:35 1:29

Iterative PEM Method

17:70 0:10
0:26 14:68

322.95 290:92
221:23 269.75

2:38 1:86
1:77 1:69

Table IV .5: Identified mass, damping, and stiffness matrices of the system, using partial instrumentation
recordings: Case 4
Methodn Matrix M C K
(10
5
)
Linear System Method

10:56 1:72
1:10 11:72

356.80 346:07
331:42 393.60

1:21 0:93
1:19 1:19

Sym. Linear System Method

10:56 1:72
1:72 11:34

356.80 346:07
346:07 389.24

1:21 0:93
0:93 0:97

RFS Method for Chain-Like Systems

15:04 0:00
0:00 13:59

454.52 404:18
404:18 404.18

1:73 1:37
1:37 1:37

Model Updating Method

10:57 1:70
1:09 11:76

356.71 345:17
331:40 393.89

1:22 0:94
1:20 1:20

Sub-Space Method

15:98 0:02
0:48 12:01

365.85 405:81
340:41 411.16

2:15 1:73
1:45 1:41

Iterative PEM Method

17:11 0:27
0:21 15:05

449.51 481:20
405:32 495.60

2:29 1:83
1:72 1:68

88
400mm
30mm
300mm
5mm
P 400X300X10 L
P 5X30X1200 L
300mm
300mm
300mm
300mm 300mm
300mm
300mm
300mm
400mm 300mm
X
Y
(a) f1' 5:6Hz
400mm
30mm
300mm
5mm
P 400X300X10 L
P 5X30X1200 L
300mm
300mm
300mm
300mm 300mm
300mm
300mm
300mm
400mm 300mm
X
Y
(b) f2' 16:6Hz

(c) f3' 26:5Hz

(d) f4' 35:8Hz
Figure IV .9: Four identified frequencies and corresponding mode-shapes of the tested building structure
in the excitation direction for Case 0.
tables, there is a reasonable agreement among the obtained values from different system identi-
fication methods. It is also noticeable that the nonlinear behavior of the MR damper is mainly
manifested in the damping value of the third dominant mode of the system. As shown in Figure
IV .9, the top two floors (location of the MR damper) have the highest participation in the third
mode of the system.
Table IV .8 to IV .9 show two identified classical frequencies and corresponding damping values
of the structure using partial instrumentation recordings for case 1 to 4, respectively. Note that
the two identified modes using partial instrumentation recordings correspond to the first and the
third dominant modes of the structure. This is due to the fact the third floor of the structure is
minimally excited by the second dominant mode of the system (see Figure IV .9). This reveals one
of main potential disadvantages of model order reduction in system identification applications.
Again, the increasing damping value of the second identified mode (third mode of the structure)
shows a significant change in the damping properties of the system.
89
Table IV .6: Identified frequencies and damping ratios: Case 0
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.62 1.623 16.70 1.324 26.56 1.165 35.89 0.765
Sym. Linear System Method 5.72 2.024 16.51 1.479 26.81 1.083 35.73 0.759
RFS Method for Chain-Like Systems 5.70 0.279 17.66 0.740 26.18 0.998 32.82 1.564
Model Updating Method 5.62 1.604 16.69 1.369 26.57 1.141 35.90 0.771
Sub-Space Method 5.65 1.768 16.65 1.114 26.54 1.065 35.83 0.629
Iterative PEM Method 5.64 1.497 16.58 0.781 26.38 0.794 35.75 0.560
Table IV .7: Identified frequencies and damping ratios: Case 4
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.55 1.974 16.68 5.003 25.60 11.894 34.64 5.119
Sym. Linear System Method 5.50 1.750 16.50 6.248 26.87 13.504 35.18 4.881
RFS Method for Chain-Like Systems 5.54 0.829 16.76 4.897 24.79 11.194 30.09 6.340
Model Updating Method 5.46 2.045 16.68 5.017 25.61 11.895 34.64 5.111
Sub-Space Method 5.42 2.193 16.70 4.296 25.95 13.149 34.59 4.262
Iterative PEM Method 5.45 1.829 16.51 4.776 25.66 11.010 34.63 4.097
Table IV .8: Identified frequencies and damping ratios (partial instrumentation): Case 1
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%)
Linear System Method 5.44 4.751 24.68 14.740
Sym. Linear System Method 5.50 4.489 23.05 16.992
RFS Method for Chain-Like Systems 5.39 3.337 22.10 12.327
Model Updating Method 5.42 5.534 24.70 14.651
Sub-Space Method 5.41 3.206 24.65 10.505
Iterative PEM Method 5.44 3.668 24.68 11.122
90
Table IV .9: Identified frequencies and damping ratios (partial instrumentation): Case 4
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%)
Linear System Method 5.45 3.888 24.30 24.089
Sym. Linear System Method 5.51 3.630 23.62 26.115
RFS Method for Chain-Like Systems 5.47 2.572 22.73 20.373
Model Updating Method 5.45 3.965 24.36 23.921
Sub-Space Method 5.41 3.346 24.43 17.548
Iterative PEM Method 5.39 3.649 24.50 18.695
IV .4.3 Restoring Forces in Chain-Like System
To evaluate the dependence of the restoring forces on basis functions, the parameters of Equa-
tion IV .29 is evaluated by means of the method developed by Masri and Caughey (1979). The
approach uses information about the state variables of non-linear systems to express the system
characteristics in terms of two-dimensional Chebyshev polynomials as follows:
G(z
0
;
_
z
0
) =
m
X
i=0
n
X
j=0
C
ij
T
i
(z
0
)T
j
(
_
z
0
) (IV .44)
in which T
i
represents the i
th
Chebyshev polynomial of the first kind and z
0
and
_
z
0
are the
displacement and velocity vectors, normalized through the following equation:
z
0
=
z
zmax+z
min
2
zmaxz
min
2
(IV .45)
The orthogonal nature of the Chebyshev polynomials and their equal ripple characteristics make
them convenient to use in least-squares approximations.
91
The fidelity of the method is first evaluated using the synthetic data obtained from two analyt-
ical models. The first case is a linear 4-DOF system while a nonlinear damper is added to the
second model between the 3
rd
and 4
th
DOFs. The nonlinearity is assumed to be proportional to
the odd powers of velocity and displacement and their products as follows:
M x(t) + C _ x(t) + Kx(t)
+f
NL

x(t); x
3
(t); _ x(t); _ x
3
(t); x(t) _ x(t); x
3
(t) _ x(t); x(t) _ x
3
(t); x
3
(t) _ x
3
(t)

= f(t) (IV .46)
Tables IV .10 and IV .11 present the corresponding parameters of the Chebyshev polynomials in the
approximation of the inter-story restoring forces for the linear and nonlinear cases, respectively.
Comparison of the two tables shows that the values in the last row of Table IV .11 identify the
damping (increase in the value corresponding to _ z) and its nonlinearity (non-zero values for odd
powers of _ z andz and their products) between the 3
rd
and 4
th
DOF of the system.
For the experimental data, Table IV .12 to IV .16 illustrate the corresponding parameters of Cheby-
shev polynomials of each basis function in the inter-story restoring forces for case 0 to 4, respec-
tively. As shown in these tables, the inter-story forces are primarily determined by the relative
displacement in each floor with much less dependence upon other basis functions. The high values
representating the relative displacement for the restoring forces in the first three floors show no
significant change throughout the experiment while the contribution of the corresponding param-
eter for the restoring force in the 4
th
floor gradually decreases during the experiment. Meanwhile,
the contribution of the relative velocity of the 4
th
floor (where nonlinear damper is employed)
increases as the current in the MR damper is intensified. The nonlinearity in the system, imposed
by MR damper, is also identified by gradual escalation in contribution ofz
3
and _ z
3
. Note that
92
Table IV .10: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Analytical Linear System
G
i
z _ z z
2
_ z
2
z
3
_ z
3
z_ z z
2
_ z z_ z
2
z
2
_ z
2
z
3
_ z z_ z
3
z
3
_ z
2
z
2
_ z
3
z
3
_ z
3
1
G
1
425.82 0.00 -0.23 -0.08 -4.44 -0.01 -0.51 -0.01 -4.56 0.12 0.62 0.38 4.50 0.04 -0.52 -0.03
G
2
400.50 -0.30 0.31 0.13 -4.03 0.20 -3.65 0.52 -5.66 -0.22 3.48 7.79 6.03 -0.27 -8.18 0.39
G
3
319.88 0.03 0.02 -0.01 1.29 0.02 4.55 -0.01 -2.15 -0.03 -4.39 -1.70 2.81 0.00 1.50 -0.08
G
4
292.20 0:06 -0.26 -0.06 5:49 0:03 1:90 0.14 0.00 0.11 2:21 2:30 0.94 -0.06 2:55 -0.09
Table IV .11: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Analytical Nonlinear System
G
i
z _ z z
2
_ z
2
z
3
_ z
3
z_ z z
2
_ z z_ z
2
z
2
_ z
2
z
3
_ z z_ z
3
z
3
_ z
2
z
2
_ z
3
z
3
_ z
3
1
G
1
329.43 -0.03 -0.01 -0.03 -7.00 0.12 0.92 0.05 2.32 -0.02 -0.65 -1.08 -4.13 -0.05 0.82 -0.02
G
2
295.83 -0.02 -0.04 0.00 -4.90 0.05 -0.03 -0.09 -2.61 -0.02 0.10 0.10 3.46 0.12 -0.19 -0.05
G
3
267.80 0.09 0.24 0.25 -1.60 -0.05 -0.07 -0.22 -2.42 -0.46 0.07 0.08 2.91 0.30 -0.08 -0.12
G
4
182.48 155:83 -0.47 0.03 68:75 10:42 25:29 13.01 -1.55 0.25 12:48 30:31 -2.33 -34.11 14:14 0.03
93
Table IV .12: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Case 0
G
i
z
0 _
z
0
z
02 _
z
0
2
z
03 _
z
0
3
z
0 _
z
0
z
02 _
z
0
z
0 _
z
0
2
z
02 _
z
0
2
z
03 _
z
0
z
0 _
z
0
3
z
03 _
z
0
2
z
02 _
z
0
3
z
03 _
z
0
3
1
G
1
225.25 1.13 -4.07 -1.37 6.88 -1.32 -20.57 -2.32 10.49 3.02 21.01 17.84 -12.19 2.78 -18.83 1.82
G
2
249.91 0.06 -0.36 -0.11 -5.49 0.08 -2.94 0.03 -3.56 0.21 2.99 3.24 3.13 -0.05 -3.31 0.15
G
3
275.74 -0.02 0.54 0.06 11.41 0.23 -0.27 -0.03 -6.82 -0.25 1.40 2.14 1.39 0.13 -2.75 -0.28
G
4
290.05 0:04 -1.90 -0.37 21.36 0:01 0.40 -0.08 6.72 0.74 -0.20 -0.39 -10.24 0.07 0.21 0.99
Table IV .13: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Case 1
G
i
z
0 _
z
0
z
02 _
z
0
2
z
03 _
z
0
3
z
0 _
z
0
z
02 _
z
0
z
0 _
z
0
2
z
02 _
z
0
2
z
03 _
z
0
z
0 _
z
0
3
z
03 _
z
0
2
z
02 _
z
0
3
z
03 _
z
0
3
1
G
1
304.22 2.29 1.03 1.04 -13.76 -1.37 53.57 -3.18 -38.83 -0.75 -50.03 -22.86 27.14 1.77 26.53 -0.88
G
2
299.25 1.05 -0.03 0.56 -7.22 -0.43 0.93 0.13 -22.37 -0.89 0.47 -0.01 23.29 -0.63 -1.10 -0.07
G
3
296.72 0.23 -8.55 -0.30 52.20 -1.17 7.14 1.83 -14.07 -0.13 -7.01 -3.09 10.50 0.10 4.07 4.34
G
4
150.17 24:93 -14.36 -0.55 40.56 19:85 -11.19 -4.76 59.59 0.49 9.50 12.00 -59.86 6.38 -10.29 7.25
Table IV .14: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Case 2
G
i
z
0 _
z
0
z
02 _
z
0
2
z
03 _
z
0
3
z
0 _
z
0
z
02 _
z
0
z
0 _
z
0
2
z
02 _
z
0
2
z
03 _
z
0
z
0 _
z
0
3
z
03 _
z
0
2
z
02 _
z
0
3
z
03 _
z
0
3
1
G
1
272.46 0.29 0.18 -0.38 -20.96 0.30 9.62 -1.06 -14.00 -0.18 -2.03 6.89 11.58 0.03 -9.96 0.14
G
2
251.78 1.14 -0.01 0.24 -17.67 -0.50 -0.10 -0.45 2.70 -0.52 0.13 -0.04 3.97 0.34 0.02 0.01
G
3
224.09 0.07 -3.59 -0.21 13.26 -0.16 -0.43 0.23 -33.56 0.56 2.02 3.56 41.05 0.06 -5.05 1.76
G
4
173.39 35:86 -4.24 -0.14 73.81 23:52 1.13 -5.30 33.61 0.11 -0.99 -1.88 -32.68 5.10 1.67 2.16
Table IV .15: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Case 3
G
i
z
0 _
z
0
z
02 _
z
0
2
z
03 _
z
0
3
z
0 _
z
0
z
02 _
z
0
z
0 _
z
0
2
z
02 _
z
0
2
z
03 _
z
0
z
0 _
z
0
3
z
03 _
z
0
2
z
02 _
z
0
3
z
03 _
z
0
3
1
G
1
344.73 2.62 1.12 0.71 -25.75 -2.71 30.00 -3.81 -27.79 -2.22 -26.55 -8.60 19.15 3.91 12.71 -0.37
G
2
300.94 0.83 -0.35 0.03 -28.69 -0.16 2.54 -0.16 -7.56 -0.27 -2.22 -2.44 9.29 -0.25 2.02 0.22
G
3
305.91 0.16 -8.12 0.45 4.51 -0.42 3.94 0.45 -30.79 -0.38 -4.57 -3.05 38.61 0.38 3.67 3.91
G
4
218.00 51:92 -7.02 -0.32 45.52 33:11 -2.15 -4.32 13.82 -1.18 2.62 -0.14 -7.87 1.56 -0.56 3.95
Table IV .16: Corresponding parameters of basis functions in restoring forces using Chebyshev polynomials: Case 4
G
i
z
0 _
z
0
z
02 _
z
0
2
z
03 _
z
0
3
z
0 _
z
0
z
02 _
z
0
z
0 _
z
0
2
z
02 _
z
0
2
z
03 _
z
0
z
0 _
z
0
3
z
03 _
z
0
2
z
02 _
z
0
3
z
03 _
z
0
3
1
G
1
269.45 -0.06 -1.49 -1.36 -11.00 0.67 6.04 0.09 -3.19 1.30 -3.20 2.97 -2.58 -1.20 -1.45 1.11
G
2
255.08 0.50 -0.27 -0.07 -29.11 -0.03 0.13 0.20 -10.21 -0.60 -0.07 0.06 15.59 -0.39 -0.09 0.33
G
3
270.07 0.28 -15.19 -0.32 15.14 -0.72 -1.70 0.86 -41.95 0.83 2.11 3.33 50.41 0.04 -3.88 7.56
G
4
176.71 62:44 -9.73 -1.72 56.70 39:62 3.40 -14.65 10.00 3.32 -0.18 -6.36 -7.17 10.38 3.17 4.91
94
the negative sign of parameters corresponding to _ z
3
shows nonlinear softening with respect to the
velocity whereas the positive sign of parameters for toz
3
is a sign of nonlinear hardening with
respect to the displacement.
IV .5 Concluding Remarks
Based on evaluation of the results from six different system identification methods using experi-
mentally recorded data from a 4-story model building, the following conclusions can be made:
(i). All six methods could successfully identify, locate and quantify the increasing damping
imposed to the system by MR damper throughout the experiment when using full instru-
mentation recordings. The identified values in the damping matrix corresponding to the
location of the MR damper (between the third and fourth floors) monotonically increase as
the input current to the MR damper is intensified.
(ii). Identified system matrices from different methods were in acceptable agreement with each
other when using full instrumentation recordings. However, identified mass values by Sub-
space and Iterative PEM methods are generally higher (8% 13%) than real measured
values in the lab. For other four system identification methods, the identified mass values
are very close to measured mass in all case studies except for case 0, where these values are
less (14% 16%) than real values. The identified of the stiffness of floors by RFS method
is generally less (15% 20%) than corresponding identified values by other five methods.
(iii). Identified system matrices using partial instrumentation recordings were in less agreement
with each other due to the short length of recordings; however, all identification methods
could successfully detect the change in the damping properties of the system.
95
(iv). In addition, the RFS method could identify the nonlinearity of the damping in the system
and its behavior (softening or hardening) with respect to different vibrational signatures of
the structure.
96
Chapter V
IDENTIFICATION OF NONLINEAR STRUCTURAL
MODELS USING ARTIFICIAL NEURAL NETWORKS
V .1 Introduction
T
HIS chapter explores the potential of using artificial neural networks for the detection of
changes in the characteristics of the structure-unknown non-linear dynamic systems.
V .1.1 Background
An Artificial Neural Network (ANN) or simply Neural Network (NN) is a mathematical or com-
putational model of an interconnected group of artificial neurons, that mimics the behavior of
biological nervous systems to model complex relationships between inputs and outputs or find
patterns in data. A neural network processes information using a connectionist approach to com-
putation and adapts its structure based on external or internal information that flows through the
network. Neural networks have been extensively employed to solve problems in numerous fields
of engineering and science and have opened up new possibilities in the various domains such as
signal processing, control systems, robotics, pattern recognition, speech recognition, medicine,
97
business and financial analysis, and gaming.
Of particular relevance to the field of SHM, there has been increasing interest in recent decades
toward using neural networks for the identification of mathematical models of physical struc-
tures on the basis of experimental measurements. Non-parametric identification methods can be
used when the model structure is not clearly known. These methods try to provide the parame-
ters of a mathematical model which fits the input/output data rather than to identify the physical
parameters of the system. Some inherent properties of artificial neural network models such as
computational-efficiency, fault-tolerance, and adaptation make them a superior tool for this pur-
pose in comparison to traditional computational models. Furthermore, a neural network with
proper architecture can can treat both linear and nonlinear systems with the same formulation.
This is a plausible property when dealing with civil engineering structure where nonlinearity is
usually present.
These potentials have motivated many researchers to study neural network based approaches for
signature analysis of the system response in system identification and damage detection fields.
Kudva et al. (1992) applied a BP neural network to detect and localize large area damage in an
analytical plate model. Wu et al. (1992) trained a Back-Propagation (BP) Neural Network to de-
tect individual member damage (loss of stiffness) from the measured response of a three-story
building modeled by a two-dimensional ”shear building” driven by earthquake excitation. Masri
et al. (1993) developed a procedure based on the use of ANN for the identification of nonlinear
dynamic systems and applied it to the damped Duffing oscillator under deterministic excitation.
Worden et al. (1993) trained a three-hidden-layer BP neural network to identify damage in a
twenty-member framework structure using simulated data and later tested it on an experimental
98
model of the same geometry. Elkordy et al. (1993) proposed a structural damage diagnostic sys-
tem based on a BP Neural Networks using both experimental and analytical data. They concluded
that NN could diagnose complicated damage patterns and handle noisy and partially incomplete
data sets. Szewczyk and Hajela (1994) formulated detection of damage in analytical models of
frame and truss structures as an inverse problem and utilized a modified Counter-Propagation
(CP) Neural Networks to solve it. Stephens and VanLuchene (1994) used a single-layer BP neu-
ral network to identify damage and condition assessment in multistory buildings using response
data from an experimental one-tenth scale reinforced concrete structure. Tsou and Shen (1994)
developed an on-line damage identification methodology for damage characterization (location
and severity) of two spring-mass systems through a BP neural network. The neural network was
constructed by three multi-layer sub-nets and used to identify the map from the change in modal
frequencies to the change in the spring stiffness values. Rhim and Lee (1995) examined the poten-
tial of using a Multi-Layer Perceptron (MLP) artificial neural network in conjunction with system
identification techniques to detect and characterize the damage in composite structures. Pandey
and Barai (1995) compared a two-hidden-layer to a single-layer BP neural network architecture
for damage detection of steel-truss bridge structures through mapping from various nodal time
histories to changes in stiffness. A non-parametric neural network-based approach is presented
by Masri et al. (2000) for the detection of changes in the characteristics of structure-unknown
systems. They showed that proposed damage detection methodology was robust in detecting rel-
atively small changes in the structural parameters, even in presence of noise in vibration measure-
ments. A structural damage detection method based on parameter identification using an iterative
neural network (NN) technique was proposed by Chang et al. (2000) and verified both numeri-
cally and experimentally using a clamped-clamped T beam. Zang and Imregun (2001) studied a
structural damage detection using measured frequency response functions (FRFs) as input data to
99
artificial neural networks. Zapico et al. (2001) studied a vibration-based procedure for damage
assessment in a two-storey steel frame and steel-concrete composite floors structure using neural
networks. Two natural frequencies and mode shapes were used as inputs to the neural network,
and three different definitions of damage (sections, bars and floors) were predicted as outputs.
Zubaydi et al. (2002) used neural network for damage identification in the side shell of a ship’s
structure. The input to the network was the auto-correlation function of the vibration response
of the structure while the output was a single function formed by adding together the damping
and a part of the restoring forces. The function was used to identify, quantify, and locate the
damage in the model. Wu et al. (2002) introduced a decentralized parametric damage detection
approach based on neural networks. Yam et al. (2003) studied a vibration-based damage detec-
tion for composite structures using wavelet transform and neural network identification. Kao and
Hung (2003) introduced a methodology to detect structural damage via free vibration responses
generated by approximating artificial neural networks. The extent of changes to the system was
assessed through comparing the periods and amplitudes of the free vibration responses of the dam-
aged and undamaged states. Quantification and localisation of damage in beam-like structures for
location and severity prediction of damage in beam-like structures was studied and experimen-
tally validated by Sahin and Shenoi (2003) by using a combination of global (changes in natural
frequencies) and local (curvature mode shapes) vibration-based analysis data as input in artifi-
cial neural networks. Maity and Saha (2004) used neural networks for damage assessment in
structure from changes in static parameter. They applied the idea on a simple cantilever beam,
where strain and displacement were used as possible candidates for damage identification by a
BP neural network. A neural networks-based damage detection method using the modal proper-
ties was studied by Lee et al. (2005) for modelling errors in the baseline finite element model of
bridges. In this model, the differences or the ratios of the mode shape components between before
100
and after damage were used as the input to the neural networks since they were found to be less
sensitive to the modelling errors than the mode shapes themselves. Fang et al. (2005) explored
the structural damage detection using frequency response functions (FRFs) as input data to the
back-propagation neural network (BPNN). Bakhary et al. (2007) proposed a statistical approach
to take into account the effect of uncertainties in developing an ANN model for damage detection
purposes. In this model, the probability of damage existence (PDE) was calculated based on the
probability density function of the existence of undamaged and damaged states. Jiang and Adeli
(2007) developed a non-parametric system identification-based approach for damage detection
of high-rise building structures subjected to seismic excitations using the dynamic fuzzy wavelet
neural network (WNN) model. The model could work for a partially instrumented system, where
the structure was divided into a series of sub-structures around a few pre-selected instrumented
floors. Li et al. (2008) proposed a damage identification method based on the combination of
artificial neural network, Dempster-Shafer (D-S) evidence theory-based information fusion and
the Shannon entropy, to form a weighted and selective information fusion technique to reduce the
impact of uncertainties on damage identification. Comprehensive literature reviews on the subject
of using neural networks in damage identification and health monitoring of structural systems can
be found in Doebling et al. (1996), Adeli (2001), and Sohn et al. (2004).
V .1.2 Scope
An overview of the concept behind the approach is presented in Section V .2. Three different neu-
ral network architectures for the identification method are proposed. In Section V .3, the usefulness
of the approach for the detection of structural changes is demonstrated in a physical systems for
four different levels of imposed nonlinear damping. Two formulations of error between actual
and predicted output are presented and the correlation between the level of change and predic-
101
Step 2: Damage Detection by the Trained Network
Acceleration
Velocity
Displacement
Excitation
Structure (Damaged)
input
Trained
Neural
Network
output from the
neural network
output from the structure
+
-
prediction error
= damage of
the structure
Σ
Step 1: Training of the Neural Network
Acceleration
Velocity
Displacement
Excitation
Structure (Undamaged)
input
Neural
Network
output
(a) Training stage
Step 2: Damage Detection by the Trained Network
Acceleration
Velocity
Displacement
Excitation
Structure (Damaged)
input
Trained
Neural
Network
output from the
neural network
output from the structure
+
-
prediction error
= damage of
the structure
Σ
Step 1: Training of the Neural Network
Acceleration
Velocity
Displacement
Excitation
Structure (Undamaged)
input
Neural
Network
output
(b) Detection stage
Figure V .1: Schematic diagram of damage detection using neural networks. (Nakamura et al. (1998))
tion errors are studied. The potential advantages as well as limitations of the methodology are
discussed. The concluding remarks are highlighted in Section V .4.
V .2 Formulation
V .2.1 Methodology
Figure V .1 shows the schematic diagram of the neural network based approach for the damage
detection. The overall procedure is conducted in two steps:
(i). Training stage: As shown in Figure V .1(a), a neural network is trained by the data obtained
from the undamaged structure.
(ii). Detection stage: As illustrated in Figure V .1(b), the trained network is fed input data, which
is the same input to the system (reference structure) and the output from the network and
102
Figure V .2: Diagram of the neural network in MATLAB
R
(Adapted from the software manual).
the output from the system are compared to each other.
If the network is well trained, both the intact system and the network should have reasonably
matching outputs. However, if the properties of the system have changed, the output from the
trained network will deviate from the system output, resulting in a quantitative measure of the
changes (error) in the physical system relative to undamaged condition. Using this methodology,
changes (damage) in the system can be detected just by observing the output error of the trained
network. It is worth to note that the proposed approach requires the training the neural network
only once for the reference system. The scheme is appealing for field implementation due to its
simplicity; however, attributing the quantified of the changes in the neural network output with
respect to changes in the physical system parameters will still be a challenging issue.
V .2.2 Neural Network Architecture
The nftool module in the Neural Network Toolbox of MATLAB
R
was employed for this study.
The neural network used is a two-layer feed-forward network with sigmoid hidden neurons and
linear output neurons (Figure V .2). The network will be trained with Levenberg-Marquardt back-
propagation algorithm, unless there is not enough memory, in which case scaled conjugate gra-
dient back-propagation (trainscg) will be used. According to the software manual, “the network
can fit multi-dimensional mapping problems arbitrarily well, given consistent data and enough
103
Input Hidden Layer
Output
X
1
X
2
X
3
X
4
X
1
.
X
2
.
X
3
.
X
4
.
X
1
X
2
X
3
X
4
F
1
F
2
F
3
F
4
..
..
..
..
Figure V .3: Neural Network (NN) with Input of Measured Displacements, Velocities, and Excitations and
Output of System Accelerations.
Input Hidden Layer
Output
X
1
X
2
X
3
X
4
X
1
.
X
2
.
X
3
.
X
4
.
G
1
G
2
G
3
G
4
Figure V .4: Neural Network (NN) with Input of Measured Relative Displacements, and Relative Veloci-
ties, and Output of Restoring Forces on All Stories
104
Input Hidden Layer
Output
X
i-1
X
i
X
i-1
.
X
i
.
G
i
Figure V .5: Neural Network (NN) with Input of Measured Relative Displacements, and Relative Veloci-
ties, and Output of Restoring Forces on Each Story
neurons in its hidden layer.”
A significant feature of the present study is the development of a sufficiently general neural
network approach which will be adequate to handle linear as well as nonlinear system with-
out any modifications. Two different architectures for the neural network are considered. As
shown in Figure V .3, the first system model approximates the unknown function H in the equa-
tion x = H(x; _ x; f). This function will be approximated by a neural network whose outputs
are the four system accelerations, while the inputs to the network are the four force excitations,
displacements and velocities recorded on all floors. This model can be easily incorporated with
a numerical differential equation solution (e.g., Runge-Kutta method) for prediction and control
purposes (Masri et al. (2000)). The representative neural network used in this study has 12 input
and 4 output nodes with 12 hidden neurons.
One classification of the non-destructive evaluation (NDE) of structural systems is local vs global
methods or, alternatively, micro or macro methods (Masri et al. (1996)). Global methods attempt
to simultaneously assess the condition of the whole system, whereas local methods are designed
to provide information about a relatively small region of the structure by using local measure-
105
ments. The two approaches are complementary to each other, with the optimum choice of method
highly dependent on the scope of the problem at hand and the nature of the sensor network.
The topology of the experimental model building which can be modeled as a chain-like sys-
tem also provides the opportunity to develop and investigate global and local NDE methods. In
the second model studied herein, the relative displacement and the relative velocity of floors are
selected as the input to the network and the restoring force is selected as the output of the network.
For this model, two different scenarios are studied. In the first case (Figure V .4), all the input data
are fed to a single neural network to predict four restoring forces in the floors. In the second case
(Figure V .5), four neural networks are trained, each individual representing a physical system cor-
responding to a specific storey of the building. The number of neurons in the hidden layer of the
neural network are selected as 8 and 4 for global and local models, respectively.
V .2.3 Performance Criteria
To evaluate the performance of the proposed method, two definitions of error are used to measure
the deviation of predicted output of the neural network models (
^
Y ) from the corresponding actual
data (Y ).
(i). Root Mean Square (RMS): A normalized error which is the average squared difference
between outputs and targets. Lower values are better. Zero means no error.
RMS =
v
u
u
u
t
P
N
i=1

Y
i

^
Y
i

2
P
N
i=1
Y
2
i
(V .1)
(ii). Regression value (R): R measures the correlation between outputs and targets. An R value
106
of one means a close relationship, while zero shows a random relationship. For ease of
comparison, the value of 1R is measured and presented in the study.
1R = 1:0
P
N
i=1

Y
i

Y
i

^
Y
i

^
Y
i

r
P
N
i=1

Y
i

Y
i

2
P
N
i=1

^
Y
i

^
Y
i

2
(V .2)
V .3 Results and Discussion
V .3.1 Performance in Detection of Change
As explained in previous section, the measured data from case 0 (no MR damper) was used to
train the neural network. Then the network was used as a model for prediction of the system’s
responses in cases 1 to 4, where various levels of nonlinearity were induced to the system by the
installed MR damper between the third and the fourth floors. To evaluate the trained network
performance, the previously two discussed indices, Root Mean Square (RMS) and Regression (1
- R) errors between the output of the network and the corresponding actual data, were calculated
for the training data set itself (case 0), as well as the recorded data after imposing nonlinearity to
the system (cases 1 to 4).
It is worth pointing out that the network size is an important factor in the final performance. If the
net is too small, then it is not able to store all the training patterns, due to an insufficient number
of parameters. On the other hand, if the net is too large, then it does not “generalize”, meaning it
tends to simply store the training patterns rather than performing interpolation. For the application
presented here, number of neurons in the hidden layer of each neural network is selected to meet
these criteria. The results for all three neural network based approaches are illustrated in Figure
V .6. Each of the panels shown in this Figure V .6 corresponds to a bar chart of the output error for
107
1 2 3 4
0
0.02
0.04
0.06
0.08
0.1
0.12
1.0 − R
General NN Model

Case 0: No MR damper
Case 1: I = 0.00A
Case 2: I = 0.05A
Case 3: I = 0.10A
Case 4: I = 0.15A
1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Floor Number
RMS
1 2 3 4
0
0.02
0.04
0.06
0.08
0.1
0.12
Global NN Model for Chain−Like System
1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Floor Number
1 2 3 4
0
0.02
0.04
0.06
0.08
0.1
0.12
Localized NN Model for Chain−Like System
1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Floor Number
Figure V .6: Performance of different neural networks models for detection of change in the system.
different floors throughout the experiment. For example, the first panel shown on the top of LHS
exhibits the output 1 - R error for the general neural network model that predicts the acceleration
of the floors. The abscissa shows the floor number. At each floor, the first bar illustrates the error
for case 0 and the rest of the bars show the corresponding error for cases 1 to 4 in the same order.
Therefore, the height of the bars shown in the panel is directly proportional to the extent of the
deviation error between the measured system response and the predictions on the basis of the
behavior of the reference model. Three major observations in these plot are conspicuous:
(i). Damage Localization: All three approaches clearly indicate higher values of error for both
RMS and 1 - R where the MR is installed (between the third and fourth floors) in cases 1
to 4.
(ii). Damage Quantification: The value of errors are ascending as the increase of current in the
108
MR damper introduces higher levels of nonlinearity to the system. Thus, both qualitatively
and quantitatively, it is seen that as the extent of the parameter changes is increased, a
corresponding increase in the deviation error of the network is observed.
(iii). For a given measured and predicted data sets in each case study, RMS generally gives
higher values of discrepancy while 1 - R index shows higher relative sensitivity toward
change in the system.
Note that both approaches which model the structure as a chain-like system are more successful in
pinpointing the location of change (fourth floor) compared to the general neural network method.
This behavior is attributable to the fact that no extra information about the physical system was
given to the net in the first approach, other than the input/output sequences, and that no model of
the system was assumed during the learning phase; whereas the neural networks that predict the
inter-story restoring forces of the building are actually benefiting from a prior information about
the topology of the system.
Without any knowledge of the characteristics of the structure being identified, the accuracy of
a well-trained neural network can be quantitatively gauged by means of its output performance
indices. In other words, dimensionless deviation errors (such as RMS or 1 - R) can be reliably
used as an straight-forward index to select an appropriate network for authentic representation of
the reference system. The results also indicate that, by selecting a threshold level relative to the
deviation error associated with the undamaged system response, all the well-trained networks can
be utilized to provide a sensitive indicator of the presence of potentially serious damage in the
system being monitored.
109
V .3.2 General Neural Network Model Combined with ODE Solvers
The general neural network model of this study can be easily incorporated with a numerical
differential equation solution (e.g., Runge-Kutta method) for prediction and control purposes.
For a system with the governing equations of motion in the form of:
m x(t) +G( _ x(t); x(t)) = f(t) (V .3)
, which can be rewritten as:
x(t) =
1
m
(f(t)G( _ x(t); x(t))) =H( _ x(t); x(t); f(t)) (V .4)
the trained neural network can replace function H to predict the response of the system. This
procedure is schematically shown in Figure V .7. After training five neural network for all the
linear and nonlinear cases, the predicted response of the structure on all floors are numerically
computed in the manner discussed above and plotted versus the corresponding measured data.
The results are illustrated in Figures V .8 to V .12 for cases 0 to 4, respectively. The ode45 solver in
MATLAB
R
which implements fourth-fifth-order formulas of the Runge-Kutta-Fehlberg method
was employed for this study. The accuracy of the predictions is so high as to make the two distinct
curves in each of the plots practically indistinguishable. This also reveals that the neural network
predictor gives promising results for both linear and nonlinear systems. It should be noted that
the predicted response will include the effects of both the network identification errors as well as
error propagation effects associated with the computational scheme (Runge-Kutta) used to obtain
the numerical solution.
110
x
k
˙ x
k
x
(k+1)
˙ x
(k+1)
¨ x
k
ODE
SOLVER
NEURAL
NETWORK
UPDATE
x
0
, ˙ x
0
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
˙ yn
yn
f1
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
f
k
Figure V .7: General neural network model combined with an ODE solver to predict the response of the
system.
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
x 10
−3 1−R = 0.07 RMS = 0.38
Time (sec)
Dis
1
(m)
−5
0
5
x 10
−3 1−R = 0.03 RMS = 0.24
Dis
2
(m)
−5
0
5
x 10
−3 1−R = 0.01 RMS = 0.15
Dis
3
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.19
Dis
4
(m)

Measured
Predicted
Figure V .8: Measured vs. predicted displacement of the structure at each floor for Case 0 using RK4.5
111
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
x 10
−3 1−R = 0.008 RMS = 0.12
Time (sec)
Dis
1
(m)
−5
0
5
x 10
−3 1−R = 0.005 RMS = 0.11
Dis
2
(m)
−5
0
5
x 10
−3 1−R = 0.005 RMS = 0.10
Dis
3
(m)
−5
0
5
x 10
−3 1−R = 0.005 RMS = 0.10
Dis
4
(m)

Measured
Predicted
Figure V .9: Measured vs. predicted displacement of the structure at each floor for Case 1 using RK4.5
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.23
Time (sec)
Dis
1
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.22
Dis
2
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.22
Dis
3
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.22
Dis
4
(m)

Measured
Predicted
Figure V .10: Measured vs. predicted displacement of the structure at each floor for Case 2 using RK4.5
112
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
x 10
−3 1−R = 0.04 RMS = 0.28
Time (sec)
Dis
1
(m)
−5
0
5
x 10
−3 1−R = 0.04 RMS = 0.27
Dis
2
(m)
−5
0
5
x 10
−3 1−R = 0.04 RMS = 0.27
Dis
3
(m)
−5
0
5
x 10
−3 1−R = 0.04 RMS = 0.27
Dis
4
(m)

Measured
Predicted
Figure V .11: Measured vs. predicted displacement of the structure at each floor for Case 3 using RK4.5
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
x 10
−3 1−R = 0.03 RMS = 0.25
Time (sec)
Dis
1
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.21
Dis
2
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.20
Dis
3
(m)
−5
0
5
x 10
−3 1−R = 0.02 RMS = 0.20
Dis
4
(m)

Measured
Predicted
Figure V .12: Measured vs. predicted displacement of the structure at each floor for Case 4 using RK4.5
113
V .3.3 Estimation of Structural Mass for Chain-Like Systems
m
2
1
m
n
m
(1)
G
G
(2)
G
(3)
G
(n)
F (t)
1
2
F (t)
F (t)
n
x (t)
i
Figure V .13: Model of a MDOF chain-like system.
As discussed in previous chapter, for the MDOF chain-like structure shown in Figure V .13 which
consists ofn elements, each with a lumped massm
i
, the nonlinear restoring functionG
(i)
at each
floor can be calculated as:
G
(i)
(z
i
; _ z
i
) =
n
X
j=i
(F
j
(t)m
j
x
j
) i = 1; 2;:::;n (V .5)
where z
i
and _ z
i
are the relative displacement and velocity between two consecutive floors, respec-
tively. Thus, starting from the tip of the chain, one can sequentially determine the time-histories of
all the inter-story restoring forces within the chain. However, as shown in Equation V .5, this will
require knowing the mass either through measuring the structural masses at the floors or calculat-
ing them based on design information. The objective of the study reported herein is to combine
a neural network that predicts the restoring forces from given displacement and velocities of the
floor with an optimization package to eliminate the requirement of knowing the structural mass.
114
First, restoring forces will be computed based on an arbitrary estimation of mass at each floor.
Then, a neural network will be trained to predict these restoring forces from the measured veloc-
ities and displacements. If the assumed masses deviate from accurate values, there will be a poor
relationship between the relative displacement and velocity at each floor and the corresponding
calculated restoring force. This will consequently result in a poor-trained neural network that
reflect itself in high values of performance criteria (RMS or 1 - R). Therefore, a cost function
based on these errors can be defined and optimized by adjusting the assumed structural masses to
archive a well-trained neural network:
J(
e
) =
n
X
i=1
W
i
RMS
i
(V .6)
As shown in Equation V .6, the importance of accuracy of the estimated values for each floor can
also manifested with weights (W
i
) in the cost function. The final optimized set of values will
represent the actual mass of each floor in the structure. Figure V .14 shows a graphical interpreta-
tion of the process. The mass of the experimental structure under investigation is estimated using
the proposed method and representative result of optimization process is shown in Figure V .15.
Initial population is randomly selected from a uniform distribution in the range of 0 25kg and
equal weights for all floors are considered in the definition of the cost function. Note that esti-
mated final values are in close agreement with the measured structural mass of each floor in the
lab (52:4=4 = 13:1kg).
The results of estimated structural mass for cases 0 to 4 are tabulated in Table V .1 and also illus-
trated in a bar plot format in Figure V .16. As shown in this figure, the final estimated values in all
115
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
˙ yn
f1
yn
¨ yn
Create NN Model
Step 1: System Identification
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
˙ yn
f1
yn
¨ yn
NN Model
Step 2: Damage Detection
Inputs NN Outputs
Targets
X
-
+
Change in
structure
net 2
Optimization
Method
Gn
G1
G
2
Error
m
1
m
2
net 1
net n
Neural Network
f
1
x
1
x
2
x
n
˙ x
n
˙ x
2
˙ x
1
Update
m
n
f
n
f2
Figure V .14: Flowchart of structural mass estimation for chain-like systems using optimization methods.
0 500 1000 1500 2000 2500
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
f=0.559962489601463 (0.559962489601463)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 500 1000 1500 2000 2500
0
5
10
15
20
25
M1
M4
M2
M3
Object Variables (4D)
Functon evaluation
Mass of floor
0 500 1000 1500 2000 2500
10
−1
10
0
10
1
4
2
3
1
Standard Deviations of All Variables
function evaluations
0 500 1000 1500 2000 2500
10
−3
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(a) Case 0
0 200 400 600 800 1000 1200 1400 1600 1800
10
−6
10
−4
10
−2
10
0
10
2
f=0.627190357581607 (0.627190357581607)
abs(f) (blue), f−min(f) (cyan), Sigma (green), Axis Ratio (red)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
5
10
15
20
25
M1
M2
M4
M3
Object Variables (4D)
Function evaluation
Mass of floor
0 200 400 600 800 1000 1200 1400 1600 1800
10
−1
10
0
10
1
4
2
3
1
Standard Deviations of All Variables
function evaluations
0 200 400 600 800 1000 1200 1400 1600 1800
10
−2
10
−1
10
0
10
1
Scaling (All Main Axes)
function evaluations
(b) Case 3
Figure V .15: Estimation of mass of the floors using CMA-ES for Case 0 and Case 3. Initial population is
randomly selected from on a uniform distribution in the range of 0 25kg and equal weights for all floors
are considered in the definition of the cost function.
116
1 2 3 4
0
2
4
6
8
10
12
14
16
18
20
Floor Number
Floor Mass (kg)

Case 0: No MR damper
Case 1: I = 0.00A
Case 2: I = 0.05A
Case 3: I = 0.10A
Case 4: I = 0.15A
Measured mass in the lab = 13.2kg
Figure V .16: Estimated mass of the floors using CMA-ES for Cases 0 to 4.
Table V .1: Estimated mass of the floors using CMA-ES for Cases 0 to 4 (M
measured
= 13:2kg).
Floor Number
Mass (kg)
Case 0 Case 1 Case 2 Case 3 Case 4
1
st
Floor 12.59 13.33 13.81 13.70 13.83
2
nd
Floor 13.32 13.62 14.05 13.96 14.04
3
rd
Floor 14.40 15.26 15.45 15.45 15.41
4
th
Floor 12.76 14.23 14.35 14.46 14.28
floors are very close the actual measured structural mass for all linear and nonlinear cases. The
deviation of these values from the actual measured mass of each floor in the lab is bounded to just
5% 16%. This is highly superior and more consistent when compared to the estimation of
structural mass by other system identification methods in the previous chapter.
V .4 Concluding Remarks
A time-domain non-parametric approach using artificial neural networks is presented for the de-
tection of changes in the characteristics of structure-unknown systems. The neural network is
117
trained, using vibration measurements from a “healthy” (reference) structure. The trained net-
work is subsequently fed with vibration measurements data from the slightly damaged (changed)
structure to monitor its health. The method requires no detailed knowledge about the topology or
the nature of the underlying structure, or of the failure modes of the system. This is a significant
advantage since the method can potentially cope with unforeseen failure scenarios. However, a
major disadvantage of the approach is that detectable changes can not be directly attributable to
a specific failure mode, but simply indicate that damage has occurred. For systems with certain
topologies (e.g., chain-like systems), the method can also provide useful information about the
damaged region of the structure. The approach is applied to recorded data from the 4-story exper-
imental model building that was studied in the previous chapter.
Through this study, it is shown that the proposed non-parametric approach is capable of pro-
viding a relatively sensitive indicator of changes (damage) in the structural parameters and can
be utilized as a high-fidelity tool for assessing the condition of structures. A technique based on
combination of neural networks with optimization methods to estimate the structural mass of the
floors in chain-like systems is also proposed. Results of the study show that the estimated final
values are in close agreement with the measured structural mass of each floor in the lab.
118
Chapter VI
Conclusion
T
HE main goal of the study reported herein is to investigate and evaluate different vibration-
signature-based methods for system identification, damage detection and health monitor-
ing of civil structures. A brief introduction to Structural Health Monitoring is presented in the
first chapter. The following chapter reports the performance of two stochastic methods of global
optimization for a subset of well-known benchmark functions. The application of these meth-
ods in finite element model updating approaches for damage detection purposes is investigated in
Chapter 3. The case study is a quarter-scale, two-span bridge system, experimentally tested at the
University of Nevada, Reno. Chapter 4 reports the performance of different system identification
approaches for experimentally recorded data of a 1/20 scale 4-story building equipped with smart
devices of magneto-rheological (MR) damper. This case study is also studied in Chapter 5 to in-
vestigate the application of artificial neural networks for the identification of nonlinear structural
models.
Considering the promising performance of various system identification and damage detection
methods and approaches under discussion, the general conclusions from this study are useful in
providing guidelines for the application of vibration-signature-based methods to real-world prob-
119
lems, especially in the implementation of structural health monitoring for complex, nonlinear
distributed systems.
VI.1 Overview of dissertation
VI.1.1 Chapter II
In the second chapter, the performance of two global optimization methods are numerically in-
vestigated on a subset of well-known test functions. The global optimization algorithms under
discussion were Genetic Algorithm (ga modules in MATLAB
R
) and an evolutionary strategy
called CMA-ES. A suit of five standard test functions with a search space of dimensionality
n = 5; 10; 25; 50; 100 was considered to study the effects of the problem-order on the per-
formance of the optimization methods. In addition, the effects of population size on the per-
formance of evolutionary methods are investigated for a subset of population sizes of Pop =
5; 10; 25; 50; 100; 250; 500; 1000. For each case, an ensemble of 100 simulations was generated
to reach a reliable statistical data set. Based on the comparison of these results, the following
conclusions can be made:
Evolutionary stochastic optimization methods are generally successful in solving high-
dimensional problems.
As expected, both optimization methods require significantly more function evaluations to
reach the solution for higher problem-orders.
Increasing the population size significantly improves the performance of these methods at
the expense of higher number of function evaluations. The study shows that the optimal
population size takes a wide range of values, depending on the cost function. For a given
120
objective function, the optimum population size may be tuned through calibration process
with the help of a statistical analysis.
For multi-modal functions, CMA-ES shows better performance than GA in the sense that
it returns smaller final function value with less average number of required function evalu-
ations to reach the solution. For instance, while CMA-ES outperform GA on Ackley and
Rastrigin functions (as shown in Figures II.7 and II.8), it significantly falls behind GA on
Rosenbrock function. Noting that Rosenbrock is the only uni-modal non-separable test
function of this study, this indicates that the performance of these optimization packages
varies with the topography of the functions. This conclusion also agrees with the findings
of the developers of CMA-ES, reported in Hansen and Kern (2004).
VI.1.2 Chapter III
The underlying objective of the study in chapter 3 is to evaluate the performance of two global
optimization methods in the finite element model updating approaches for damage detection in
dispersed structural systems, which usually deals with minimization of a complex, non-linear,
non-convex, high-dimensional cost function. The case study was a two-span reinforced concrete
bridge, experimentally tested at the University of Nevada, Reno. The Subspace method for sys-
tem identification was used to extract the modal parameters (natural frequencies, mode shapes,
and modal damping) of the bridge system. A NASTRAN
R
computer model was developed based
on the previous SAP2000 model provided by the NEES@Reno team, and validated with the sys-
tem identification results from the measured data. A simple on-line damage detection method,
using an ARMA model, was proposed and employed to trigger the finite element model updating
process. Two scenarios, assuming the availability of limited or large number of sensors were in-
121
vestigated for the finite element model updating procedure. The feasibility of the proposed finite
element model updating algorithm to accurately detect, localize, and quantify the damage in the
columns of the tested bridge throughout the experiment was investigated and validated by com-
parison to experimental measurements and visual inspections.
Based on the comparison of the results from the application of the finite element model updat-
ing algorithm under discussion with the strain gauge measurements and visual observations, the
following conclusions can be made:
(i). The simple ARMA model proposed for preliminary on-line damage detection can signifi-
cantly increase the efficacy of the model updating process.
(ii). The finite element model updating algorithm presented and applied in this study could
accurately detect and quantify the overall damage in the tested bridge bents throughout the
experiment.
(iii). The proposed method also showed very promising results for damage detection in the sys-
tem using output-only data. This reveals the potential of the technique to provide a useful
tool for SHM purposes in conjunction with promising methods for the identification of
modal properties using available ambient vibration data.
(iv). The finite element model updating algorithm used in this study was shown to be robust and
accurate to detect, localize and quantify the damage in the columns in synthetic simulations;
however, the experimental results could not be completely validated. The reliability of these
results highly depends upon the accuracy of the identified (equivalent) modal properties of
the (damaged, nonlinear) structure in different stages of the experiment.
122
(v). Detected damage values are highly correlated (
cmaes
= 0:956;
ga
= 0:946) with the
damage index developed by Park and Ang (1985), which is a practical measure of damage
based on dissipated hysteretic energy and ductility demand.
(vi). Both CMA-ES and GA converge to pretty close global minimums; however, GA may take
more computational effort to reach the solution, especially for higher-order problem.
VI.1.3 Chapter IV
Based on evaluation of the results from six different system identification methods using experi-
mentally recorded data from a 4-story model building, the following conclusions can be made:
(i). Identified system matrices from different methods were in acceptable agreement with each
other when using full instrumentation recordings.
(ii). All six methods could successfully identify, locate and quantify the increasing damping
imposed on the system by the MR damper throughout the experiment, when using full
instrumentation recordings. The identified values in the damping matrix corresponding to
the location of the MR damper (between the third and fourth floors) monotonically increase
as the input current to the MR damper is intensified.
(iii). Identified system matrices using partial instrumentation recordings were in less agreement
with each other due to the short length of recordings; however, all identification methods
could successfully detect the change in the damping properties of the system.
(iv). In addition, the RFS method could identify the nonlinearity of the damping in the system
and its behavior (softening or hardening) with respect to different vibrational signatures of
the structure.
123
VI.1.4 Chapter V
In chapter 5, a time-domain non-parametric approach using artificial neural networks is presented
for the detection of changes in the characteristics of structure-unknown systems. The neural
network is trained, using vibration measurements from a “healthy” (reference) structure. The
trained network is subsequently fed with vibration measurements data from the slightly damaged
(changed) structure to monitor its health. The method requires no detailed knowledge about the
topology or the nature of the underlying structure, or of the failure modes of the system. This is
a significant advantage since the method can potentially cope with unforeseen failure scenarios.
However, a major disadvantage of the approach is that detectable changes can not be directly
attributable to a specific failure mode, but simply indicate that damage may have occurred. For
systems with certain topologies (e.g., chain-like systems), the method can also provide useful in-
formation about the damaged region(s) of the structure. The approach is applied to recorded data
from the 4-story experimental model building that was studied in the previous chapter.
Through this study, it is shown that the proposed non-parametric approach is capable of pro-
viding a relatively sensitive indicator of changes (damage) in the structural parameters, and can
be utilized as a high-fidelity tool for assessing the condition of structures. A technique based on
combination of neural networks with optimization methods to estimate the structural mass of the
floors in chain-like systems is also proposed. Results of the study show that the estimated final
mass values are in close agreement with the measured structural mass of each floor in the lab.
124
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Appendix A
136
Table A.1: Identified mass, damping, and stiffness matrices of the system: Case 0
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
11:10 0:74 0:17 0:71
0:63 11:99 0:27 0:29
0:63 0:49 12:83 0:58
0:88 0:54 0:55 11:67
3
7
7
5
2
6
6
4
42:74 2:85 16:22 4:11
8:20 29:61 9:55 7:29
15:17 7:47 48.23 5:33
0:96 3:69 8:70 12.05
3
7
7
5
2
6
6
4
2:60 1:44 0:02 0:14
1:63 3:27 1:80 0:11
0:06 1:72 3:08 1:45
0:15 0:07 1:58 1:43
3
7
7
5
Sym. Linear System Method
2
6
6
4
11:10 0:74 0:17 0:71
0:74 11:06 0:38 0:30
0:17 0:38 12:13 0:53
0:71 0:30 0:53 10:44
3
7
7
5
2
6
6
4
42:74 2:85 16:22 4:11
2:85 29:60 10:26 6:55
16:22 10:26 48.45 5:80
4:11 6:55 5:80 9.61
3
7
7
5
2
6
6
4
2:60 1:44 0:02 0:14
1:44 2:96 1:64 0:09
0:02 1:64 2:97 1:39
0:14 0:09 1:39 1:25
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
11:34 0 0 0
0 11:15 0 0
0 0 13:06 0
0 0 0 12:55
3
7
7
5
2
6
6
4
39:60 20:83 0 0
20:83 47:78 26:95 0
0 26:95 38.05 11:10
0 0 11:10 11.10
3
7
7
5
2
6
6
4
2:53 1:27 0 0
1:27 2:79 1:52 0
0 1:52 3:07 1:55
0 0 1:55 1:55
3
7
7
5
Model Updating Method
2
6
6
4
11:12 0:72 0:21 0:74
0:60 11:94 0:28 0:30
0:65 0:52 12:83 0:59
0:85 0:51 0:54 11:65
3
7
7
5
2
6
6
4
44:00 2:63 16:23 3:96
7:94 29:87 10:59 7:57
15:03 6:55 47.25 4:83
2:66 2:71 7:56 11.69
3
7
7
5
2
6
6
4
2:60 1:44 0:01 0:14
1:63 3:26 1:80 0:11
0:05 1:71 3:08 1:45
0:15 0:07 1:58 1:43
3
7
7
5
Sub-Space Method
2
6
6
4
14:93 0:13 0:56 0:28
0:01 14:10 0:02 0:17
0:29 0:03 13:81 0:04
0:15 0:10 0:02 12:56
3
7
7
5
2
6
6
4
49:99 2:91 21:66 4:20
10:07 29:36 10:09 7:75
16:39 7:61 50.46 7:16
2:36 3:02 5:95 12.07
3
7
7
5
2
6
6
4
3:60 2:14 0:25 0:06
2:10 3:98 2:13 0:09
0:23 2:05 3:49 1:64
0:01 0:14 1:77 1:60
3
7
7
5
Iterative PEM Method
2
6
6
4
14:50 0:10 0:50 0:18
0:05 14:09 0:18 0:02
0:25 0:10 14:15 0:14
0:11 0:04 0:09 12:78
3
7
7
5
2
6
6
4
46:97 6:92 26:79 1:39
16:36 24:06 5:17 6:10
0:41 8:17 34.47 11:28
5:53 0:99 0:64 6.06
3
7
7
5
2
6
6
4
3:47 2:08 0:26 0:05
2:06 3:92 2:13 0:12
0:24 2:07 3:56 1:69
0:03 0:13 1:78 1:61
3
7
7
5
137
Table A.2: Identified mass, damping, and stiffness matrices of the system: Case 1
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
13:33 0:54 0:08 0:20
0:65 13:04 0:52 0:08
0:31 0:71 13:78 0:96
0:18 0:01 0:72 12:46
3
7
7
5
2
6
6
4
40:48 10:81 18:84 1:86
19:33 38:01 4:70 16:75
10:47 18:08 207.38 200:01
9:18 4:53 170:12 217.96
3
7
7
5
2
6
6
4
3:09 1:82 0:22 0:04
1:76 3:46 1:82 0:04
0:16 1:73 3:10 1:48
0:02 0:04 1:50 1:41
3
7
7
5
Sym. Linear System Method
2
6
6
4
13:33 0:54 0:08 0:20
0:54 13:18 0:51 0:09
0:08 0:51 13:95 0:99
0:20 0:09 0:99 12:65
3
7
7
5
2
6
6
4
40:48 10:81 18:84 1:86
10:81 35:78 8:75 14:54
18:84 8:75 220.23 209:24
1:86 14:54 209:24 255.56
3
7
7
5
2
6
6
4
3:09 1:82 0:22 0:04
1:82 3:53 1:84 0:04
0:22 1:84 3:20 1:51
0:04 0:04 1:51 1:42
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
13:01 0 0 0
0 12:02 0 0
0 0 14:06 0
0 0 0 13:68
3
7
7
5
2
6
6
4
129:46 65:23 0 0
65:23 94:96 29:74 0
0 29:74 241.53 211:79
0 0 211:79 211.79
3
7
7
5
2
6
6
4
2:59 1:24 0 0
1:24 2:59 1:35 0
0 1:35 2:66 1:31
0 0 1:31 1:31
3
7
7
5
Model Updating Method
2
6
6
4
13:34 0:54 0:07 0:20
0:66 13:03 0:52 0:08
0:31 0:72 13:78 0:96
0:17 0:02 0:73 12:47
3
7
7
5
2
6
6
4
41:06 11:49 17:25 0:78
18:30 37:49 4:38 17:46
11:23 18:00 207.31 200:32
9:36 4:85 170:14 217.65
3
7
7
5
2
6
6
4
3:09 1:82 0:22 0:04
1:75 3:45 1:81 0:04
0:16 1:73 3:10 1:48
0:02 0:04 1:50 1:41
3
7
7
5
Sub-Space Method
2
6
6
4
14:37 0:18 0:08 0:21
0:01 13:92 0:19 0:00
0:07 0:16 14:83 0:57
0:11 0:14 0:51 13:93
3
7
7
5
2
6
6
4
48:30 11:34 34:94 11:55
14:27 35:53 0:27 18:18
3:04 5:84 256.83 253:10
1:29 3:05 206:75 256.06
3
7
7
5
2
6
6
4
3:38 2:06 0:32 0:00
2:04 3:83 2:04 0:09
0:41 2:20 3:59 1:68
0:07 0:18 1:79 1:62
3
7
7
5
Iterative PEM Method
2
6
6
4
14:40 0:11 0:34 0:07
0:10 14:00 0:14 0:03
0:12 0:33 14:84 0:18
0:21 0:11 0:52 14:33
3
7
7
5
2
6
6
4
24:58 6:35 0:49 14:84
11:91 34:75 24:27 2:46
22:11 18:21 224.38 228:63
25:81 4:69 176:41 236.86
3
7
7
5
2
6
6
4
3:40 2:07 0:29 0:03
2:04 3:88 2:10 0:11
0:36 2:08 3:58 1:74
0:09 0:18 1:84 1:67
3
7
7
5
138
Table A.3: Identified mass, damping, and stiffness matrices of the system: Case 2
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
13:49 0:62 0:01 0:17
0:56 13:37 1:00 0:28
0:21 1:26 12:40 1:98
0:12 0:20 1:45 11:58
3
7
7
5
2
6
6
4
53:16 5:17 27:80 1:10
25:52 60:35 5:76 43:92
0:96 59:57 249.99 217:35
4:34 32:58 218:71 265.50
3
7
7
5
2
6
6
4
3:10 1:81 0:23 0:01
1:83 3:48 1:66 0:07
0:11 1:37 2:46 1:14
0:02 0:12 1:14 1:21
3
7
7
5
Sym. Linear System Method
2
6
6
4
13:49 0:62 0:01 0:17
0:62 13:34 1:02 0:29
0:01 1:02 13:11 2:14
0:17 0:29 2:14 12:22
3
7
7
5
2
6
6
4
53:16 5:17 27:80 1:10
5:17 54:47 1:61 39:23
27:80 1:61 263.18 252:55
1:10 39:23 252:55 322.71
3
7
7
5
2
6
6
4
3:10 1:81 0:23 0:01
1:81 3:46 1:66 0:07
0:23 1:66 2:74 1:23
0:01 0:07 1:23 1:28
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
13:53 0 0 0
0 12:59 0 0
0 0 14:38 0
0 0 0 13:78
3
7
7
5
2
6
6
4
128:56 69:26 0 0
69:26 81:18 11:92 0
0 11:92 286.35 274:43
0 0 274:43 274.43
3
7
7
5
2
6
6
4
2:73 1:33 0 0
1:33 2:79 1:46 0
0 1:46 2:96 1:50
0 0 1:50 1:50
3
7
7
5
Model Updating Method
2
6
6
4
13:49 0:63 0:00 0:16
0:56 13:37 1:01 0:28
0:21 1:27 12:38 1:97
0:11 0:20 1:43 11:57
3
7
7
5
2
6
6
4
53:78 4:56 28:06 0:85
26:03 60:42 4:47 43:49
0:83 59:81 248.76 216:25
4:50 32:25 218:88 265.58
3
7
7
5
2
6
6
4
3:10 1:81 0:23 0:01
1:83 3:47 1:66 0:07
0:11 1:36 2:45 1:14
0:02 0:11 1:14 1:21
3
7
7
5
Sub-Space Method
2
6
6
4
14:37 0:13 0:25 0:14
0:03 15:12 0:24 0:00
0:13 0:14 14:69 0:17
0:02 0:12 0:04 14:19
3
7
7
5
2
6
6
4
81:84 4:86 55:67 11:61
15:03 44:60 3:18 29:93
7:38 60:62 325.13 301:52
22:36 46:78 263:30 329.88
3
7
7
5
2
6
6
4
3:39 2:09 0:36 0:01
2:21 4:12 2:15 0:06
0:41 2:07 3:45 1:67
0:05 0:12 1:80 1:67
3
7
7
5
Iterative PEM Method
2
6
6
4
14:31 0:13 0:12 0:41
0:30 14:86 0:07 0:47
1:00 0:68 13:50 2:18
0:30 0:80 0:64 14:41
3
7
7
5
2
6
6
4
44:91 1:09 30:67 6:48
21:46 42:52 6:28 36:78
15:02 60:34 322.61 319:47
13:81 37:14 278:96 340.93
3
7
7
5
2
6
6
4
3:40 2:14 0:34 0:04
2:11 4:04 2:09 0:03
0:50 1:87 3:34 1:76
0:01 0:09 1:62 1:64
3
7
7
5
139
Table A.4: Identified mass, damping, and stiffness matrices of the system: Case 3
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
13:47 0:48 0:04 0:20
0:47 13:08 0:63 0:01
0:22 0:89 13:18 1:51
0:16 0:02 1:55 12:02
3
7
7
5
2
6
6
4
45:54 3:45 25:91 1:95
26:90 49:65 8:00 30:50
18:48 59:54 291.05 279:24
12:06 38:94 263:11 309.75
3
7
7
5
2
6
6
4
3:11 1:85 0:24 0:03
1:79 3:44 1:78 0:03
0:17 1:58 2:80 1:33
0:00 0:08 1:17 1:23
3
7
7
5
Sym. Linear System Method
2
6
6
4
13:47 0:48 0:04 0:20
0:48 13:27 0:63 0:02
0:04 0:63 13:70 1:63
0:20 0:02 1:63 12:45
3
7
7
5
2
6
6
4
45:54 3:45 25:91 1:95
3:45 43:02 3:36 23:77
25:91 3:36 303.63 313:37
1:95 23:77 313:37 386.18
3
7
7
5
2
6
6
4
3:11 1:85 0:24 0:03
1:85 3:52 1:81 0:03
0:24 1:81 3:04 1:42
0:03 0:03 1:42 1:37
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
13:61 0 0 0
0 12:70 0 0
0 0 14:69 0
0 0 0 13:91
3
7
7
5
2
6
6
4
131:99 67:39 0 0
67:39 91:48 24:08 0
0 24:08 352.23 328:15
0 0 328:15 328.15
3
7
7
5
2
6
6
4
2:70 1:30 0 0
1:30 2:72 1:42 0
0 1:42 2:88 1:45
0 0 1:45 1:45
3
7
7
5
Model Updating Method
2
6
6
4
13:47 0:48 0:03 0:19
0:48 13:09 0:63 0:01
0:22 0:88 13:17 1:51
0:16 0:00 1:56 12:02
3
7
7
5
2
6
6
4
45:54 3:05 25:65 2:78
26:78 50:26 7:91 30:69
18:20 59:00 290.68 278:81
12:00 40:20 264:02 309.19
3
7
7
5
2
6
6
4
3:11 1:85 0:24 0:03
1:79 3:44 1:78 0:03
0:17 1:58 2:80 1:33
0:01 0:07 1:17 1:23
3
7
7
5
Sub-Space Method
2
6
6
4
14:67 0:01 0:02 0:75
0:01 14:33 0:03 0:02
0:02 0:47 14:80 1:03
0:14 0:33 0:59 13:54
3
7
7
5
2
6
6
4
72:37 16:02 90:74 55:60
24:14 41:57 17:06 33:18
17:62 39:37 419.28 409:60
15:36 36:53 378:25 425.72
3
7
7
5
2
6
6
4
3:44 2:08 0:35 0:03
2:09 3:92 2:10 0:10
0:35 2:05 3:45 1:65
0:07 0:10 1:65 1:57
3
7
7
5
Iterative PEM Method
2
6
6
4
14:67 0:37 0:38 0:19
0:08 14:59 0:19 0:05
0:48 0:07 15:53 0:32
0:27 0:19 0:43 14:50
3
7
7
5
2
6
6
4
9:72 44:68 78:85 30:51
17:40 22:12 71:58 60:23
41:37 5:38 319.90 340:25
6:89 34:06 332:61 394.66
3
7
7
5
2
6
6
4
3:42 2:09 0:25 0:06
2:01 3:77 1:95 0:05
0:26 1:96 3:56 1:78
0:05 0:14 1:78 1:66
3
7
7
5
140
Table A.5: Identified mass, damping, and stiffness matrices of the system: Case 4
Methodn Matrix M C K
(10
5
)
Linear System Method
2
6
6
4
13:72 0:53 0:14 0:37
0:55 13:00 0:67 0:12
0:43 1:06 13:17 2:27
0:27 0:16 1:43 10:84
3
7
7
5
2
6
6
4
54:46 9:98 36:75 12:84
32:06 52:49 15:33 46:34
25:80 84:01 326.03 295:32
19:19 55:89 295:02 323.97
3
7
7
5
2
6
6
4
3:17 1:85 0:16 0:08
1:75 3:39 1:70 0:01
0:10 1:47 2:62 1:22
0:03 0:14 1:01 1:12
3
7
7
5
Sym. Linear System Method
2
6
6
4
13:72 0:53 0:14 0:37
0:53 13:26 0:67 0:14
0:14 0:67 13:68 2:53
0:37 0:14 2:53 11:93
3
7
7
5
2
6
6
4
54:46 9:98 36:75 12:84
9:98 47:69 1:55 38:42
36:75 1:55 337.10 342:98
12:84 38:42 342:98 424.73
3
7
7
5
2
6
6
4
3:17 1:85 0:16 0:08
1:85 3:50 1:74 0:02
0:16 1:74 2:93 1:33
0:08 0:02 1:33 1:33
3
7
7
5
RFS Method for Chain-Like Systems
2
6
6
4
13:60 0 0 0
0 12:74 0 0
0 0 14:68 0
0 0 0 13:67
3
7
7
5
2
6
6
4
119:17 59:31 0 0
59:31 80:42 21:11 0
0 21:11 397.79 376:68
0 0 376:68 376.68
3
7
7
5
2
6
6
4
2:73 1:32 0 0
1:32 2:76 1:44 0
0 1:44 2:95 1:51
0 0 1:51 1:51
3
7
7
5
Model Updating Method
2
6
6
4
13:73 0:53 0:13 0:34
0:56 12:99 0:66 0:13
0:43 1:07 13:17 2:28
0:27 0:16 1:44 10:84
3
7
7
5
2
6
6
4
54:92 9:71 34:79 10:80
31:46 51:52 15:28 46:07
25:58 83:75 327.39 295:90
18:87 54:36 293:76 323.73
3
7
7
5
2
6
6
4
3:18 1:85 0:17 0:07
1:75 3:38 1:70 0:01
0:10 1:47 2:62 1:22
0:03 0:14 1:01 1:12
3
7
7
5
Sub-Space Method
2
6
6
4
14:86 0:13 0:27 0:42
0:03 14:29 0:11 0:03
0:06 0:34 14:49 0:17
0:19 0:16 0:43 14:30
3
7
7
5
2
6
6
4
87:93 24:87 86:41 49:44
27:47 45:98 22:36 42:95
0:65 80:53 453.49 429:76
22:58 85:22 394:62 431.57
3
7
7
5
2
6
6
4
3:50 2:11 0:34 0:02
2:08 3:89 2:04 0:07
0:37 1:98 3:40 1:68
0:12 0:08 1:69 1:64
3
7
7
5
Iterative PEM Method
2
6
6
4
14:72 0:29 0:19 0:10
0:13 14:30 0:34 0:00
0:25 0:82 14:96 0:02
0:32 0:11 1:08 14:06
3
7
7
5
2
6
6
4
44:01 7:69 23:05 1:12
15:72 41:06 18:23 11:80
2:17 70:85 395.98 386:26
8:31 50:06 375:62 419.98
3
7
7
5
2
6
6
4
3:45 2:05 0:23 0:06
2:05 3:84 1:98 0:04
0:36 1:94 3:38 1:68
0:15 0:08 1:52 1:51
3
7
7
5
141
Table A.6: Identified mass, damping, and stiffness matrices of the system, using partial instrumentation
recordings: Case 1
Methodn Matrix M C K
(10
5
)
Linear System Method

9:00 1:00
0:85 12:83

243.18 199:32
191:80 236.73

1:10 0:84
1:45 1:40

Sym. Linear System Method

9:00 1:00
1:00 10:77

243.18 199:32
199:32 222.34

1:10 0:84
0:84 0:86

RFS Method for Chain-Like Systems

14:93 0:00
0:00 13:65

296.55 227:11
227:11 227.11

1:64 1:29
1:29 1:29

Model Updating Method

9:02 0:96
0:84 12:88

250.66 194:03
192:14 233.61

1:11 0:85
1:46 1:41

Sub-Space Method

14:77 3:04
0:34 11:33

276.80 274:35
177:99 213.74

2:36 1:91
1:35 1:29

Iterative PEM Method

17:70 0:10
0:26 14:68

322.95 290:92
221:23 269.75

2:38 1:86
1:77 1:69

Table A.7: Identified mass, damping, and stiffness matrices of the system, using partial instrumentation
recordings: Case 2
Methodn Matrix M C K
(10
5
)
Linear System Method

9:55 1:38
1:01 12:41

302.81 278:96
252:23 320.27

1:16 0:89
1:36 1:34

Sym. Linear System Method

9:55 1:38
1:38 11:17

302.81 278:96
278:96 317.53

1:16 0:89
0:89 0:92

RFS Method for Chain-Like Systems

14:78 0:00
0:00 13:70

369.14 307:79
307:79 307.79

1:76 1:40
1:40 1:40

Model Updating Method

9:56 1:38
1:00 12:44

303.96 279:68
252:59 320.79

1:17 0:90
1:37 1:34

Sub-Space Method

15:89 0:08
0:25 12:10

299.48 315:93
247:97 315.58

2:20 1:75
1:50 1:45

Iterative PEM Method

17:43 0:18
0:07 15:09

372.44 374:36
304:21 391.21

2:40 1:89
1:83 1:76

142
Table A.8: Identified mass, damping, and stiffness matrices of the system, using partial instrumentation
recordings: Case 3
Methodn Matrix M C K
(10
5
)
Linear System Method

9:60 1:02
1:10 12:64

323.17 298:93
309:31 371.76

1:03 0:79
1:32 1:30

Sym. Linear System Method

9:60 1:02
1:02 11:36

323.17 298:93
298:93 343.00

1:03 0:79
0:79 0:82

RFS Method for Chain-Like Systems

15:88 0:00
0:00 13:78

437.65 377:82
377:82 377.82

1:70 1:33
1:33 1:33

Model Updating Method

9:85 0:95
1:03 12:74

346.98 315:60
302:65 363.48

1:03 0:78
1:34 1:32

Sub-Space Method

16:53 0:32
0:46 15:10

457.98 430:65
374:14 442.64

1:86 1:45
1:70 1:65

Iterative PEM Method

16:81 0:38
0:25 16:07

442.44 438:23
432:21 512.71

1:81 1:40
1:84 1:79

Table A.9: Identified mass, damping, and stiffness matrices of the system, using partial instrumentation
recordings: Case 4
Methodn Matrix M C K
(10
5
)
Linear System Method

10:56 1:72
1:10 11:72

356.80 346:07
331:42 393.60

1:21 0:93
1:19 1:19

Sym. Linear System Method

10:56 1:72
1:72 11:34

356.80 346:07
346:07 389.24

1:21 0:93
0:93 0:97

RFS Method for Chain-Like Systems

15:04 0:00
0:00 13:59

454.52 404:18
404:18 404.18

1:73 1:37
1:37 1:37

Model Updating Method

10:57 1:70
1:09 11:76

356.71 345:17
331:40 393.89

1:22 0:94
1:20 1:20

Sub-Space Method

15:98 0:02
0:48 12:01

365.85 405:81
340:41 411.16

2:15 1:73
1:45 1:41

Iterative PEM Method

17:11 0:27
0:21 15:05

449.51 481:20
405:32 495.60

2:29 1:83
1:72 1:68

143
Table A.10: Identified frequencies and damping ratios: Case 0
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.62 1.623 16.70 1.324 26.56 1.165 35.89 0.765
Sym. Linear System Method 5.72 2.024 16.51 1.479 26.81 1.083 35.73 0.759
RFS Method for Chain-Like Systems 5.70 0.279 17.66 0.740 26.18 0.998 32.82 1.564
Model Updating Method 5.62 1.604 16.69 1.369 26.57 1.141 35.90 0.771
Sub-Space Method 5.65 1.768 16.65 1.114 26.54 1.065 35.83 0.629
Iterative PEM Method 5.64 1.497 16.58 0.781 26.38 0.794 35.75 0.560
Table A.11: Identified frequencies and damping ratios: Case 1
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.37 1.920 16.45 3.326 25.93 6.148 35.34 2.802
Sym. Linear System Method 5.46 1.629 16.38 3.856 26.09 7.093 35.39 2.724
RFS Method for Chain-Like Systems 5.45 0.867 16.25 4.109 23.93 7.443 30.03 5.073
Model Updating Method 5.39 1.891 16.46 3.348 25.92 6.125 35.32 2.809
Sub-Space Method 5.42 1.878 16.43 3.266 26.07 7.163 35.27 2.577
Iterative PEM Method 5.38 1.714 16.47 3.225 25.93 5.077 35.39 2.680
Table A.12: Identified frequencies and damping ratios: Case 2
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.49 2.393 16.59 4.280 25.58 9.233 35.01 4.151
Sym. Linear System Method 5.50 1.690 16.53 5.706 25.84 10.256 35.42 3.741
RFS Method for Chain-Like Systems 5.56 0.807 16.71 3.914 24.76 8.530 30.65 5.310
Model Updating Method 5.44 2.380 16.56 4.287 25.56 9.246 34.99 4.149
Sub-Space Method 5.36 2.700 16.58 4.408 25.67 8.953 34.91 3.216
Iterative PEM Method 5.44 2.293 16.56 4.061 25.64 8.410 34.80 3.235
144
Table A.13: Identified frequencies and damping ratios: Case 3
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.43 2.286 16.49 4.686 25.61 9.751 34.94 4.281
Sym. Linear System Method 5.36 1.824 16.40 5.915 26.16 11.079 35.23 3.836
RFS Method for Chain-Like Systems 5.54 0.885 16.52 4.819 24.36 10.117 29.98 6.074
Model Updating Method 5.42 2.284 16.50 4.694 25.58 9.773 34.93 4.275
Sub-Space Method 5.47 2.361 16.52 4.222 26.14 13.253 34.86 3.897
Iterative PEM Method 5.48 2.146 16.34 4.474 26.00 9.829 33.54 1.073
Table A.14: Identified frequencies and damping ratios: Case 4
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%) f
3
(Hz)
3
(%) f
4
(Hz)
4
(%)
Linear System Method 5.55 1.974 16.68 5.003 25.60 11.894 34.64 5.119
Sym. Linear System Method 5.50 1.750 16.50 6.248 26.87 13.504 35.18 4.881
RFS Method for Chain-Like Systems 5.54 0.829 16.76 4.897 24.79 11.194 30.09 6.340
Model Updating Method 5.46 2.045 16.68 5.017 25.61 11.895 34.64 5.111
Sub-Space Method 5.42 2.193 16.70 4.296 25.95 13.149 34.59 4.262
Iterative PEM Method 5.45 1.829 16.51 4.776 25.66 11.010 34.63 4.097
Table A.15: Identified frequencies and damping ratios (partial instrumentation): Case 1
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%)
Linear System Method 5.44 4.751 24.68 14.740
Sym. Linear System Method 5.50 4.489 23.05 16.992
RFS Method for Chain-Like Systems 5.39 3.337 22.10 12.327
Model Updating Method 5.42 5.534 24.70 14.651
Sub-Space Method 5.41 3.206 24.65 10.505
Iterative PEM Method 5.44 3.668 24.68 11.122
145
Table A.16: Identified frequencies and damping ratios (partial instrumentation): Case 2
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%)
Linear System Method 5.52 4.446 24.87 19.353
Sym. Linear System Method 5.51 4.273 23.54 22.180
RFS Method for Chain-Like Systems 5.49 2.992 23.03 15.681
Model Updating Method 5.47 4.506 24.93 19.320
Sub-Space Method 5.53 3.803 24.81 13.397
Iterative PEM Method 5.50 4.390 24.86 14.266
Table A.17: Identified frequencies and damping ratios (partial instrumentation): Case 3
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%)
Linear System Method 5.33 4.232 23.59 22.325
Sym. Linear System Method 5.29 4.996 21.64 24.565
RFS Method for Chain-Like Systems 5.45 2.902 22.04 19.130
Model Updating Method 5.44 4.193 23.37 22.577
Sub-Space Method 5.43 3.329 23.15 19.594
Iterative PEM Method 5.50 4.390 24.86 14.266
Table A.18: Identified frequencies and damping ratios (partial instrumentation): Case 4
Methodn Mode f
1
(Hz)
1
(%) f
2
(Hz)
2
(%)
Linear System Method 5.45 3.888 24.30 24.089
Sym. Linear System Method 5.51 3.630 23.62 26.115
RFS Method for Chain-Like Systems 5.47 2.572 22.73 20.373
Model Updating Method 5.45 3.965 24.36 23.921
Sub-Space Method 5.41 3.346 24.43 17.548
Iterative PEM Method 5.39 3.649 24.50 18.695
146

Abstract (if available)

Abstract Civil infrastructures play a vital role in human societies. Recent catastrophic events due to the deficiency, failure or malfunction of these systems, claiming many lives and resulting in substantial economic loss, have attracted extensive attention focused on reviewing and amending the design and maintenance procedures of civil infrastructures. In addition to the possible failure of structural components, long-term forms of damage due to deterioration or fatigue may also necessitate regular monitoring of civil structures. Therefore, depending on the importance, use and risk, the structure of interest needs to be equipped with inspection, monitoring and maintenance systems. Structural Health Monitoring (SHM) is generally associated with any engineering methodology whose aim is to detect, locate and quantify the damage in the target system. Vibration-based techniques, as the most conventional SHM approaches, acquire and analyze the structural response using a variety of sensors mounted at different locations on the structure. The main goal of the study reported herein is to investigate and evaluate different vibration-signature-based methods for system identification, damage detection and health monitoring of civil structures. Various well-known techniques such as finite element model updating approach and damage detection methods based on artificial neural networks are studied and evaluated. Experimental data from two case studies, a quarter-scale two-span bridge system, tested at the University of Nevada, Reno, and a 1/20 scale 4-story building equipped with smart devices of magneto-rheological (MR) damper, are used for investigation and validation purposes. Guidelines are established for the optimum selection of the dominant control parameters involved in the application of some of the robust SHM approaches for achieving reliable SHM results under realistic conditions.

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Asset Metadata

Creator Jafarkhani, Reza (author)

Core Title Studies into vibration-signature-based methods for system identification, damage detection and health monitoring of civil infrastructures

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering (Structural Engineering)

Publication Date 04/18/2013

Defense Date 01/18/2013

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

Tag Damage Detection,finite element model updating,neural networks,nonlinear systems,OAI-PMH Harvest,optimization,Structural dynamics,system identification

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Masri, Sami F. (committee chair), Ghanem, Roger G. (committee member), Johnson, Erik A. (committee member), Wellford, L. Carter (committee member)

Creator Email jafarkha@usc.edu,r_jafarkhani@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-239394

Unique identifier UC11293887

Identifier etd-Jafarkhani-1568.pdf (filename),usctheses-c3-239394 (legacy record id)

Legacy Identifier etd-Jafarkhani-1568.pdf

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Document Type Dissertation

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Rights Jafarkhani, Reza

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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...

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