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Wave method for structural system identification and health monitoring of buildings based on layered shear beam model

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Wave method for structural system identification and health monitoring of buildings based on layered shear beam model

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WAVE METHOD FOR STRUCTURAL SYSTEM
IDENTIFICATION AND HEALTH MONITORING OF BUILDINGS
BASED ON LAYERED SHEAR BEAM MODEL

by
Mohammadtaghi Rahmani

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CIVIL ENGINEERING)

Committee in charge:
Maria I. Todorovska, Chair
Mihailo D. Trifunac
Vincent W. Lee
Hung Leung Wong
Firdaus Udwadia, Outside Member
May 2014

Copyright © 2014
Mohammadtaghi Rahmani

ii

DEDICATION

To my mother, Ozra Toorani

iv

ACKNOWLEDGEMENTS
This dissertation research was supported by the U.S. National Science Foundation (NSF) via
grant CMMI-0800399 (PI: M. Todorovska). Additional support for my Ph.D. study was provided
by the University of Southern California (USC) through Teaching Assistantship in the Civil and
Environmental Engineering Department and in the Department of Physics and Astronomy. I am
indebted for this support, which made my studies at USC possible.
I would like to express my deepest gratitude to my adviser Professor Maria I. Todorovska.
This dissertation was possible through her guidance, valuable comments, and support during my
PhD study. Professor Todorovska is unique in dedicating her time to her students to help them do
high quality research. I am very much indebted to her and never am able to thank her enough.
I also would like to cordially thank Professor Mihailo D. Trifunac and Professor Vincent W.
Lee for their valuable advice and support during my PhD study. It has truly been an honor and a
pleasure to work with them in past five years. I also would like to thank my committee members,
Professor Firdaus E. Udwadia and Professor Hung L. Wong for their constructive comments on
my works.
The strong motion data used in this dissertation was provided by the California Strong
Motion Instrumentation Program (CSMIP) via the Center for Engineering Strong Motion Data
(www.strongmotioncenter.org/).
I would like to thank all the faculty and staff members of the Department of Civil and
Environmental Engineering at the University of Southern California for their generous help
during my stay at USC.
Finally, I wish to convey my appreciation to my loving family for their unconditional
support in the past five years.
v

LIST OF JOURNAL PAPERS BASED ON THIS THESIS

1. Todorovska MI, Rahmani M (2013). System identification of buildings by wave travel time
analysis and layered shear beam models - spatial resolution and accuracy, Structural Control
and Health Monitoring, 20(5): 686–702, DOI: 10.1002/stc.1484, first published online on
3/6/2012.
2. Rahmani M, Todorovska MI (2013). 1D system identification of buildings from earthquake
response by seismic interferometry with waveform inversion of impulse responses – method
and application to Millikan Library, Soil Dynamics and Earthquake Engrg, Jose Roësset
Special Issue, E. Kausel and J. E. Luco, Guest Editors, 47: 157-174, DOI:
10.1016/j.soildyn.2012.09.014.
3. Rahmani M, Todorovska MI (2014). 1D system identification of a 54-story steel frame
building by seismic interferometry, Earthquake Engng Struct. Dyn., 43:627–640, DOI:
10.1002/eqe.2364, first published online of 9/13/2013.
4. Rahmani M, Todorovska MI (2014). Structural health monitoring of a 54-story steel frame
building using a wave method and earthquake records, Earthquake Spectra, DOI:
10.1193/112912EQS339M, in press, fist published online on 10/15/2013.
5. Rahmani M, Ebrahimian M, Todorovska MI (2014). Wave dispersion in high-rise buildings
due to soil-structure interaction, Earthq. Eng. Struct. Dyn., DOI: 10.1002/eqe.2454.
6. Ebrahimian M, Rahmani M, Todorovska MI (2014). Nonparametric identification of wave
dispersion in high-rise buildings by seismic interferometry, Earthq. Eng. Struct. Dyn., DOI:
10.1002/eqe.2453.
7. Rahmani M, Ebrahimian M, Todorovska M. I. (2014). Automated time-velocity analysis for
early earthquake damage detection in buildings: Application to a damaged full-scale RC
building and comparison with other methods, Earthq. Eng. Struct. Dyn., Special issue on:
Earthquake Engineering Applications of Structural Health Monitoring, CR Farrar and JL
Beck Guest Editors, submitted for publication, 01/16/2014.

Final drafts of these publications can be downloaded from the project web page at
http://www.usc.edu/dept/civil_eng/Earthquake_eng/Earthquake_damage_detection_NSF_2008/
vi

ABSTRACT
Monitoring the integrity of a building based on instrumental data and detecting damage
during or soon after an earthquake or some other natural or manmade disaster may significantly
reduce loss of life and injuries caused by potential collapse of a weakened structure during
shaking from aftershocks and facilitate emergency response in large cities. Likewise, it would
help prevent or reduce loss of function of critical facilities and monetary losses caused by
needless evacuation of a structure that is safe. This dissertation presents a wave method for
structural system identification and structural health monitoring (SHM) of high-rise buildings.
The method is robust when applied to real structures and large amplitude response, and is not
sensitive to the properties of the underlying soil and its changes. It is intended for use in seismic
alert systems as well as for general condition monitoring.
The method uses data from an array of accelerometers, identifying the velocities of waves
propagating vertically through the structure, and detects changes in these velocities, possibly
caused by damage. The identification is based on fitting a layered shear beam model of the
buildings, such that the layers correspond, in general, to a group of floors. The fitting involves
matching propagating pulses in impulse response functions computed at different levels in the
building. Three identification algorithms are presented: 1) the direct (ray) algorithm, which is
based on reading the pulse arrival time, 2) the time shift matching algorithm, which is an iterative
version of the direct algorithm, and 3) the waveform inversion algorithm, which involves
nonlinear least squares (LSQ) fit of the pulses as waveforms over a selected time interval. The
latter is extended to an automated moving window analysis, i.e. time-velocity analysis, for
detecting changes during an earthquake.
Detailed applications of the method are presented for three full-scale buildings: (1) a
typically instrumented tall steel building during six earthquakes over a period of 19 years, none of
which caused damage (Los Angeles 54-story Office Building), (2) a densely instrumented high-
vii

rise reinforced concrete (RC) building during one smaller local earthquake (Millikan library in
Pasadena), and (3) a damaged high-rise RC building (Sherman Oaks 12-story Office Building,
during the San Fernando earthquake of 1971). Also, analyses of models are presented, which
demonstrate the insensitivity of the method to the effects of soil-structure interaction, the effects
of the slabs on the wave propagation, and the modeling error when the method is applied to
buildings with significant bending deformation (e.g. buildings with shear walls).
It is concluded that the presented method is robust, accurate and sensitive to damage, being
able to detect considerable changes in an RC building that has been only lightly damaged, and
permanent changes over time in a tall steel building that has shown no signs of damage. With
careful consideration of the frequency bands for the fit, it can be applied to many high rise
buildings typical of metropolitan areas, such as, e.g. Los Angeles and San Francisco in California,
and urban areas in general. Directions for research on further development of the method are
identified.

viii

EXECUTIVE SUMMARY
This dissertation presents a major development of a wave method for structural health
monitoring (SHM) introduced recently in conceptual and proof of concept studies. The method is
intended for use in seismic alert systems as well as for general condition monitoring. It is based
on identifying the velocities of waves propagating vertically through the structure and detecting
changes in these velocities, during particular strong earthquake shaking or between events.
Inferences about the state of damage are made based on calibration data, considering the
estimation error and the variability of the wave velocities due to factors other than damage. The
method uses data from an array of vibrational sensors, such as accelerometers, which exist in
many high-rise buildings located in seismic areas in the world, such as e.g. metropolitan Los
Angeles and San Francisco in California.
The most favorable features of the wave method are its (i) robustness when applied to large
amplitude response data and real full-scale buildings, damaged or undamaged, and its (ii)
insensitivity to the effects of soil-structure interaction. The robustness is a problem for many of
the published methods, while the latter is a problem for the most wide spread robust method,
which is based on detecting changes in the fundamental frequency of vibration. The method is
also (iii) local in nature. However, higher resolution and high accuracy of the identification are
required to practically realize the local nature of the method.
In this method, the wave velocities are identified essentially from the phase of the response
using propagating pulses in the building impulse response functions for virtual source at the roof.
In particular, propagating pulses in low pass filtered impulse response functions of a
mathematical model of the building are fitted in such pulses computed from the observed
response. In this dissertation work, a simple continuous model is used for the fit, such as a shear
beam with piecewise constant material properties along the height. The effect of dispersion is
ix

controlled by carefully choosing the cut-off frequency for the low pass filter. Propagation of
waves related to both lateral and torsional deformation of the structure is considered.
The methodological developments in this dissertation include several fitting algorithms,
which are critically examined. Two simpler algorithms (one direct and one iterative), based on
the Ray Theory approximation of wave propagation, match only the time shifts of the pulses,
while the more accurate nonlinear least squares fit algorithm matches the pulses as functions of
time over selected time windows, therefore taking into account wave phenomena such as
scattering. The Lavenberg-Marquardt algorithm is used for the fit, which was shown to be stable
for well chosen initial conditions. In addition, an algorithm for automated time-velocity analysis
is presented, which consists of identification of the model velocities in moving time windows,
assuming that the response is linear within each window.
The analysis of Los Angeles 54-story Office Building, by a 4-layer model above ground
level, showed that a layered shear beam is a good physical model for this building over a band
that includes the first five modes of vibration. The average velocity over the height was found to
be about 140 m/s for the NS and EW responses, and about 265 m/s for the torsional response.
The study revealed about 10-12% permanent change in the overall stiffness, caused mainly by the
Landers and Big Bear, 1992 sequence and the Northridge, 1994 earthquake, even though the
building showed no signs of damage. This change is well beyond the estimation error and the
standard deviation of the variability of the identified velocities during the six earthquakes.
The analysis of the densely instrumented Millikan library, which has external shear walls
and a central core, showed evidence of dispersion in the pulse propagation, apparently caused by
bending deformation. Because the fitted model does not account for bending, its effects are
controlled by identification in a band that contains only the first two modes of vibration. The
identified velocities are representative of the higher frequencies in the band. The average
velocities were found to be about 400 m/s for the NS response, about 240 m/s and 340 m/s in two
x

frequency bands for the EW response, and about 460 m/s for the torsional response. The analysis
showed that there is a tradeoff between resolution and accuracy of the identification. It also
showed that the spatial resolution of the identification is controlled by the bandwidth of the
impulse responses, rather than by the density of sensors, and that the velocity profile is aliased if
the bandwidth is too small for the number of layers in the fitted model. This is contrary to the
popular belief that the density of sensors alone is sufficient for higher resolution identification.
The number of sensors, however, helps reduce the aliasing error, in statistical sense by increasing
the number of observation points. The theoretical minimum layer thickness that can be resolved is
one quarter of the shortest wavelength in the band.
The Sherman Oaks 12-story Office Building was analyzed in moving time windows by the
automatic time-velocity algorithm, using a two layer model. The initial average wave velocities
were found to be about 140 m/s for the NS and about 110 m/s for the EW response. Even though
the observed damage has been described as light, significant changes in the wave velocities were
detected, well beyond the estimation error. For the NS response, reduction of wave velocity of
about 30% in the lower half and about 23% in the upper half of the building was detected. For the
EW response, the corresponding changes were 37% and 27%. The average interstory drifts
reached about 0.7% for the NS and about 1.3% for the EW response. Time frequency analysis
was also conducted for the fundamental apparent frequency, which showed reduction of about
44% for the NS and about 48% for the EW response. The larger reduction of the apparent
frequency than the reduction of wave velocity was interpreted to be due to contribution from
nonlinearity in the response of the soil-foundation system.
A model with slabs, in which the floor slabs are considered as separate very stiff layers, was
also analyzed and used for fitting. The analysis showed that wave propagation in a model with
slabs is dispersive, due to scattering from the slabs, and results in velocities lower than the shear
wave velocity in the soft layers. The existence of pass and stop frequency bands in such a model,
xi

revealed in a previously published analytical study, is also discussed and their relation to the
typical recording range in buildings. The nature of the dispersion, however, is such that the phase
velocity changes slowly for lower frequencies, with little evidence of it in low pass filtered
impulse response functions. As shown in this dissertation, low pass filtered impulse response
functions and transfer-functions of a model with slabs can be closely matched by a model without
slabs, but with reduced velocity as compared to the velocity of the soft layers in the model with
slabs. Therefore, in identifying real buildings from low pass filtered impulse responses, there is
no practical advantage of fitting a model with slabs, which is computationally much more
intensive.
This dissertation also includes a study of a simple soil-structure interaction (SSI) model with
coupled horizontal and rocking response. The analysis shows that wave propagation in the
system is dispersive, characterized by lower phase velocity near the frequency of the fundamental
mode, which is the one most affected by the interaction. It also explains why the measured
velocity of the pulse in broader band impulse responses (i.e. over a band that includes at least one
of the higher modes of vibration) is affected little by the SSI, reflecting the properties of the
structure itself.
Finally, the modeling error is investigated due to the fact that the fitted model does not
account for bending deformation. Such deformation is significant for shear wall structures and
for structures with a combined moment frame and shear wall lateral resisting system. For that
purpose, a layered shear beam model is fitted in response of uniform Timoshenko beam models
of Millikan library NS and EW responses. The analysis shows that the identified velocity profile
is distorted in a way that it shows decreasing shear stiffness towards the top even though the
Timoshenko beam model has uniform properties. The lack of bending flexibility in the fitted
model is, therefore, compensated for by increasing shear flexibility near the top.
xii

It is concluded that the presented wave method for SHM is robust, efficient, accurate,
sensitive even to smaller changes in the structural stiffness and not sensitive to changes in the
soil. These are all favorable features for its practical implementation in real structures as a
decision making tool. More similar studies of other damaged and undamaged real buildings,
complemented with studies of full-scale physical models in shake table experiments, are
necessary to further calibrate the method. Directions for further development of the method are
also identified, such as fitting more realistic models.

xiii

xiv

TABLE OF CONTENTS

Dedication ....................................................................................................... iv
Acknowledgements ........................................................................................... v
List of Publications ......................................................................................... vi
Abstract .......................................................................................................... vii
Executive Summary ........................................................................................ ix
List of Tables ................................................................................................. xix
List of Figures ............................................................................................. xxiii
Chapter 1: Introduction ................................................................................... 1
1.1 Problem Statement and Dissertation Objectives ................................................... 2
1.2 Review of Structural Health Monitoring Methods ................................................. 7
1.3 Wave Methods for Structural Health Monitoring ................................................ 18
1.4 Selection of Case Studies for this Dissertation ................................................... 25
1.5 Organization of this Dissertation ......................................................................... 30
Chapter 2: Methodology ................................................................................ 33
2.1 Impulse Response Functions of a Linear System ................................................... 35
2.2 The Forward Problem ............................................................................................ 38
2.2.1 Modeling of the Translational Response ......................................................................... 38
2.2.2 Analytical Low-Pass Filtered Impulse Response Functions .......................................... 44
2.2.3 Extension to Torsional Response .................................................................................... 47
2.3 The Inverse Problem .............................................................................................. 51
2.3.1 The Direct (Ray) Algorithm ........................................................................................... 52
2.3.2 The Nonlinear Least Squares Algorithm ........................................................................ 54
2.3.3 The Time Shift Matching Algorithm .............................................................................. 56
2.3.4 Extension to Moving Window Analysis - Time-Velocity Graphs ................................... 56
2.3.5 Choice of Time Windows ................................................................................................ 58
2.3.6 Choice of Frequency Band .............................................................................................. 59
2.3.7 Resolution and Accuracy ................................................................................................ 60
2.3.8 Model Validation ............................................................................................................ 65
Chapter 3: Critical Assessment of the Direct (Ray) Algorithm and Travel
Time Analysis ................................................................................................. 67
3.1 Analysis of Simulated Impulse Response Functions .............................................. 68
xv

3.1.1 Model .............................................................................................................................. 68
3.1.2 Pulse Travel Time .......................................................................................................... 70
3.1.3 Pulse Amplitudes ............................................................................................................ 73
3.2 Case Study - Millikan Library N-S Response ......................................................... 75
3.3 Demonstration of the Identification Error ............................................................. 82
3.4 Summary and Conclusion ...................................................................................... 83
Chapter 4: Structural System Identification and Health Monitoring of a Tall
Steel Frame Building – Los Angeles 54-Story Office Building ....................... 85
4.1 System Identification of LA 54-Story Office Building during Northridge
Earthquake of 1994 ...................................................................................................... 87
4.1.1 Building Description and Earthquake Data ................................................................... 89
4.1.2 Apparent Frequencies of Vibration and Pulse Arrival Times ........................................ 94
4.1.3 Equivalent Uniform Shear Beam Models ........................................................................ 97
4.1.4 Equivalent 4-Layer Shear Beam Models ......................................................................... 97
4.1.5 Summary and Conclusion ............................................................................................. 105
4.2 Structural Health Monitoring of LA 54-story Office Building during Six
Earthquakes between 1992 and 2010 .......................................................................... 106
4.2.1 Strong Motion Records in Building .............................................................................. 109
4.2.2 Identified Parameters and Sample Statistics ................................................................ 111
4.2.3 Trends and Permanent Changes .................................................................................. 119
4.2.4 Resolution and Sensitivity to Localized Damage .......................................................... 128
4.2.5 Summary and Conclusion ............................................................................................. 130
Chapter 5: Identification of a Densely Instrumented High-Rise Building –
Millikan Library 9-Story RC Structure ....................................................... 133
5.1 Building Description ............................................................................................. 134
5.2 Earthquake Records of Yorba Linda Earthquake of 2002 ..................................... 136
5.3 Equivalent Uniform Models .................................................................................. 141
5.4 Multilayer Model Fitting ....................................................................................... 143
5.5 Summary and Conclusion ..................................................................................... 157
Chapter 6: Earthquake Damage Detection in an RC High-Rise Building -
Sherman Oaks 12-Story Office Building Lightly Damaged by San Fernando
Earthquake of 1971 ...................................................................................... 159
6.1 Building Description ............................................................................................. 160
6.2 Structural Damage Description ............................................................................. 162
6.3 Strong Motion Data and Correction for Synchronization Errors .......................... 164
6.4 Time-Velocity Graphs and Interstory Drifts ......................................................... 167
6.5 Time-Frequency vs. Time-Velocity Analysis ......................................................... 172
6.6 Summary and Conclusion ..................................................................................... 175
Chapter 7: Identification by Fitting a Model with Slabs ............................. 177
7.1 Identification of Millikan Library N-S Response ................................................... 179
7.1.1 Model Description ......................................................................................................... 179
7.1.2 Results and Discussion ................................................................................................. 181
7.2 Investigation of Wave Dispersion in a Model with Slabs ...................................... 190
xvi

7.2.1 The Model .................................................................................................................... 190
7.2.2 Identification Results .................................................................................................... 191
7.2.3 Analytical Dispersion for a Periodic Structure ............................................................. 193
7.2.4 Comparison of Identification Results with Analytical Predictions ............................... 195
7.2.5 Effect of the Stop Band on the Model Impulse Responses ........................................... 197
7.2.6 Assessment of the Goodness of the Adjusted Model .................................................... 198
7.3 Summary and Conclusion ..................................................................................... 202
Chapter 8: Dispersion due to Soil-Structure Interaction and Why the
Method is not Sensitive to the SSI Effects .................................................. 203
8.1 Interferometric Identification of a Soil-Structure Interaction Model ..................... 204
8.1.1 Models and Parameters ................................................................................................ 205
8.1.2 Analysis and Results ..................................................................................................... 206
8.2 Discussion and Conclusion .................................................................................... 213
Chapter 9: Modeling Error Due to Bending Deformation ........................... 215
9.1 Benchmark Timoshenko Beam Models ................................................................. 216
9.2 Modeling Error in Fitting a Layered Shear-Beam ................................................. 217
9.3 Conclusion ............................................................................................................. 220
Chapter 10: Summary and Conclusions ....................................................... 221
10.1 Brief Summary of Findings of this Dissertation Research ................................... 222
10.2 Conclusions and Recommendations for Future Work ......................................... 226
Bibliography ................................................................................................. 227
About the Author ........................................................................................ 237

xvii

xviii

LIST OF TABLES

Table 1.1 Structural performance levels and damage with respect to the maximum Inter-story
Drift, Steel and Concrete frames, taken from FEMA356 (ASCE 41, 2000), table C1-3 ............... 10
Table 1.2 A list of all considered buildings for our preliminary analyses ..................................... 28
Table 3.1 Input parameters for the equivalent uniform model ...................................................... 69
Table 3.2 Input parameters for the 3-layer model .......................................................................... 69
Table 3.3 Shear wave velocities of Millikan Library NS response identified by the direct
algorithm based on a 1-layer model, using accelerations of Yorba Linda earthquake of 2002
recorded at west wall (0-15 Hz) ..................................................................................................... 79
Table 3.4 Shear wave velocities of Millikan Library NS response identified by the direct
algorithm based on a 3-layer model, using accelerations of Yorba Linda earthquake of 2002
recorded at west wall (0-15 Hz) ..................................................................................................... 79
Table 3.5a Shear wave velocities of Millikan Library NS response identified by the direct
algorithm based on a 8-layer model and using accelerations of Yorba Linda earthquake of 2002
recorded at west wall (0-15 Hz) ..................................................................................................... 81
Table 3.5b Shear wave velocities of Millikan Library NS response identified by the direct travel
time method based on a 8-layer model and using accelerations of Yorba Linda earthquake of
2002 recorded at west wall (0-25 Hz) ............................................................................................ 81
Table 4.1 Los Angeles 54-story office building: observed (apparent) modal frequencies during
Northridge, 1994 earthquake, and their ratios. The ratios of the fixed-base frequencies for a
uniform shear-beam are also shown for comparison ..................................................................... 95
Table 4.2 Los Angeles 54-story office building: observed pulse arrival times (average for the
causal and acausal pulses) during Northridge earthquake of 1994. ............................................... 95
ave
i
t
Table 4.3 Identified equivalent uniform model wave velocity
eq
β , apparent quality factor Q and
apparent damping ratio ζ during Northridge earthquake of 1994 ................................................ 98
Table 4.4 Los Angeles 54-story office building: identified 4-layer model wave velocities during
Northridge, 1994 earthquake for the NS, EW and torsional responses ........................................ 98
xix

Table 4.5 Test of accuracy no. 1: 4-layer model is fitted in simulated IRFs for a uniform beam (
ζ = 2%, sampling interval 0.005 s) in the band 0-1.8 Hz (contains 6 modes) ............................ 101
Table 4.6 Test of accuracy no. 2: 4-layer model is fitted in simulated IRFs for a 4-layer beam with
non-monotonically varying wave velocity ( ζ = 2%, sampling interval 0.005 s) in the band 0-1.7
Hz (contains 5 modes). ................................................................................................................ 101
Table 4.7 Los Angeles 54-story office building: observed (apparent) frequencies and fitted
model (fixed-base) frequencies
i,app
f
i
f frequencies during Northridge, 1994 earthquake ................ 104
Table 4.8 List of earthquakes recorded in the building (CSMIP Station 24629; 34.048 N, 118.26
W) ................................................................................................................................................ 107
Table 4.9 Identification results for equivalent uniform model during six earthquakes .............. 115
Table 4.10 Identification results for equivalent 4-layer model during six earthquakes .............. 117
Table 4.11 Sample statistics for the six earthquakes ( μ=sample mean, s=sample standard
deviation, / s μ =sample coefficient of variation). ....................................................................... 118
Table 5.1 Observed pulse arrival times (average for the causal and acausal pulses) at each
floor in Millikan library response during the Yorba Linda earthquake of 2002, and identified (by
the direct algorithm) equivalent wave velocities,
ave
i
t
eq
β , factors and damping ratios Q 1/ 2Q ζ =
for the NS, EW and torsional motions ......................................................................................... 142
Table 5.2 Equivalent (over the height of the structure) wave velocity,
eq
β , and apparent quality
factor, Q, and damping ratio, ζ , of Millikan Library NS, EW and torsional responses during
Yorba Linda earthquake of 2002, as identified by the direct algorithm ...................................... 142
Table 5.3 Comparison of shear wave velocities of Millikan Library NS response during Yorba
Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-15 Hz) ........................................................ 144
Table 5.4 Comparison of shear wave velocities of Millikan Library EW response during Yorba
Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-7.5 Hz) ....................................................... 145
Table 5.5 Comparison of shear wave velocities of Millikan Library EW response during Yorba
Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (7-15 Hz) ........................................................ 145
Table 5.6 Comparison of torsional wave velocities of Millikan Library EW response during
Yorba Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-25 Hz) ........................................................ 154
xx

Table 5.7 Comparison of shear wave velocities of Millikan Library NS response during Yorba
Linda earthquake of 2002, as identified using 9-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-15 Hz) ........................................................ 156
Table 5.8 Comparison of shear wave velocities of Millikan Library EW response during Yorba
Linda earthquake of 2002, as identified using 9-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-7.5 Hz) ....................................................... 156
Table 5.9 Apparent and fixed-base frequencies of vibration of Millikan Library NS, EW and
torsional responses identified by this study (nonlinear LSQ fit algorithm) from records of Yorba
Linda earthquake of 2002, and identified by Luco et al. (1987) from forced vibration tests
conducted in 1975 (Foutch et al. 1975) ....................................................................................... 157
Table 7.1 Illustration of significant scattering phenomena in models with inserted slabs.
Comparison of shear wave velocities and pulse arrival times of Millikan Library NS response, as
identified using a fitted 8-layer model without the slabs, with post inserted slabs, and adjusted
model with slabs (0-15 Hz, 2% ζ = ) ............................................................................................ 184
Table 7.2 Comparison of shear wave velocities of Millikan Library NS response during Yorba
Linda earthquake of 2002, as identified using 8-layer model with and without the slabs by
nonlinear LSQ and Adjusting of velocity profile (0-15 Hz, 3.6% ζ = ) ........................................ 184
Table 7.3 Illustration of significant scattering phenomena in models with inserted slabs.
Comparison of shear wave velocities and pulse arrival times of model with no slab, model with
inserted slabs, and two models with Adjusted velocity profiles. AF denotes the adjusting factor
for each layer. (0-9 Hz, 2% ζ = ) ............................................................................................... 191
Table 8.1 Identified wave velocity in the SSI system from low-pass and band-pass filtered
impulse response functions in different bands ( are estimates obtained directly from pulse
travel time and and
D
eq
c
LSQ
eq
c σ are waveform inversion estimates and the corresponding standard
deviation). Comparison with the true velocity of the superstructure is also shown ................ 209
S
c
Table 8.2 Identified wave velocities, ()
res
i
c f , of the SSI system from the system frequencies,
i
f ,
and comparison with the true shear wave velocity of the superstructure, .............................. 210
S
c
Table 9.1 Properties of an approximate uniform TB modelof Millikan library NS and EW fix-
base responses (Ebrahimian et al. 2014). .................................................................................... 217
Table 9.2 Identified shear wave velocities,
D
j
c and
LSQ
j
c , of an equivalent 3-layer shear
beam fitted in the response of uniform TB model NS (top) and EW (bottom) by the direct
and least squares fit algorithms ............................................................................................... 218

xxi

xxii

LIST OF FIGURES

Figure 1.1 A view of downtown Los Angeles from USC university park campus (Courtesy of
Lance Hill) ....................................................................................................................................... 3
Figure 1.2 Main elements of Structural Health Monitoring (SHM) [drawn by M. Rahmani] ......... 6
Figure 1.3 A schematic diagram of Max. Drift and associated performance levels in an Steel
frame, presented in FEMA356 (drawn by M. Rahmani) ............................................................... 10
Figure 1.4 A sample fragility function for three damage levels [Naeim et al. 2006] .................... 11
Figure 2.1 Virtual source function in the frequency (left) and time (right) domains, for a box
function .......................................................................................................................................... 37
Figure 2.2 The model. a) Layered shear-beam representing a building. b) Layered half-space .... 38
Figure 2.3 A stack of rigid disks twisting on each other, illustrating the lowest mode of torsional
wave in a solid rod (shaft) .............................................................................................................. 49
Figure 2.4 (a) Two adjacent sections of the rod (i.e. two disks) twisting due to a propagating
torsional wave. (b) Cross-section of the rod and shear stress distribution ..................................... 50
Figure 2.5 Illustration of the Time-Velocity algorithm for two records ........................................ 57
Figure 3.1 Profiles of mass density and shear wave velocity distributions for the 3-layer (solid
line) and the uniform model (dashed line). .................................................................................... 69
Figure 3.2 Impulse response functions at each floor for the 3-layer model, for virtual source (a) at
ground floor and (b) at roof ........................................................................................................... 71
Figure 3.3 Pulse arrival times for the 3-layer model as measured from simulated impulse response
functions ......................................................................................................................................... 72
Figure 3.4 Attenuation of pulse amplitudes measured from IRFs (virtual source at roof) versus
bandwidth
max
f . a) Attenuation on the way up, , on the way down, , and
total, , for the 3-layer model. b) Total attenuation, , for the 3-layer and uniform
models (fixed-base), and for a uniform model with foundation rocking (Todorovska, 2009a) ..... 74
/
top a
AA /
cto
AA
p
a
/
c
A
a
A /
c
AA
xxiii

Figure 3.5 Millikan library NS response during Loma Linda, 2002 earthquake observed at West
wall. a) Accelerations. b) Impulse responses for virtual source at roof. c) Transfer-function of
roof acceleration w.r.t. ground floor .............................................................................................. 76
Figure 3.6 Identification of Millikan library NS response during Yorba Linda earthquake of 2002.
a) Identified shear wave velocity profiles. b) Agreement of pulse arrival times at the different
floors. Note: the arrival at the 9
th
floor could not be resolved. c) Agreement of transfer-functions
....................................................................................................................................................... 78
Figure 3.7 Pulse arrival time at each floor: as measured from IRF of 1-layer model vs. theoretical
....................................................................................................................................................... 82
Figure 4.1 Los Angeles 54-story office building (CSMIP 24629): a) photo; b) vertical cross-
section, and c) typical floor layouts (redrawn from www.strongmotioncenter.org). ..................... 88
Figure 4.2 The layered shear beam model .................................................................................... 88
Figure 4.3 Los Angeles 54-story office building: NS impulse response at roof, Northridge, 1994
earthquake, 0-1.7 Hz. ..................................................................................................................... 89
Figure 4.4 Observed NS acceleration at the West wall, EW accelerations at the North wall, and
NS accelerations at the East wall due to torsion during Northridge earthquake of 1994 .............. 91
Figure 4.5 Fourier transform amplitudes of absolute NS accelerations at West wall, NS
accelerations at penthouse level due to rigid body rocking, EW accelerations at North wall, and
NS accelerations at East wall due to torsion, observed during Northridge earthquake of 1994.
Also transfer-functions are shown between penthouse and P4 level accelerations. ...................... 92
Figure 4.6 Fourier transform amplitudes of absolute NS displacement at West wall, NS
displacement at penthouse level due to rigid body rocking, and EW displacement at North wall.
....................................................................................................................................................... 93
Figure 4.7 Observed and fitted impulse response functions for NS at west wall, EW and torsional
responses ........................................................................................................................................ 96
Figure 4.8 Observed pulse arrival time at the layer boundaries , assumed mass density profile
i
t
() z ρ , the identified wave velocity profiles () z β during Northridge, 1994 earthquake, and
average layer peak drift () z γ for the NS, EW and torsional responses ..................................... 100
Figure 4.9 Observed and fitted transfer functions (TF) between accelerations at penthouse and
ground/P4 levels .......................................................................................................................... 102
Figure 4.10 Model mode shapes for NS (top) and EW (bottom) responses. ............................... 104
Figure 4.11 Google map of the epicenters of the earthquakes recorded in the building .............. 107
xxiv

Figure 4.12 Penthouse displacements and base accelerations observed during the six earthquakes.
..................................................................................................................................................... 110
Figure 4.13 Average drifts observed during the six earthquakes ................................................ 110
Figure 4.14 Transfer functions of observed NS, EW and torsional responses during six
earthquakes. ................................................................................................................................. 112
Figure 4.15 Impulse response functions of observed NS and EW responses during six
earthquakes. ................................................................................................................................. 113
Figure 4.16 Identified wave velocities in the layers for the six events ....................................... 120
Figure 4.17 Peak stress (~
2
eq
β γ ) vs. peak strain ( γ ) relations for the six events ...................... 121
Figure 4.18 Roof displacement during the six earthquakes ........................................................ 121
Figure 4.19 Identified global parameters during the six earthquakes vs. weighted peak drift,
w
γ .
..................................................................................................................................................... 123
Figure 4.20 Reduction of global stiffness, measured by the reduction of
eq
β ,

and .
..................................................................................................................................................... 124
1,app
f
2,app
f
Figure 4.21 Same as Fig. 4.18, but for the layer velocities vs. layer peak layer drift,
i
γ . .......... 126
Figure 4.22 Reduction of local (layer) stiffness, measured by the reduction of , vs.
peak layer drift,
,1,...,4
i
i β =
i
γ ...................................................................................................................... 127
Figure 4.23 Change of equivalent layer velocity due to local change within the layer. .............. 129
Figure 5.1 Millikan library: a) photo (courtesy of M. Trifunac); b) vertical cross-section and c)
typical floor layout (redrawn from Snieder and Safak 2006); (d) sensor locations at basement . 135
Figure 5.2 Millikan library NS response during Loma Linda, 2002 earthquake observed at West
wall (Fig. 3.1). a) Accelerations. b) Impulse responses for virtual source at roof. c) Transfer-
function of roof acceleration w.r.t. ground floor .......................................................................... 137
Figure 5.3 Millikan library EW response during Loma Linda, 2002 earthquake observed at West
wall (Fig. 3.1). a) Accelerations. b) Impulse responses for virtual source at roof. c) Transfer-
function of roof acceleration w.r.t. ground floor .......................................................................... 138
Figure 5.4 Millikan library torsional response during Loma Linda, 2002 earthquake. a) Difference
of NS accelerations recorded at East and West wall. b) Impulse responses for virtual source at
roof. c) Transfer-function of roof torsional acceleration w.r.t. ground floor ............................... 139
xxv

Figure 5.5 Evidence of dispersion in the observed IRFs of EW response of Millikan library during
Loma Linda, 2002 earthquake. a) 0-15 Hz (solid line) and 0-7.5 Hz (dashed line). b) 0-7.5 Hz. c)
7.5-15 Hz ..................................................................................................................................... 140
Figure 5.6 Identified velocity profiles, obtained by different algorithms, of Millikan library during
the Yorba Linda EQ, 2002. a) NS response (0-15 Hz), b) EW response (0-7.7.5 Hz), EW
response (7-15 Hz), and d) torsional response (0-25 Hz). The algorithms used are: LSQ=nonlinear
least squares fit of transmitted pulses, TSM=time shift matching, and Direct=the direct algorithm
..................................................................................................................................................... 146
Figure 5.7 Agreement of IRFs of models identified by the nonlinear LSQ algorithm with the
observed ones. a) NS response, 0-15 Hz. b) EW response, 0-7.5 Hz. c) Torsional response, 0-
25 Hz . 2% ζ = was assumed .................................................................................................... 148
Figure 5.8 Illustration of non-linear least square estimation convergence. Shear wave velocity of
layer 8+9 (top two layers) of Millikan Library, N-S component (see table 3.4) ......................... 148
Figure 5.9 Other measures of agreement of the identified models of the NS response of Millikan
library with the observations during Yorba Linda earthquake of 2002. a) Agreement of pulse
arrival times at the different floors. b) Agreement of transfer-functions. Note: the arrival at the
9th floor could not be resolved .................................................................................................... 150
Figure 5.10 Other measures of agreement of the identified models of the EW response of
Millikan library with the observations during Yorba Linda earthquake of 2002. a) Agreement of
pulse arrival times at the different floors. b) Agreement of transfer-functions. Note: the arrival at
the 9th floor could not be resolved .............................................................................................. 152
Figure 5.11 Other measures of agreement of the identified models of the Torsional response of
Millikan library with the observations during Yorba Linda earthquake of 2002. a) Agreement of
pulse arrival times at the different floors. b) Agreement of transfer-functions. Note: the arrival at
the 9th floor could not be resolved, and the observed torsion is not available at the 2nd and 8th
floors ............................................................................................................................................ 153
Figure 5.12 Identified velocity profiles, obtained by LSQ algorithms for 8-layer and 9-layer
models, of Millikan library during the Yorba Linda, 2002, earthquake. a) NS response (0-15 Hz),
b) EW response (0-7.7.5 Hz), using LSQ algorithm .................................................................... 155
Figure 6.1 Sherman Oaks 12-story Office Building: plans of the instrumented floors (middle) and
vertical cross section (drawn based on John Blume Associates (1973)). Instrument locations are
marked. ........................................................................................................................................ 161
Figure 6.2 Map of San Fernando Valley showing the location of Sherman Oaks 12-story Office
Building relative to the local freeways and the ruptured area of the San Fernando, 1971
earthquake (drawn after (Trifunac, 1974). A photo of the building is also shown (courtesy of M.
Trifunac) ...................................................................................................................................... 162
xxvi

Figure 6.3 Distribution of structural damage along the height in Sherman Oaks 12-story Office
Building during the San Fernando, 1971 earthquake (left; drawn based on John Blume
Associates, 1973) and the 2-layer shear beam model to be fitted (right) ..................................... 163
Figure 6.4 Comparison of IRFs before and after correction for synchronization ....................... 165
Figure 6.5 Recorded motions in Sherman Oaks 12-story Office Building during the San
Fernando, 1971 earthquake: a) acceleration time histories and b) Fourier Transform amplitudes
of acceleration at roof, 7
th
and Ground floors .............................................................................. 166
Figure 6.6 Comparison of Impulse Response Functions (part a)) and Transfer-Functions (part b))
for two 10 s time windows, one in the beginning of shaking (2 – 12 s) and the other one near the
end of shaking (22 - 32 s). Changes in wave travel time and frequencies of vibration can be seen
..................................................................................................................................................... 168
Figure 6.7 a) Time-velocity graphs and interstory drifts for Layers 1 and 2. b) Measures of input
energy () E t and power , estimated from the ground floor velocity ............................... 170 () Pt
Figure 6.8 Estimation of loss of stiffness with time by different methods: a) Instantaneous
frequency (estimated from the ridge of the Gabor Transform of relative roof acceleration); b)
Equivalent uniform shear beam velocity ..................................................................................... 174
Figure 7.1 Illustration of shear beam models of Millikan library including floors’ slabs. a) vertical
section of building (E-W elevation), b) fitted layered shear beam model to N-S response (results
taken from chapter 5), c) the same model as b) including 9 post-inserted slabs, d) adjusted model
with slabs (adjusting factor, α ), e) fitted shear beam model including 9 slabs (a total of 18
lay ers) ........................................................................................................................................... 180
Figure 7.2 Phenomena illustration - Comparison of observed IRFs of Millikan library during
Yorba Linda 2002 EQ with those of 1) identified models using LSQ algorithm without slabs, 2)
LSQ model with post-inserted slabs ............................................................................................ 185
Figure 7.3 Phenomena illustration - Comparison of observed IRFs of Millikan library during
Yorba Linda 2002 EQ with those of 1) identified models using LSQ algorithm without slabs, 2)
the Adjusted model with slabs ...................................................................................................... 186
Figure 7.4 Comparison of observed TF of Millikan library during Yorba Linda 2002 EQ with
identified models using LSQ algorithm without slabs and model with post-inserted slabs ........ 187
Figure 7.5 Comparison of observed TF of Millikan library during Yorba Linda 2002 EQ with
identified models using LSQ algorithm without slabs and the Adjusted model with slabs ......... 187
Figure 7.6 Comparison of observed IRFs of Millikan library during Yorba Linda 2002 EQ with
those of identified models using LSQ algorithm with and without slabs .................................... 188
Figure 7.7 Comparison of observed TF of Millikan library (N-S) during Yorba Linda 2002 EQ
with identified models using LSQ algorithm with and without slabs .......................................... 189
xxvii

Figure 7.8 a) The 9-layer shear beam model, b) shear beam model with 9 added slabs at the
layers’ boundaries ....................................................................................................................... 190
Figure 7.9 IRFs of models with and without slabs; Pulse arrival time elongation illustration .... 192
Figure 7.10 Comparison of TFs of models with and without slabs (Fig. 4.7 and 4.8) ................. 192
Figure 7.11 Dispersion curves for uniform shear beam with added lumped mass. Comparison of
analytical solution with a slab model. Velocities are estimated using low-pass filtered IRFs. ... 196
Figure 7.12 The same as Fig.6.11, but the wave velocities are estimated using resonance
frequencies of the slab model ...................................................................................................... 197
Figure 7.13 IRFs of model with slabs for different cut-off frequencies,
max
f ........................... 198
Figure 7.14 Comparison of IRFs of model with no slab and the adjusted models with slabs ..... 199
Figure 7.15 Comparison of TFs of model with no slab and the adjusted models with slabs ....... 199
Figure 8.1 a) Sketches of the soil-structure interaction model used to generate observed response
(left) and of the model that is fitted (right). b) Impulse response functions, for virtual source at
roof (left), and transfer functions, w.r.t. base motion (right) of the SSI model and of its
superstructure if it were fixed-base .............................................................................................. 207
Figure 8.2 Results of identification of the SSI model by fitting low-pass filtered impulse response
functions. a) Comparison of the observed and fitted model impulse response functions (left) and
transfer-functions (right) for fit in the band 0 - 15 Hz. b) Comparison of the identified velocities,
, in three low-pass bands with the true superstructure velocity, . The phase velocity of an
approximate Timoshenko beam model of the same building is shown in the background
(redrawn from Ebrahimian et al. (2014)) ..................................................................................... 211
LSQ
eq
c
S
c
,
1
TB ph
c
Figure 8.3 Same as Fig. 3.9 but for identification by fitting band-pass filtered impulse response
functions. The dots in part b) show estimates of wave velocity from the SSI model apparent
frequencies ................................................................................................................................... 212
Figure 9.1 Assessment of modeling error in identified equivalent 3-layer shear-beam velocity
profile fitted in low-pass filtered IRFs of TB model NS (top) and TB model EW (bottom) of
Millikan Library (Table 4). Comparison of fitted and observed impulse responses (left), transfer-
function (center) and mode-shapes (right). .................................................................................. 219

xxviii

Chapter 1
Introduction
The first section of this chapter presents the motivation for this dissertation, the problem
statement and the objectives of this study. The second section presents literature review of s
structural health monitoring methods other than the wave method, while the third section presents
a detailed review of the wave method. In the forth section, the process of selection of case studies
for detailed study is presented, along with a list of buildings that were considered and for which
preliminary analysis was performed. Finally, the organization of this dissertation is presented.

1

1.1 Problem Statement and Dissertation Objectives
1.1.1 Structural Health Monitoring – Definition, Benefits and Challenges
Structural health monitoring (SHM) refers to the process of determining and tracking the
structural integrity and assessing the nature of damage in a structure (Chang et al., 2003). Its
great potential benefits to society have been recognized since the beginning of the deployment of
strong motion instruments in structures in the 1930s, and structural system identification and
health monitoring has been a very active area of research for many decades (Doebling et al.,
1996; 1998; Chang et al., 2003; Sohn et al., 2004). However, it is also a very challenging
problem, in view of the difficulties in building a system that is practically useful.
Detecting earthquake damage in an instrumented structure, as it occurs or minutes after the
earthquake shaking is over, based on information obtained from the recorded response would
reduce injuries and loss of life caused by earthquakes, facilitate emergency response and help
speed up the recovery following the disaster. For instance, a timely decision can be made to
evacuate an unsafe building, avoiding loss of life and injuries caused by potential collapse of a
weakened structure during shaking from aftershocks (Todorovska and Trifunac, 2008a,b,c).
The timeliness of the information, even when the damage is obvious or there is no structural
damage, is also useful, as physical inspection takes time after a disaster, when the demand for
inspection is large and the supply of inspection teams is limited (BORP – Building Operation
Resumption Program, 2001; Naeim et al., 2005). Timely assessment that the structure is safe
would help avoid needless evacuation, which is important for critical facilities such as hospitals
that are much needed after a disaster. A safe building can even be used as a shelter after an
earthquake when commute is disrupted (Hisada et al., 2012). Avoiding needless evacuation of
businesses will help reduce monetary losses due to downtime, and will shorten the period of
recovery of the affected area, in general. That is particularly important for metropolitan areas
2

such as Los Angeles (Figure 1.1) and many such areas in the world. Another use of SHM is for
general condition monitoring of the aging infrastructure.

Figure 1.1 A view of downtown Los Angeles from USC university park campus (Courtesy of
Lance Hill)
To be practically useful, a structural health monitoring method must be robust when applied
to real data, sensitive even to smaller and local damage, time-efficient, and accurate. It should
neither miss significant damage nor suggest false alarms causing needless evacuation and costly
service interruptions. This is particularly important for critical facilities, e.g. hospitals and
emergency centers. Satisfying all these requirements has been a challenge. The majority of
vibrational methods found in structural health monitoring literature are not robust when applied to
real structures. Further, the damage sensitive parameters vary due to causes other than damage,
e.g. amplitude dependency, nonlinear soil response (via the mechanism of soil-structure
interaction), which is the case for the natural frequencies of vibration, and environmental effects
(e.g. temperature). A more detailed discussion on these challenges is provided in section 1.2.
3

1.1.2 Objectives and Scope of this Dissertation Research
The objective of this dissertation research is to develop a robust and automated methodology for
structural health monitoring of buildings that is based on identifying the velocities of vertically
propagating waves through the structure, which depend on the structural stiffness, and detecting
their changes. The method uses data from typical strong motion instrumentation in buildings, i.e.
an array of accelerometers with sensors at two or more levels in the structure, which exist in
many buildings in seismic areas. Therefore, many structures can benefit from this method. Also,
although limited, data for calibration of the method already exist.
The method identifies the distribution of the wave velocities along the building height
associated with lateral and torsional deformation of the structure by fitting a model of the
structural response in recorded motions. Their changes during an earthquake are monitored by
comparison with values estimated during an initial time window when the response is small or by
analyzing a small earlier event. Smaller changes due to deterioration with time are monitored by
detecting changes between events using data from smaller local or larger distant earthquakes,
many of which are typically recorded during the life of a structure. Examples are presented of
application of the method to both earthquake damage detection and general condition monitoring
of a tall building that has shown no signs of damage (in Chapters 4 and 6).
Within the scope of this dissertation is to fit only one dimensional (1D) models of the
building response, which are applicable to relatively symmetric structures. Further, it is assumed
that the structure deforms only in shear, which minimized the number of parameters to be fitted.
Such a model may be appropriate for moment resisting frame structures (according to the ratios
of the frequencies of vibration, which are close to those for a uniform shear beam). The variation
of structural stiffness along the height is modeled by layering, such that the layers comprise of
group of floors, in general. The modeling error in the identified wave velocity profile due to
ignoring bending deformation is investigated in Chapter 9.
4

5

The emphasis of this dissertation research is on the identification algorithms and their
demonstration on real buildings using small and large amplitude response records. These
applications, along with forward simulations using the model, also provide an opportunity to get
insight into the nature of wave propagation in real buildings, which will be useful for further
development of the wave method. Along these lines, dispersion is analyzed, as well as the
influence of the slabs on the wave propagation and on the identification.
The presented applications to real buildings, undamaged and damaged, provide calibration
points for the variability of the damage sensitive parameters not related to damage and for the
changes associated with a particular level of damage. Such information is necessary for
probabilistic description of the state of damage by this method, which is out of the scope of this
dissertation research. The developed algorithms are applied to two reinforced concrete (RC) and
one steel structure. These structures were selected following a preliminary analysis of about 15
buildings (a listed of these buildings and the selection process are presented in section 1.4 of this
chapter). Comprehensive calibration of the method is an open ended process and is out of the
scope of this dissertation research.
An important feature of the presented SHM method is that it is not sensitive to the effects of
soil-structure interaction, even in the more general case when rocking is present. This is
demonstrated on identification of a shear beam model on embedded foundation in Chapter 8.
The scope of this dissertation work is illustrated schematically in Fig. 1.2, which shows the
elements of a SHM system, with the part that of the problem that is addressed by this dissertation
being highlighted. The system consists of an array of sensors and data acquisition and
transmission system, processor and decision making system. This dissertation deals with the part
that converts the recorded response in useful information for decision making. Such information
would be in probabilistic form, due to the uncertainties in the estimation and the variability of the
damage sensitive parameters due to causes other than damage.
6

Data
S
a Acquisition
Sensing & R Recording

n Unit
Data Tran
Cab
& Se
Figure 1.2
nsmission

bles
ensors
Reporting
Main elements o
Pro
of a Structural He
Part
addre
disse
Receiving a
ocessing Raw
Ide
C
ealth Monitoring
of the problem
essed in this
ertation
and
w Data
Structur
entification a
Change in Da
Para
Results Int
Decis
(SHM)
ral System
and Detectio
amage Sensit
ameter

terpretation
ion Making
and
on of
tive
6

1.2 Review of Structural Health Monitoring Methods
Structural damage due to extreme events is classified as Light, Moderate, or Severe. A Light or
Moderate level of damage is inflicted to the structural elements partially, and may be observed as
loosened connections, fractures in the shear walls and some of the frames, cracks at the base of
columns and joints, etc (NOAA, 1971). Severe damage is characterized by demolished/collapsed
walls or slabs, severe cracks at the base of columns and exposure of reinforcements, or distorted
columns and beams (in steel frame buildings). A structural health monitoring (SHM)
methodologies aim to relate changes in the monitored response to possible damage or
deterioration in the structure over time. A common input to majority of the SHM methods, which
deal with real structures, is the recorded structural response. This response is required to develop
and calibrate the SHM method. In the past four decades, scientists have adopted algorithms and
methodologies to detect the level of damage and its spatial distribution along the structure.
Change in the observed response may be caused by various factors. Some factors refer to the
changes in the structural frame and elements which result in change in the structural stiffness. The
change in the structural stiffness may be due to 1) Extreme Events, or 2) Structural Degradation.
Examples of extreme events include earthquakes, landslides, tsunami, typhoon, ocean waves (for
offshore platforms), or explosion. Two main sources of structural degradation are (a) corrosion of
elements and joints, and (b) fatigue under cyclic loads. Moreover, change in the vibrational
response can be due to non-structural or environmental factors. These factors include Soil-
Structure Interaction (SSI), temperature change (i.e. through the seasons or daily), rainfall
(Todorovska and Al-Rjoub, 2006), floors plan modification (e.g. removal of partition walls), and
changes in gravitational loads (dead and live). The less sensitive the SHM algorithm is to these
non-structural sources, the more robust it will be in detecting structural damage. Following this
section, the methodologies for SHM will be reviewed. We classify the methods and highlight
their respective advantages and limitations.
7

1.2.1 Visual Inspection
Visual Inspection has been the very first mean of assessing structural damage (Naeim et al. 2005).
In 1989, the Applied Technology Council (ATC) released a standardized procedure for these
evaluations named “Procedure for Post-Earthquake Safety Evaluation of Buildings” (ATC-20,
1989). This procedure requires inspectors to complete a “Rapid Safety Assessment Form” during
a building inspection. Based on their evaluation, inspectors assign one of the following tags for
the building: Green, Yellow, or Red. The Green tag signifies that the building has minor damage
and is safe to occupy. The Yellow tag indicates observed structural damage, such that the
building is unsafe and has restricted entrance (ATC-20, 1989). The Red tag denotes that severe
damage was observed, such that the building is unsafe and no entrance is permitted.
Nowadays, visual inspections are inevitable after a major event; however, when used alone,
they are limited and have practical disadvantages. For instance, the waiting times for an
inspection may be several days, due to high demand from clients (Osteraas et al. 2000; Kamat and
El-Tawil, 2007). As a result, building owners must keep the damaged building unoccupied for
days which incurs monetary loss due to downtime of the structure. In many cases, the assigned
tags to the buildings are changed after a secondary inspection. In other words, these evaluations
are subjective and often over-estimate the actual damage (NAHB report, 1994). In addition, some
damage in buildings cannot be seen by the naked eye (e.g. fractured beams or joints obscured by
the ceiling or partitions). Therefore, this method might be useful for detecting Severe damage in
buildings, but for a Light or Moderate damage, the inspection procedure fails to provide a precise
evaluation. In this respect, SHM based on sensory data presents an opportunity to reduce the wait
time and detect hidden damage.

8

1.2.2 Regulatory and Design-Based Performance Assessment
The Design-based method relies on exceeding deterministic ratios of some design parameters,
such as: base or story shear, drift, peak ground or floor acceleration, or response spectra (Naeim,
et al. 2005, 2006). Given a design parameter, the ratio of the observed design parameter and the
expected design capacity is calculated. Naeim et al. (2006) demonstrate that exceedance of such
ratio along the height of the building does not necessarily mean damage. The authors believe that
the designed capacity of the structure is affected by some reduction factors (e.g. ductility of the
frame) which may be markedly different from reality. They also state that calculating the design
capacity is possible if structural details are available (e.g. design load, sections drawings, etc).
Therefore, the drawbacks of this SHM methodology are the uncertain damage indication and the
time-consuming modeling of the structural details.
Code-based damage assessment is another damage indicator adopted by the Federal
Emergency Management (FEMA) and the American Society of Civil Engineers (ASCE),
essentially for Performance-based design of structures. The idea originated from design-based
damage assessment, so that the structure is designed for loads to meet a desired performance
level. The damage sensitive parameter is the observed inter-story drift, whose exceedance from a
certain critical value indicates a certain state of damage. FEMA356 (superseded by ASCE 41,
2000) lists these critical values along with their corresponding damage levels. Table 1.1 presents
one of these tables (FEMA356, C1-3) which includes the maximum inter-story drifts and the
performance levels for both Steel and Concrete structures. The table is illustrated by a step
function in Fig. 1.3.

9

D
T
F
fr

Drift, Steel and
Table 1.1
Figure 1.3 A
rame, presente
Structural per
d Concrete fra
schematic dia
ed in FEMA3
rformance lev
ames, taken f
agram of Max
356
vels and dama
from FEMA3
x. Drift and as
age with resp
56 (ASCE 41
ssociated perf

ect to the max
1, 2000), table
ximum Inter-
e C1-3
-story
formance lev

els in an Stee el
10
In reality, the design-based damage sensitive measures (i.e. drift ratio, floor accelerations,
floor shear, etc.) are not deterministic quantities. They are affected by the uncertainty in structural
motion and structural capacity estimation. In other words, there is no sharp transition between
two consecutive performance levels as presented by FEMA356 (see Figure 1.3). The Probability-
based measures convey the probability of achieving certain performance level, provided the
magnitude of the damage sensitive parameter. The fragility functions originate from the same
bases. Porter and Kiremidjian (2001) proposed fragility functions and an algorithm to calculate
them for different structural and non-structural elements. Yun et al. (2002) suggested a close-
form solution for generating fragility curves of a steel frame. Subsequently, Aslani and Miranda
(2005) described an algorithm for calculating the fragility curves (i.e. the probabilities) based on
the Maximum Likelihood method. The authors claim that the fragility curves for the majority of
structural elements closely follow a log-normal distribution (Aslani and Miranda 2005; Naeim et
al. 2006).

Figure 1.4 A sample fragility function of a structural element for three damage levels [Naeim et
al. 2006]
Figure 1.4 presents a sample fragility function. The vertical axis shows the probability of
occurrence of a certain damage level. One drawback of this method is that the inter-story drift
11

includes the displacement from rigid-body rocking of the building as well. Hence, large inter-
story drift does not necessarily indicate damage in the structure (Todorovska, 2009b).
1.2.3 Modal Methods for SHM
The structural response can be represented as a superposition of its modes of vibration. Each
mode is characterized by its frequency and shape, which depend on the structural stiffness and are
therefore sensitive to damage. The Modal methods have been the most widely used methods for
structural system identification and parameter based structural health monitoring, as evidenced by
the number of publications in literature (Doebling et al. 1996). They majority of the modal
methods can be classified into two groups, one based on detecting changes in the frequencies of
vibration and the one of, and the other one based on detecting changes in the mode shapes. In
this section, selected publications are reviews, with emphasis on the original articles.
Udwadia and Trifunac (1974) studied and identified the apparent fundamental frequency of
two buildings, the 9-story RC Millikan library building and a 10-story steel frame building,
located in Pasadena, California. For this purpose, the authors used the recorded response in the
full-scale buildings during forced vibration test, ambient vibration test, and small and large
amplitude earthquakes. The authors presented the variability of the apparent frequency as a
function of time. They also observed that during a moderate earthquake, the fundamental
frequency can drop by 50% while there is no observable structural damage, and that there is
frequency recovery after the earthquake. The reduction of apparent frequency is interpreted to be
due to nonlinearity of the structural elements or the soil beneath. Later, Trifunac et al. (2001a,b)
investigated similar phenomena in the Van Nuys 7-story hotel building.
Udwadia and Marmarelis (1976) used Wiener’s theory for characterization of nonlinear
systems and studied its applicability to identification of a full-scale building. The authors
employed the theory to identify the 9-story Millikan library during San Fernando 1971
earthquake. Possible signs of nonlinearity in the building were illustrated and discussed.
12

Vandiver (1975, 1977) investigated the change in the first two modes of an offshore
platform. Wojnarowski et al. (1977) and Coppolino and Rubin (1980) analyzed numerical models
of offshore platforms with inflicted damage to various elements of the model, and they measured
the change in the modal frequencies. The authors recognized that change in soil-foundation
properties caused the greatest change in the fundamental frequency. Kenley and Dodds (1980)
investigated the change in frequency under various levels of damage for a Finite Element model
of an offshore platform. They argue that the change in global stiffness of the structure must
exceed 5% to cause a change in the frequency.
Luco et al. (1988) and Wong et al. (1988) used a soil-structure interaction model to identify
the 9-story Millikan library using a set of Forced vibration tests. Using this model, the authors
decomposed the resonant frequency associated with the superstructure (i.e. the fixed-base
frequency) and that of foundation rocking and sway motion.
Ismail et al. (1990) and Man et al. (1994) proposed a relation for the change in frequencies
of a beam as a function of the crack length and location. They state that an opening-closing crack
causes less change in frequency than an open crack with the same length and location.
The researchers state known limitations for the method: 1) Low sensitivity to detect light
damages (less than 5% overall change); 2) The fundamental frequency is affected by Soil-
Structure Interaction (SSI), such that a portion of the change in the frequency is imposed by SSI,
depending on the soil-foundation characteristics; 3) The fundamental frequency is a global
characteristic of the structure, so an observed change only determines existence of damage in the
structure, and does not provide spatial distribution of damage.
Clinton et al. (2006) studied the observed variability of the Millikan library’s system
frequency considering a history of identifications. The authors correlated the observed changes in
the frequency with factor other than stiffness degradation, such as environmental effects (e.g.
temperature and change in floor plan and partitions). Mikael et al. (2013) also investigated the
13

variability of system frequency and damping for multiple buildings. They used the Random
Decrement Technique (RDT) and ambient vibration data to identify the buildings over period of
one month. The authors discussed the range of variability due to each considered environmental
factor (e.g. wind, rain, and temperature), and concluded that the maximum observed wandering in
frequency and damping is caused by temperature.
The second group of Modal method involves detecting change in the Mode Shapes. The
majority of researchers evaluated the change in the mode shapes of a damaged beam analyzed by
Finite Element Method (FEM) (Doebling et al., 1996). Yuen (1985) defined an eigen-parameter
to evaluate the change in the mode shape and the mode slope for multiple states of damage. Kim
et al. (1992) and Srinivasan and Kot (1992) investigated the change in mode shapes of a cracked
beam. They concluded that the mode shape is a slightly more sensitive damage indicator than the
fundamental frequency. They believe that this method may be utilized, under certain assumptions,
to determine the location of the crack as well. Pandey et al. (1991) showed that mode shape
curvature can be used as a damage indicator. They concluded that the curvature of mode shapes
can provide the spatial distribution of damage (cracks) in the beam, because curvature and strain
are proportional. Yet, the application of the change in mode shape/mode shape curvature is
limited among earthquake engineers. The main challenge is the uncertainty of their estimations
and the ability to detect small and localized damage (Chang et al. 2003).
1.2.4 Finite Element Model Updating
Finite Element Method (FEM) is a powerful approach to analyze a detailed and complex
structural model sustaining certain load combinations. Inversely, FEM may be adopted as a tool
to identify a structure given the structural response and the base excitation. Finite Element Model
Updating has been suggested by researchers as a methodology for structural identification and
damage detection in buildings. Lombaert et al. (2009) and Moaveni et al. (2010) presented a
Bayesian FE model updating to examine the inflicted damage to a 7-story RC test structure on a
14

shake table. They divided the FE model into substructures whose stiffness was the updating
parameter. The reduction in the stiffness of these substructures designates damage in a particular
part of the structure.
He and Ewins (1986) suggested a stiffness error matrix which relates the change in stiffness
of a damaged FE model to the undamaged stiffness matrix. They showed that the stiffness matrix
present significantly more information than the mass matrix for damage identification purposes.
Peterson et al. (1993) propose a recursive approach to construct the stiffness and mass matrices
by updating the FE model from one degree of freedom to the next, which illuminates any damage
between two adjacent members. DeVore et al. (2012) developed a similar methodology to that of
Peterson’s (1993) to identify a shear building structure, recursively, from the top floor toward the
base by identifying the stiffness and damping ratio of each DOF separately. The authors showed
that the method precisely detects the local inflicted damages, given the response at all the levels.
It is noteworthy that the recorded response in the structure is the result of linear and nonlinear
deformation of the structural frame, flexible foundation, and the soil beneath – known as system
response. Therefore, FE-based methodologies update the response of the superstructure with the
system response, which can be a source of error in detecting light and moderate damage. Another
challenge that most of FEMs face is the non-uniqueness problem due to inadequate modal data
(Chang et al. 2003). Yang and Lee (1999) try to resolve this issue by first devising the structure
into multiple regions. Considering the region with highest potential damage, they perform they
damage detection search. Furthermore, analyzing a Finite Element model of building, despite
having efficient optimization algorithms in hand, is a time-consuming task, and these
methodologies are not suitable for a post-event early safety warning system.
1.2.5 Damage Detection Using Neural Network
The Neural Network approach for damage identification in structure has received increasing
attention in recent years (Doebling et al., 1996, Saadat et al. 2004). For an arbitrary complex
15

system, the Neural Network (NN) is trained using the inputs and outputs of the system, so that it
can predict correctly the response of an undamaged system driven by an arbitrary input. In other
words, NN are used as a function approximator, or time series predictor (Lippmann 1987; Pao
1989). Wu et al. (1992) and Elkordy et al. (1993) used back-propagation NN to detect damage in
a 3-story and a 5-story 2-D building FEM model, respectively. The damage was inflicted by
reducing the stiffness of members. The NN has been utilized to identify the map from the
structural response (e.g. Fourier spectra, or mode shape) to the level of damage inflicted to the
floors. One of the drawbacks of this method, especially in complex models, is its high
computational effort. Also, Neural Network based methodologies fail to reflect the actual state of
damage if it is not included in the training domain (Saadat et al. 2004).
1.2.6 Wavelet Transform and Novelties
Wavelet Transform is another tool for detecting damage in a structure. Wavelet analysis is a
Time-frequency analysis technique which can be utilized to analyze and highlight any abrupt
change in a time domain response (Vill 1947). For the analysis, one has to decide on the type of
wavelet and the order of decomposition. In structural health monitoring applications, Daubechies
and Bi-orthogonal wavelet families are mostly employed. During the decomposition process, a
signal is halved into two signals: Approximation and Detail. Detail contains the high-frequency
components of the signal, and is used to detect any discontinuity, abrupt change, or singularity in
the signal. Rezai et al. (1998) used wavelet transform to detect damage in the 7-story Van Nuys
building severely damaged during Northridge 1994 earthquake. Hou et al. (2000) demonstrated
the method on a numerically simulated response of a structure with inflicted damage.
Todorovska and Trifunac (2010) used bi-orthogonal wavelet expansion to examine damage
in the Imperial County Service building motion during the Imperial Valley EQ in 1979. They
decomposed and investigated the high-frequency detail of motion (12.5-25 Hz), and plotted the
square wavelet coefficients versus their central time – known as “Novelties”. The authors found
16

that the damage state and time suggested by novelties are consistent with the time of large inter-
drifts. According to Todorovska and Trifunac (2010), the abrupt change in the wavelets is
detectable only if the spikes are above the noise level of the signal. Also, the method is accurate
to detect damage if it is close to the damaged element; hence, the method is promising if a dense
instrumentation is deployed.
1.2.7 Non-Destructive Testing of Structural Elements
Excitation Methods are alternative methodologies for damage detection. These methods are a
subset of Nondestructive testing, which encompasses a wide range of tests and analyses in science
and technology to identify the characteristics of an object (Doebling et al., 1996). The Excitation
methods can be categorized into two major groups: 1) Ambient excitation methods, and 2) Force
excitation methods. The input to the ambient test is not measured, while the input to the force
excitation is needed for analysis. In the case of force excitation, the force can be applied locally,
to a desired element, or globally, to the structure as a whole (e.g. exciting the roof). The source of
the excitation force can be shakers, actuators, step relaxation, or impact (Doebling et al., 1996).
Luco et al. (1987) performed multiple forced vibration tests on a 9-story RC building to examine
the changes in the apparent resonant frequency over time. Furthermore, the authors investigated
the relative contribution of soil-structure interaction to the observed changes in the frequency
(Wong et al. 1988; Luco et al. 1988).
To detect a possible local damage, the vibration test can be performed on a specific element
as well, but special type of actuator is required. The Piezoelectric transducer (PZT) is an example
of such actuator, which is used for both actuation and sensing simultaneously (Doebling et al.,
1996). PZT can excite the element at a significantly high frequency, typically more than 30kHz.
Different damage detection methods, such as the impedance-based method (Park et al., 1999a)
and the electro-mechanical impedance technique (Giurgiutiu and Rogers, 1997) can be employed
17

to detect the flaws in a structural element. The local excitation tests are costly and time-
consuming, and are not yet practical for automated damage detection in buildings.

1.3 Wave Methods for Structural Health Monitoring
The SHM method being developed in this dissertation belongs to the class of wave methods that
use data from recorded seismic response (frequency range typically up to 25 Hz; Trifunac and
Todorovska, 2001) and analyze wave propagation in the building as a whole rather than in the
individual elements. The method is based on the assumption that structural damage would
produce a change in the velocity of wave propagation through the structure. E.g., if the building
is represented as a uniform shear beam, its shear wave velocity / β μρ = , where ρ is the mass
density and μ is the shear modulus. As β is directly related to the stiffness, loss of stiffness due
to damage would lead to reduction of the wave velocity in the damaged part, and an increase in
wave travel time. This provides a basis for inference about the presence and location of damage
in the structure. This concept was first proposed by Şafak (1989, 1990) who analyzed a building
model (described later in this section), and was demonstrated on damaged full-scale buildings
(Ivanovi ć et al., 2001; Oyunchimeg and Kawakami, 2003; Todorovska and Trifunac, 2008a,b).
The methodological development in this dissertation builds directly on the proof of concept
studies of Todorovska and Trifunac (2008a,b). The remaining part of this section presents a
review of studies on wave propagation in building, which is the underlying process for the SHM
method, seismic interferometry, as a means to measure the wave velocity, and its applications to
damage detection. The major milestones are reviewed in more detail.
The building response can be alternatively represented as superposition of modes or as
superposition of propagating waves. Kanai and Yoshizawa (1963) and Kanai (1965) appear to
have been the earliest publications that present the idea of treating the problem of seismic
18

vibration of buildings as the multiple reflection phenomenon of waves in an elastic layer. They
present derivation of the shear displacement, u, as a sum of reflected waves at the foundation
boundary, z = 0, and at the free surface, z = H, as follows
0
2
22
()| () ( ) ( )
44
() ( )
z
HH
u t Ft Ft Ft
VV
HH
Ft Ft
VV
γβ
ββ
=
⎡
⎧⎫
=+ − + −
⎨⎬
⎢
⎩⎭
⎣
⎤ ⎧⎫
+− + − +
⎨⎬
⎥
⎩⎭
⎦
" (1.1)
and
2
35
()| 2 ( ) ( ) ( )
zH
HH H
u t Ft Ft F t
VV V
γβ β
=
⎡⎤
=− + − + − +
⎢⎥
⎣⎦
" (1.2)
where is the incident wave, () Ft γ is the transmission coefficient of waves from the foundation
into the structure, and β is the reflection coefficient of waves from the flexible foundation. The
material damping is assumed to be zero. Assuming that building response resembles that of a
uniform shear beam in terms of deformation and mass and stiffness distribution, Kanai estimated
the equivalent shear wave velocity V in a building using the relation , where
1
4/ VHT = H is
the height and is the fundamental period of the building.
1
T
Subsequently, wave propagation in buildings has been studied by a number of authors. E.g.,
wave propagation in 2D models of buildings and wave dispersion due to nonhomogeneities were
was studied by Todorovska (1988); Todorovska and Trifunac (1989, 1990), and Todorovska and
Lee (1989). Fukuwa and Matsushima (1994) studied wave dispersion in a building model with
slabs, modeled as a periodic structure (this paper is reviewed in detail in Chapter 7). Later, Iwan
(1997) proposed drift spectra for the demand estimation of buildings shaken by earthquakes,
based on a shear beam model and representation of the motion as a combination of upgoing and
downgoing waves, introduced earlier by Kanai and Yoshizawa (1963)..
19

Şafak (1998, 1999) proposed a simplified 1D model for soil-foundation structure, extending
the soil layers to the superstructure and considering the building floors as elastic homogeneous
layers. The seismic response is expressed in terms of wave travel times between the layers and
wave reflection and transmission coefficients at the layer interfaces. The author argues that the
method is simple, accurate, and could be a better tool for identification and damage detection in
buildings.
Wave propagation in the Van Nuys 7-story hotel was studied on a model and using recorded
earthquake response by Todorovska et al. (2001a,b). Further, damage in the same building was
detected from changes in wave travel time (Ivanovi ć et al., 2001) and from changes in
wavenumbers (Trifunac et al., 2003).
Kawakami and Oyunchimeg (2003, 2004) and Oyunchimeg and Kawakami (2003) studied
wave propagation in full-scale buildings, both undamaged and damaged, using earthquake
records and “Normalized Input-Output Minimization” (NIOM) method. This method was
proposed by earlier by Kawakami and Haddadi (1998), who used it to analyze soil borehole data.
The method transforms the recorded motions in waveforms with spikes, shifted in time relative to
a virtual source impulse, which represent essentially regularized impulse response functions.
These spikes are used to measure wave travel time from base to roof. Kawakami and
Oyunchimeg (2004) used the methodology to measure the travel time ( τ ) of a propagating wave
in a 10-storey frame. They also used
1
1/ 4 f τ = (fixed-base frequency of a uniform shear beam)
to evaluate the analytical fixed-base frequency and compare it with their model’s fundamental
frequency. They concluded that 1/ 4 τ results in approximately the same value as the fixed-base
frequency (modal analysis) for a uniform shear building, and underestimates the fundamental
frequency if the floors’ stiffness is increasing from base to roof.
Nonlinear wave propagation in building models was studied by Gi čev (2005) and later by
Gi čev and Trifunac (2007; 2009a,b). Gi čev and Trifunac (2012) introduced a damage detection
20

algorithm based on predetermined damage scenarios (PDSs) for a building of interest. The
scenarios are determined using nonlinear response analysis of the building. 1D wave propagation
in a shear building model and the finite differences method are used to solve the nonlinear
response of the model. The authors showed that the algorithm could accurately detect damage
location in a 7-story RC building, damaged during Northridge 1994 earthquake.
Snieder and Şafak (2006) developed the idea that was presented by Şafak (1999) by 1)
proposing a deconvolution scheme for calculating the waveform of observed response in a
building, and 2) simulating the building as a uniform layer over soil half-space. Deconvolving the
observed response of building, () u ω , at some level with respect to motion of a reference level z
ref
() u ω results in the system Impulse Response Function (IRF), which physically represents an
impulse propagating from the source.
{}
1
()
ˆ
() () ()
()
FT
u
hhtFT
u
ω
h ω ω
ω
−
=⇔ =
ref
(1.3)
In equation (1.3), () u ω and () u ω
ref
t () h
are the Fourier transform of response at the level ( )
and at the reference ( ),
z () ut
() u
ref
ω is the Transfer Function of the response with respect to the
reference level, and is the Impulse response function at level . Another physical
justification for

is that it physically represents the response at the location where was
recorded to a unit impulse at the location where was recorded. Snieder and Şafak (2006)
propose regularization of the form
( ht
() ht
) z
() ut
() ut
ref
2
ˆˆ () ( )
ˆ
(, ; )
ˆ()
uz u z
hz z
uz
ω
ε
=
+
ref
ref
ref

(1.4)
where the regularization parameter ε is introduced to prevent division by accidental zeros in
ˆ() uz
ref
. The authors also present solution for propagation of shear waves in a shear model of a
building. They include the quality factor Q (Courant and Hilbert, 1962; Aki and Richards, 1980)
21

to model the attenuation. They finally apply the method to Millikan library, a 9-storey RC
building in Pasadena, California, and estimate the phase velocity and the quality factor for the
uniform shear beam model.
The study of Snieder and Şafak (2006) was followed by other authors (e.g. Kohler et al.,
2007; Todorovska and Trifunac, 2008a,b; Todorovska, 2009a,b; Trifunac et al., 2010; Prieto et
al., 2010).
Kohler et al. (2007) analyzed the Factor building, a 17-storey moment resistant steel frame
structure with a dense instrumentation located in Westwood of the Los Angeles metropolitan
area, using observed response during 20 earthquakes. Further, they presented impulse response
functions for a finite element model of the building and studied pulse propagation in the model.
They concluded that the finite element model gives very close impulse response functions, mode
shapes, and modal frequencies to the actual ones.
Todorovska and Trifunac (2008a,b) investigated observed impulse response functions in two
damaged buildings in Southern California. The authors considered the buildings as a uniform
shear beam. They showed that the detected changes in shear wave velocities are consistent with
the location and degree of the observed damage and concluded that the method is promising and
should be further developed.
Todorovska (2009a) presented a soil-structure model consisting of shear beam mounted on a
semi-cylindrical foundation, called later “SSI model”. The SSI model takes into account the
coupling of the shear beam and the horizontal and rocking motion of foundation. The model
resulted in important conclusions on the observed system IRFs in buildings. The author concludes
that 1) observed travel time τ is not affected by SSI, and 2) the SSI model has practically the
same vibrational frequencies for the higher modes (i.e. second and third mode) as the fixed-base
shear beam model. Based on these findings, Todorovska (2009b) further investigated and
resolved the degree to which the observed wandering of the apparent NS frequency of Millikan
22

library, shown earlier by Udwadia and Trifunac (1974), during four earthquakes was caused by
softening of the soil as opposed to recoverable and permanent softening of the structure.
Prieto et al. (2010) first showed that impulse response functions for a building can be
obtained from ambient vibration data, but many days of recording is required for good quality
results. They analyzed the Factor building.
Trifunac at al. (2010) studied by impulse response functions the Borik-2 building, a
prefabricated 14-story reinforced concrete building located in Banja Luka, Republic of Srpska
(Bosnia, former Yugoslavia) using data from 20 earthquakes, one of which could have caused
damage. The study aimed to help determine the threshold change associated with damage.
Interferometry encompasses all the methods used to investigate pairs of signals in order to
obtain useful information on the propagation medium (Curtis et al. 2006). Seismic interferometry
is being used by geophysicists to identify the subsurface characteristics of the earth crust via
propagating waves (Tolstoy, 1961; Gilbert and Backus, 1966; Aki and Richards, 1980; Bolt et al.
1982; Kausel and Pais, 1987; Trampert et al. 1993). Two main interferometric approaches that
are used by structural engineers are cross-correlation and impluse response analysis. It is
noteworthy that these methods have to be used with care as the nature and scale of models in
geophysics and buildings are significantly different in some aspects (e.g. distances between
sensors and wavelengths of the propagating waves).
Shear stress and displacement in a horizontally layered half-space, excited by incident SH
waves has been solved in geophysics using the propagator matrix approach (Gilbert and Backus,
1966). The propagator operator is a generalization of the Thompson-Haskell method for surface
waves in a layered half-space (Thompson, 1950; Haskell, 1953). Courant and Hilbert (1962)
introducing the quality factor (Q) to include damping in a non-dispersive shear wave. In 1980,
Aki and Richards published a book “Quantitative Seismology” which was a major milestone in
23

integrating the methods in seismology and geophysics available at that time (Aki and Richards,
1980). We will discuss the details of propagator matrix in Chapter 2 (i.e. methodology).
Kausal and Pais (1987) used cross-correlation method to estimate characteristics of motion
at some levels in a layered soil half-space based on information provided at the ground level. The
authors define cross-correlation as “cross-correlation function R
AB
relating the motions at A and
B is equal to the autocorrelation function for the motion at A, but shifted by some time-
lag/advance”. Kausal and Pais conclude that dominant characteristics of actual recorded motion
cannot be estimated because 1) the cross-correlation magnifies some wave velocities than others,
and 2) the actual earthquake waves contain different types of waves (not only plane) propagating
with different phase velocities. Ivanovi ć et al. (2001) utilized cross-correlation to identify shear
wave velocity of one dimensional building models (uniform or layered shear beam) using
earthquake response records, from the time lag τ between the motions recorded by two sensors
and their distance ( h / h β τ = ). At the time lag,

τ , the amplitude of the cross-correlation
function is maximum. They examined the method using the recorded data in a damaged 7-storey
RC building, and showed that the time lag in the autocorrelation function is consistent with the
location of actual damage.

24

1.4 Selection of Case Studies for this Dissertation
An illustration of the United States’ Faults map, provided by United States Geological Survey
(USGS), reveals that the faults are concentrated in the western states, namely California and
Nevada (USGS official website
1
). Of a great economic and geographic importance, California
became the cradle of earthquake engineering evolvement in the US in the 20
th
century. According
to Blume (1972), natural periods of buildings were measured for the first time in 1912 by Elmer
Hall (1870-1932), an associate professor of Physics at UC Berkeley. Elmer used early generation
seismic instruments to measure the response of a 6-storey building in San Francisco. Research on
the damage assessment of buildings subjected to strong motion gained popularity in late 1930s.
In 1932, the USGS developed its strong motion monitoring network in a certain number of
buildings in San Francisco and Los Angeles. Especially interesting was the Long Beach great
earthquake recorded in less than a year on March 10, 1933 (M
w
6.4). According to Binder and
Wheeler (1960), the first seismic provision in the US was adopted in 1933, following the Long
Beach earthquake.
Seismic instrumentation and monitoring networks have been increasing around the globe. In
California, this task is overseen by the California Geological Survey (CGS), formerly named
CDMG
2
. CGS is one of the agencies in the California Department of Conservation. In 1972,
CGS established the California Strong Motion Instrumentation Program (CSMIP) by California
Legislation to obtain vital earthquake data for engineering and scientific communities via the
seismic instrumentation networks. The program has installed over 900 stations, including 650
ground-response stations, 170 buildings, 20 dams, and 60 bridges.
To investigate and develop our identification and damage detection algorithms, we have
analyzed a set of recorded data in buildings, mostly located in southern California. A set of
criteria were considered to decide on the buildings for further detail analysis and presentation in

1
United States Geological Survey (www.usgs.gov)
2
California Division of Mines and Geology
25

this dissertation. In this section, we briefly will explain our criteria for selected case studies.
Among all buildings, 1) the Millikan library in Pasadena, which is a densely instrumented high-
rise reinforced concrete (RC) building, with sensors at each floor and 2) the LA 54-story steel
frame tall building with six available earthquake records were selected for the identification study
phase. Furthermore, to test and develop our damage detection methodology, 3) we have
considered a 12-story RC building lightly damaged during San Fernando earthquake in 1971. The
LA 54-sotry tall building and 12-story lightly damaged RC building have six and three sensors in
height respectively. These are typical numbers of sensors in the high-rise buildings. Also, in the
latter case, detailed description of damage is available.
The main source of data for this study is from available seismic network deployed by
California Geological Survey (CGS) under California Strong Motion Instrumentation Program
(CSMIP). The data and buildings’ information is available to public through Center for
Engineering Strong Motion Data
3
(CESMD).
A set of 16 buildings were selected for the preliminary analysis. These buildings are listed in
table 1.2. The analysis was comprehensive in some cases, e.g. Hollywood storage, a 14-story RC
building, has three arrays of fifteen accelerograms located at N-S center, E-W center, and N-S
west wall of building. Thirteen earthquakes were recorded in the building over a period of 76
years (starting from Southern California EQ. in 1933). Over fifty sets of IRFs and TFs were
calculated in order to investigate the wave propagation in the building. The most important
criteria for selecting these case studies are as follows
- Available earthquakes: We chose those buildings that have possibly experienced at least one
large magnitude earthquake in their life. It is desired that a station has records of both large and
small amplitude earthquakes, and also both close and distant earthquakes are available.

3
Website: ww.strongmotioncenter.org
26

27

Buildings that were lightly or moderately damaged during a major earthquake would be very
valuable for testing our damage detection algorithms.
- Number of stories and structural material: Buildings with at least five stories above ground
level are considered. These are those residential or office buildings with mostly concrete or
steel frames. The structural elements of selected buildings have reinforced concrete frames
(RC), steel frame, or composite sections. Structures with timber elements or masonry structures
were excluded.
- Buildings with previous studies: It was desired to carry out our detailed analysis on buildings
which are well known and have their dynamical characteristics studied by other researchers.
Therefore, we decided to work on buildings such as Millikan library, which is a classic case
study among researchers.
- Structural details: More details of structural and non-structural elements, such as plan of shear
walls, moment resistant frames, and foundation type, can be very helpful in interpreting out
results. We considered those structures which have available structural details and sketches. For
those buildings subjected to major earthquakes, a visual assessment of structural damage would
be valuable for our study. Sherman Oaks 12-story RC building is an example which has
structural details and damage description after San Fernando 1971 earthquake (M=6.6).

A list of all considered buildings for our preliminary analyses
No. Name of station
Type of
structural/Lateral force
resisting system
Number of stories
above/below
ground
Site geology
Number of
available EQ.s
Table 1.2
Comments
1
Hollywood Storage 14-story
Building
(CSMIP station #24236)
-/Reinforced concrete 14/1 Deep alluvium 14
Non-symmetric pile
foundation
2
Burbank - 6-story
Commercial Building
(CSMIP station #24370)
Steel beams and
columns; concrete slab
/Moment resisting steel
frames
6/0 Alluvium 5 -
3
Pasadena - 9-story
Commercial Building
(CSMIP station #24571)
Concrete columns and
slab/Moment resisting
concrete frames
9/1 Deep alluvium 4
Concrete spread
footing foundation
4
Los Angeles - 52-story
Office Building
(CSMIP station #24602)
Concrete slabs supported
by steel frames/Braced
steel frames at the core
52/5
Alluvium over
sedimentary
rock
4 -
5
Burbank - 10-story
Residential Building
(CSMIP station #24385)
Precast concrete bearing
walls/Precast concrete
shear walls
10/0 Alluvium 5 -
6
Sherman Oaks - 13-story
Commercial Building
(CSMIP station #24322)
Concrete beams and
columns/Moment
resisting concrete frames
13/2 Alluvium 5
Concrete pile
foundation
7
Sherman Oaks 12-story
Office Building
(USGS station #0466)
Concrete frame/Moment
resisting concrete frames
12/0 Silty sand 2
Many aftershocks
are recorded.
8
Santa Susana ETEC Bldg
462 (USGS station #5108)
Inverted chevron-type
braced steel frames
8/0
Sandstone
bedrock
1
Many aftershocks
are recorded.
9
9-Story Millikan Library
(USGS station #264)
Reinforced
concrete/Shear wall and
RC central shear core
9/0 Alluvium 5 -
28

28

29

No. Name of station
Type of
structural/Lateral force
resisting system
Number of stories
above/below
ground
Site geology
Number of
available EQ.s
Comments
10
Los Angeles - 54-story
Office Building
(CSMIP station #24629)
Concrete slabs supported
by steel frames/Moment
resisting perimeter steel
frame
54/4
Alluvium over
sedimentary
rock
4 -

-
11
Torre Central Building –
Chile
Reinforced
concrete/Shear wall in
both directions
9/2
Dense gravel –
Class “C”
41
The Chile great
earthquake 2010 is
recorded, plus 16
pre-event and 24
post-event shakings
12
UCLA 17-Story Factor
Building
(USGS station #5405)
Steel frame/Moment-
resisting frame
17/0 - 1
Many small eq.s are
available. Soil type
not mentioned.
13
Los Angeles 32-story
Residential Building
(CSMIP station #24288)
Concrete slab supported
by steel beams and
columns/Perimeter
moment resisting steel
frames
32/0 Soft rock 6
14
San Francisco 47-Story
Building
(CSMIP station #58532)
Steel frames/Moment
resisting steel frame
47/2 Fill 1
Loma-prieta is
available.
15
Imperial County Services
Building
(CSMIP station #01260)
Reinforced
concrete/Shear wall and
moment resisting frame
6/0 - 1
Analyzed by
Todorovska and
Trifunac 2008a
3 aftershocks are
available. Analyzed
by Todorovska and
Trifunac 2008b
16
Van Nuys Hotel
(CSMIP station #24386)
Reinforced
concrete/Interior
column-slab frames and
exterior column spandrel
beam frames
7/0
Holocene
alluvium
9

Table 1.2 continued
29
1.5 Organization of this Dissertation
Chapter 2 presents the methodology, in two parts: (1) the forward problem and (2) the inverse
problem. In the first part, the mathematical model is presented and analytical solutions for its
Transfer-Functions and Impulse Response Functions. In the second part, three identification
algorithms are presented: (i) the direct algorithm, (ii) the nonlinear least squares algorithm and
(iii) the time shift matching algorithm, and their generalization to moving window analysis, called
time-velocity algorithm.
Chapter 3 presents an analysis of the nature of pulse propagation in a building with variable
properties along the height using simulated broader-band impulse response functions for a model
of Millikan Library NS response. Of particular interest are the internal reflections from the layer
boundaries. It also presents a critical assessment of the direct (ray) algorithm and wave travel
time analysis for application to building response. In view of the high accuracy requirement for
decision making on evacuation, the emphasis is on the spatial resolution and accuracy of
identification, demonstrated on the example of the NS response of Millikan Library.
Chapter 4 presents a system identification and health monitoring analysis of a tall steel frame
building in downtown Los Angeles with density of sensors typical for instrumented buildings.
The building is known as Los Angeles 54-story Office Buildings. Detailed identification analysis
is presented for one earthquake (Northridge, 1994) and health monitoring analysis using data
from six earthquakes over a period of 19 years since construction, none of which caused
structural damage.
Chapter 5 presents a system identification analysis of a densely instrumented building, with
sensors on every floor. The case study is Millikan Library, which is a 9-story RC building with
external shear walls and a central core. A Comparative analysis of identification results by the
three algorithms is presented, and the effect of the data bandwidth on the spatial resolution and
accuracy is analyzed.
30

Chapter 6 presents an application of the time-velocity algorithm to a damaged building with
typical density of instruments. The case study is Sherman Oaks 12-story Office Building, which
is a RC building that was lightly damaged during San Fernando, 1971 earthquake. It also presents
a comparison with results of inter-story drift analysis and instantaneous frequency analysis.
In chapter 7, pulse propagation in a building model with slabs (modeled as thin, very stiff
and heavy layers) is analyzed. Also, results of identification by fitting a model with slabs are
presented and compared with results of identification by fitting a model in which the slabs are not
considered explicitly. The case study is Millikan Library NS response.
In chapter 8, the problem of wave dispersion in buildings due to soil-structure interaction is
presented. Identification results for fitting an SSI model and their trends for different frequency
bands are discussed.
Chapter 9 presents our investigation of modeling error due to dispersion caused by bending
deformation. For this purpose, we fit a non-dispersive layered shear beam into a fixed-base
uniform Timoshenko beam simulated based on the Millikan library’s response. The identified
velocity profile and important findings are presented subsequently.
Finally, Chapter 10 presents the summary and conclusions of this study, as well as
recommendations for future studies.

31

32

Chapter 2

Methodology
This Chapter consists of three parts. The first part presents an outline of the method and
preliminaries. The second part is on the forward problem and presents the mathematical model of
the building, based on which the model is identified. In this dissertation, the model used is a
layered shear beam. Analytical, recursive solution of the state of the model is derived using the
propagator matrix approach, as well as recursive analytical expressions for the model low-pass
filtered impulse response functions. An extension to torsional waves is also presented. The third
part is on the inverse problem and presents several identification algorithms. Three fitting
algorithms are presented for a given time window, and an extension to moving window analysis,
which produces time-velocity maps for detecting changes. The goodness of fit is discussed in
detail, with special attention to the spatial resolution and accuracy, and the choice of low pass
filter and its effects.

33

This chapter is based on materials published in:
Todorovska, M, Rahmani, M. “System Identification of Buildings by Wave Travel Time Analysis
and Layered Shear Beam Models - Spatial Resolution and Accuracy”, Journal of Structural
Control and Health Monitoring, 20(5) 686-702, DOI: 10.1002/stc.1484 (2012).

Rahmani M, Todorovska MI (2013). 1D system identification of buildings from earthquake
response by seismic interferometry with waveform inversion of impulse responses – method and
application to Millikan Library, Soil Dynamics and Earthquake Engrg, Jose Roësset Special
Issue, E. Kausel and J. E. Luco, Guest Editors, 47: 157-174, DOI: 10.1016/j.soildyn.2012.09.014.

Rahmani M, Ebrahimian M, Todorovska M. I. (2014). Automated time-velocity analysis for
early earthquake damage detection in buildings: Application to a damaged full-scale RC building
and comparison with other methods, Earthq. Eng. Struct. Dyn., Special issue on: Earthquake
Engineering Applications of Structural Health Monitoring, CR Farrar and JL Beck Guest Editors,
submitted for publication, 01/16/2014.

34

2.1 Impulse Response Functions of a Linear System
Essential part of the SHM methodology is the system identification, which consists of fitting a
layered shear beam model (without base rocking) into observed system response, which may be
affected by base rocking. The model is fitted by matching the propagating pulses in the observed
impulse response functions (IRF) after low pass filtering. This section presents the basic
definitions and concepts the method is based on.
Let () ω be the transfer-function between two signals , at some level () h ut z
,
and , at
a reference level in a building. Deconvolving the observed response,
ref
() ut
() u ω , with respect to the
motion of the reference level
ref
( u ) ω results in the system Impulse Response Function, which
physically represents an impulse propagating from a virtual source
{}
1
ref
()
( ) ( ) ( )
()
u
hhtFT
u
ω
FT
h ω ω
ω
−
=⇔ =
(2.1)
In equation (2.1), () ω and
ref
() u u ω are the Fourier transforms of the response at some level

and at the reference level respectively. is the Impulse response function (IRF) at level
z
() ht z ,
that is the inverse Fourier transform of the transfer function (TF) of recorded response.
1
FT
−

indicates inverse Fourier transform, for which we follow the convention
1
ˆˆ
() () () ()
2
it it
ff te dt f t f ed
ωω
ω ωω
π
−
−∞ −∞
=⇔ =
∫∫
∞∞
() ht
t
(2.2)
Another physical justification for is that it physically represents the response at the
location where was recorded to a unit impulse at the location where was recorded.
At the reference level, hz
() ut
ref
() u
ref ref
ˆ
(, ;) z ω 1 = and
ref ref
(, ;) ( hz z t t) δ = = Dirac delta-function, which
represents a virtual source, whereas represents the response at
ref
(, ; z ) t hz z to a virtual source at
35

ref
z . When the virtual source does not coincide with the physical source, there will be both
causal and acausal pulses in the impulse response functions (Snider and Şafak, 2006).
In this dissertation, the observed IRF,
max
(,0, ; ) hz t ω , at a particular level z , for a virtual
source at the roof , and in the frequency band (0 z =) ( )
max
20, f ωπ ω =∈
() uz
, is computed from a
regularized transfer function (TF) between the motions at that level, , and the motion at roof,
( Snieder and Şafak, 2006), (0) u
2
ˆˆ () (0)
ˆ
(, hz 0; )
ˆ(0)
uz u
u
ω =
+

(2.3)
ε
as its band-limited inverse Fourier transform
max
max
max
1
ˆ
(, hz 0, ; ) (,0; )
2
it
t hz e d
ω
ω
ω
ωω
π
−
−
=
∫
ω
(2.4)
In eqn. (2.3), ε is a regularization parameter chosen to avoid division by a very small number
(Snieder and Şafak, 2006).
The cut off frequency,
max
ω , is an important parameter, because it determines the width of
the pulses in the IRF, and the uncertainty in their time localization. If
max
ω <∞ , the source pulse
is a box function in the frequency domain and the sinc function
max
t
t
sin ω
π
in the time domain.
The relation between the two functions can be written as
max max
max
1,
sin
ˆ
() ω ( )
0, otherwise
FS
t
SSt
t
ωωω
ω
π
−<< ⎧
=⇔=
⎨
⎩
(2.5)
36

Fig. 2.1 illustrates the source functions (pulse) for box function filters. The half-width of the
main lobe of the sinc function
max max
/1/(2 t )f π ω Δ== , and is a measure of its spread in time.
As
max
ω →∞ , and the source function approaches the Dirac 0 t Δ→ δ -function. The accuracy
with which the central time of the pulses in IRFs represents the true pulse arrival time increases
with decreasing pulse width, i.e. with increasing
max
ω . Also, for real data, there is an effective
max
ω , which is smaller than the capability of the sensors (typically 25 Hz or 50 Hz; Trifunac,
1971, 1972; Trifunac and Todorovska, 2001), beyond which the TF has very small amplitudes,
and increasing the bandwidth does not decrease the width of the pulses. In contrast, the model TF
(or constant Q value, e.g.) has amplitudes that decrease less with frequency, and the pulse width
continues to decrease with increasing
max
ω . This leads to a disparity in the pulse shape in the
observed and fitted IRFs for
max
ω much larger than the effective data bandwidth. Controlling
max
ω enables one not only to control the accuracy of the fit, but also to filter out from the
observed IRFs possible effects from phenomena not captured by the model (e.g. bending
deformation and dispersion).

Figure 2.1 Virtual source function in the frequency (left) and time (right) domains, for a box
function
37

2.2 The Forward Problem
2.2.1 Modeling of the Translational Response
The building is modeled as an elastic layered shear beam, supported by a half-space, and excited
by vertically incident plane shear waves (SV) (Fig. 2.2). The layers may correspond to individual
floors, or to group of floors, depending on the density of sensors and desired spatial resolution.
Within each layer, the medium is assumed to be homogeneous and isotropic, and that perfect
bond exists between the layers. The building is assumed to move only horizontally, in the -
direction, the foundation rocking due to soil-structure interaction being neglected.
y

Figure 2.2 The model. a) Layered shear-beam representing a building. b) Layered half-space.

38

The layers, numbered from top to bottom, are characterized by thickness , mass density
i
h
i
ρ
, and shear modulus
i
μ , where 1, , i n = … for the layers in the building, which implies shear wave
velocities /
i i i
β μ = ρ . The displacements at the roof and at the consecutive layer interfaces are
, , . Amplitude attenuation due to material friction is introduced via the quality
factor , which is equivalent to damping ratio
1
u
2
u …
1 n
u
+
Q 1/ (2 ) Q ζ = (Courant and Hilbert, 1962). The
motion of this model is identical to that of a layered half-space excited by vertically incident SH-
waves (Fig. 2.2b) (Trampert et al. 1993; Mehta et al. 2007).
The motion of model in Fig. 2.2b is mathematically identical to that of a horizontally layered
half-space, excited by vertically incident SH waves, which has been solved in geophysics using
the propagator matrix approach (Gilbert and Backus, 1966). Let be the y-
component of displacement in the x-z plane (Fig. 2.2b). Equation of motion of medium (i.e. wave
equation) can be written as
( , ; )
y
uUxzt =
22 2
2
22
uu
uu
tz x
ρμ ρ μ
⎛⎞
∂∂ ∂
=∇ ⇔ = + ⎜⎟
⎜⎟
∂∂ ∂
⎝⎠

2
u
(2.6)
where ρ and μ are the density and shear modulus of the medium. Shear stress in the medium as
the constitutive relation can be written as
yz
U
z
τμ
∂
=
∂
(2.7)
yx
U
x
τμ
∂
=
∂
(2.8)

39

where
yz
τ is the corresponding shear stress on a plane normal to y and in the z direction.
Solution of the wave equation (2.6) by the method of separation of variables gives
0
(, ; )
x z
ix iz
c c it
Ux zt Be e e
ω ω
ω
−
−
=
(2.9)
where
x
c and are the horizontal and vertical phase velocities,
z
c ω is the circular frequency, and
is an amplitude factor. Separating the dependency on (x, t), can also be written as
0
B (, Ux ; ) zt
( ) (, ; ) ( )exp Ux zt u z i px t ω =⎡ −
⎣⎦
⎤ (2.10)
where 1/
x
pc =

= horizontal slowness. Combining the solution of the equation of motion (eqn.
(2.10)) and the constitutive equation (eqn. (2.7)) and writing in matrix form gives
() ( ) zz
z
∂
=
∂
fAf (2.11)
where () z f is a displacement-shear stress vector: 2 1 ×
()
()
()
zy
uz
z
z τ
⎧⎫
⎪⎪
=
⎨⎬
⎪⎪
⎩⎭
f (2.12)
and A is a 2 matrix 2 ×
12
34
A=
aa
aa
⎡⎤
⎢⎥
⎣⎦

(2.13)
The entries of A are derived as follows. Substitution of eqn. (2.12) into eqn. (2.11) gives
12
34
() ()
=
() ()
zy zy
uz uz aa
zz aa z
ττ
⎧⎫ ⎧⎫ ⎡⎤ ∂⎪⎪ ⎪⎪
⎨⎬ ⎨⎬
⎢⎥
∂
⎪⎪ ⎪⎪ ⎣⎦ ⎩⎭ ⎩⎭
(2.14)
40

Considering eqn. (2.7) and (2.9), eqn (2.14) can rewrite as
12
34
( ) ( ( )) ( ( ))
(()) (()) ((
uz a u z a u z
zz
uz a uz a uz
zz z
μ
μμ
∂∂ ⎧
=+
⎪
⎪∂∂
⎨
∂∂ ∂
⎪
=+
⎪
∂∂ ∂ ⎩
))

(2.15)
Further, the first and second derivative of displacement can be written as
() exp( )
()()
z
uz i z
zz c
iuz
ω
ωη
∂∂
=−
∂∂
=−
(2.16)
and
2
22
2
() ( ) () uz uz
z
ωη
∂
=−
∂
(2.17)
where
22
1/ p ηβ =− = vertical slowness ( 1/
z
c = , =vertical phase velocity). By
substituting eqn.
z
c
(2.16) and (2.17) into eqn. (2.15) and considering the real and imaginary
coefficients, matrix A is determined as follows
22
01/
0
μ
μω η
⎡⎤
=
⎢⎥
−⎢⎥
⎣⎦
A
(2.18)
It is seen that A is a function of the material properties, the vertical slowness η , and frequency
. ω
The propagator operator (called Matricant in mathematics), introduced to geophysics by
Gilbert and Backus (1966) as a matrix method for solving the layered half-space problem, is a
generalization of the Thompson-Haskell method for surface waves (Thompson, 1950; Haskell,
1953). According to Gilbert and Backus (1966), if matrix

in eqn. A (2.18) is a continuous
41

function of z, as is the case within each layer, the solution at any point z in the layer can be
projected from the solution at another point in the same layer as
0
z
00
() ( , ) ( ) zzz z = fP f (2.19)
where propagator from . The solution of the displacement and stress, f(
0
(, ) zz = P
0
z ) z , in a
semi-infinite homogeneous layer can be written in matrix form as
{ } [ ] { } () F() w zz = f (2.20)
where F, for an SH wave, is a 2 2 × matrix known as “Layer matrix” (Aki and Richards, 1980)
and, w is column vector that weights the columns of F based on the initial condition given for the
layer. The columns of F belong to either an upward or downward wave in the layer provided that
both waves exist initially. The layer matrix can be written as a product of two simpler matrices,
the eigenvectors of matrix A in eqn. (2.18) (E) and Phase factor matrix ( ) (Aki and Richards,
1980)
Λ
exp( ( )) 0
11
F=E. =
0 xp( ( ))
ref
ref
iz z
ii izz
ωη
ωμη ωμη ω η
−⎛⎞
⎛⎞
⎜⎟
⎜⎟
⎜⎟
−−
⎝⎠
⎝⎠
Λ
e−
P
(2.21)
Combining eqn. (2.19) and (2.20), the propagator can be determined as
0
(, ) zz
1
00
(, ) ( ) ( ) zz z z
−
= PFF (2.22)
and has entries
42

11 0 0
12 0 0
21 0 0
22 0 0
P( , ) cos ( )
P( , ) sin ( )
P(, ) ( )sin ( )
P(, ) cos ( )
zz z z
i
zz z z
i
zz i i z z
zz z z
ωη
ωη
ωημ
ωημ ωη
ωη
=−
=−
=−
=−

(2.23a)
(2.23b)
(2.23c)
(2.23d)
As () z f
0
z =
must be continuous at the layer interfaces, if it is known at one point, e.g. at the
surface ( ), where the stress is zero and unit displacement can be assumed, it can be
projected to any other point in any of the layers. Then, starting from the top with ,
0
{}
T
(0) 1 0 = f
() z f can be computed recursively at the layer interfaces
1
zz = , , , and from there
propagated inside the layers, which completely defines the displacements and stresses. Then,
anywhere in the layered medium
2
z …
1
() ( ,0) (0), , 1, ,
ii
zz z zz i
−
=≤≤ fP f …n= (2.24)
11 2 21 1
(,0) (, ) ( , ) ( , ) ( ,0)
ii i
zzz zz zz z
−− −
= PP P P P (2.25)
where (,0) z P = propagator directly from 0 z = to z , and
1
(,
jj
zz )
−
P = propagator for the
layer (eq.
-th j
(2.23a)).
For the 1D building model (Fig. 2.2a), 0 p = , and 1/ η β = . Further, as the first entry of
() z f is , gives the Transfer Function (TF) of displacement at () uz
11
(,0) z P z with respect to the
displacement at (the top), and the inverse Fourier transform of

gives the impulse
response functions (IRF) at
0 z =
11
) z P (,0
z for virtual source at 0 z = . This also defines completely the TF
ref
; )
ˆ
(, hz z ω

and IRF

with respect to any reference point as
ref
(, ; ) hz z t
ref
z
ref 11 11 ref
ˆ
(, ; ) ( ,0)/ ( ,0) hz z z z ω =PP (2.26)
43

{ }
1
ref ref
ˆ
(, ; ) (, ; ) hz z t FT h z z ω
−
=

(2.27)
To account for amplitude attenuation due to material friction, quality factor Q is introduced,
describing the amplitude reduction of a propagating wave or a vibrating volume over one cycle. If
0
A is the initial amplitude, then
[]
0
(; ) exp /(2 ) At A t Q ωω =− (2.28)
and
[]
0
(; ) exp /(2 ) Az A z Q ω ωβ =− (2.29)
High implies small attenuation, and constant Q implies frequency proportional
attenuation (increasing with frequency) (Aki and Richards, 1980). It is incorporated in the
solution by replacing the real valued vertical slowness
Q
1/ η β = by complex slowness
cm
η
plx
1
2
cmplx
i
Q
η ηη =+
(2.30)
Q is related to the damping ratio ζ (ratio of damping and critical damping of viscously
damped oscillators) by 1/ (2 ) Q ζ = . Then, 1% ζ =

is equivalent to , which is value
typical for sediments. Though usually assumed to be frequency independent, Q and
50 Q =
ζ are in
general strong functions of frequency (Trifunac, 1994). As a consequence of Q , the pulse
propagation is dispersed, however, for lightly damped structures, this effect is small.
2.2.2 Analytical Low-Pass Filtered Impulse Response Functions
In practice, the impulse response functions (IRF) are computed from band-limited transfer-
functions over
max
ωω ≤ , for which
44

max
max
ref ref
1
ˆ
(, ; ) (, ; )
2
it
hz z t h z z e d
ω
ω
ω
ω ω
π
−
−
=
∫

(2.31)
An important benefit of eqn. (2.31) is the convergence of the integral for , which is not the
case when
0 Q >
max
ω →∞ (Trampert et al. 1993).
The integration in eqn. (2.31) can be carried out both numerically, e.g. using Fast Fourier
Transform, and analytically (Trampert et al. 1993). While the former is more straightforward, the
latter is more useful for devising and analyzing identification algorithms, as shown later. In the
following, we derive and summarize the analytical IRFs presented by Trampert et al. (1993).
When

(the top), the kernel in eqn.
ref
0 z = (2.31) is the propagator
11
(,0) P z (eqn. (2.25)),
and analytical integration of eqn. (2.31) gives the following expression for band-limited IRF at
some level m normalized to unit amplitude at the source (Trampert et al. 1993) n ≤
() (
1 ()
2
() () () ()
()
1
(,0;)
m m
mm m m i
m ii i i
m
i
a
hz t SC t SA t
b
ττ
−
=
⎡⎤
=−+ ∑
⎢⎥
⎣⎦
)
+ (2.32)
where and are inverse Fourier transforms of attenuated box functions for
causal and acausal waves given by
()
()
m
i
SC t
()
()
m
i
SA t
()
( )
()
() ()
()
()
()
() ()
22
() (
max
max max
ma
) 22
x
exp
1
cos sin
i
m i
ii
m
m
m
mm
ii
SC t t t t
tt
αω
α
αω ω
ω
αα
⎧⎫
−
⎪⎪ −
⎡⎤
=− + −
⎨⎬
⎣⎦
⎪⎪
++
⎩⎭

(2.33)
()
()
()
() ()
()
max
max
()
()
() ( )
22
()
max
ma
( 22
x
)
exp
1
cos sin
i
m i
ii
m
m
m
m
i
m
i
SA t t t t
tt
αω
α
αω ω
ω
αα
⎧⎫
⎪⎪
⎡⎤
=+
⎨⎬
⎣⎦
⎪⎪
++
⎩⎭
−
(2.34)
45

In eqn. (2.32),
() m
i
τ are time shifts such that
( )
() ( )
() ( )
0
1
1
21
1
2
0
mm
mm
mm
mm
i i
i i
h
h
η
η
τ
ττ
ττ
−
−
−
=+
=−
=

(2.35)
and the coefficients and are functions of the reflection and transmission coefficients
() m
i
a
(m
b
)
R and T such that
(0)
1
() ( 1) () ( 1)
21 2
() ( 1) () ( 1)
21 2
1
, - odd
, - eve
,
,n
ii
mm m m
m
mm
ii
i
mm
i m i i
a
aa a a R i
aa R a a i
−−
−
−−
−
=
==
==
(2.36)
()
2
2(
m
m
)
j
j
bT
=
= ∏
(2.37)
where is the reflection coefficient of waves in m-th layer reflected from the interface with the
- th layer and
m
R
1) (m −
j
T is the transmission coefficient for waves from the j -th layer into the
-th layer defined as ( j −1)
11
11
mm m m
m
mm m m
R
ημη μ
ημ η μ
−−
−−
−
=
+

(2.38)
11
2
jj
j
jjj j
T
η μ
ημ η μ
−−
=
+

(2.39)
In equations (2.33) and (2.34), are amplitude attenuation factors
() m
i
α
46

() ( )
() ( )
(0)
1
1
21
1
2
0
m
mm m
mm
m
i i
m i i
mm
h
h
α
αα η ς
αα η ς
−
−
−
=
=+
=−
(2.40)
The other parameters are the same as defined earlier, i.e. is the thickness,
m
h 1/
mm
η β = is
the vertical slowness, 1/ (2 )
mm
Q ζ = and
m
μ is the shear modulus, all of these of the -th
layer. As shown by eqn.
m
(2.32), the impulse response function at each interface represents an
assembly of shifted in time sinc functions, which include the transmitted causal and acausal
pulses, and reflections from the layer interfaces. The TFs in eqn. (2.26) and the derived IRFs in
eqn. (2.27) and (2.32) are exact.

2.2.3 Extension to Torsional Response
In manmade structures, torsional motion has been the cause of many reported failures after major
earthquakes. Torsional motions in buildings have been taken into account in building codes and
provisions by increasing the design load for the structural element, depending on the amount of
lateral load eccentricity from the center of stiffness. In this dissertation, we further develop our
identification process to identify the torsional stiffness of our case studies as well. Accordingly,
the shear wave velocities of vertically propagating torsional waves in building are identified,
where the structure is considered as a cylindrical rod twisting around its central axis.
The equation of motion for a prismatic rod rotating about its central axis (Fig. 2.3) can be
written as follows
M
J
x
θ
∂
=
∂

(2.41)
47

where M (N.m) is the applied torque and J (Kg.m
2
/m) is the mass moment of inertia per unit
length with respect to the rotational axis normal to the surface. J for an arbitrary shape surface is
given by:
2
A
J rdA ρ =
∫

(2.42)
where ρ is the density of rod and is the distance of an element on the surface from rotational
axis. For a homogeneous cylinder of radius a ,
r
J can be calculated as
2
2
00
4
()
/2
a
J rrd dr
a
π
ρθ
ρπ
=
∫∫
=
(2.43)
It is noteworthy that considering a building as a circular shaft is plausible if two assumptions
hold. The first assumption is that the height of building (H) is significantly larger that its length
(l) (i.e. H l >> ). Second, we assume that the wavelength of the wave ( cf λ = ) in the building is
significantly larger than its length (l) (i.e. l λ >> ). These assumptions hold for the case studies in
this dissertation.
According to Achenbach (1973), torsional wave propagates in a finite rod (shaft) in infinite
number of modes. The lowest torsional mode in the rod is non-dispersive while all other modes
are dispersive (Achenbach, 1973). In the lowest mode the displacements are proportional to the
radius such that each cross-section twists as a rigid circular disk. Figure 2.3 illustrates the lowest
mode of torsional wave, using a stack of rigid circular disks twisting on each other around the
central axis, x. This motion is purely in shear (transverse) (Achenbach, 1973). Nevertheless, we
derive the wave equation for this model (Fig. 2.3) and find the equation for the torsional wave
velocity, which clarifies the meaning of our estimated velocities.

48

M
x

Figure 2.3 A stack of rigid disks twisting on each other, illustrating the lowest mode of torsional
wave in a solid rod (shaft)

Figure 2.4a shows two adjacent cross-sections (disks) in the shaft (Fig. 2.3). Equation (2.44)
relates longitudinal shear angle ( γ ) and relative torsional angle ( θ ) presented in this figure
dx r d γ θ =
(2.44)
The shear stress (
x θ
τ ) at the distance from the central axis can be calculated as follows r
x
dx
θ
d
GGr
θ
τγ == (2.45)
where ( ) is the shear modulus of the rod. Equation G
2
N/m (2.45) implies that the shear stress is
directly proportional to the distance ( ) from the axis of symmetry, r x. Figure 2.4b presents
shear stress distribution on a cross-section of the rod. The amount of torque ( M ) induced by this
stress filed can be calculated using the double integral in eqn. (2.42) and the shear stress eqn.
(2.45):
3
0
()
2
x
a
M x rdrd
Grd
x
θ
r
τ θ
θ
π
=
∫∫
∂
=
∫
∂
(2.46)
which results in
49

4
()
2
Mx a G
x
π θ ∂
=
∂
(2.47)
Substituting eqn. (2.47) into the equation of motion (2.41) yields to the torsional wave equation as
follows
22
2
22
T
c
tx
θ θ ∂∂
=
∂∂
(2.48)
where
2
T
cG ρ = is the phase velocity of a pure shear wave implying that the lowest mode of
torsional wave in a rod is a pure shear wave. The shear wave velocities of torsional waves in
buildings are further estimated for structural identification and damage detection purposes.

Figure 2.4 (a) Two adjacent sections of the rod (i.e. two disks) twisting due to a propagating
torsional wave. (b) Cross-section of the rod and shear stress distribution.

x Δ
θ Δ
r
γ
M Δ
x
() r τ
a
r
d θ
(b)
(a)
50

2.3 The Inverse Problem
Achieving high accuracy of identification for high spatial resolution models (with one story per
layer), which is necessary to enable detection of smaller and localized damage is a challenging
task. We address the challenge by detailed error analysis of several identification algorithms, all
involving fitting the model impulse response functions (IRF) in impulse responses computed
from observed response. The first algorithm is the direct algorithm (DA), which is based on
reading the propagating pulse arrival time. The second proposed algorithm is the time shift
matching algorithm (TSM), which fits only the time shifts of the transmitted pulses in the IRFs,
recursively from top to bottom.
The most elaborate algorithm involves nonlinear least squares (LSQ) fit of the IRF,
computed based on complete linear wave propagation theory, at all observation points within time
windows containing the transmitted pulse, which further constrains the fit by including, in
addition to the time shift, information on the pulse amplitudes, the spatial variation of which
depends primarily on the impedance contrast between the layer interfaces. The three algorithms
are:
• The Direct (Ray) Algorithm
• The Non-Linear Least Squares Algorithm
• The Time Shift Matching Algorithm
• Time-Velocity Algorithm: An Extension to Moving Window Analysis
In this section, details of each algorithm including its advantages and limitations will be
discussed.

51

2.3.1 The Direct (Ray) Algorithm
The Ray theory (Aki and Richards, 1980) interpretation of the IRFs in eqn. (2.32) leads to a
simple direct identification algorithm for the layers (Fig. 2.2b), as follows. It is an approximate
theory, based on the assumption that body waves travel through an inhomogeneous medium as a
wave front, with the local propagation speeds, obeying the laws of geometric optics, i.e. along ray
paths determined by Snell’s law (Aki and Richards, 1980). If the variations of material properties
is smooth so that the effects of scattering from inhomogeneities along the path are small, the
approximate solution of the wave equation implies that the wave front travel time between points
A and B, T is the line integral
AB
1
s
()
AB
AB
Td
s β
=
∫
(2.49)
Equation (2.49) implies that the pulse travel time over the height of the building (Fig. 2.2) is
a sum of the travel time through the individual layers, and the travel time through the individual
layers can be obtained directly from the pulse time shifts. This provides the basis for a simple
identification algorithm, which we term direct, derived from the analytical IRFs in the previous
section as follows. The velocities
m
β can be obtained from the time shifts of the pulses
(relative to source time), and the quality factors can be obtained from the ratios of the peak
amplitudes of the causal and acausal pulses, and (eqn.
() m
τ
() ( 1)
/
mm
h ττ
i
1 =
m
Q
(
()
0
m
) SC
i
()
()
0
m
SA
i
(2.32)) (Trampert et al.
1993). If there is a recording at every layer interface, the transmitted pulse ( ) is sufficient to
identify all layers from the relationships
i
11 mm
β
−
−= (2.50)
52

()
()
()
()
()
() (
() ( 1
1)
)
max 11
11
() ( 1)
11
00
n
0
lln
0
m
m
m
m
m
m
m
SA SA
SC SC
ττ ως
−
−
−
−= −
(2.51)
where 1/ (2 )
m
Q
m
ς = . Eqn. (2.50) and (2.51) imply
( )
() ( 1)
11
mm
mm
h βτ τ
−
=−
(2.52)
()
()
()
()
() ( 1)
11
() ( 1)
max
() ( 1)
11 11
11 1
ln ln
00
00
2
m
m
m m
m
m
m
SA S
Q
A
SC SC τ
ω
τ
− −
−
⎛⎞⎡
⎛⎞
⎜⎟
⎤
⎢ ⎥ =−
⎜⎟
⎜⎟
⎢ ⎥ −
⎝⎠
⎝⎠⎣ ⎦
(2.53)
and overall Q
()
()
()
ma
(
x 1
)
1
()
1
11
ln
0
0
2
n m
n
SA
SC
Q
ωτ
=
(2.54)
where
()
1
m
τ is the travel time from base to roof. If there is not a sensor at each layer interface, it
is still possible, in principal, to identify all the layers, using the reflected pulses. However, the
reflected pulses are practically useful only if they can be resolved. In chapter 3, the effect of
scattering from the boundaries will be discussed, when the stiffness variations among layers are
significant. To investigate the effect of scattering on our identification results, we simulate a shear
beam model of Millikan Library including buildings’ slabs as separate layers.

53

2.3.2 The Nonlinear Least Squares Algorithm
This algorithm involves fitting, in the least squares (LSQ) sense, of pulses in the IRFs, as
functions of time, over predefined time windows. We use the Levenberg-Marquardt method for
nonlinear least squares estimation (Levenberg, 1944; Marquardt, 1963), which is a fixed
regressor, small residual algorithm, considered to be generally robust, and is implemented in
Matlab (Seber and Wild, 2003). Let
12
(, , , )
L
βββ = β …
obs
(, ) hzt
jji
be the vector of the velocity profile,
where is the number of layers. Further, let (eqn. L (2.3) and (2.4)) be the observed
IRFs in time window j at point i of that window, and let
( )
mod
,;
jji
hzt β be the model IRF at
the same point in space and time, for velocity profile β

(eqn. (2.32)). Here
j
z is the z -
coordinate of that window, and
ji
t is the time of the i-th point of the j-th window. The estimation
is based on the representation
()
obs mod
(, ) , ;
j ji j ji ji
hzt h zt ε
∗
=+ β
(2.55)
where is the true value of β, and
∗
β
ji
ε is the error, assumed to be identically distributed,
statistically independent, zero mean random variable. Then, the least squares estimate of , , is
such that it minimizes the sum of the squares of the error at all observation points
β
ˆ
β
() () ( )
2
obs mod
11
,,
j
N
J
jji j ji
ji
Shzthzt
==
⎡⎤
=− ∑∑
⎣⎦
ββ;
(2.56)
where is the number of points in the j-th window, and N
j
J is the total number of windows. As
a measure of the speed of convergence and error of the fit, the standard deviation of ,
ˆ
β σ , is
obtained from the scatter of the successive estimates.
The value of in the model is assumed, as the variation of the pulse amplitudes along the
height is mostly governed by the impedance of the layers (see eqn.
Q
(2.54)). Also, our experience
54

showed that, for reasonably chosen values of Q , is not sensitive to . Consequently, we use
either the apparent

estimated by the direct method (section 2.2.1), or some value that
approximately matches the observed apparent damping. We leave detailed study of for the
future.
ˆ
β Q
Q
Q
The time windows over which IRFs are fitted are chosen so that they enclose only the main
lobes of both the acausal and causal transmitted pulses. The reason for this is to maximize the
signal to noise ratio, as, in practice, the reflected pulses from the layer boundaries have much
smaller amplitudes than the transmitted pulses, and are difficult to distinguish from the ripples of
the sinc functions. The algorithm requires initial value of , for which we use the estimate by the
direct algorithm, obtained from the ratio of distance traveled and pulse time shift (
β
/
ii
h β =
i
τ )
(section 2.2.1). Because the Levenberg-Marquardt method is a small residual, iterative method, to
insure convergence, the initial value has to be reasonably close to the true value. Our study of
several buildings showed that such initial values are appropriate.
The two main advantages of the nonlinear LSQ algorithm over the direct algorithm are that
(1) it uses also information about the pulse amplitudes, which serves as additional constraint for
the fit, and (2) it is not limited, at least in principle, by the assumption of ray theory that the
variation of material properties is smooth.
In this dissertation, Simulated Annealing (SA) has been used, as well, as an alternative
optimization method for the Levenberg-Marquardt (L-M) method. The main advantage of SA
over L-M method is that SA does not trap in local minima, and therefore, is more convergent than
L-M method. On the other hand, SA is essentially a brute-force search method, which categorizes
it as a slow and time-consuming optimization method.

55

2.3.3 The Time Shift Matching Algorithm
This iterative algorithm involves progressive adjustment of the layer velocities until the time
shifts of the transmitted pulses in the model IRFs are close enough to those in the observed IRFs.
The time shift is measured from the time of the peak of the pulse. The β profile is estimated
recursively starting from the top layer, and the average of the peak time for the causal and acausal
transmitted pulses is matched, which eliminates errors due to poor synchronization of motions
recorded by different channels, and effects of pulse distortion. The starting value for the iteration
is the estimate by the direct method, and the process is terminated when the times of the peaks
becomes smaller than the time step of the data.
By interpolation of the recorded time series, the time step can be made arbitrarily small.
However, the actual error in the identified β profile is larger than what is implied by a close
match of the peak times of the pulses. The reason for this is that there is an error in which the
peak time of the pulses, as measured from the IRFs, corresponds to the true pulse arrival time,
and that this error is different for the IRFs of the data and of the model, because they do not have
exactly the same shape of spectrum. Hence, the estimated using TSM model does not represent a
true stiffness profile for the layers.
2.3.4 Extension to Moving Window Analysis – Time-Velocity Graphs
The Time-Velocity algorithm is a generalization of the waveform inversion algorithm for
identification of buildings, to an automated moving window analysis. In each time window, the
vertical phase velocity of the building is identified by fitting an equivalent layered shear beam
model in the recorded acceleration response of the building by matching, in the least squares
(LSQ) sense, amplitudes of selected pulses in computed impulse response functions (IRF),
simultaneously at all levels where motion was recorded. The identified vertical phase velocities of
the layers are mapped on time-velocity graphs, which depict their temporal variation during the
56

earthquake shaking. Inference on loss of stiffness, possibly due to damage, is made based on the
changes in the identified velocities relative to the values in the initial time window of smaller
response, assuming that it represents the response of the undamaged structure. The Time-Velocity
algorithm is illustrated in Fig. 2.5 for the case when only two records, at base and roof, are used
for the identification. The Time-Velocity algorithm is illustrated in Fig. 2.5 for a single layer
model and using records at the roof and base only, to avoid clutter. It shows the acceleration
records (left), their transformation into IRFs (middle), and the time-velocity graph. It shows a
representative moving time window, of length and central time , the time windows
win
w
c
t
pulse
w over which pulses of the model IRFs are fitted, and the resulting wave velocity for the
window, mapped at .
c
tt =

Illustration of the Time-Velocity algorithm for two records.
Figure 2.5
The model that is fitted in this study is an elastic, viscously damped, shear beam, with
piecewise uniform material properties, excited by vertically incident shear waves. The layers are
characterized by thickness , mass density
i
h
i
ρ , and shear modulus
i
μ , which gives shear wave
57

velocities /
ii
v
i
μ ρ = and quality factor (the damping ratio Q 1/ (2 ) Q ζ = ; 2% ζ = was
assumed in this study). If desired, the slabs can be modeled as separate layers (thin, heavy and
stiff). In this study, the layers comprise of the part of the building between the instrumented
floors. Layering is introduced to enable finding the spatial distribution of the severity of damage.
For this generic model, both TFs and low-pass filtered IRFs can be computed analytically, from
the propagator of the medium (details in section 1). The model IRFs are fitted in the observed
IRFs over selected time windows with length
pulse
w , chosen to contain the acausal and causal
pulses (Fig. 2.5).
2.3.5 Choice of Time Windows
The estimate of wave velocity is a weighted average over the time window . As
it is the case with moving window analyses, there is a tradeoff between the time resolution and
the quality of the estimate, due to the Heisengebg-Gabor uncertainty principle for signals, and a
critical window length below which an estimate cannot be obtained. E.g., the average wave
velocity over the building height cannot be estimated if the time window is shorter than the wave
travel time over the building height, with correction for the pulse width, i.e.
(/
cwin
w2)
f
tt ∈±
max
0.5 / 1/ 2
win pulse
ww Hc τ >+ = + .
To capture both the acausal and causal pulses (Fig. 2.5), the time window must contain two
traversals over the height, i.e.
max
/1
win
ww Hc 22 /f
pulse
τ >+=+ . Therefore, taller and softer
buildings would require longer time windows than shorter and stiffer buildings, and must be
chosen carefully for the particular case study. Practically, longer and tapered time windows
should be used than the theoretical minimum to avoid Gibbis effect in the Fourier transform, and
to account for lengthening of the wave travel time due to damage.
win
w
58

The critical value of to include both acausal and causal pulses can be roughly
estimated from the building fundamental period of vibration as
(assuming uniform shear beam and
win
w
1
T
1m
/2 1/
win
wT f >+
ax
1
/4 T τ ≈ ). Finally, the width of the time windows over which
the acausal and causal pulses are fitted (see Fig. 2.5) is chosen to be approximately equal to the
width of the source pulse, i.e.
ma
1/
se
wf
x pul
≈ .
2.3.6 Choice of Frequency Band
The choice of the frequency band for the fit is very important. As shown in section (2.4.1), the
resolving power of the identification using IRFs is not only limited by the density of sensors but
also by the bandwidth of the data, which determines the width of the source pulse. More
broadband IRFs result in narrower pulses and afford higher resolving power and better accuracy
of the identification, for given density of observation points. However, in practice, limiting the
maximum frequency for the fit may be necessary, to ensure that the fit is meaningful. E.g., the
identification based on a layered shear beam model, which is characterized by nondispersive
wave propagation (except for small dispersion resulting from material damping), is not
meaningful if there is evidence of significant dispersion in the observed response.
Causes for dispersion not accounted for in this model are, e.g., deformation of the building
as a whole to some degree in bending, which is more pronounced in buildings with shear walls
(Ebrahimian and Todorovska, 2013), as well as lateral reflections from the external boundaries
and impedance contrasts (Todorovska and Lee, 1989), which results in vertical phase velocity
that varies with frequency and in multiple wave modes. Therefore, we first check for wave
dispersion in the data by inspecting the propagating pulses for distortion of their shape and by
comparing pulse travel times in band (0,
max
ω ) for different
max
ω and in disjoint frequency
bands, noting systematic differences that appear to be well beyond uncertainty of the estimation.
59

In case of significant differences, a cut off frequency
max
ω is determined such that for the band
as the largest frequency such that pulses are not deformed and the pulse time shift is not affected
by incremental change of
max
ω . The identification is also carried out in higher disjoint band(s) if
transmitted pulses can be clearly identified in these bands.
Another reason to limit the bandwidth for the fit is that the model is continuous and can
represent well a building only for longer wavelength compared to the height. We also define
effective data bandwidth as the maximum frequency beyond which the width of the main lobe of
the pulses in the observed IRFs does not decrease. The effective bandwidth of observed IRFs is
limited because of the nature of the excitation and of the structural response, which results in
small amplitudes of the observed transfer function at higher frequencies, and because of the
regularization parameter, ε (eqn. (2.3)). A more detailed discussion will be presented in chapter
3 using simulated IRFs and observed IRFs and TFs of Millikan Library N-S response.

2.3.7 Resolution and Accuracy
( ) (; sin /
max ref
hz t t ω π As discussed in section 2.1, the source pulse ) t = is localized both in time
and in frequency, around time 0 and frequency /2 t =
ma
ωω =
x
, but not perfectly. Its spread in
frequency is . Let the half-width of the main lobe /2
max s
ωω Δ= /1/(2
max max
tf
s
) π ω Δ==
be a measure of its spread in time, where /(2 )
max
f
max
ω π = . (The second moment, usually
used to define the spread in time cannot be used, because it is infinite for this source function.)
This implies that smaller
ma
ω
x
results in a wider source pulse, which worsens the time
localization of the IRFs, and, consequently, the spatial localization of the wave front. The product
of
s
ω Δ

and
s
t Δ

for the source pulse is always constant
60

2
ss
t
π
ω ΔΔ = (2.57)
Equation (2.57) represents the Heisenberg-Gabor uncertainty principle for signals (Gabor,
1946, 1953), according to which a signal cannot be perfectly localized both in time and in
frequency, and increased localization in one domain is at the expense of decreased localization in
the other domain (Its equivalent in physics states that both the position and velocity of a particle
cannot be determined exactly). An important consequence of this principle is the finite spatial
resolution of the identification of the variations of the layer properties, and the uncertainty in the
identified properties at a particular location in the building. It is demonstrated further in this
section that the limited resolving power leads to some minimum thickness of a layer that can be
resolved by the IRFs, even if there exists a recording within that layer.
The related uncertainties and z Δ β Δ can be derived from the Heisenberg-Gabor uncertainty
principle and interpreted as follows. The disturbance at the virtual source will propagate through
the medium as a pulse in space with half-width . Further,
/
max
zt
s
βπβω Δ= Δ =
z Δ can be
expressed as z β τ Δ=Δ , where β Δ is uncertainty in β and τ is travel time, which gives
//( )
max
β βπ Δ= ω τ
. This means that the point estimate of velocity ( ) z β , measured from the
time shift of a pulse in the IRF, is a weighted average over the interval [] , and that
the identified value of
, zzz −Δ + Δz
β

is an estimate, the error of which has distribution with spread β Δ .
Recalling that
/
max
t
s
πω =Δ
, it follows that
// t
s
ββτ Δ=Δ
, which means that the relative
error in the estimate of β will be proportional to the ratio of the source pulse half-width and the
time lag τ used to identify β .
The above discussion implies that detecting localized variations in () z β from readings of
the pulse shift requires high enough signal bandwidth
max
ω , and that, for given
max
ω , the error
61

of the identified () z β

can be reduced if it is identified from longer travel time τ , i.e. from larger
distances. It also implies that the error in the identified β

will be smaller in more flexible
buildings, and, within the building, in its softer parts. As an example, if β ≈ 200 m/s,
max
f = 25
Hz 4 m , and ⇒ z Δ≈ h ≈
max
f =10 Hz ⇒ z Δ ≈ 10

m 2.5h ≈ , where is the story height. This
uncertainty in measuring
h
() z β is similar to the uncertainty in identifying instantaneous
frequency of signals using wavelet transform, Gabor transform or any other type of moving
window technique, where instant refers to a time window, and there is uncertainty in the
identified frequency, which is larger if the time window is shorter (Todorovska and Trifunac,
2007).
An important consequence of the localization in time is the minimum resolvable thickness in
the building model termed as resolving power of the IRFs. The resolving power of IRFs is not
limited only by the density of sensors, but also by the effective bandwidth of the data, which
determines the width of the pulses in the IRFs. Recalling the notation for the spread of the pulse
in time (i.e. half-width of the source pulse,
t
s
Δ
) and the spread in frequency (i.e. half-width of
the Box function,
f
s
Δ
), one can formulate the minimum resolvable height ( h ) for a given cut-
off frequency. This is based on the fact that a layer involving one or multiple floors can be
resolved if the causal and acausal pulses at the layer’s boundaries slide at least
min
t
s
Δ
in time apart
from each other. In other words, the shear wave has to propagate a minimum distance of
0.5 h
n
t
s
β = ×Δ , where β
mi
is the identified shear wave velocity of the layer. This is provided
that amplitudes of the side lobes of the source pulse are relatively small. As mentioned at the
beginning of this section, for a lowpass filtered IRF (i.e. [0,
max
f ]), the source pulse is a sinc
function which is spread in time with

(half width of the source pulse).
Consequently, the minimum resolvable thickness for a lowpass filtered IRF would be:
1/ 2
max
f t
s
Δ=
62

min
min
max
.
24 4
t
h
f
λ β β Δ
== =
(2.58)
where
min
λ is the shortest wavelength propagating in that layer.
Further, bandpass filtered IRF (i.e. [
1
f ,
2
f ]) can be simply achieved by subtracting two
lowpass filtered signals with cut-off frequencies at
1
f and
2
f . Equation (2.59) presents the
bandpass filtered IRF at the source.
21
ref
sin( ) sin( )
(,)
tt
hz t
tt
ω ω
π π
=− (2.59)
where
1
ω and
2
ω are the angular frequencies at the edges of the Box function ([
12
, ω ω ]). This
equation can be further simplified using trigonometric functions relationships as follows.
12
ref
2sin( .)
(,) cos
2
t
hz t t
t
ωω ωω
πω
+ ΔΔ ⎛⎞
=
⎜⎟
Δ
⎝⎠
(2.60)
where
21
()/2 ω ωω Δ= − . Equation (2.60) implies that the bandpass filtered IRF is a cosine
function modulated by a sinc function. This fairly wide sinc function can be emerged by plotting
the envelope of calculated IRF. The resolving power of a bandpass IRF ought to be calculated
based on half-width of the central envelope at the source. Equation (2.61) and (2.62) show the
spread of the envelope in time at the source and the minimum resolvable thickness for this IRF.
21
11
2
s
s
t
f ff
π
ω
Δ= = =
Δ− Δ

(2.61)
min
min
4( ) 4
s
h
f
λ β
==
Δ

(2.62)
63

It is noteworthy that the bandwidth for strong motion data is typically 25 Hz or 50 Hz
(Trifunac, 1971, 1972). However, the effective
max
f , which determines the width of the pulses
in observed IRFs in real buildings, is considerably smaller, because of the physical nature of the
response, which becomes very small beyond some frequency, even smaller than the regularization
parameter ε in eqn. (2.3) used to compute transfer functions from observed response (Snieder
and Şafak, 2006). The regularization parameter, ε ,introduced to prevent division by accidental
zeros in
ref
ˆ( uz ) , also sets to zero the transfer function beyond some frequency, resulting in
pulses in the IRFs that have width wider than theoretically expected for chosen
max
f . Another
reason to limit
max
f in the identification is to exclude in the observed IRFs effects from details
(elements) not captured by the model (e.g. effects of bending deformation).
Another source of error in the direct algorithm identification process is the pulse arrival time
reading precision error. This error is proportional to the sampling interval of the IRFs, and can
be tabulated as follows. In Physics, measurement error for a tool is defined as half of the least
count. Considering the propagating pulse arrival times, our least count would be the sampling
interval,
sam
t Δ . Consequently, the reading precission would be less than or equal to half of the
sampling interval (E ). Assuming the Ray theory holds for a shear beam model (i.e. /2
sam
t
τ
≤Δ
/ h β τ = ), we can relate the reading error to the error in the estimated phase velocities using
fractional error measurement principles as follows.
2
sam
Et
τ
ββ
β τβ τ
Δ ΔΔ
=± ⇒ ≤±
(2.63)
Equation (2.63) reveals that the measurement error falls in a range, whose maximum value is
directly proportional to the sampling interval. Moreover, similar to the identification error due to
Heisenberg-Gabor uncertainty principle, a longer travel time τ (e.g. a larger distance or a more
flexible beam) results in smaller measurement error. In the next section, it is shown that the
64

measurement error is a small portion of the identification error in the direct identification
algorithm.
2.3.8 Model Validation
The goodness of fit is examined by analysis of several measures, such as σ of the nonlinear
LSQ estimate, and the agreement of the pulse arrival times at different floors, as measured from
the time of the peaks of the transmitted pulses in the IRF. Further, the model is validated by
examining the agreement of the transfer-function (TF) between roof and base motion of the
model. This test examines how realistic is the model. In particular, the frequencies of the modes
that fall within the identification subband are compared, with consideration of the fact that the
model TF is that of a fixed-base structure, while the observed TF is that of a structure on flexible
base (with some constrains). For the horizontal responses, the foundation cannot translate but can
rock, and for the torsional response, it cannot twist but can translate (Todorovska, 2009).
Consequently, the frequencies of the peaks in the TFs are the apparent frequencies, and depend
on the foundation stiffness, the fundamental mode being most affected. The frequencies of the
higher modes, however, which are affected less by the soil-structure interaction, are expected to
agree, to a degree afforded by the spatial resolution of the model. The latter refers to the fact that
the frequencies of vibration are sensitive to the distribution of stiffness along the building height
(while the travel times are not). Consequently, for coarse models, the agreement of the
frequencies may not be very good. For high resolution models, however, checking for such
agreement is necessary, because of the larger identification error. In chapter four and five, we
discuss the goodness of our estimated models while presenting the results of identification of two
buildings in greater Los Angeles area.

65

66

Chapter 3
Critical Assessment of the Direct (Ray)
Algorithm and Travel Time Analysis
In this chapter, we investigate structural system identification from wave travel time (Direct
algorithm) for use in structural health monitoring, with emphasis on the spatial resolution and
accuracy of the identification.
Simulated broadband (0-50 Hz) IRFs for layered building models are presented, and the
trade-off between resolution and accuracy of identification are demonstrated. These results along
with derivations in the methodology chapter are published in:
Todorovska MI, Rahmani M (2013). System identification of buildings by wave travel time
analysis and layered shear beam models - spatial resolution and accuracy, Structural Control and
Health Monitoring, 20(5): 686–702, DOI: 10.1002/stc.1484, first published online on 3/6/2012.

67

3.1 Analysis of Simulated Impulse Response Functions
Analysis of accuracy of our identification requires a predictive theoretical model for the response
that is derived from the complete wave propagation theory. Exact analytical solutions for model’s
transfer-function and impulse response function are presented in chapter 2. In this section, we
show simulated broadband (0-50 Hz) IRFs for layered building models and examine their features
relevant for identification, such as reflected pulses and pulse amplitude attenuation. In the next
section, we demonstrate the trade-off between resolution and accuracy on a case study of a real
building excited by an earthquake - Millikan Library NS response during Yorba Linda earthquake
of 2002, which was recorded by a dense network of sensors.
Results are shown of simulated IRFs (0-50 Hz) of a model of a 9-story RC building, and of
identified shear wave velocity profiles of Millikan Library NS response from recorded response
during Yorba Linda earthquake of 2002, using the direct algorithm. The model IRFs aim to
illustrate the effect of layering on the pulse propagation and IRFs, and the use of reflected pulses
for identification. Further, it is examined if the direct algorithm, which is based on simplifying
assumptions, is valid for buildings, and if the pulse amplitudes can be used to identify the
structural damping. The results of the identification aim to show the magnitude of the
identification error for the direct algorithm, for a real building and data, as function of the detail
of the model fitted.
3.1.1 Model
To avoid unnecessary complexity, a 3-layer model of a 9-story RC building, with three
stories per layer, is chosen, with mass density and shear wave velocity profiles as shown in Fig.
3.1 (the solid lines). The dashed lines show the profiles for an equivalent uniform model
approximation, which has same travel time from level ground to roof. Without loss of generality,
the model corresponds to Millikan Library NS response. Tables 3.1 and 3.2 show the layer
68

parameters models, ordered from top to bottom. The theoretical domain travel time /
ii
h
i
τ β = is
also shown, as well as the sum of the domain travel times
1
n
i
i
τ
=
∑
. The quality factor 25, which
corresponds to critical damping ratio
Q =
2 ζ = %.
Table 3.1 Input parameters for the equivalent uniform model
Layer Floors
i
h
i

[m]
ρ

[kg/m
3
]
i
β
[m/s]
Domain travel time
/
ii i
h τ β = [s]
1 1-9 39 496 390 0.1

Table 3.2 Input parameters for the 3-layer model
Layer Floors
i
h
i

[m]
ρ

[kg/m
3
]
i
β
[m/s]
Domain travel time
/
ii i
h τ β = [s]
1 7-9 12.8 526 242 0.053
2 4-6 12.8 473 569 0.0225
3 1-3 13.4 490 536 0.025
1
1
i
i
=0.103 s
τ
=
∑

Profiles of mass density and shear wave velocity distributions for the 3-layer (solid
line) and the uniform model (dashed line).
Figure 3.1
69

3.1.2 Pulse Travel Time
Impulse response functions for the 3-layer model are shown in Fig. 3.2. In part a), the virtual
source is at ground level (point of energy entry), and all pulses are causal. Most prominent is the
pulse that is transmitted through all of the layer interfaces. It arrives at the roof first at t = 0.1 s,
where it is reflected with doubled amplitude, and propagates back to ground level. The pulses
reflected from the interface between the top and middle layer can also be clearly seen, especially
the one reflected back to the top layer. It has opposite sign because it reflects from a stiffer layer,
and arrives at the roof at 0.2 s. The top layer can be identified from its time shift t = τ relative
to the first arrival of the transmitted pulse at the roof, which is 0.1 s, and equals twice the travel
time through the top layer, / h β
11
2 . This gives
1
2 12.8 / 0.1 256 β =×= m/s, which is close to
the input value (242 m/s).
In part b) of Fig. 3.2, the virtual source is at the roof, not coinciding with the physical source.
Therefore, the transmitted pulse on the way up is acausal, while on the way down is causal
(Snieder and Şafak, 2006). As in the case of a uniform model, the transmitted pulse does not
reflect from the base (because all reflections from the roof are suppressed) (Snieder and Şafak,
2006). Fig. 3.2b shows that the pulses from the internal reflections all propagate only downwards
and are never reflected back. The reflections of the acausal pulse, which physically propagate
downwards, are all causal. In contrast, the reflections of the causal pulse, which physically
propagate upwards, are all acausal. E.g., the pulse that is physically reflected back into the top
layer now is an acausal pulse propagating through the middle layer, and can be used to identify
the middle layer, as follows. In the IRF at the interface between bottom and middle layers, its
time shift relative to the arrival of the transmitted causal pulse is
22
2/ h τ β = , and can be used to
resolve the middle layer. Another observation is that the pulses from the internal reflections all
occur in the IRFs between the transmitted acausal and causal pulses.
70

a)

b)
Impulse response functions at each floor for the 3-layer model, for virtual source
(a) at ground floor and (b) at roof.
Figure 3.2
71

As shown on the model IRFs, the top and middle layer can be resolved as separate layers
even if there is no recording at their interface. However, in practice, this can be done only if the
reflected pulses can be distinguished from the “noise”, i.e. from the side lobes of the transmitted
and other reflected pulses, as well as pulses due to foundation rocking (Todorovska, 2009a).
Fig. 3.3 shows the pulse arrival times at the layer interfaces, as measured from the IRFs.
The differences between the arrival times at different floors closely match the domain travel times
and the total travel time (see Table 3.1 and 3.2). This demonstrates that the direct identification
algorithm, which is based on the approximate ray theory interpretation of the IRFs, is valid for
layered building models in which the layers correspond to a group of floors (such a used in
(Todorovska and Trifunac, 2008a,b)). However, this is not the case for layered models with slabs
(modeled as thin and heavy layers between soft layers). Our analysis shows that, for such
models, the pulse travel time (measured from the IRFs) is larger than the sum of the domain
travel times, indicating additional phase delays due to scattering of the pulse from the slabs. The
direct algorithm does not work for this case because the assumption of smoothness of the
variation of material properties across the model is violated.

Pulse arrival times for the 3-layer model as measured from simulated impulse
response functions.
Figure 3.3
72

3.1.3 Pulse Amplitudes
Next, we analyze the attenuation of pulse amplitudes as function of the IRF bandwidth
max
f .
Let
a
A ,
top
A and
c
A be the amplitudes of the acausal pulse at ground level, half of the
amplitude of the source pulse at the roof, and the amplitude of the causal pulse at ground level.
Fig. 3.4a shows ratios /
top a
A A , /
ctop
A A and /
c a
A A , versus
max
f for the 3-layer model,
which are measures of the pulse attenuation on the way, on the way down, and the total
attenuation on its way up and down. It can be seen that the pulse is strongly amplified on its way
up ( ) and deamplified on its way down ( /
a
AA 1
top
> /
cto
AA 1
p
< ), which indicates that the
pulse amplitude is governed primarily by the impedance of the layers, and to a minor degree by
the material attenuation. The ratio /
ca
A A , however, is a measure of the pulse attenuation due
to , because the effect of the impedance cancels out over the total path (up and down), and the
model has no foundation rocking (Todorovska, 2009a). Fig.
Q
3.4b shows /
c a
A A versus
max
f

for the 3-layer, and its equivalent uniform models, which are fixed-base, and also for the uniform
model with foundation rocking. It can be seen that the pulse attenuation due to Q is practically
identical for the uniform and 3-layer models, and increases with
max
f , as expected from the
assumed attenuation model. However, the attenuation for the model with rocking, which has
same in the structure, is much larger, demonstrating that the pulse amplitudes in IRFs reflect
the combined attenuation - due to Q in the building and due to radiation damping via foundation
rocking. This makes it impossible, in general, to identify Q of the fixed-base structure from IRFs,
except for building with very large foundation rocking stiffness.
Q

73

Attenuation of pulse amplitudes measured from IRFs (virtual source at roof) versus
bandwidth
max
f . a) Attenuation on the way up, /
top a
A A , on the way down, /
ctop
A A , and
total, /
ca
A A , for the 3-layer model. b) Total attenuation, /
c
Figure 3.4
a
A A , for the 3-layer and uniform
models (fixed-base), and for a uniform model with foundation rocking (Todorovska, 2009a).

74

3.2 Case Study - Millikan Library N-S Response
Millikan library is a 9-story reinforced concrete building in Pasadena, California, instrumented
over a period of 40 years. The building vertically extends 43.9 m above grade and 48.2 m above
basement level. Details on lateral resistance systems and soil condition will be provided in
chapter 5. Published work suggests uniform mass distribution over the first three, middle three
and top three stories, which we use to construct our models (Jennings and Kuroiwa, 1968).
Yorba Linda earthquake of September 3, 2002 (M=4.8, epicentral distance R=40 km) was
recorded by a dense network of sensors. The building response was small, with maximum rocking
angle of 0.012 10
-3
rad. Fig. × 3.5 shows the observed NS response (at West wall): a) the
recorded acceleration time histories (low pass filtered at 25 Hz), b) the corresponding IRFs for
max
f =15 Hz and 25 Hz (solid and dashed lines), and c) the transfer function (TF) between roof
ground floor accelerations. As seen in part c), mostly the first two modes contribute to the
recorded response, and the TF amplitudes beyond 15 Hz are very small. This reflects on the
widths of the pulses in part b), which are practically the same for the different bandwidths, while
theoretically should have differed by factor of 1.7, and suggests that the effective bandwidth
max
f for these records is about 15 Hz, which is much smaller than the capability of the recording
instruments (typically 25 to 50 Hz (Trifunac and Todorovska, 2001)). This effective
max
f is
critical for the resolving power of the IRFs. As seen in Fig. 3.5b), the transmitted causal and
acausal pulses are too wide relative to their time shifts, and cannot be resolved at the 9
th
floor, and
the top floor cannot be resolved as a separate layer, even though there is a sensor there. It is also
seen that no reflected pulses can be resolved in the observed IRFs.
75

Millikan library NS response during Loma Linda, 2002 earthquake observed at
West wall. a) Accelerations. b) Impulse responses for virtual source at roof. c) Transfer-function
of roof acceleration w.r.t. ground floor.
Figure 3.5

76

Three models with different spatial resolution were fitted using the direct algorithm: 1-layer,
3-layer (with three floors per layer), and 8-layer model in which the top layer consists of the 8
th

and 9
th
floors. The assumed mass distribution is same as in Fig. 3.1, and Q 25 ( = ζ =
app
2%), which
is close to the first mode apparent (soil-structure interaction system) damping ζ =1.74% as
identified from the transfer-function. The average of the time shifts for the causal and acausal
pulses was used, as our study of several buildings showed that it leads to consistently smaller
identification error. The goodness of fit is assessed from the general agreement between the
predicted by the identified model and the observed pulse arrival times at the different levels, and
travel times through the layers. In particular, the relative error in pulse travel time was computed,
( )
pred obs obs
// τττ τ τ Δ= − , where
pred
τ and
ob s
τ are the predicted by the model and the
observed travel times through the layer. This enabled to estimate the relative error in the
identified β

as / / ββ Δ= τ −Δτ (see the methodology section). The different resolution
models were fitted for data and model bandwidth
max
f =15 Hz. The highest resolution model (8-
layer) was also fitted for
ma
f
x
=25 Hz. The results are shown in Tables 3.3-3.5 and Fig. 3.6.
The different columns in the tables show the layer number (top to bottom), width , average z-
coordinate
i
h
i
z , observed
i
τ , identified
i
β , predicted
i
τ , and the errors / τ τ Δ and / β β Δ .
Fig. 3.6 shows the identified β -profiles (part a)), and the agreement of the observed and
predicted pulse arrival times (part b)), and transfer-functions (part c)). The agreement of transfer-
functions is an indicator of how realistic the fitted model is.
The results confirm that the identification error / β β Δ is larger for more detailed models
(8-layer in this case) than for coarser models (1-layer and 3-layer in this case) as predicted
theoretically in the methodology section (Fig. 3.6b). The error for the identified 8-layer model is
considerable (Tables 3.5a,b), and the 3-layer model is the optimal for the direct algorithm and this
77

Identification of Millikan library NS response during Yorba Linda earthquake of
2002. a) Identified shear wave velocity profiles. b) Agreement of pulse arrival times at the
different floors. Note: the arrival at the 9
th
floor could not be resolved. c) Agreement of transfer-
functions.
Figure 3.6

78

Table 3.3 Shear wave velocities of Millikan Library NS response identified by the direct
algorithm based on a 1-layer model, using accelerations of Yorba Linda earthquake of 2002
recorded at west wall (0-15 Hz)
Layer Floors
i
h
[m]
i
z
i

[m]
Observed
τ [s]
Identified
i
β [m/s]
Predicted
i
τ [s]
/ τ τ Δ /
%
β β Δ
%
1 1-9 38.9 36.45 0.096 405.2 0.097 1.04 -1.04

Table 3.4 Shear wave velocities of Millikan Library NS response identified by the direct
algorithm based on a 3-layer model, using accelerations of Yorba Linda earthquake of 2002
recorded at west wall (0-15 Hz)
Layer Floors
i
h
[m]
i
z
i

[m]
Observed
τ [s]
Identified
i
β [m/s]
Predicted
i
τ [s]
/ τ τ Δ /
%
β β Δ
%
1 7-9 12.75 6.38 0.048 265.63 0.045 -6.3 6.3
2 4-6 12.75 19.13 0.025 510 0.0285 14 -14
3 1-3 13.4 32.2 0.023 582.61 0.0215 -6.5 6.5

79

building, both in terms of detail and accuracy. Further development of this method of
identification by fitting IRFs is necessary to improve its accuracy for detailed models, e.g. by
introducing iteration and/or least squares fit.
In examining the agreement of transfer-functions (Fig. 3.6c), it is important to note that the
observed TFs are those of the soil-structure system (without foundation translation but with
foundation rocking), while the model TFs are those of the fixed-base structure. Further, the
modal frequencies are sensitive to the variation of shear wave velocity and mass density with
height, due to which more detailed models would predict better the modal frequencies then
coarser models, but only if they are accurate. (In contrast, the total travel time depends on the
properties of the layers, but not on their order and would be the same for any order of the layers.).
Fig. 3.6c shows that the frequency of the fundamental mode is significantly higher for all the
models (2.5-3 Hz) than for the observed response (~1.7 Hz), as can be expected for this building
NS response for which it is known that foundation rocking contributes considerably to the total
response (about 30%) and the effects of the soil-structure interaction are significant. The
predicted fixed-base fundamental frequency agrees generally with other studies and disagrees
with (Todorovska, 2009a). The frequencies of the higher modes are not affected much by soil-
structure interaction, and their agreement can be used as a measure of how realistically the
identified models represent the building. Fig. 3.6c shows that the predicted frequency of the
second mode differs between the three models, but is close to the observed one, which indicates
that the models, identified solely from wave travel times, represent realistically the dynamic
behavior of the fixed-base building.

80

Table 3.5a Shear wave velocities of Millikan Library NS response identified by the direct
algorithm based on a 8-layer model and using accelerations of Yorba Linda earthquake of 2002
recorded at west wall (0-15 Hz)
Layer Floor
i
h
[m]
i
z
i

[m]
Observed
τ [s]
Identified
i
β [m/s]
Predicted
i
τ [s]
/ τ τ Δ /
%
β β Δ
%
1
9 4.25
4.25 0.04 212.5 0.042 5 -5
8 4.25
2 7 4.25 10.63 0.008 531.3 0.0015 -81 81
3 6 4.25 14.88 0.0055 772.7 0.002 -64 64
4 5 4.25 19.13 0.008 531.3 0.019 137 -137
5 4 4.25 23.38 0.0115 369.6 0.014 22 -22
6 3 4.25 27.63 0.0055 772.7 0.0015 -73 73
7 2 4.25 31.88 0.005 850.0 0.001 -80 80
8 1 4.9 36.45 0.0125 392.0 0.025 100 -100

Table 3.5b Shear wave velocities of Millikan Library NS response identified by the direct
travel time method based on a 8-layer model and using accelerations of Yorba Linda earthquake
of 2002 recorded at west wall (0-25 Hz)
Layer Floor
i
h
[m]
i
z
i

[m]
Observed
τ [s]
Identified
i
β [m/s]
Predicted
i
τ [s]
/ τ τ Δ /
%
β β Δ
%
1
9 4.25
4.25 0.0395 215.2 0.04 1 -1
8 4.25
2 7 4.25 10.63 0.0075 566.7 0.003 -60 60
3 6 4.25 14.88 0.006 708.3 0.0055 -8.3 8.3
4 5 4.25 19.13 0.0085 500.0 0.0155 82 -82
5 4 4.25 23.38 0.011 386.4 0.007 -37 37
6 3 4.25 27.63 0.0055 772.7 0.004 -27 27
7 2 4.25 31.88 0.005 850.0 0.0055 10 -10
8 1 4.9 36.45 0.013 376.9 0.0155 19 -19

81

3.3 Demonstration of the Identification Error
Fig. 3.7 further clarifies how the uncertainty in the localization of the pulses in the IRFs affects
the accuracy of the identification. It compares the observed pulse arrival times at each floor in the
IRFs of the 1-layer model (Table 3.3), (the symbols) with the physical arrival time (the straight
line). The properties are as in Table 3.3, and
max
f =15 Hz. As it can be seen, the symbols do not
lie on the straight line but are scattered around it. The error is considerable even for this simple
model with constant impedance and no internal reflections, and is much larger than what is
inferred from the precision of the readings of the time of the peaks of the pulses. In principle, this
error occurs because of the uncertainty in the localization of the IRFs for finite data bandwidth
(Heisenberg-Gabor uncertainty principle), in the sense that the time of the peaks of the pulses do
not coincide with the true, physical pulse arrival time. In this case, it is practically realized by the
interference of the main lobe of each pulse with the side lobes of the other pulse, which leads to
some distortion of the main lobes and shift of the pulse peak time. In general, pulses from internal
reflections and foundation rocking further contribute to the error. These errors are present in IRFs
of real data, and lead to significant relative error in the identified shear wave velocities if the layer
thickness is small and if the data effective bandwidth is small.

Pulse arrival time at each floor: as measured from IRF of 1-layer model vs.
theoretical.
Figure 3.7
82

3.4 Summary and Conclusion
The most important findings of the first two sections of this chapter are the following. (1) The
resolving power of IRFs is not limited only by the density of sensors, but also by the effective
bandwidth of the data, which determines the width of the pulses in the IRFs. This is a
consequence of the Heisenberg-Gabor uncertainty principle. (2) There is a trade-off between
resolution and accuracy of the identified velocity profile, which also follows from the same
principle. The error is smaller for smoother models (with smaller number of thicker layers).
(3) The pulse amplitudes are affected by the soil-structure interaction (through foundation
rocking), and cannot be used alone to determine the structural damping. (4) In principle, reflected
pulses can also be used for identification, compensating for lack of sensors at some floors.
However, in practice, this would be possible only if the effective data bandwidth is sufficiently
large. (5) The application of the direct identification algorithm to Millikan library NS response
produced robust and realistic results. (6) For this building, the direct algorithm is optimal for a 3-
layer model, in terms of both spatial resolution and accuracy (the error / β β Δ ≈ 6% in the top
and bottom layers and 14% in the middle layer). (6) The error of this algorithm in identifying the
high resolution model (one story per layer) is large in some of the layers, and is unacceptable for
structural health monitoring.

83

84

Chapter 4
Structural System Identification and Health
Monitoring of a Tall Steel Frame Building –
Los Angeles 54-Story Office Building
In this chapter, firstly, a 54-story steel, perimeter-frame building in downtown Los Angeles,
California, is identified using seismic interferometry for the Northridge earthquake of 1994
(M
L
=6.4, R=32 km). A layered shear beam, characterized by the corresponding velocities of
vertically propagating waves through the structure, is considered. The previously introduced
waveform inversion algorithm is used. This section concludes that the layered shear beam and
torsional shaft models are valid for this tall and relatively soft building, within bands that include
the first five modes of vibration for each of the NS, EW and torsional responses.
Next, the variations of identified wave velocities are investigated over a period of 19 years
since construction (1992-2010), using records of six earthquakes. The set includes all significant
earthquakes that shook this building, which produced maximum transient drift ~0.3%, and caused
no reported damage. The results suggest variations larger than the estimation error, with
coefficient of variation about 2-4.4%. About 10% permanent reduction of the building stiffness is
detected, caused mainly by the Landers and Big Bear earthquake sequence of June 28, 1992, and
the Northridge earthquake of January 17, 1994.

85

This chapter is based on the following two journal papers:
Rahmani M, Todorovska MI (2014). 1D system identification of a 54-story steel frame building
by seismic interferometry, Earthquake Engng Struct. Dyn., 43:627–640, DOI: 10.1002/eqe.2364,
first published online of 9/13/2013.

Rahmani M, Todorovska MI (2014). Structural health monitoring of a 54-story steel frame
building using a wave method and earthquake records, Earthquake Spectra, DOI:
10.1193/112912EQS339M, in press, fist published online on 10/15/2013.

86

4.1 System Identification of LA 54-Story Office Building
during Northridge Earthquake of 1994
This section presents a system identification study of a 54-story steel-frame building in
downtown Los Angeles, California (Fig. 4.1), from earthquake records in terms of its velocities of
vertically propagating waves through the structure. The velocities are identified waveform
inversion of pulses in low-pass filtered impulse response functions (Fig. 4.2). Similarly, the
torsional wave velocities are also identified, using data from pairs of sensors at different floors.
The models that are fitted account only for dispersion due to material damping, and do not
account for bending deformation, which produces dispersion. The bending deformation is
believed to be significant for very tall and flexible buildings, like the one analyzed in this paper,
even though the observed ratios of the frequencies of vibration are closer to the values for a shear
beam. The study presented in this chapter investigates if the identification algorithm, which is
based on fitting a layered shear beam model, is valid for this building and for similar structures.
The pulse propagation can be traced even visually, as shown in Fig. 4.3 for the NS response
of the Los Angele 54-story building during the Northridge, 1994 earthquake. Fig. 4.3 shows the
impulse response at the roof (low-pass filtered at 1.7 Hz) computed for a virtual source at the
ground floor. In this figure, τ is the pulse travel time over the building height. At least ten
consecutive arrivals of the pulse can be seen, resulting from multiple reflections from the roof and
the base. The first arrival is at time t 1.5 =τ ≈ s, and the consecutive arrivals occur at time
intervals 2τ , with alternating sign (Trampert et al. 1993; Snieder and Şafak, 2006).
In this section, firstly, the building and earthquake data are described. Then, identification
analysis and results are presented and critically discussed. We have published the results of this
section in the journal of Earthquake Engineering and Structural Dynamics (Rahmani and
Todorovska, 2013c).
87

a) b) c)
Los Angeles 54-story office building (CSMIP 24629): a) photo; b) vertical cross-
section, and c) typical floor layouts (redrawn from www.strongmotioncenter.org).
Figure 4.1

The layered shear beam model.
Figure 4.2

88

Los Angeles 54-story office building: NS impulse response at roof, Northridge,
1994 earthquake, 0-1.7 Hz.
Figure 4.3

4.1.1 Building Description and Earthquake Data
The Los Angeles 54-story office building is located in downtown Los Angeles, California (Fig.
4.1a). It has 54 stories (210.2 m) above and 4 stories (14 m) below ground level. It has a
rectangular base with two rounded sides, 196 ft × 121 ft (59.7 m × 36.9 m) up to the 36
th
floor,
decreasing in the EW direction to 156 ft × 121 ft (47.5 m × 36.9 m) (Fig. 4.1b). The building
was designed in 1988 following the 1985 Los Angeles City Code and Title 24 of the California
Administrative Code. The lateral force resisting system is moment resisting perimeter steel frame
(framed tube) with 10 ft (3 m) column spacing. It has Virendeel trusses and 48 inch (1.22 m)
deep transfer girders at the 36
th
and 46
th
floors where vertical setbacks occur. The vertical load
carrying system consists of 2.5 inch (6.35 cm) concrete slabs on 3 inch (7.6 cm) steel decks with
welded metal studs, supported by steel frames. The building is supported by a concrete mat
foundation, 7 ft (2.1 m) and 9.5 ft (2.9 m) thick, and 6 inches (15.2 cm) concrete slab on grade.
The site geology is alluvium over sedimentary rocks. The building was instrumented in 1991 by
89

the Strong Motion Instrumentation Program of the California Geological Survey (CSMIP Station
No. 24629) with 20 accelerometers distributed on 6 levels: basement (P4), ground, 20
th
, 36
th
, 46
th
,
and Penthouse (54
th
floor) (Fig. 4.1b,c). The instruments are 12-bit resolution SSA-1 recorders
with FBA-11 accelerometers.
6.4
L
M The Northridge earthquake of January 17, 1994 ( = ) occurred 32 km to the North-
West from the building, which showed no signs of damage (Naeim et al. 2006). Fig. 4.4 shows
the corrected (Trifunac, 1971) transverse (NS at the West wall), longitudinal (EW at the North
wall), and torsional accelerations (computed from the NS accelerations at the East and West
walls) recorded during this earthquake. Fig. 4.5 shows the Fourier amplitudes of the acceleration
at penthouse and basement levels, and the corresponding transfer-function, for the NS, EW and
torsional motions, between 0 and 3.5 Hz. The peaks in the transfer-functions, corresponding to
the first 5-6 modes of vibration are indicated. The Fourier amplitudes of the NS rocking,
multiplied by the building height, is also shown. It can be seen that the motion at the top due to
rigid body rocking is a small fraction of the total roof motion, but is not negligible.
Fig. 4.6 illustrates the spectral characteristics of the displacements at penthouse and
basement levels. Comparison of figures 4.5 and 4.6 shows that the modes contribute differently
to the roof accelerations and displacements. While the second mode contributes most and the
higher modes contribute considerably to the roof acceleration, the fundamental mode contributes
most to the roof displacement.

90

Observed NS acceleration at the West wall, EW accelerations at the North wall,
and NS accelerations at the East wall due to torsion during Northridge earthquake of 1994.
Figure 4.4
91

Fourier transform amplitudes of absolute NS accelerations at West wall, NS
accelerations at penthouse level due to rigid body rocking, EW accelerations at North wall, and
NS accelerations at East wall due to torsion, observed during Northridge earthquake of 1994.
Also transfer-functions are shown between penthouse and P4 level accelerations.
Figure 4.5
92

Fourier transform amplitudes of absolute NS displacement at West wall, NS
displacement at penthouse level due to rigid body rocking, and EW displacement at North wall.
Figure 4.6

93

4.1.2 Apparent Frequencies of Vibration and Pulse Arrival Times
Table 4.1 shows the first six apparent frequencies of vibration
,
,1, ,
iapp
fi 6 = … for the NS, EW,
and torsional motion, identified from the peaks of the transfer-function amplitudes (Fig. 4.5), and
their ratios. NS response, from now on, refers to the average of the NS responses recorded at the
East and West walls. It can be seen that the observed frequency ratios are close to those for a
uniform shear beam (1:3:5...), also shown in the table, and much different from those for a
uniform bending beam (1:6.27:17.5...). This suggests that, within a frequency band spanned by
these modes (up to ~2 Hz for NS and EW responses, and ~3.5 Hz for torsion), the building
vibration is close to that of a shear beam. The lack of signs of dispersion in the impulse response
function for the NS response, shown in Fig.4.3, is also in favor of the appropriateness of a shear
beam model for that response within the band 0-1.7 Hz.
We chose to fit the impulse responses in the bands 0-1.7 Hz for the NS and EW responses,
and 0-3.5 Hz for the torsional response, which contain the first five longitudinal and transverse,
and the first six torsional modes of vibration. The corresponding spreads of the source pulses in
time are
s
t Δ=
i
0.3 s for NS and EW responses and 0.15 s for the torsional response. The observed
impulse responses are shown in Fig. 4.7 by solid lines. Despite the larger pulse widths, the causal
and acausal pulses are well separated at all the levels. Table 4.2 shows the pulse arrival times
at levels
ave
i
t
z , obtained by averaging the readings for the causal and acausal pulses. The travel time
from basement to roof is about 1.5 s for both NS and EW responses, and 0.9 s for the torsional
response.

94

Los Angeles 54-story office building: observed (apparent) modal frequencies
during Northridge, 1994 earthquake, and their ratios. The ratios of the fixed-base frequencies for
a uniform shear-beam are also shown for comparison.
Table 4.1

Uniform
shear beam
Observed
NS EW Torsion
Mode
i
1
f
f

i,app
f
[Hz]
i,app
1,app
f
f
i,app
f

[Hz]
i,app
1,app
f
f
i,app
f

[Hz]
i,app
1,app
f
f

1 1 0.162 1.0 0.192 1.0 0.360 1.0
2 3 0.500 3.1 0.527 2.7 0.936 2.6
3 5 0.822 5.1 0.842 4.4 1.483 4.1
4 7 1.168 7.2 1.196 6.2 2.042 5.7
5 9 1.510 9.3 1.530 8.0 2.508 7.0
6 11 1.828 11.3 1.96 10.2 3.2 8.9

Los Angeles 54-story office building: observed pulse arrival times (average for
the causal and acausal pulses) during Northridge earthquake of 1994.
ave
i
t
Table 4.2
Level
z
[m]

ave
t

[s]
NS
(0-1.7 Hz)
EW
(0-1.7 Hz)
Torsion
(0-3.5 Hz)
Penthouse 0 0 0 0
46
th
27.9 0.335 0.355 0.17
36
th
67.7 0.595 0.648 0.323
20
th
131.3 1.01 1.005 0.588
Gnd 210.2 1.503 1.49 -*
P4 224.2 1.52 1.505 0.865
* no sensor

95

Observed and fitted impulse response functions for NS at west wall, EW and
torsional responses.
Figure 4.7
96

4.1.3 Equivalent Uniform Shear Beam Model
Table 4.3 shows the parameters of a fitted uniform model (ground floor to penthouse for the NS
and EW, and P-4 level to penthouse for the torsional response). The wave velocities
eq
β were
identified by the LSQ algorithm, while the apparent Q was identified by the direct algorithm, i.e.
from the relation
max
12 ln
SA
C
Q
S
ω τ =
, where and SC are the peak amplitudes of the acausal
and causal pulses, and
SA
τ is the pulse travel time over the layer width. The apparent damping ratio
1/ (2 )
app app
Q ζ = is also shown. The fit gave
eq
β ≈ 140 m/s for the NS and EW response, and
259 m/s for the torsional response, and
ap
Q
p
=25 (
app
ζ = 2%) for the NS and torsional
responses, and 14 (
ap
Q
p
= ζ =3.6%) for the EW response. For comparison, the apparent 1
st

mode damping, as identified from the transfer-function by the half-power method, is about 1.5%
for the NS response, and 2.6% for the EW response. For the 2
nd
mode, it is about 2% for both NS
and EW. The difference between the modal damping ratios and those derived from Q is because
the latter is an interval value, representative for the band, while the modal damping ratios are
point values.
4.1.4 Equivalent 4-Layer Shear Beam Model
Table 4.4 summarizes the results of the fitted 4-layer models, using the LSQ fit algorithm
(2% = was assumed). In this table, ζ τ is the pulse travel time through the layer, computed from
the differences of the pulse arrival times, t in Table
ave
i
4.3, /
ini
h β τ = is the initial value, β is
the final value, and /
β
σ β is the normalized spread of β .
The peak drift in the layers γ is also listed, as well as the minimum layer width that can be
resolvedh , in meters and in number of floors. It can be seen
min min m
/4 /(4f λβ ==
ax
)
97

Identified equivalent uniform model wave velocity
eq
β , apparent quality factor Q
and apparent damping ratio ζ during Northridge earthquake of 1994.
Table 4.3

Component
Frequency
band
h [m] τ [s]
eq
β

[m/s]
1/ 4 τ
[Hz]
Q
1/ 2Q

ζ =

[%]
NS
1
0-1.7 Hz 210.2 1.503 139.8 0.166 25 2
EW
1
0-1.7 Hz 210.2 1.49 141.1 0.168 13.8 3.6
Torsion
2
0-3.5 Hz 224.2 0.865 259.2 0.29 25 2

Los Angeles 54-story office building: identified 4-layer model wave velocities
during Northridge, 1994 earthquake for the NS, EW and torsional responses.
Table 4.4
NS at west wall, 0-1.7 Hz
Layer Floors
γ

[cm/m]
β
[m/s]
h
[m]
τ
[s]
ini
β
[m/s]
β
σ
[%]
β
min
h
min
h

[m]

[flrs]

1 46-Penth 27.9 0.156 0.335 83.3 92.5 0.75 13.6 3.9
2 36-46 39.8 0.1 0.26 153.1 153 1.8 22.5 5.6
3 20-36 63.6 0.097 0.415 153.3 141.6 1.2 20.8 5.2
4 Gnd-20 78.9 0.073 0.493 160 167.7 1.3 24.7 6
EW at north wall, 0-1.7 Hz
Layer Floors
h
[m]
γ

[cm/m]

τ
[s]
ini
β
[m/s]
β
[m/s]
β
σ
β
[%]
min
h

[m]
min
h

[flrs]

1 46-Penth 27.9 0.22 0.355 78.6 78.2 0.77 11.5 3.3
2 36-46 39.8 0.15 0.293 136 140.7 1.2 20.7 5.2
3 20-36 63.6 0.098 0.358 178 166.5 1 24.5 6.2
4 Gnd-20 78.9 0.084 0.485 162.7 178.4 0.82 26.2 6.3
Torsion, 0-3.5 Hz
Layer Floors
h
[m]
6
10 γ
−
×
[rad/m]
τ [s]
ini
β
[m/s]
β
[m/s]
β
σ
β
[%]
min
h

[m]
min
h

[flrs]

1 46-Penth 27.9 6.70 0.17 164 166.4 1.5 11.9 3.3
2 36-46 39.8 4.81 0.153 261 251.3 2.5 18.0 4.5
3 20-36 63.6 2.58 0.265 240 250 1.4 17.9 4.5
4 P4-20 92.9 1.98 0.278 334.8 352 1.3 25.1 6
98

that the layer thicknesses are at least twice
mi
h and that the resolution increases from bottom to
top (from about 6 floors in the bottom layer to about 3-4 floors in the top layer). The results are
shown graphically in Fig.
n
,
4.8, the rows corresponding to the NS, EW and torsional responses, and
the columns corresponding to the pulse arrival times,
i
t the mass density, , ρ (assumed to be
300 kg/m
3
), the layer velocities, β , and the peak layer drifts γ . The variation of β across the
layers is 92-168 m/s for the NS, 78-178 m/s for the EW, and 166-352 m/s for the torsional
response. As it can be seen in Fig. 4.8, for the NS and EW responses, the variation is small,
except near the top, and is slightly larger for the EW response, consistent with narrowing of the
building.
In the NS velocity profile, the velocity does not decrease monotonically towards the top as
for the other two profiles (Fig. 4.8). We conducted two numerical experiments, consisting of
fitting a 4-layer shear beam in simulated impulse responses, to find out if that effect is due to
estimation error. Tables 4.5 and 4.6 show the results. The velocity profiles were identified by the
direct and by the LSQ algorithm. In the first experiment, the “observed” impulse responses are
those for a uniform beam with β =120 m/s. As its can be seen from Table 4.5, the direct
algorithm identified a nonuniform profile, the largest error occurring in the top layer (15%). The
LSQ algorithm, however, identified practically a uniform beam, the error being less than 0.5% in
all layers. In the second experiment, the “observed” impulse responses were simulated by a 4-
layer beam, with nonmonotonic variation of stiffness. As it can be seen from Table 4.6, the direct
algorithm identified a profile with monotonically decreasing stiffness towards the top, the largest
error being about 12%, while the LSQ algorithm correctly identified the variation of stiffness, the
error being less than 1%. This analysis suggests that the estimation by the LSQ fitting algorithm
is accurate and the nonmonotonic velocity profile for the NS response is due to a physical cause.

99

Observed pulse arrival time at the layer boundaries , assumed mass density
profile
i
t
() z ρ , the identified wave velocity profiles () z β during Northridge, 1994 earthquake, and
average layer peak drift () z γ for the NS, EW and torsional responses.
Figure 4.8

100

Test of accuracy no. 1: 4-layer model is fitted in simulated IRFs for a uniform beam
(ζ =2%, sampling interval 0.005 s) in the band 0-1.8 Hz (contains 6 modes).
Table 4.5
Layer
h
[m]
True
value
Direct algorithm
estimate
LSQ algorithm
estimate
/
β
σ β β
[m/s]
Error
%
β
[m/s]

[m/s]
β
%
Error
%
1 27.9 120 101.5 -15.4 120.1 0.03 0.1
2 39.8 120 129.4 7.8 120.5 0.04 0.4
3 63.6 120 122.3 1.9 120.4 0.03 0.3
4 78.9 120 122.8 2.3 120 0.02 0.0

Test of accuracy no. 2: 4-layer model is fitted in simulated IRFs for a 4-layer beam
with non-monotonically varying wave velocity (ζ =2%, sampling interval 0.005 s) in the band
0-1.7 Hz (contains 5 modes).
Table 4.6
Layer
h
[m]
True
value
Direct algorithm
estimate
LSQ algorithm
estimate
/
β
σ β β
[m/s]
Error
%
β
[m/s]

[m/s]
β
%
Error
%
1 27.9 90 82.7 -8.1 89.8 0.05 -0.2
2 39.8 150 143.4 -4.4 151.3 0.07 0.9
3 63.6 140 157 12.1 140.4 0.05 0.3
4 78.9 170 162.7 -4.3 170.5 0.05 0.3

101

Observed and fitted transfer functions (TF) between accelerations at penthouse and
ground/P4 levels.
Figure 4.9
102

Finally, we examine the agreement of the observed and model impulse responses and
transfer-functions. The agreement of impulse responses, shown in Fig. 4.7, is very good as it can
be expected because the model was identified by fitting the impulse responses. The transfer-
functions, however, shown in Fig. 4.9, also agree quite well, within the bands of the fit, except for
some discrepancy for the 1
st
and 2
nd
modes of the NS response. The agreement of transfer-
functions is not automatic. E.g., the presence of bending deformation in the real building could be
compensated by the layering structure, by lower model velocity towards the top. However, the
frequencies of vibration, which depend on the distribution of stiffness and mass along the height,
would not be matched.
The readings of the observed apparent frequencies and the model fixed-base
frequencies
, iapp
f
i
f are shown in Table 4.7. It can be seen that their relative difference
,
(
iiapp
)
i
ff f − is small for the higher modes, generally not exceeding 2.5%. The difference is
the largest for the NS response 1
st
and 2
nd
mode (13.4% for the 1
st
mode and 6% for the 2
nd

mode), and is consistent with difference due to bending deformation in the real building, not
accounted for in the model. For the EW and torsional responses, the difference is larger only for
the 1
st
mode (5% for the EW and 6.5% for the torsional response). Such difference is consistent
both with presence of bending and frequency shift due to dynamic soil-structure interaction. The
degree to which each of these effects contributed to the difference in the model and observed
fundamental mode frequencies (of all three components of motion) cannot be determined by this
analysis. Fig. 4.9 also shows transfer-functions for models fitted on two other bands, with
max
f
=1.3 Hz and 2.7 Hz. It can be seen that the transfer functions are close. Fig. 4.10 shows plots of
the first five mode shapes for the NS and EW.

103

Los Angeles 54-story office building: observed (apparent) frequencies and
fitted model (fixed-base) frequencies
i,app
f
i
Table 4.7
f frequencies during Northridge, 1994 earthquake.
NS EW Torsion
Mode
i,app
f
i

[Hz]
f
[Hz]
ii,app
i
ff
f
−
i,app
f
i
%

[Hz]
f
[Hz]
ii,app
i
ff
f
−
%
i,app
f
i

[Hz]
f
[Hz]
ii,app
i
ff
f
−
%
1 0.162 0.187 13.4 0.192 0.202 4.95 0.360 0.338 6.5
2 0.500 0.532 6.0 0.527 0.532 0.9 0.936 0.922 1.5
3 0.822 0.833 1.3 0.842 0.832 -1.2 1.483 1.447 2.5
4 1.168 1.156 -1.0 1.196 1.167 -2.5 2.042 2.040 0.1
5 1.510 1.514 0.3 1.530 1.551 1.4 2.508 2.658 -6.0

Model mode shapes for NS (top) and EW (bottom) responses.
Figure 4.10

104

4.1.5 Summary and Conclusion
1D models, assuming shear deformation only, were fitted in the lateral and torsional responses of
a 54-story steel-frame building recorded during the Northridge earthquake of 1994. The
frequency bands for the fit were 0-1.7 Hs for the NS and EW responses, and 0-3.5 Hz for the
torsion, which contain the first five modes of vibration. The pulse travel time over the building
height, measured from the impulse responses, is τ ≈1.5 s for NS and EW responses, and τ ≈0.9
s for torsion. The fitted uniform model velocities are
eq
β ≈ 140 m/s for NS and EW responses,
and 260 m/s for torsion, and the apparent Q ≈ ≈ 25 for the NS and torsional, and 14 for the EW
response (Table
≈
4.3). The fitted 4-layer model velocities were in the range
i
β ≈ 90-170 m/s for
the NS, 80-180 m/s for the EW, and ≈ ≈170-350 m/s for the torsional responses (Table 4.4 and
Fig. 4.8).
Based on the apparent frequency ratios (Table 4.1), the pulse shapes in the impulse response
functions (Fig. 4.3 and 4.7), and the agreement of the model and observed frequencies of
vibration (Table 4.7, Fig. 4.9), it is concluded that a layered shear model is a good physical model
for the EW and torsional responses, and a reasonable model for the NS response, within the bands
of the fit. Extending the frequency band of the fit would require fitting more detailed models.

105

4.2 Structural Health Monitoring of LA 54-story Office
Building during Six Earthquakes between 1992 and 2010
To be effective, Structural Health Monitoring (SHM) methods must work with real buildings and
larger amplitude response, and be reliable, sensitive to damage and accurate. Ideally, they should
also be able to detect localized damage, which is challenging, and smaller changes due to
structural degradation with time, which are difficult to separate from identified changes due to
other factors, such as identification error and changes in the operating and environmental
conditions (Doebling et al., 1996; Chang et al., 2003; Clinton et al., 2006; Boroschek et al., 2008;
Todorovska and Trifunac, 2008b; Herak and Herak, 2010; Mikael et al., 2013). While the rare
records in damaged full-scale buildings remain invaluable for relating changes in the damage
sensitive parameters to levels of damage of concern for safety (e.g., Todorovska and Trifunac,
2008a,b), the much more frequent records of smaller and distant earthquakes are also very
valuable. (1) Analyses of multiple earthquake records in full-scale buildings, over longer periods
of time, can provide knowledge about the variability of the damage sensitive parameters due to
factors other than damage and permanent changes due to structural degradation (for a particular
structure or type of structures) in the most realistic conditions.
Knowledge of this variability is useful for making inferences about the state of damage from
detected changes. (2) Such analyses also provide opportunities to test the capabilities of SHM
methods being developed. This section presents such an analysis for the LA 54-story steel-frame
building (Fig. 4.1) and a wave method for SHM. The data consists of records of six earthquakes,
over a period of 19 years since construction, none of which caused reported damage (Fig.4.11,
Table 4.8). The results of this section are published in the journal of Earthquake Spectra
(Rahmani and Todorovska, 2013d).

106

.
Google map of the epicenters of the earthquakes recorded in the building.
Figure 4.11

List of earthquakes recorded in the building (CSMIP Station 24629; 34.048 N,
118.26 W)
Table 4.8
Event
name
Code
Date/
Time
(PDT)
Epicenter
H
[km]
L
M
Epic
distance
[km]
Rec.
length
[s]
Gnd
ma x
a

[g]
Struc.

[g]
ma x
a
Landers LA
06/28/1992
04:57:31
34.22N
116.43W
1 7.3 170/158 87 0.040 0.130
Big Bear BB
06/28/1992
08:05:31
34.20N
116.83W
10 6.5 133 87 0.030 0.067
Northridge NO
01/17/1994
04:30:00
34.21N
118.54W
19 6.4 32/28 180 0.140 0.190
Hector
Mine
HM
10/16/1999
02:46:45
34.60N
116.27W
6 7.1 193 106 0.019 0.082
Chino
Hills
CH
07/29/2008
11:42:15
33.95N
117.77W
14 5.4 47 83 0.063 0.086
Whittier
Narrows*
WN
03/16/2010
04:04:00
34.00N
118.07W
18 4.4 18 - 0.020 0.022
Calexico CA
04/04/2010
15:40:39
32.26N
115.29W
32 7.2 355/286 87 0.009 0.038
* Data not available; H=focal depth
107

The analysis in this section aims to assess the general variability of the identified wave
velocities in this building, and detect possible permanent changes in the structure by the wave
method. A recently proposed waveform inversion algorithm for the identification of the wave
velocities is applied, which is much more accurate than the ones used previously. The detected
changes are compared with those of the first two apparent frequencies of vibration, which are
also identified. This is the first such analysis with the waveform inversion algorithm, which
examines its capability to detect permanent changes from the scatter. Also, to the knowledge of
the authors, this is the first analysis of the variability of damage sensitive parameters for this
building using any method.
Similar to previous section, a layered shear beam/torsional shaft is fitted in recorded
earthquake response, in a carefully chosen low-pass frequency band, by waveform inversion of
pulses in impulse response functions. The study in this section presents the first attempt to detect
by this algorithm, and by the wave SHM method, in general, small changes due to stiffness
degradation in a steel frame building. This study also provides an insight into and a measure for
the variability of the vertical wave velocities in steel-frame buildings, from one earthquake to
another, none of which has caused observed damage. For comparison, the first two apparent
frequencies are also analyzed.
Observed fundamental frequency of vibration of steel buildings excited by multiple
earthquakes have been reported, e.g. by Çelebi et al. (1993), Li and Mau (1997), Rodgers and
Çelebi (2006), and Liu and Tsai (2010). To the knowledge of the authors, only the Northridge,
1994 earthquake data in this building has been analyzed (e.g. Naeim, 1997; Todorovska and
Rahmani, 2012). Further in this section, the recorded data is presented, and the results of the
identification for the six earthquakes and the sample statistics are summarized, followed by
exploratory analysis of trends as function of interstory drift. Finally, the conclusions drawn are
presented.
108

4.2.1 Strong Motion Records in Building
The building was instrumented in 1991 with a 20-channel digital accelerometer array distributed
on 6 levels: basement (P4), ground, 20
th
, 36
th
, 46
th
, and Penthouse (54
th
floor) (Fig. 1). Fig. 4.11
shows a map of the building site and the epicenters of the seven earthquakes, reported to have
been recorded in this building, over a period of 19 years (1992 to 2010). Six of them, for which
data are available, are analyzed. No damage has been reported from any of these earthquakes.
Table 4.8 shows the earthquake name, a two-letter code assigned in this study, date and time,
epicentral coordinates and depth, magnitude, record length, and peak ground and structural
accelerations. Three of these earthquakes were distant but large (Landers, 1992, Hector Mine,
1999, and Calexico, 2010), one was moderate but near (Northridge, 1994), one was moderate and
distant (Big Bear, 1992), and one was small but relatively close (Chino Hills, 2008). The
processed data were made available equally spaced at 0.01 s. For the computation of impulse
response functions, we interpolated the data to 0.005 s. The Northridge data have been band-pass
filtered by Ormsby filter with ramps at 0.06 - 0.12 Hz and 46 - 50 Hz (Lee and Trifunac, 1990).
The other data have been band-pass filtered with Butterworth filter, with 3 dB pts at 0.08 Hz and
40 Hz for Landers, and with 3 dB pts at 0.1 Hz and 40 Hz for the other events.
Figs 4.12 and 4.13 illustrate the variety of the base excitations and building responses they
produced. Fig. 4.12 shows pairs of P-4 level (basement) acceleration and penthouse displacement,
and Fig. 4.13 shows the transient drift, computed from the difference in displacements between at
penthouse and P-4 level. Fig. 4.12 shows that the building response is poorly correlated with the
ground acceleration, and is sensitive to the frequency content of the excitation. While the
Northridge earthquake produced the largest base acceleration, the more distant Landers and
Hector Mine earthquakes produced the largest response (roof displacement ~55 cm and 50 cm,
and average drift of ~0.2%, for EW motions; see Fig. 4.13). The Chino Hills earthquake produced
the second largest base acceleration, but very small response.
109

Penthouse displacements and base accelerations observed during the six
earthquakes.
Figure 4.12

Average drifts observed during the six earthquakes.
Figure 4.13

110

4.2.2 Identified Parameters and Sample Statistics
Figs 4.14 and 4.15 show the observed TF amplitudes and IRFs for the six events, for the NS, EW
and torsional responses. NS response indicates the average of the NS responses at the East and
West sides of the building. The torsion was computed from the difference of these motions. The
TFs were computed from the ratio of the complex Fourier transforms of the motions at penthouse
and P4 levels. The IRFs were computed for virtual source at penthouse level. It can be seen that
the TFs are very similar, except that, for the Chino Hills, 2008 earthquake, the peaks
corresponding to the fundamental modes are small or lost. While the high-pass filter might have
affected the amplitudes of the first peaks for all earthquakes, the very small peak amplitude for
the Chino Hills earthquake is likely due to the small signal to noise ratio at low frequencies for
this earthquake, which did not excite much the fundamental mode. The impulse response
functions are also very close.
Table 4.9 summarizes the identified global parameters: wave travel time τ over the height of
the building (ground floor to penthouse for the NS and EW, and P4 level to penthouse for the
torsional response), and the wave velocity
eq
β , quality factor , and fixed-base frequency Q
/( ) τ 14

of the fitted equivalent uniform model. While
eq
β was identified by the waveform
inversion algorithm, Q was identified from the pulse amplitudes by the direct algorithm, and
represents the apparent damping, which depends on the structural damping and rocking radiation
damping (Todorovska, 2009a). The corresponding apparent damping ratio is /( ) Q ζ =1 2 . The
fixed-base frequency of the fitted uniform model, /( ) τ 1 4 , in general differs from the actual
fixed-base frequency, which depends on the distribution of stiffness and mass along the height,
but can be used as a proxy of the actual fixed-base frequency to follow its changes (Trifunac and
Todorovska, 2008a,b). Table 4.9 also shows the apparent frequencies for the first two modes,
and , identified from the transfer functions, and
,app
f
1 ,app
f
2 w
γ = weighted peak transient drift.
111

Transfer functions of observed NS, EW and torsional responses during six
earthquakes.
Figure 4.14

112

Impulse response functions of observed NS and EW responses during six
earthquakes.
Figure 4.15

113

The apparent frequencies were estimated manually, based on visual analysis of the shape of
the corresponding peak in the TF. While more elaborate automatic algorithms could have been
used (e.g., as in Carreño and Boroschek, 2011), we believe that the conclusions of this paper
would not have changed. The weighted peak drift,
w
γ , was computed as the weighted average of
the peak layer drifts, with weights proportional to the layer heights. We use
w
γ instead of the
peak drift between roof and base, because the drift varied differently along the height for different
earthquakes, and the latter represented poorly the overall deformations of the building for some of
the events. It can be seen that, for the NS response, the largest
w
γ occurred during Calexico,
2010 and Landers, 1992 earthquakes (0.1008 cm/m and 0.0987 cm/m), while, for the EW and
torsional responses, it occurred during the Landers, 1992 earthquake (0.265 cm/m and 0.00627
mrad/m).
Table 4.10 shows the identified local parameters, i.e. layer velocities
i
β , ,, i =14 …
estimated by the waveform inversion algorithm and the corresponding normalized standard
deviation /
β
σ β , and the peak layer drifts
i
γ . The largest NS drift occurred in the top layer
during Northridge, and, in the bottom two layers - during Landers and Calexico earthquakes. The
largest EW drifts occurred during Landers and Hector Mine earthquakes in all layers. The largest
torsional drift occurred, in the bottom layer - during Hector Mine, while, in the other three layers
- during the Landers earthquake. The mass density was assumed to be uniform throughout the
building, with , for all the models. The layer widths are 27.9 m,
3
300 kg/m ρ =
1
h =
2
h = 39.8
m, 63.6 m and 78.9 m (NS and EW) and 92.9 m (torsion).
3
h =
4
h =

114

Table 4.9 Identification results for equivalent uniform model during six earthquakes.
NS at west wall, 0-1.7 Hz; h=210.2 m
Event
τ

[s]
eq
β

[m/s]
β
σ
β
(%)
1/ 4 τ

[Hz]
Q

1
2Q
ζ =
1,ap p
f
2,a pp
f
w

[Hz]

[Hz]
γ

[cm/m]
LA 1.4175 148.3 0.42 0.176 17.5 2.9 0.17 0.53 0.0987
BB 1.4600 144.0 0.46 0.171 20 2.5 0.17 0.52 0.0439
NO 1.5025 139.9 0.48 0.166 25 2 0.165 0.502 0.0964
HM 1.4925 140.8 0.49 0.168 16.7 3 0.16 0.498 0.0739
CH 1.5125 139.0 0.44 0.165 18.5 2.7 --* 0.50 0.0277
CA 1.4725 142.8 0.49 0.170 30.7 1.63 0.165 0.498 0.1008
EW at north wall, 0-1.7 Hz; h=210.2 m
Event
τ

[s]
eq
β

[m/s]
β
σ
β
(%)
1/ 4 τ

[Hz]
Q

1
2Q
ζ =
1,ap p
f
2,a pp
f
w

[Hz]

[Hz]
γ

[cm/m]
LA 1.3875 151.5 0.5 0.180 12.7 3.9 0.2 0.56 0.2653
BB 1.4250 147.5 0.53 0.175 19.2 2.6 0.19 0.56 0.0371
NO 1.490 141.1 0.55 0.168 13.9 3.6 0.185 0.53 0.1188
HM 1.4825 141.8 0.65 0.169 17.2 2.9 0.18 0.528 0.2351
CH 1.4525 144.7 0.49 0.172 14.7 3.4 0.185 0.54 0.0148
CA 1.4350 146.5 0.53 0.174 16.1 3.1 0.19 0.53 0.0839
Torsion, 0-3.5 Hz; h=224.2 m
Event
τ

[s]
eq
β

[m/s]
β
σ
β
(%)
1/ 4 τ

[Hz]
Q

1
2Q
ζ =
1,ap p
f

[Hz]
2,a pp
f

[Hz]
6
10
w
γ
−
×
[rad/m]
LA 0.810 276.8 1.35 0.309 100 0.5 0.37 1 6.27
BB 0.833 269.1 1.20 0.300 20 2.5 0.37 0.98 1.18
NO 0.865 259.2 1.50 0.289 25 2 0.36 0.935 3.24
HM 0.864 259.5 1.22 0.289 - 0 0.35 0.935 6.03
CH 0.857 261.6 1.17 0.292 33 1.5 0.355 0.965 0.73
CA 0.852 263.1 1.21 0.293 - 0 0.35 0.93 2.17
*The first mode is not readable
115

The key parameter in the estimation is the choice of cut-off frequency,
max
f , which controls
the spatial resolution and the effects of dispersion. A higher value of
max
f enables higher
resolution, but too high value leads to distortion of the pulses caused by dispersion. The optimal
value chosen carefully for this study was found to be
max
f = 1.7 Hz for the NS and EW responses,
and
max
f = 3.5 Hz for the torsional response, which encloses the first 5-6 modes of vibration. Up
to this frequency, the building behaves close to a shear beam/torsional shaft, as shown in the
second chapter.
Finally, Table 4.11 summarizes the sample statistics: sample mean
μ
, sample standard
deviation s

and sample coefficient of variation / s μ. They suggest small variability of
eq
β ,
and during these six earthquakes, with
,app
f
1 ,app 2
f / s μ

not exceeding 3.6%, and also small
variability of
i
β in the layers, not exceeding 4.4%. The range of the peak drifts are also
specified in the last column, where
w
γ γ ≡ for the global parameters, and
i
γγ ≡

for the local
parameters.

116

Table 4.10 Identification results for equivalent 4-layer model during six earthquakes.
NS at west wall, 0-1.7 Hz
EW at north wall, 0-1.7
Hz
Torsion, 0-3.5 Hz
Event %
γ

[cm/m]
%
γ

[cm/m]
β
[m/s]
β
[m/s]
β
[m/s]
β
σ
β
β
σ
β
β
σ
6
10 γ
−
%
β
×
[rad/m]
Layer 1

LA 98.8 0.4 0.087 80.2 0.8 0.286 169.1 1.5 13.0
BB 95.5 0.7 0.085 79.4 0.7 0.072 165 1.3 3.56
NO 92.5 0.75 0.156 78.2 0.77 0.22 166.4 1.5 6.70
HM 96.4 0.7 0.11 78 0.8 0.2 162.7 1.2 11.67
CH 88.2 0.8 0.073 80.1 1.1 0.041 159.8 1.4 2.16
CA 94.2 0.7 0.075 77.9 0.5 0.084 163 1.2 5.66
Layer 2

LA 163.2 0.7 0.104 153.1 1.2 0.259 257.6 2.4 11.4
BB 158.3 1.1 0.063 143 1 0.037 262.2 2.0 1.71
NO 153 1.8 0.1 140.7 1.2 0.15 251.3 2.5 4.81
HM 154.5 1.1 0.07 148 1.3 0.23 254 2.0 10.16
CH 156.7 1.1 0.035 134.8 1.8 0.015 250 2.2 0.91
CA 158.5 1.1 0.094 146.4 0.8 0.081 252.7 1.9 4.38
Layer 3
LA 150.4 0.5 0.11 174.5 1.2 0.304 263.1 1.5 4.9
BB 148 0.8 0.03 169.3 1 0.037 258.4 1.2 0.77
NO 141.6 1.2 0.097 166.5 1 0.098 250 1.4 2.58
HM 144 0.8 0.079 166.2 1.1 0.26 250 1.1 4.62
CH 145.4 0.8 0.019 171.1 1.6 0.013 249.2 1.3 0.58
CA 142.5 0.7 0.11 167 0.7 0.093 249.6 1.1 0.97
Layer 4
LA 174.6 0.5 0.091 197.3 1.1 0.23 377.2 1.4 3.00
BB 170 0.8 0.031 192.7 0.8 0.025 368.3 1.1 0.51
NO 167.7 1.3 0.073 178.4 0.82 0.084 352.1 1.3 1.98
HM 166.6 0.7 0.059 181.8 0.9 0.23 369 1.1 3.52
CH 164 0.7 0.015 185.4 1.2 0.007 367 1.3 0.33
CA 172.3 0.8 0.106 191 0.6 0.078 376.7 1.1 1.00
117

Sample statistics for the six earthquakes ( μ=sample mean, s =sample standard
deviation, / s μ =sample coefficient of variation).
Table 4.11
s / μ
(%)

μ (m/s) s (m/s) [cm/m] γ

NS at west wall, 0-1.7 Hz
4-layer model
1
β

94.3 3.65 3.9 0.073-0.156
2
β

157.4 3.58 2.3 0.035-0.104
3
β

145.3 3.36 2.3 0.019-0.110
4
β

169.2 3.88 2.3 0.015-0.106
Equivalent uniform
model
eq
β

142.5 3.40 2.4
0.0277-0.1008
1/4 τ

0.169 0.004 2.4
1,ap p
f

0.166 0.0042 2.5
2,a p p
f

0.51 0.014 2.7

s/ μ
(%)
μ (m/s) s (m/s) [cm/m] γ

EW at north wall, 0-1.7 Hz
4-layer model
1
β

79 1.1 1.3 0.041-0.286
2
β

144.3 6.3 4.4 0.015-0.259
3
β

169.1 3.3 1.9 0.013-0.304
4
β

187.8 7.1 3.8 0.007-0.230
Equivalent uniform
model
eq
β

145.5 3.9 2.7
0.0148-0.265
1/4 τ

0.173 0.0044 2.5
1,ap p
f

0.188 0.0068 3.6
2,a p p
f

0.54 0.015 2.8

118

Table 4.11 continued.
6
10 γ
−
μ (m/s) s (m/s) s/ μ (%)
×
[rad/m]
Torsion, 0-3.5 Hz
4-layer model
1
β

164.3 3.2 2 2.2-13
2
β

254.6 4.53 1.8 0.9 -11.4
3
β

253.4 5.9 2.3 0.6-4.9
4
β

368.4 9.1 2.5 0.3-3.5
Equivalent uniform
model
eq
β

264.9 6.9 2.6
0.7-6.3
1/4 τ

0.295 0.0078 2.7
1,ap p
f

0.36 0.0092 2.55
2,a p p
f

0.958 0.0288 3

4.2.3 Trends and Permanent Changes
Fig. 4.16 shows graphically the layer velocities along the building height, the bars being ordered
(top to bottom) in chronological order of the earthquake. The variations in the layer velocities
seem erratic at first sight, possibly due to the fact that the largest drifts along the height were not
necessarily caused by the same event, which we explore later.
Fig. 4.17 shows
w
β γ
2
(~ peak stress) vs.
w
γ for the equivalent uniform model for the six
earthquakes, which suggests essentially linear behavior for the (transient) drift levels this building
experienced (< 0.0265%). The earthquakes are identified by their chronological order number, as
in the remaining figures. Fig. 4.18 shows plots of the roof displacement for all events, which was
within 60 cm. In the following discussion, we explore possible trends in the small variations of
the parameters, as function of peak drift.
Fig. 4.19 shows scatter plots of
eq
β , and vs. peak drift
,app
f
1 ,app
f
2 w
γ . The horizontal
bands show the s μ± interval for the sample, while the bars show the β σ ±

interval for the
119

individual fits. The corresponding coefficients of variation (
var
/ cs μ ≡ ) are shown. It can be
seen that the variations among the earthquakes are greater than the estimation error for each
earthquake, and physical causes are likely, which we examine further. The corresponding
reduction in stiffness can be read directly from Fig. 4.20, as inferred from the ratios
,
,
(/ )
eq eq
ββ
2
0 ,,,
(/
app app
ff )
2
11 0
and (/
,, app app
ff
,
)
2
2 2
eq
0
, where the reference values are
those for the Landers earthquake. The changes in β suggest overall change in stiffness of
~12%. In the following, we examine the degree and possible causes for the detected variations in
eq
β , and compare them with the variations of and .
,app
f
1 ,
f
2 app

Figure 4.16 Identified wave velocities in the layers for the six events.

120

Peak stress (~
2
eq
β γ ) vs. peak strain (γ ) relations for the six events.
Figure 4.17

Roof displacement during the six earthquakes.
Figure 4.18
121

The changes in
eq
β seen in Figs 4.19 and 4.20 suggest that permanent reduction of stiffness,
of about 5%, occurred after the Landers- Big Bear sequence. The two earthquakes occurred
within 3 hours from each other, in the morning of July 28, 1992. Between the two earthquakes,
any significant changes in temperature, mass or human-caused alterations of structural stiffness
are unlikely to have occurred, and the identified changes in
eq
β were likely to have been caused
by the earthquakes. Interestingly, the Landers earthquake caused much larger response but the
reduction of stiffness is detected during the subsequent Big Bear earthquake. A moving window
estimation of
eq
β over the released 87 s of response showed that the change did not occur during
the released portion of the recorded motion.
As it can be seen from Figs 4.12 and 4.13, the released length of the Landers records was too
short to capture the significant response of this building. It is possible that the change in stiffness
occurred during the unreleased portion of the Landers record. It is also possible that the detected
permanent change occurred gradually and was a cumulative effect of the many cycles of response
the building experienced during both earthquakes (Nastar et al., 2010). Another permanent
reduction of stiffness appears to have occurred in 1994 during the Northridge earthquake, as
suggested by
eq
β for the EW and torsional responses, as well as additional recoverable reduction.
The torsional response reveals ~5% permanent reduction and ~2% recoverable reduction, while,
in the EW response it is the opposite. This difference may be due to higher sensitivity of the
torsional response to the permanent changes at smaller drift levels. The variations of and
also suggest permanent and recoverable changes of stiffness with comparable magnitudes.
This differs from what has been found by similar analyses for RC buildings (Todorovska, 2009b),
for which the fluctuations were greater for than for
,app
f
1
,app
f
2
,app
f
1 eq
β . Such difference in behavior is
122

consistent with greater sensitivity of to nonlinearities in the soil behavior for the RC
structures, which are stiffer than steel frame structures (Todorovska, 2009b).
,app
f
1

Identified global parameters during the six earthquakes vs. weighted peak drift,
w
Figure 4.19
γ .

123

Reduction of global stiffness, measured by the reduction of
eq
β ,

and
.
1,app
f
2,app
Figure 4.20
f

Next, we look for such changes in the layer velocities. Figs 4.21 and 4.22 show scatter plots
for
i
β and , vs. the peak layer drift, similar to those in Figs
,
(/ )
ii
ββ
2
0
/
,, i =1 4 … 4.19 and
4.20. The resolution of the method and error are important issues in fitting layered models. As
shown in chapter 2, the minimum layer width that can be resolved is roughly
min
h
mi min
h
n
λ= 4, where
min
/
max
f λ β =

is the shortest wavelength in the data. For this
building, and for the choice of
max
f in this study (1.7 Hz for NS and EW motions and 3.5 Hz for
torsion), the layer thicknesses are larger than the (theoretical) resolution by a factor of 2-3. For
the middle two layers, e.g., is about 6 stories. For given
min
h
max
f , the estimation error is larger
for thinner and stiffer layers, and therefore is larger for the identified
i
β than for the identified
124

eq
β . This is evident in the more noisy appearance of the scatter plots for
i
β than for
eq
β , which,
nevertheless, clearly show permanent reduction of stiffness. The points for the Chino Hills
earthquake (No. 5) in Layer 1 (NS and torsion), Layer 2 (EW) and Layer 1 (NS) appear to be
outliers.
Outliers excluded, Fig. 4.22 suggests overall permanent change in stiffness typically
between 5% and 10%. The changes are larger in Layers 2 and 4 for EW motions, and Layer 4 for
torsion, but do not exceed 15%. The largest reduction in these cases, though not all permanent,
occurred during Northridge earthquake (No. 3). An open question remains why the Chino Hills
earthquake estimates show greater deviation from the observed trends. This was a small local
earthquake, which occurred around noon in midsummer, and practically did not excite the first
mode. The temperature at the time of the earthquake was about 80˚F (weathersource.com), and
was likely higher than during the other earthquakes, judging by the season and time of the day. In
depth investigation of the degree to which the temperature, the nature of the excitation and other
factors (environmental and operating conditions) contributed to the more “noisy” estimate for this
earthquake is out of the scope of this paper (Clinton et al., 2006; Boroschek et al., 2008; Herak
and Herak, 2010; Mikael et al., 2013).

125

Same as Fig. 4.19, but for the layer velocities vs. layer peak layer drift,
i
γ . Figure 4.21
126

Reduction of local (layer) stiffness, measured by the reduction of ,1,...,
i
i 4 β = ,
vs. peak layer drift,
i
Figure 4.22
γ .

127

4.2.4 Resolution and Sensitivity to Localized Damage
In the second chapter, it was mentioned that the resolution of the wave method is determined by
the bandwidth of the IRFs. The smallest layer width that can be resolved is
min min max
/4 /(4 ) h f λ β == , where
min
λ is the shortest wavelength in the data. It is the
layer width for which the time shift between a pulse entering a layer and its reflection from the
other layer boundary equals the half-width of the pulse. This rule is critical for resolving the top
layer, as the direct acausal and causal pulses in the IRF of the motion at the bottom of that later
have comparable amplitudes. For the interior layers, a finer structure can be defined than if
the instrumentation is dense but aliasing errors occur (more details in chapter 2 and 3).
min
h
Whereas a layer comprises of many floors, localized damage, e.g. in only few of the floors,
will produce smaller change in the layer velocity, which may be comparable with the estimation
error or the variability due to factors other than damage, and difficult to detect. In view of these
uncertainties, the state of damage can be described only probabilistically, e.g. using hypothesis
testing. For such a task, the following relationship between the local change, in part of the layer,
and the corresponding global change in the equivalent uniform layer velocity, derived from ray
theory, may be useful. Let
g
h be the total layer thickness, be the thickness of the damaged
part,
l
h
0
β be the initial wave velocity everywhere in the layer,
l
β be the final wave velocity in
the damaged part, and
g
β be the final equivalent wave velocity for the layer. The reduction of
equivalent layer velocity
0
/
g
β β due to local reduction of wave velocity
0
/
l
β β over portion
of the layer can be computed from the condition that the global change in travel time
equals the local change in travel time
/
l
h
g
h
128

00
gg
ll
gl
hh
hh
ββ β β
−= −
(5.1)
which gives
0 0
1
11
g
l
gl
h
h
β
β β
β
=
⎛⎞
+−
⎜⎟
⎝⎠

(5.2)
Fig. 4.23 shows
0
/
g
β β vs.
0
/
l
β β for different values of . Equation (5.2) implies
that, if in 10% of the layer the wave velocity has dropped to 30% of its initial value (70% local
change), the global velocity will reduce to 81% of the initial value (19% global change). If the
same change has occurred in 20% of the layer, the global change will rise to 32%, and if it has
occurred in 50% of the layer, the global change will rise to 54%.
/
lg
hh

Change of equivalent layer velocity due to local change within the layer. Figure 4.23

129

4.2.5 Summary and Conclusions
System identification and health monitoring analysis of a 54-story steel-frame building in
downtown Los Angeles was presented using recorded accelerations during six earthquakes over
period of 19 years (1992-2010). The set included all significant earthquakes that shook this
building since its construction in 1991. The transient apparent drift, determined from
displacements obtained by double integration of the recorded accelerations, did not exceed
~0.3%, which is less than half of the maximum transient drift for immediate occupancy (0.7%),
and is much smaller than the transient drift of concern for structural safety (2.5%) for steel
moment-frame buildings, as specified in ASCE guidelines (ASCE 2000; ASCE/SEI, 2007). The
largest drift occurred during the distant Landers, 1992 and Hector Mine, 1999 earthquakes, while
the local Northridge, 1994 earthquake, caused the largest damage in the area (Table 4.8 and Fig.
4.11). No damage was reported from any of these earthquakes.
The identified wave velocities suggest that the response of the structure was essentially
linear. Nevertheless, they suggest that permanent change in the overall structural stiffness of
~10-12% occurred, mainly caused by the Landers-Big Bear sequence and the Northridge
earthquake. These changes were widespread throughout the structure. The method used in this
study cannot determine the mechanism of the changes. The permanent changes in wave velocity
are comparable with those of the first two apparent frequencies of vibration, which is consistent
with smaller effects of the soil on the variations of the apparent frequency for more flexible
structures, as compared to the stiffer RC structures. While this study, of small amplitude response
of a steel-frame building, did not demonstrate obvious advantages of the wave method over
monitoring changes in the frequencies of vibration, as was the case for the stiffer RC structures
(Todorovska and Trifunac, 2008b; Todorovska, 2009b), it does not exclude possible advantages
in the case of stronger shaking, softer soil, and damage. Also, the agreement of results by
different SHM methods, in general, is useful because of increased confidence in the results.
130

Statistical analysis of the estimates for the six earthquakes gave average vertical velocity of
142.5 m/s for the NS, 145.5 m/s for the EW and 265 m/s for the torsional responses. The
identified variation along the height was larger for the EW response, consistent with the
narrowing of the building with height. The average observed apparent frequency for the
fundamental mode was 0.166 Hz for NS, 0.188 Hz for the EW and 0.36 Hz for torsional
responses, and of the second mode was 0.51 Hz for the NS, 0.54 Hz for the EW and 0.96 Hz for
the torsional responses. The coefficient of variation was small, typically less than 2.5% and at
most ~4.5%, but larger than the estimation error.
The detected variability of the properties of this building can be compared with similar
studies for other steel buildings only in terms of the variations of the apparent frequency of
vibration. For example, Rodgers and Celebi (2006) analyzed the variability of the apparent
frequencies of a 13-story steel building in Alhambra, ~15 km North-East of the 54-story building,
over 16 earthquake in 32 years (four of which were also recorded by the 54-story building), none
of which caused reported damage. Based on their results, we obtained sample standard deviation
of 5-5.6% over 32 years, which is about twice larger, over twice longer period, from what we
obtained for the 54-story building (2.5-3.6% over 19 years). Rodgers and Celebi (2006), who
estimated the frequencies from the Fourier spectra of the recorded response, found large
variations especially at low amplitudes (total variation of ~20%), but no clear trends in the
variations both with time and with peak base acceleration. We believe that estimation of the
frequencies from transfer-functions rather than Fourier spectra, and correlation with peak drift
rather than peak base acceleration would have reduced the scatter and may have revealed some
trends in their analysis. (Recall that, in this study, the Northridge earthquake produced the largest
peak ground acceleration, but the third largest response, and the Chino Hills earthquake produced
the second largest peak acceleration but the smallest response; see Fig. 4.12) Analysis of changes
in the wave velocities, which are not sensitive to the effects of soil-structure interaction, may
131

132

have further reduced the scatter and revealed permanent changes, like those we found for the 54-
story building, and earlier for Millikan Library (Todorovska, 2009b).
The general conclusions of this study, about the capabilities of the wave method for SHM, is
that, with the waveform inversion algorithm, it was able to detect permanent changes in stiffness
in the 54-story steel building, although no damage was observed and the overall variation of wave
velocities was small. Therefore, it is a promising method for SHM of buildings. The method can
be further improved by extending the analysis to higher frequencies, which would improve its
accuracy and spatial resolution, and to two and three dimensions, which would enable analysis of
less regular structures and coupled lateral and torsional responses. We leave such tasks for the
future.
It is also concluded that the records of multiple earthquake excitation, even though small,
were very useful, both for the development of the wave SHM method, providing an opportunity
to test its capabilities, as well as for providing new information about the changes in stiffness of
this building. Such records exist for many buildings in California, instrumented by the owner or
by the federal and state strong motion instrumentation programs, and can be used for SHM.
Although many records in buildings have been released by the federal and state government
programs, and can be conveniently accessed on the web, the sets for a particular building are
incomplete, often missing significant records, and, therefore, not useful for SHM research to their
full potential.

Chapter 5
Identification of a Densely Instrumented
High-Rise Building – Millikan Library 9-Story
RC Structure
This chapter presents our findings for our second detailed study, Millikan library. The
building and data are described, and identification results and their analysis are presented for
equivalent uniform models, for which the fit is most robust, and for multilayer models by the
three fitting algorithms (Direct, nonlinear LSQ fit, and Time Shift Matching).
The analysis includes (1) comparison of the performance of the three algorithms, (2)
demonstration that the results are not affected by soil-structure interaction (including foundation
rocking), (3) identification of the fixed-base frequencies and mode shapes, and the rigid body
rocking frequencies, and their comparison with published results using forced vibration tests data
and independent identification method (Luco et al. 1987).
This Chapter is based on the following journal paper:
Rahmani M, Todorovska MI (2013). 1D system identification of buildings from earthquake
response by seismic interferometry with waveform inversion of impulse responses – method and
application to Millikan Library, Soil Dynamics and Earthquake Engrg, Jose Roësset Special
Issue, E. Kausel and J. E. Luco, Guest Editors, 47: 157-174, DOI: 10.1016/j.soildyn.2012.09.014.

133

5.1 Building Description
Millikan library (Figure 5.1) is a 9-story reinforced concrete building in Pasadena, California,
instrumented over a period of 40 years, and tested extensively, in particular for soil-structure
interaction studies (Jennings and Kuroiwa, 1968; Udwadia and Trifunac, 1974; Foutch et al.
1975; Luco et al. 1987, 1988; Todorovska, 2009). It is a rare example of a densely instrumented
building for which data is publically available and can be used for testing system identification
methods. The building is 21× 23 m in plan, and vertically extends 43.9 m above grade and 48.2 m
above basement level (Fig. 5.1). Resistance to lateral forces in the NS direction is provided by RC
shear walls on the east and west sides of the building. The RC central core houses the elevators
and provides resistance to lateral forces in the EW direction.
The local soil can be characterized as alluvium, with average shear wave velocity in the top
30 meters of about 300 m/s, and depth to “bedrock” of about 275 m (Foutch et al. 1975; Luco et
al. 1987, 1988; Wong et al. 1988). Published work also suggests uniform mass distribution over
the top three, middle three and bottom three stories (Jennings and Kuroiwa, 1968), and we used ρ

=526 kg/m
3
, 473 kg/m
3
, and 490 kg/m
3
respectively.

134

(a) (b)

(c) (d)
Millikan library: a) photo (courtesy of M. Trifunac); b) vertical cross-section and
c) typical floor layout (redrawn from Snieder and Safak 2006); (d) sensor locations at basement.
Figure 5.1

135

5.2 Earthquake Records of Yorba Linda Earthquake of 2002
Yorba Linda earthquake of September 3, 2002 (M=4.8, epicentral distance R=40 km) was
recorded by a dense network of sensors (Fig. 5.1c and d) operated by the U.S. geological Survey
(Snieder and Safak, 2006). The building response was small, with maximum rocking angle of
0.012× 10
-3
rad (NS response). Figs 5.2, 5.3 and 5.4 illustrate the observed NS (at the West wall),
EW and torsional responses. Parts a) show the acceleration time histories (low pass filtered at 25
Hz), parts b) show the corresponding IRFs for virtual source at roof, and parts c) show the
transfer functions between the roof and ground floor acceleration accelerations. (The NS motions
at the West wall were used instead of the average of the motions at NS and East walls because
they were recorded at all levels.) The acceleration data was made available low pass filtered at 25
Hz and sampled at 0.05 s. t Δ=
In parts b) of Figs 5.2-5.4, the solid and dashed lines correspond to cut off frequencies
max max
/ f π ω =2
ma x
=15 Hz and 25 Hz. The similarities of pulse width for both values, which
should have differed by a factor of 1.7, and the very small TF amplitudes beyond 15 Hz, suggest
effective f around 15 Hz. Parts b) also show that the effective data bandwidth is not
sufficient to resolve any pulses reflected from the internal boundaries, as well as the transmitted
causal and acausal pulses at the 9
th
floor.

136

Millikan library NS response during Loma Linda, 2002 earthquake observed at
West wall (Fig. 5.1). a) Accelerations. b) Impulse responses for virtual source at roof. c)
Transfer-function of roof acceleration w.r.t. ground floor.
Figure 5.2

137

Millikan library EW response during Loma Linda, 2002 earthquake observed at
West wall (Fig. 5.1). a) Accelerations. b) Impulse responses for virtual source at roof. c)
Transfer-function of roof acceleration w.r.t. ground floor.
Figure 5.3

138

Millikan library torsional response during Loma Linda, 2002 earthquake. a)
Difference of NS accelerations recorded at East and West wall. b) Impulse responses for virtual
source at roof. c) Transfer-function of roof torsional acceleration w.r.t. ground floor.
Figure 5.4

139

a) b) c)

Evidence of dispersion in the observed IRFs of EW response of Millikan library
during Loma Linda, 2002 earthquake. a) 0-15 Hz (solid line) and 0-7.5 Hz (dashed line). b) 0-7.5
Hz. c) 7.5-15 Hz.
Figure 5.5

We found evidence of dispersion only in the EW response, and identified two subbands in
which it was possible to fit our non-dispersive model. Fig. 5.5 shows the observed IRFs for the
whole effective band, 0-15 Hz (the solid line is part a), and for the two subbands, 0-7.5 Hz (part
b) and 7-15 Hz (part c), in which we fit our model.
Table 5.1 summarizes the observed pulse arrival times at each level (average of the
absolute values for the acausal and causal pulses), measured from the time of the peak of the
pulse computed from interpolated data with
ave
i
t
t Δ = 0.001 s.

140

5.3 Equivalent Uniform Models
Table 5.2 shows the identified equivalent wave velocity /
eq
H β τ = for the structure (τ = pulse
travel time from ground floor to roof, = building height), the apparent quality factor
computed from the peak amplitudes of the acausal and causal pulses, and , as
H Q
SA SC
max
ln
SA
Q
SC
ω τ 12 =

and the corresponding damping ratio /( Q) ζ =12 (more details in chapter
2). It can be seen that τ is the range 0.1- 0.16 s, and
eq
β is in the range 240-460 m/s, being the
largest for the tosional motions (463 m/s) and the smallest for the EW motions 1
st
subband (243
m/s). Comparison of
eq
β for NS and EW motions suggests much stiffer structure in the NS
direction (405 m/s as compared to 243 m/s and 347 m/s), which can be attributed to the presence
of the external shear walls at the East and West ends of the building (Fig. 5.1c).
Table 5.2 further shows that the apparent , which is representative for the frequency
interval, is in the range 14–19, implying apparent
Q
ζ = 2.6 to 3.6%. These values are higher than
the apparent ζ for the fundamental mode obtained from the transfer-functions (Fig. 5.2, 5.3, and
5.4). For example, for NS motions, the apparent ζ

is 3.6 %, and near the apparent frequency
=1.72 Hz it is 1.7-1.8%. This suggests that the attenuation of response amplitudes (due
combined structural, soil and radiation damping) is larger for higher frequencies than predicted by
constant (frequency proportional damping). This is also supported by the difference in
apparent
,app
f
1
Q
ζ for the two subbands of EW motions (2.6 Hz for the subband 0-7.5 Hz vs. 3.5 Hz for
the subband 7-15 Hz). Table 5.2 also suggests that the apparent damping is the largest for the
NS motions (ζ =3.6% vs. 2.6% for the first subband of EW motions and for torsion), which is
consistent with significant radiation of energy into the soil via foundation rocking, known to be
considerable for this component of motion (Foutch et al. 1975; Luco et al. 1987, 1988).
141

Observed pulse arrival times (average for the causal and acausal pulses) at
each floor in Millikan library response during the Yorba Linda earthquake of 2002, and identified
(by the direct algorithm) equivalent wave velocities,
ave
i
t
eq
Table 5.1
β , Q factors and damping ratios
1/ 2Q ζ = for the NS, EW and torsional motions.
Level
i
z
[m]
NS EW Torsion
(0-15 Hz)
ave
i
t

[s]
(0-25 Hz)
ave
i
t

[s]
(0-7.5 Hz)
ave
i
t

[s]
(7-15 Hz)
ave
i
t

[s]
(0-25 Hz)
ave
i
t

[s]
Roof 0 0 0 0 0 0
9 4.25 -* -* -* -* -*
8 8.5 0.04 0.0395 0.0745 0.0415 -
7 12.75 0.048 0.047 0.0885 0.047 0.039
6 17 0.0535 0.053 0.095 0.056 0.046
5 21.25 0.0615 0.0615 0.101 0.0685 0.053
4 25.5 0.073 0.0725 0.111 0.083 0.059
3 29.75 0.0785 0.078 0.1305 0.0965 0.064
2 34 0.0835 0.083 0.1485 0.102 -
Gnd 38.9 0.096 0.096 0.16 0.112 0.084
*Some values are missing where the causal and acausal pulses could not be resolved or where
motion was not recorded.

Equivalent (over the height of the structure) wave velocity,
eq
β , and apparent
quality factor, Q, and damping ratio,
Table 5.2
ζ , of Millikan Library NS, EW and torsional responses
during Yorba Linda earthquake of 2002, as identified by the direct algorithm
Component Frequency band
i
τ [s]
eq
β

[m/s]
Q 1/ 2Q ζ =

NS 0-15 Hz 0.096 405 13.9 3.6%
EW 0-7.5 Hz 0.16 243 19.2 2.6%
EW 7-15 Hz 0.112 347 14.3 3.5%
Torsion 0-25 Hz 0.084 463 19.2 2.6%

142

5.4 Multilayer Model Fitting
The results of the identification based on detailed models are summarized in Tables 5.3-5.7. The
first group (Tables 5.3, 5.4a,b, 5.5) shows results for “determined” models, that have as many
layers as there are recordings that can be used for the fit. Because the causal and acausal pulses
could not be resolved at the 9
th
floor, the top two stories were combined in one layer. For the
torsional response, the top three layers were combined in one layer, and the first and second
floors were also combined in one layer. The different tables in this group correspond to different
components of motion or subbands. The second group (Tables 5.6 and 5.7) shows results for
“underdetermined” (9 layer) models, for the NS component and for the first subband of the EW
component.
In each table, Column 1 shows the layer number (Fig. 5.1), Column 2 shows the floors
contained in the layer, and Column 3 shows the layer height, h
1
. The next group of three
columns (4-6) shows results related to identification of the layer velocity by the direct algorithm,
which were used as initial values for the nonlinear LSQ and the TSM algorithms. Column 4
shows the observed travel time through the layer,
i
τ (determined from the pulse arrival times in
Table 5.1), Column 5 shows the identified velocity /
i
h
i
β τ
1
= , and Column 6 shows the relative
error
()
pred obs obs
/ / τττ τ τ Δ= − , where
pred
τ is the travel time through the layer measured
from model IRFs, and
ob s
τ is the observed travel times, measured from observed IRFs (same as
in column 4). This ratio is also a measure of / / ββ ττ Δ =−Δ (which follows from / h β τ =
and differential calculus). However, it is a rough estimate of / β β Δ , because it is obtained from
two readings that both have error. The next group of four columns (7-10) shows results of
identification by the nonlinear LSQ algorithm. Columns 7, 8 and 9 show the identified β , the
143

standard deviation of the estimate,
β
σ , the ratio /
β
σ β , and / τ τ Δ . Finally, Column 11
shows the identified β by the Time Shift Matching (TSM) algorithm.
β The identified - profiles for the first group of models (Tables 5.3, 5.4a,b, 5.5) are shown
graphically in Fig. 5.6. The different columns show results by the nonlinear LSQ, TSM and
Direct algorithms (left to right). Each bar corresponds to a different floor, and when some floors
were merged in a single layer, the bars show the equivalent value for the floors in the layer. In
the plots for the nonlinear LSQ algorithm, the 2
β
σ interval is shown as a solid line at the right
end of the bars. The sketch in the last column shows the layering structure.

Comparison of shear wave velocities of Millikan Library NS response during
Yorba Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-15 Hz)
Table 5.3
Layer #
Floor
i
h

[m]
Direct LSQ* TSM
i
τ [s]
i
β
[m/s]
/ τ τ Δ
[%]
i
β
[m/s]
β
σ
[m/s]
/
β
σ β
[%]
/ τ τ Δ
[%]
i
β
[m/s]
1 8+9 8.5 0.04 212.5 5 281.6 2.3 0.8 11 226
2 7 4.25 0.008 531.3 81 226.6 3.6 1.6 37.5 292
3 6 4.25 0.0055 772.7 64 360.8 11.3 3.1 63.6 599
4 5 4.25 0.008 531.3 137 554.4 27.5 5.0 6 639
5 4 4.25 0.0115 369.6 22 516.6 21.5 4.2 13 554
6 3 4.25 0.0055 772.7 73 638 40.5 6.3 0 657
7 2 4.25 0.005 850.0 80 835 87.3 10.5 20 723
8 Gnd 4.9 0.0125 392.0 100 491.6 18.1 3.7 12 478
*Normalized_RMS error=28.8%

144

Comparison of shear wave velocities of Millikan Library EW response during
Yorba Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-7.5 Hz)
Table 5.4
Layer
Floor
i
h
[m]

Direct LSQ TSM
i
τ
/
β
σ β
[%]

[s]
i
β
[m/s]
/ τ τ Δ
[%]
i
β
[m/s]
β
σ
[m/s]
/ τ τ Δ
[%]
i
β
[m/s]
1 8+9 8.5 0.0745 114 11 173 0.86 0.5 28 132.6
2 7 4.25 0.014 303.6 82 172 2.3 1.3 85 151.8
3 6 4.25 0.0065 654 84 141 1.5 1.1 192 269.7
4 5 4.25 0.0065 654 84 397 27 6.8 23 392.3
5 4 4.25 0.0095 447 31 344 13 3.8 26 461.3
6 3 4.25 0.0195 218 215 328 9.5 2.9 5 408.6
7 2 4.25 0.018 236 69 331 9.5 2.8 2.8 378
8 Gnd 4.9 0.0115 426 78 384 17 4.4 13 362
*Normalized_RMS error=24%
Comparison of shear wave velocities of Millikan Library EW response during
Yorba Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (7-15 Hz)
Table 5.5
Layer
Floor
i

h
[m]
Direct LSQ* TSM
/
β
i
τ

[s]
i
β

[m/s]
/ τ τ Δ

[%]
i
β

[m/s]
β
σ

[m/s]
σ β

%
/ τ τ Δ
[%]
i
β

[m/s]
1 8+9 8.5 0.0415 205 3.6 247 1.83 0.7 3.6 218
2 7 4.25 0.0055 773 100 294 5.56 1.9 0 333
3 6 4.25 0.009 472 0 393 8.9 2.3 27.8 498
4 5 4.25 0.0125 340 96 457 12.3 2.7 8 510
5 4 4.25 0.0145 293 55 334 6.8 2 10.3 311
6 3 4.25 0.0135 315 30 327 6.5 2 3.7 287
7 2 4.25 0.0055 773 36.5 487 18.7 3.8 9 511
8 Gnd 4.9 0.01 490 15 432 11.6 2.7 5 447
*Normalized_RMS error=29%
145

Identified velocity profiles, obtained by different algorithms, of Millikan library
during the Yorba Linda EQ, 2002. a) NS response (0-15 Hz), b) EW response (0-7.7.5 Hz), EW
response (7-15 Hz), and d) torsional response (0-25 Hz). The algorithms used are: LSQ=nonlinear
least squares fit of transmitted pulses, TSM=time shift matching, and Direct=the direct algorithm.
Figure 5.6
146

Figure 5.7 shows the agreement of the IRFs for the model (identified by the nonlinear LSQ
algorithm) with the observed IRFs for the NS, EW (1
st
subband) and torsional motions (parts a),
b) and c)), computed for ζ =2%. Further, Figs 5.9, 5.10 and 5.11 show, respectively for the NS,
EW and tosrsional motions, the agreement of pulse arrival times (part a)) and of transfer-
functions (part b). The pulse arrival times for the TSM algorithm are indistinguishable from the
observed ones and are not shown. Finally, figure 5.12 compares the β - profiles for the 8-layer
and 9-layer models for NS and for the 1
st
subband of EW motions (Tables 5.6 and 5.7), all
identified by the nonlinear LSQ algorithms.
Figure 5.8 Illustrates convergence of used non-linear least square algorithm for estimating
the wave velocity of layer 8 & 9 (N-S component). Excluding the outlier in this figure, the
Standard Deviation of estimation (
β
σ ) is calculated for sequence of converging values (table
5.3). Note that the fitting and velocity estimation is carried out over all layers simultaneously. The
values for standard deviations are taken from the diagonal elements of Covariance matrix of
estimated velocities.

147

a) b) c)
Agreement of IRFs of models identified by the nonlinear LSQ algorithm with the
observed ones. a) NS response, 0-15 Hz. b) EW response, 0-7.5 Hz. c) Torsional response, 0-
25 Hz.
Figure 5.7
2% ζ = was assumed.

200 210 220 230 240 250 260 270 280 290 300
Estimation of shear wave velocity at layer 8 & 9 (m/s)
Initial value
Estimated velocity
Illustration of non-linear least square estimation convergence. Shear wave velocity
of layer 8+9 (top two layers) of Millikan Library, N-S component (see table 5.3)
Figure 5.8

148

Herein, we first discuss the results obtained by the nonlinear LSQ algorithm for the different
components of motion. The β -profiles in Fig. 5.6 (1
st
column) suggest softer top three stories for
all three components of motion, and soft first story for NS motions and torsion. The comparison
of IRFs in Fig. 5.7 shows that the agreement of the transmitted pulses is generally good. The
agreement is worse at the 7
th
and 8
th
floors, where separation of the acausal and causal pulses is
the smallest, and their interference is the strongest. The agreement is the worst for the 1
st
subband
of EW motion, for which the source pulse is the widest, and its time localization is poor. For this
reason, the variation in velocity in the top three stories for the 1
st
subband of EW motions is not
reliable (Fig. 5.6, row 2 and column 1). Comparison of
β
σ (Fig. 5.6 and Tables 5.3-5.5) shows
that /
β
σ β is the largest (10%) at the 2
nd
floor for NS motions, which is above the soft 1
st
floor,
suggesting slower convergence of the algorithm in layers with large impedance contrast relatively
to the neighboring layers.
Next, we discuss the agreement of the velocity profiles in Fig. 5.6 obtained by the different
algorithms. It can be seen that the profiles by the TSM algorithm are consistent with those by the
nonlinear LSQ algorithm, except in the top three layers for EW motion 1
st
subband, for the reason
mentioned earlier. In contrast, the profiles by the Direct algorithm differ considerably, except for
the torsional model which has fewer layers.
Comparison of pulse arrival times, measured from model and data IRFs (Figs 5.9-5.11),
shows remarkable improvement for the high-resolution models identified by the nonlinear LSQ
algorithm, as compared to the Direct algorithm. The agreement with the data arrival times is very
good, except for the first subband of EW motions above and including the 7
th
floor, for the
mentioned reasons.

149

a)

b)
Other measures of agreement of the identified models of the NS response of
Millikan library with the observations during Yorba Linda earthquake of 2002. a) Agreement of
pulse arrival times at the different floors. b) Agreement of transfer-functions. Note: the arrival at
the 9
th
floor could not be resolved.
Figure 5.9
150

For the high-resolution models identified by the Direct algorithm, the agreement revealed to
be poor for the NS motions shown in Fig. 5.9, is much worse for the 1
st
subband of EW motions
(Fig. 5.10), but is better for the torsional motions because of the smaller number of layers and
slightly larger effective data bandwidth.
Figs 5.9-5.11 show that the agreement of the pulse arrival times between model and data is
not as good for the nonlinear LSQ algorithm as it is for the TSM algorithm. However, this does
not mean that the TSM algorithm is better, because of the error in measurement of the pulse
arrival times, which is different for the data and for the models, because of differences in the
pulse shapes. We believe that the difference in pulse shapes is mostly due to differences in the
model for Q (constant for the model and increasing with frequency for the data). A comparison
of the frequency of the second mode predicted by the model, f
2
, with the observed apparent
frequency, , (Figs.
,app
f
2
5.9-5.11) is in favor of the nonlinear LSQ algorithm. For the NS
motions, the agreement is excellent for both algorithms, while for the EW motions, the agreement
is close but not as good, and is slightly better for the nonlinear LSQ algorithm. For the torsional
motions, the agreement is excellent for the nonlinear LSQ algorithm, and not so good for the
TSM algorithm.

151

a)

b)
Other measures of agreement of the identified models of the EW response of
Millikan library with the observations during Yorba Linda earthquake of 2002. a) Agreement of
pulse arrival times at the different floors. b) Agreement of transfer-functions. Note: the arrival at
the 9
th
floor could not be resolved.
Figure 5.10
152

a)

b)

Other measures of agreement of the identified models of the Torsional response
of Millikan library with the observations during Yorba Linda earthquake of 2002. a) Agreement
of pulse arrival times at the different floors. b) Agreement of transfer-functions. Note: the arrival
at the 9
th
floor could not be resolved, and the observed torsion is not available at the 2
nd
and 8
th

floors.
Figure 5.11
153

Comparison of torsional wave velocities of Millikan Library EW response during
Yorba Linda earthquake of 2002, as identified using 8-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-25 Hz)
Table 5.6
Layer
Floor
i
h
[m]

Direct LSQ* TSM
i
τ
[s]
i
β
[m/s]
/ τ τ Δ
[%]
i
β
[m/s]
β
σ
[m/s]
/ /
β
σ β
[%]
τ τ Δ

[%]
i
β
[m/s]
1 7+8+9 12.75 0.039 327 5.1 348.6 2.7 0.8 3.8 339
2 6 4.25 0.007 607 57.1 417.416.3 3.9 21.4 448
3 5 4.25 0.007 607 21.4 487 19.9 4.1 28.6 567
4 4 4.25 0.006 708.3 33.3 813.473.3 9.0 25 797
5 3 4.25 0.005 850 30 739 50.6 6.8 10 850
6 Gnd+2 9.15 0.02 457.5 0 587.4 26 4.4 15 457
*Normalized_RMS error=37%
The readings of the observed (apparent) and model (fixed-base) frequencies for the first two
modes, and their differences are shown in Table 5.8. These results suggest shift of the first mode
frequency due to soil-structure interaction of about 42% for the NS response, 33% for the EW
response, and 31% for the torsional response. Table 5.9 also shows frequencies for the NS and
EW motions published in late 1980s by Luco et al. (1987, 1988), identified from forced vibration
tests conducted in 1975 by Foutch et al. The amplitudes of response during the Yorba Linda
earthquake of 2002 and during the forced vibration test were likely of the same order. It can be
seen that the observed apparent frequencies during the 2002 earthquake were lower than in 1975
by about 4% for NS response, 7% for EW response and 17% for torsion, which is consistent with
the long term trend reported in (Clinton et al. 2006), likely caused by structural degradation. The
identified fixed-base frequencies in this study are higher than those identified in (Luco et al.
1988) by 20-30%. These differences are likely caused by biases from the differences in model
assumptions, location of the source of the excitation, and estimation errors.
154

Finally, we compare the profiles identified by fitting determined and underdetermined
models (Fig. 5.12). It can be seen that the profiles differ mainly near the top. Comparison of the
transfer-functions (Fig. 5.9 and 5.10) shows that, for NS motions, the agreement is better for the
determined model, and, for EW motions it is equally good.
Due to the spread of the source pulse in time, leading to spread of the propagating pulse in
space, the resolving power of the method is limited, even when the density of sensors is high.
When the width of this pulse (
max
/ wf β ≈ ) is larger than the layer width, the identified layer
velocities depends also on the velocities of the neighboring layers, and can be viewed as weighted
average of the velocities of the layers enclosed by the pulse, the weighing function being the
shape of the pulse.

Identified velocity profiles, obtained by LSQ algorithms for 8-layer and 9-layer
models, of Millikan library during the Yorba Linda, 2002, earthquake. a) NS response (0-15 Hz),
b) EW response (0-7.7.5 Hz), using LSQ algorithm
Figure 5.12
155

Comparison of shear wave velocities of Millikan Library NS response during
Yorba Linda earthquake of 2002, as identified using 9-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-15 Hz)
Table 5.7
Layer Floor
i
h
[m]

Direct LSQ*
i
τ
[s]
i
β
[m/s]
/ τ τ Δ
[%]
i
β
[m/s]
β
σ
[m/s]
/
β
σ β
%
/ τ τ Δ
[%]
1 9 4.25 - 212.5 - 224 4.4 2 -
2 8 4.25 0.04 212.5 5 460 22 4.8 30
3 7 4.25 0.008 531.3 81 290 7 2.4 81
4 6 4.25 0.0055 772.7 64 265 5.3 2 154
5 5 4.25 0.008 531.3 137 410 16 3.9 12
6 4 4.25 0.0115 369.6 22 419 15 3.6 17
7 3 4.25 0.0055 772.7 73 683 59 8.6 18
8 2 4.25 0.005 850.0 80 884 112 12.7 40
9 Gnd 4.9 0.0125 392.0 100 494 18 3.6 32
*Normalized_RMS error=27.7%
Comparison of shear wave velocities of Millikan Library EW response during
Yorba Linda earthquake of 2002, as identified using 9-layer model by three algorithms: Direct,
nonlinear LSQ and Time Shift Matching (TSM) (0-7.5 Hz)
Table 5.8
Layer Floor
i
h
[m]

Direct LSQ*
i
τ /
β

[s]
i
β
[m/s]
/ τ τ Δ
[%]
i
β
[m/s]
β
σ
[m/s]
σ β
%
/ τ τ Δ

[%]
1 9 4.25 - 114 - 159 3.7 2.3 -
2 8 4.25 0.0745 114 11 193 2.8 1.5 23
3 7 4.25 0.014 304 82 178 2.9 1.6 61
4 6 4.25 0.0065 654 84 142 1.5 1.1 192
5 5 4.25 0.0065 654 84 379 23 6.1 23
6 4 4.25 0.0095 447 31 343 13 3.8 26
7 3 4.25 0.0195 218 215 328 9.3 2.8 2.5
8 2 4.25 0.018 236 69 330 9.3 2.8 8.3
9 Gnd 4.9 0.0115 426 78 380 16.5 4.3 8.7
*Normalized_RMS error=24%
156

Apparent and fixed-base frequencies of vibration of Millikan Library NS, EW and
torsional responses identified by this study (nonlinear LSQ fit algorithm) from records of Yorba
Linda earthquake of 2002, and identified by Luco et al. (1987) from forced vibration tests
conducted in 1975 (Foutch et al. 1975).
Table 5.9
NS EW Torsion

This
study
Luco et
al. 1987
Diff.
This
study
Luco et
al. 1987
Diff.
This
study
Luco et
al. 1987
Diff.
,
f
1 app
1.72 1.79 -3.9% 1.12 1.21 -7.4% 2.4 2.89 -16.9%
f
1
3.0 2.33 28.7% 1.68 1.39 20.8% 3.5 2.94 19%
,app
f
11
1
ff −
42.6% 22.2% 33% 12.3% 31% 1.7%

5.5 Summary and Conclusions
Based on the presented analysis, it can be concluded that, (1) for identification of more detailed
building models, which is necessary for detecting localized damage in structures, the nonlinear
LSQ algorithm should be used. (2) The goodness of the fit is the worst near the top and improves
progressively towards the base, which is favorable for SHM because damage usually occurs in
the middle and lower part of the structure. (3) The accuracy of the identified β - profiles is better
if the effective bandwidth of the recorded response is larger (i.e. the higher modes are excited).
(4) Once the β - profiles are identified, the fixed-base frequencies and mode shapes can be easily
identified, which are useful for calibration of numerical models of the structure.
Moreover, it is shown that (5) Millikan library NS and torsional responses are predominantly
in shear and essentially nondispersive in the frequency range 0-15 Hz. (6) Millikan library EW
response is dispersive, suggesting significant contribution to the building deformation from
bending, and significant differences among the vertical waveguides in the building (elevator core
v.s. the remaining part). Nevertheless, a shear beam model could be fitted in narrower frequency
157

158

bands, 0-7.5 Hz and 7-15 Hz, which enabled measuring the dispersion. This implies that, (7) even
for structures with dispersive wave propagation, the wave travel time method can be used to
identify changes in the wave velocity for SHM, by appropriate choice of the frequency band.

Chapter 6
Earthquake Damage Detection in an RC High-
Rise Building - Sherman Oaks 12-Story Office
Building Lightly Damaged by San Fernando
Earthquake of 1971
This chapter presents an application of the automated time velocity algorithm, presented in
Chapter 2, to a damaged building. This algorithm estimates the layer wave velocities in moving
time windows using the least squares fit identification algorithm, and identifies change relative to
the values in the initial window. The case study is Sherman Oaks 12-story Office Building, an
RC structure lightly damaged by the San Fernando earthquake of 1971. The study has several
objectives. One objective is to demonstrate the stability and sensitivity of the algorithm in the
most realistic environment. Another objective is to obtain calibration data for the method, i.e.
relate percentage change in wave velocity to a level of damage. The third objective is to compare
the findings of this wave method with findings by other methods, in particular, with identified
changes in the fundamental frequency of vibration and levels of interstory drift.
This chapter is organized as follows. First, the building and the observed damage are
described, and the recorded strong motion data are presented. Next, the time-velocity graphs are
presented, identified based on a fitted two layer model and a uniform model, and are compared
with analysis of input energy and power into the structure, interstory drift and instantaneous
159

frequency of vibration. Finally, inferences are made about the occurrence and timing of damage
as well as changes in the soil-foundation system, and conclusions are drawn.
This Chapter is based on the following journal paper:
Rahmani M, Ebrahimian M, Todorovska M. I. (2014). Automated time-velocity analysis for
early earthquake damage detection in buildings: Application to a damaged full-scale RC building
and comparison with other methods, Earthq. Eng. Struct. Dyn., Special issue on: Earthquake
Engineering Applications of Structural Health Monitoring, CR Farrar and JL Beck Guest Editors,
submitted for publication, 01/16/2014.

6.1 Building Description
Sherman Oaks 12-story Office Building (station No. 466 of the National Strong Motion
Instrumentation Program) is a reinforced concrete (RC) structure located in the city of Sherman
Oaks in the North-Western part of the Los Angeles metropolitan area. Fig. 6.1 shows sketches of
its EW elevation and plans of the ground floor, 7th floor and roof (drawn based on John Blume
Associates report (1973)), and Fig. 6.2 shows a map of San Fernando Valley with the location of
the building relative to the major freeways, and relative to the rupture area of the San Fernando,
1971 earthquake. A photo of the building is also shown (courtesy of M. Trifunac).
The building has plan dimensions 18.3 m × 49 m (aspect ratio about 2.5:1) except at the
ground floor, which has 27.4 m × 41.1 m. The main roof is 48.5 m above the ground. A
mechanical penthouse occupies about 30% of the roof area. It was designed in 1969 following
the requirements of the 1968 Los Angeles Building Code, and was completed in 1970. Moment-
resistant (MR) frames provide resistance to lateral forces in both directions. In the transverse
(East-West) direction, only the exterior bays extend to the roof, while the interior frames (along
column line A, C, D, E, and G) extend only to the third floor. Above the third floor, they were
designed to carry only vertical loads. In the longitudinal (North-South) direction, only the exterior
frames (along column lines 1 and 3) were designed to carry lateral loads. These frames include
160

two shear walls, each two stories high, along column line 1. In addition, a one story separation
wall along column line 4 acted as a shear wall. A typical floor consists of a 4.5 inch slab and
supporting girders made of lightweight concrete.
The soil at the site is moderately firm, consisting mainly of silt and silty sand, with some
deposits of clay and sand. The building was founded on pile foundation, consisting of drilled and
cast in place concrete piles, 10.5 to 15 m long. Pile foundations provide a stiffer soil-foundation
system in comparison to footings and mat foundations. Nevertheless, soil-structure interaction
can still take place, the pile group rocking as a whole.

Sherman Oaks 12-story Office Building: plans of the instrumented floors (middle)
and vertical cross section (drawn based on John Blume Associates (1973)). Instrument locations
are marked.
Figure 6.1
161

Map of San Fernando Valley showing the location of Sherman Oaks 12-story
Office Building relative to the local freeways and the ruptured area of the San Fernando, 1971
earthquake (drawn after (Trifunac, 1974). A photo of the building is also shown (courtesy of M.
Trifunac).
Figure 6.2

6.2 Structural Damage Description
The building was located 27 km from the epicenter, but only 16 km from the rupture area, and in
the direction of rupture propagation of the San Fernando, 1971 earthquake (Fig. 6.2). The
building experienced both structural and nonstructural damage. The visible structural damage
was slight and consisted almost entirely of minor cracking and spalling of concrete. The most
severe damage in the building occurred at the second floor where extensive cracking of the
exposed girder stub at the exterior column was observed. The nonstructural damage consisted of
partitions pulled away from columns, some damages in the stairwells, ceilings, stairs, and
162

mechanical equipment. Fig. 6.3 shows our summary of the distribution of the structural damage
along the building height, drawn based on the more detailed description in the report (John Blume
& Associates, 1973), and the 2-layer shear beam model of the structure that is fitted in this study.
As it can be seen, the cracking was distributed over the structural frames.

Distribution of structural damage along the height in Sherman Oaks 12-story
Office Building during the San Fernando, 1971 earthquake (left; drawn based on John Blume
Associates, 1973) and the 2-layer shear beam model to be fitted (right).
Figure 6.3

163

6.3 Strong Motion Data and Correction for Synchronization
Errors
The building response was recorded by three tri-axial SMA-1, located on the ground floor, 7
th

floor and the main roof. The instrument locations are shown in Fig. 6.1 by small squares. On the
vertical elevation, the location of the instrument on the 7
th
floor differs from that in the report (J.
Blume Associates, 1973), which we believe is in error. The film records were digitized and
processed, as described in (Trifunac et al. 1974).
In addition, we corrected the acceleration records for synchronization errors based on the
symmetry of the acausal and causal pulses in the IRFs. If the effects of wave dispersion are
small, the time shifts of the acausal and causal pulses in the IRs are practically symmetric, so
significant lack of symmetry in observed IRFs is an indication of synchronization errors in the
data. Fig. 6.4 shows IRFs of the building (computed from the entire length of the traces and low-
pass filtered at 6 Hz) before the correction (the thinner lines) and after the correction (the thicker
lines). The lack of symmetry can be seen by bare eye. For each component of motion, the traces
at the 7
th
and ground floors were shifter relative to the roof record by adding zeros or truncation.
The time shift for a particular trace was determined from the difference between the time shift of
the acausal (or causal) pulse and the average time shift of the two pulses. The corrections for the
NS traces at the 7
th
and ground floors are -0.04 and -0.01 s, and for the EW traces at the 7
th
and
ground floors are -0.04 s and -0.06 s.
Such errors occur in making decision about the first point of the trace when interpreting the
bitmap of shades of gray obtained by scanning the film record, and can be avoided with the more
modern software for automatic digitization of accelerograms (Trifunac et al. 1999). It is
important to note that the correction applied in this study allows for the physical difference in
phase between the records at the different floors due to wave propagation. Time shifts minimizing
the cross-correlation of the different records would erroneously eliminate the physical shift.
164

Comparison of IRFs before and after correction for synchronization. Figure 6.4

For this study, we refiltered and interpolated the data at time step 0.01 s (Lee and Trifunac,
1990). Fig. 6.5 shows time histories of the corrected accelerations, band-pass filtered with
Ormsby filters between 0.075-0.125 to 25.00-27.00 Hz (part a)), and their Fourier transform
amplitudes (part b)).

165

Recorded motions in Sherman Oaks 12-story Office Building during the San
Fernando, 1971 earthquake: a) acceleration time histories and b) Fourier Transform amplitudes
of acceleration at roof, 7
th
and Ground floors.
Figure 6.5

166

6.4 Time-Velocity Graphs and Interstory Drifts
For the interferometric analysis, we chose
max
Hz f = 4
win
w
for the NS for the EW
response, and time windows with length of around ~10 s. Before proceeding with analysis
of the time-velocity graphs, we illustrate impulse response functions (IRFs) and transfer-functions
(TFs) in two 10 s windows, representative of the early and late part of the response. Fig.
max
. Hz f =4 5
6.6
shows such results for windows 2-12 s and 30-40 s for the NS, and 3-13 s and 22-32 s for the EW
response. The plots of the IRFs (part a)) show clearly an increase in pulse travel time by ~35%,
consistent with reduction of stiffness due to damage. Rough readings of the time shifts of the
acausal pulses suggest increase of pulse travel time from ~0.18 s to 0.24 s for the NS response,
and from ~0.22 s to 0.3 s for the EW response.
The plots of the TFs (part b)) also show reduction of the frequencies of vibration. Rough
readings of the apparent frequencies from the TFs suggest reduction from ~0.7 Hz to 0.3 Hz
(~57%) for the first, from 1.98 Hz to 1.1 Hz (~44%) for the second, and from 3.51 Hz to 2.34 Hz
(~33%) for the third vibrational mode. Such readings are more uncertain for the transverse (EW)
response, in particular for the later window and the higher modes, because of the interference of
the torsional response, which could not be separated from the records.

167

Comparison of Impulse Response Functions (part a)) and Transfer-Functions (part
b)) for two 10 s time windows, one in the beginning of shaking (2 – 12 s) and the other one near
the end of shaking (22 - 32 s). Changes in wave travel time and frequencies of vibration can be
seen.
Figure 6.6
168

The results of the automated time-velocity analysis are presented in Fig. 6.7, along with time
histories of interstory drifts, and time histories of measures of the cumulative input energy and
power pumped into the structure. Part a) shows the identified layer velocities v and
1
v
2
of the
fitted 2-layer shear beam model (Fig. 6.3) identified in moving windows with length
win
w =8, 10,
and 12 s with time shift
shift
t = 1 s, and the intestrory drifts in the layers. Part b) shows ( Et)
() ( )
t
Et v t dt
2
11
0
=
∫
(6.1)
which is proportional to the energy arriving at the site and its time rate . () Pt
For the longitudinal (NS) response, Fig. 6.7 reveals significantly larger input energy than for
the transverse response, most of which was pumped in the structure over interval of 8 s, between
12 s and 20 s from trigger time. The time velocity graphs reveal reduction of the velocities of the
top and bottom layers of about 24% and 31%, most of which occurred around t 18-22 s. This
drop coincides with larger drifts, which exceeded 0.5% in both layers at about t 19 s, reaching
about 0.7% in the bottom layer. Extensive cracking in the NS frames and shear walls appeared
to have occurred around 19-21 s, about 8 s from the beginning of the larger input power, after
which the softened structure continued to vibrate with larger amplitudes until 28 s.
=
=
=
t =
t
For the transverse (EW) response, most of the energy was pumped into the structure over a
longer time interval of 18 s, between t = 13 s and 31 s. Fig. 6.7 reveals reduction of the top and
bottom layer velocities of about 27% and 37%. Most of the reduction in the bottom layer
occurred between t 16 s and 30 s. The drift in the bottom layer exceeded 0.5% at t = ≈ 22 s,
exceeded 1% at t 30 s. It reached its maximum of 1.3% at ≈ t = 32 s, and remained large even
after the layer velocity leveled off, the damaged structure continuing to vibrate with large
amplitudes.
169

a) Time-velocity graphs and interstory drifts for Layers 1 and 2. b) Measures of
input energy and power , estimated from the ground floor velocity. () Et () Pt
Figure 6.7
170

The observed changes in layer velocities, which exceeded 24%, were much larger than the
estimation error of the individual estimates. The relative estimation error /
v
v σ , where
v
σ is
the standard deviation of the LSQ estimate of , was the largest when the layer velocity
changed rapidly. With few outliers removed, for the NS response,
eq
v
/
v
v σ was < 2.6% in the top
layer and <5.3% in the bottom layer. Similarly, for the EW response, / v
v
σ was <3.4% in the
top layer and < 5.2% in the bottom layer. In the time intervals of slower or no change in velocity,
/
v
v σ was much smaller. The error is illustrated in the time-velocity plots in Fig. 6.7 by a
rectangle (
win v
w ) σ ×2 for the = 10 s. These errors are quite smaller than the observed
changes in the Time-Velocity graphs.
win
w
Despite the smaller input energy, the transverse (EW) response experienced almost two
times larger drifts than the longitudinal response, consistent with the smaller transverse stiffness.
The reduction in wave velocity, however, was only slightly larger (< 20%).
According to ASCE 41 standard, the EW drift (1.3%) exceeded the Immediate Occupancy
performance level (1% for RC moment resistant frames), while the NS drift did not exceed that
level. According to drift limits associated with different damage levels for ductile MR frames in
Ghobarah (2004) (<0.2% for no damage; 0.4% for light damage; <1% for moderate damage; >1%
for irreparable damage; and >3% for collapse), the NS drift suggests moderate damage and the
EW drift suggests irreparable damage. The observed structural damage, however, was described
as light. This comparison shows that predictions of damage based on the published correlations
of damage levels with interstory drift may be very inaccurate when applied to a particular
building.

171

6.5 Time-Frequency vs. Time-Velocity Analysis
Case studies of instrumented buildings damaged by an earthquake present rare opportunities to
test different SHM methods and compare results by different methods. In Fig. 6.8, we present our
result of time-frequency analysis and of novelties analysis for this building. We compare the
results of the time-frequency analysis ( ()
app
f t , part a)) with results of time-velocity analysis
based on fitting an equivalent uniform shear beam model ( , part b)), both analyses
following changes in global parameters of the building. Velocity is directly related to the
fundamental frequency of vibration of the equivalent shear beam
()
eq
vt
(
eq
v
eq
)t
/( ) f v H
1
= 4 , where H is the
building height. In part b), the uncertainty of the estimate of is illustrated by the rectangle
eq
v
(
win v
w ) σ ×2 . With few outliers removed,
v
σ was smaller (~1 m/s) when did not change
much, and larger when in changed more rapidly, but did not exceed 2 m/s.
eq
v
Part a) of Fig. 6.8 shows the apparent instantaneous frequency ()
app
f t , estimated from the
projection on the time-frequency plane of the ridge of the Gabor transform of the relative roof
acceleration. The estimation follows closely (Todorovska and Trifunac, 2007), and only few
details about the method will be mentioned in this chapter. The estimation is closely related to
moving window Fourier analysis with a Gaussian window. Results are shown, with the
parameter of the transform, σ = 2, 3 and 4 s. The uncertainties of the estimate of ()
app
f t in
time and frequency are / σσ =
t
2 and /
f
σπσ =12 2 , the meaning of the estimate being
that, in the time interval t
t
σ ± ,
app
f is within the interval
f
f σ ± , where
t
σ and
f
σ are the
standard deviations of the spreads in time and in frequency. The lengths of the time windows
satisfy the minimum requirement of at least one period being contained in
t
σ 6 . The localization
rectangle ()( t )
tf
t σ σ ±× ± for σ = 3 s is also shown in the figure for convenience of the
172

interpretation of the significance of the detected changes in
app
f . The ()
app
f t
()
app
curves are
interrupted to eliminate artifacts caused by rapidly changing envelope of the signal. Such artifacts
occur because the method assumes that the envelope of the signal changes slowly within one
period of vibration, the time variation of the signal being mostly due change in phase
(Todorovska and Trifunac, 2007). Where this assumption is violated, f t exhibits large
fluctuations that are not real. The horizontal bars in Fig. 6.8a at t ≈ 40 s show the corresponding
ambient vibration frequencies of the damaged structure (Kircher, 1975), included in this figure
for completeness.
t = Fig. 6.8a reveals a sharp drop of
app
f for NS response around 11-12 s of ~20.5% which
coincides with the first strong input energy pulse, but is much earlier than the occurrence of larger
drift amplitudes. Another drop follows at about t = 21 s of about 23.5%, which coincides with
the occurrence of larger drift amplitudes and with the drop in layer velocities and v v
1 2
(Fig.
6.7a) and (Fig.
eq
v 6.8b). The total drop
app
f for NS response is ~44%, and is much larger than
the uncertainty in the estimation of
app
f . A sharp drop can be seen in
app
f for the EW
response also at t 11-12 s, which is even earlier than the occurrence of larger drifts in the EW
response, followed by a gradual decrease, concurrent with the larger drifts and the decrease in
. The total change in
=
eq
v
app
f for the EW response is ~48%. The observed changes are in
general agreement with previous studies (McVerry, 1979).

173

Estimation of loss of stiffness with time by different methods: a) Instantaneous
frequency (estimated from the ridge of the Gabor Transform of relative roof acceleration); b)
Equivalent uniform shear beam velocity.
Figure 6.8
The total drop in
app
f (~44% for the NS and ~48% for the EW response) is larger than the
total drop in (~26% for the NS and ~32% for the EW response). The fact that (i)
eq
v
app
f
depends both on the stiffness of the soil-foundation system and that of the structure, while the
wave velocities , and are not sensitive to the effects of soil-structure interaction, (ii)
the coincidence of the rapid drop in
v
1
v
2 eq
v
app
f at t = 11-12 s for both NS and EW response, much
earlier than the occurrence of larger drifts, and (iii) the coincidence of the first occurrence of
larger drifts with significant drops in v
1
, v
2
and - all suggest that the larger change in
eq
v
app
f
174

as compared to was due to softening of the soil-foundation system, rather than to smaller
sensitivity of the wave velocity to structural damage. The rapid drop in
eq
v
app
f at t = 11-12 in
both NS and EW responses appears to have been caused mainly by softening of the soil-
foundation system. Possible scenarios for this effect are softening of the soil and unlocking of the
foundation (pile group) by the formation of gaps between the piles and the surrounding soil
(Trifunac et al. 2001a,b). The following more gradual changes in
app
f , concurrent with the
changes in , appear to have been primarily caused by structural damage.
eq
v

6.6 Summary and Conclusion
An automated algorithm for time-velocity analysis of building response is presented, to be used
for rapid assessment of the structural health and decision making on evacuation, during or
immediately after a strong earthquake. The algorithm is based on interferometric estimation of
the vertical phase velocity in the building in moving time windows, by least squares fit of pulses
in impulse response functions of a layered shear beam. The robustness and sensitivity of the
method to damage was examined by application to a 12-story RC building, lightly damaged by
the San Fernando, 1971 earthquake (Sherman Oaks 12-story Office Building). The analysis also
provides calibration data for the method. Analyses of interstory drifts, instantaneous frequency
and novelties are also presented. The results by the different methods are used to critically
compare them, and to obtain a more complete and accurate history of the performance of this
building and soil-foundation system during this earthquake.
In the initial stage of smaller response, the identification analysis gave wave travel time from
ground floor to roof of ~0.18 s for the NS and ~0.22 s for the EW response, and average vertical
phase velocity of ~140 m/s for the NS and ~110 m/s for the EW response. The peak interstory
drift, assessed from the soil-structure system response, reached ~0.7% in the longitudinal (NS)
175

response, and ~1.3% in the transverse (EW) response, exceeding the immediate occupancy
performance level of 1% (ASCE 41). The time-velocity analysis detected change in wave
velocity in the NS response of ~31% in the lower and ~23.5% in the upper half of the building.
The corresponding changes for the EW response are ~37% and ~27%. The global change in
wave velocity, over the entire building height, is ~26% for the NS and 32% for the EW response.
The detected changes in apparent frequency of vibration (also a global characteristic) are 44% for
the NS and 48% for the EW response.
Our interpretation of the results by the different types of analyses is as follows. Cracking in
the building, which was for the first time exposed to strong shaking, started to occur at about t =
4.5 s (based on the novelties analysis), resulting in gradual reduction of structural stiffness. At
11 s, a rapid reduction of the soil-foundation stiffness occurred in both NS and EW responses
(based on the changes in apparent frequency of vibration and the absence of such change in the
wave velocities). Following the series larger bursts of input energy, which started to occur at t
11 s, rapid reduction in structural stiffness occurred after t
t ≈
>
= 15 s.
Based on the presented analysis, we concluded that the automated time-velocity algorithm is:
(i) robust when applied to this damaged full-scale building, (ii) sensitive even to light damage in
RC buildings, (iii) sensitive to the spatial distribution of damage, and (iv) not sensitive to
changes in the soil-foundation system. All of these features are favorable for its implementation
in structural health monitoring systems and seismic alarm systems in buildings. More similar
analyses of full-scale buildings, both damaged and undamaged, are needed to provide additional
calibration data for the threshold change associated with damage, and changes associated with
different levels of damage in different types of structures. Analyses of full-scale shake-table tests
of buildings may also provide valuable calibration data.

176

Chapter 7
Identification by Fitting a Model with Slabs
A building can be viewed as a sequence of alternating soft and very stiff and thin layers, where
the soft layers represent the columns and the other elements of the lateral resistant system, while
the stiff and thin layers represent the floor slabs. This would be a more realistic model than the
layered shear beam used earlier, in which the floor slabs are not considered explicitly.
This chapter presents results of identification of a real building (Millikan library NS
response during Yorba Linda, 2002 earthquake) by fitting such a model. The same mathematical
model for a layered shear beam is used (see Chapter 2), but with assumed properties of the thin
layers. The analysis reveals that the approximate Ray Theory of wave propagation in elastic
media, which has been successfully used in geophysics and seismology to solve numerous
problems, is not valid for a building model with slabs. It reveals that the phase velocity through
the beam with slabs is significantly lower than the shear wave velocity of the soft layers, as a
result of scattering from the slabs. It is shown that the Least Squares identification algorithm
successfully handles the scattering, producing fits with small error. The Direct Algorithm,
however, and the Ray Theory interpretation of the response identify the soft layers as softer than
they actually are, by 20% or more. However, with an adjustment by a scaling factor, which is
simple to obtain, it is shown that the Direct Algorithm can produce results with much smaller
error, involving much smaller computational effort than the Least Squares Algorithm.
177

In addition to identification of the NS response of Millikan Library, which represents a
stiffer structure, a detailed analysis of the dispersion due to the presence of slabs is presented
based on a model of a softer building. The identified dispersion is compared with published
theoretical dispersion for a 1D periodic structure (Fukuwa and Matsushima, 1994).

178

7.1 Identification of Millikan Library NS Response
7.1.1 Model Description
To investigate the effects of modeling slabs in the shear wave velocity identification and
calculated IRFs and TFs, a combination of four models are considered as follows:
1. LSQ model – No slab: This 9-layer model represents our fitted model into Millikan library’s
observed response (N-S component), presented in chapter 5. The mass of missing slabs is
distributed over the soft layers, such that the total mass of all models remains constant. Fig. 7.1b
illustrates this model. The estimated velocity vector of the beam is denoted by
LSQ
β .
2. LSQ model with post-inserted slabs: The slabs are considered as separate layers that are thin,
heavy and very stiff, and are separated by soft, thick layers. The soft layers have the same shear
wave velocity profile as our first non-slab fitted model, and have reduced mass such that the total
mass of model remains constant. The model is shown in Fig. 7.1c. The velocity profile of soft
layers remains the same as the non-slab model.
3. Adjusted model with slabs: This model resembles our second model, but the wave velocities in
the soft layers are increased uniformly by an adjusting factor such that the total wave travel time
is equal to that of observed one. Fig. 7.1d shows this model.
4. LSQ model with slabs: This model presents the identification results for a layered shear beam
of 9 soft layers and 9 slabs fitted into Millikan library’s recorded response in N-S direction. The
model is shown in Fig. 7.1e. The estimated velocity vector of the fitted model is denoted by
LSQ Slab −
β . The numerical values of the models are listed in table 7.1 and 7.2.

179

9-story Millikan library

Fitted shear beam – No slab
Velocity profile:
LSQ
β

Model at (b) with 9 inserted slabs,
Velocity profile:
LSQ
β
(a) (b) (c)
Roof Roof

Adjusted model at (b) with 9 slabs,
Adjusted velocity profile:
LSQ
α . β

Fitted shear beam model with 9 slabs,
Estimated velocity profile:
LSQ Slab −
β
(d) (e)

Illustration of shear beam models of Millikan library including floors’ slabs. a)
vertical section of building (E-W elevation), b) fitted layered shear beam model to N-S response
(results taken from chapter 5), c) the same model as b) including 9 post-inserted slabs, d) adjusted
model with slabs (adjusting factor, α ), e) fitted shear beam model including 9 slabs (a total of 18
layers)
Figure 7.1
39 m
Roof
0.3 m
3.95 m
39 m
Roof
0.3 m
3.95 m
39 m
0.3 m
3.95 m
39 m
4.25 m
GF
180

7.1.2 Results and Discussion
Table 7.1 and 7.2 summarize the models properties, identification results, and the “observed”
pulse arrival time at the top of each layer in the impulse responses, for our four models. Fig.
i
t
1105
7.2 through 7.7 illustrate the IRFs and TFs of the models. Fig. 7.2 presents the IRFs of Millikan
library’s N-S response and our first three models, LSQ model with no slab, and model with nine
post-inserted slabs. The figure shows that once the slabs are inserted between the soft layers of
the LSQ model, the primary pulse attenuates more as result of the slabs, but most importantly, it
arrives at the ground floor at time 0. t = s, which implies 13% longer travel time than the
sum of the travel times through the individual domains, which is 0.098 s. This observation
reveals that the ray theory does not give accurate estimation of the shear wave velocities of the
model with slabs.
An easy and fast treatment of the travel time elongation due to slabs is to “adjust” the
parameters of the original model with post-inserted slabs by increasing the velocities of the soft
layers by an adjusting factor of 1.128. Fig. 7.3 shows the impulse responses for the adjusted
model. It can be seen that the first arrival at the ground matches that for the non-slab LSQ model
and the observed response. On its way down, this pulse gets more and more deformed suggesting
dispersion due to scattering from the slabs.
Figure 7.4 and 7.5 present the transfer functions for the observed response and our first three
models. Fig. 7.4 reveals that the model with post-inserted slabs, with longer pulse travel times,
has noticeably different transfer-function than the other two, predicting smaller fundamental
fixed-base frequency. Fig. 7.5 shows that the TFs of the fitted non-slab model and the adjusted
model are very close. They are practically identical up to about 10 Hz, and fairly close up to
about 20 Hz. Hence, their predictions of the frequencies of the first two modes of vibration are
practically the same, and very close for the third modes. Furthermore, comparing the pulse arrival
181

times in table 7.1, it can be seen that the simple adjusted model predicts closely but not exactly
the pulse arrival times at all levels.
Our forth model, which presents a more accurate identification of Millikan library’s N-S
response, including the heavy and stiff slabs, requires fitting both travel time and amplitude of the
observed IRFs using a 9-layer shear beam with 9 deployed slabs (a total of 18 layers). The task is
carried out by the waveform inversion algorithm at all stories, simultaneously. The properties of
slabs are assumed to be constant ( and
3
2000kg/m
slab
ρ = 1000 m/s
slab
β = ), so only the shear
wave velocities of soft layers are estimated.
Figure 7.6 presents the IRFs of the recorded response during Yorba Linda 2002 earthquake
and the fitted LSQ models with and without the slabs (model #1 and #4). It can be seen that the
IRFs of the two models are coincident, even though the models are significantly different in terms
of the estimated velocity profile and the presence of slabs. Table 7.2 presents the identification
results for the models. The results of the LSQ model with slabs suggest higher values for shear
wave velocities of layers than the model with no slab. The table also reveals that the velocity
profile of the adjusted model with slabs is noticeably different at some layers (e.g. 6-th layer)
from the LSQ model with slabs. This is because the waveform inversion algorithm not only
accounts for the pulse arrival time, but also fits the amplitude of propagating pulses, which makes
this model more accurate than others models. Fig. 7.7 shows the TFs of the models. The fitted
LSQ model with and without slabs have practically identical TFs up to 10 Hz (first two modes),
similar to the results for the adjusted model.
These results, based on Millikan library’s response, suggest that the model with slabs
identified using waveform inversion of IRFs can provide us a more accurate estimate of true
floors’ shear wave velocities; however, if the effective frequency band of response is such that it
covers only the first two modes (e.g. Millikan N-S and E-W response), it does not make a
difference whether the slabs have been explicitly modeled. Nevertheless, this simple adjusting
182

can be used to identify quickly the shear wave velocities for a model with slabs. Note that the
Millikan library is a relatively stiff RC building with large density, plus additional dead loads due
to the book shelves. Moreover, the effective frequency band of recorded response (0~15 Hz)
covers only three modes of vibration. As a result, identifying a model for this building including
the slabs cannot reveal the asymptotic effects of modeling the slabs, particularly, at higher
frequencies. For this purpose, in the next section, we will investigate two analytical shear beams
that have significantly softer and lighter layers in comparison to the slabs. The effect of slabs will
be more pronounced in such a soft model.
183

Illustration of significant scattering phenomena in models with inserted slabs.
Comparison of shear wave velocities and pulse arrival times of Millikan Library NS response, as
identified using a fitted 8-layer model without the slabs, with post inserted slabs, and adjusted
model with slabs (0-15 Hz, 2% ζ = )
Table 7.1
Layer Floor
i
h
obs
i
t

[m]
Arrival
time

[s]
LSQ Model – No
slab
LSQ Model
with post-
inserted slabs
Adjusted
+
model
with slabs
mdl
i
t
mdl slab
i
t
−
i

[s]

[s]
i
β β

[m/s]

[m/s]
i
t

[s]
1 8+9 8.5 0.04 281 0.0355 0.0385 316.86 0.0365
2 7 4.25 0.048 226.6 0.0465 0.0545 255.5 0.0465
3 6 4.25 0.0535 360.8 0.056 0.0675 406.8 0.0575
4 5 4.25 0.0615 554.4 0.064 0.0725 625.1 0.065
5 4 4.25 0.073 516.6 0.0735 0.0835 582.5 0.0735
6 3 4.25 0.0785 638 0.0795 0.0925 719.4 0.0785
7 2 4.25 0.0835 835 0.0835 0.0975 941.5 0.082
8 Gnd 4.9 0.096 491.6 0.098 0.1105 554.3 0.099
+ The adjusting factor is 1.1276
Comparison of shear wave velocities of Millikan Library NS response during Yorba
Linda earthquake of 2002, as identified using 8-layer model with and without the slabs by
nonlinear LSQ and Adjusting of velocity profile (0-15 Hz,
Table 7.2
3.6% ζ = )
Layer
Floor
i
h

[m]
Arrival
time
obs
i
t [s]
Fitted LSQ model – No
slab
Fitted LSQ model with
9 slabs
Adjusted
model
i
β
[m/s]
/
β
σ β
[%]
i
t

[s]
i
β
[m/s]
/
β
σ β
[%]
i
t

[s]
i
β

[m/s]
1 8+9 8.5 0.04 281.4 0.8 0.0355 329 0.8 0.0355 317.3
2 7 4.25 0.048 230.2 1.6 0.046 249 1.6 0.046 259.6
3 6 4.25 0.0535 358.2 3.0 0.056 398 2.8 0.0565 403.9
4 5 4.25 0.0615 549.3 4.8 0.0645 600 4.7 0.065 619.4
5 4 4.25 0.073 522.4 4.2 0.0735 561 4.0 0.0745 589.1
6 3 4.25 0.0785 639.3 6.2 0.0795 639 6.2 0.080 720.9
7 2 4.25 0.0835 824 9.8 0.0835 916 11.9 0.0835 929.1
8 Gnd 4.9 0.096 490.2 3.5 0.098 573 3.5 0.099 552.7
184

Phenomena illustration - Comparison of observed IRFs of Millikan library during
Yorba Linda 2002 EQ with those of 1) identified models using LSQ algorithm without slabs, 2)
LSQ model with post-inserted slabs
Figure 7.2

185

Phenomena illustration - Comparison of observed IRFs of Millikan library during
Yorba Linda 2002 EQ with those of 1) identified models using LSQ algorithm without slabs, 2)
the Adjusted model with slabs
Figure 7.3

186

Comparison of observed TF of Millikan library during Yorba Linda 2002 EQ with
identified models using LSQ algorithm without slabs and model with post-inserted slabs
Figure 7.4

Comparison of observed TF of Millikan library during Yorba Linda 2002 EQ with
identified models using LSQ algorithm without slabs and the Adjusted model with slabs
Figure 7.5
187

Comparison of observed IRFs of Millikan library during Yorba Linda 2002 EQ
with those of identified models using LSQ algorithm with and without slabs
Figure 7.6

188

Comparison of observed TF of Millikan library (N-S) during Yorba Linda 2002
EQ with identified models using LSQ algorithm with and without slabs
Figure 7.7

189

7.2 Investigation of Wave Dispersion in a Model with Slabs
7.2.1 The Model
To better investigate the effect of modeling the slabs in our identification results and in our
calculated IRFs and TFs, we consider two analytical shear beam models as follows
1. The first model is a 9-layer beam with no slab. Fig. 7.8 presents the model which consists
of 9 layers with shear wave velocities ranging from 80 m/s at the top layer to 120 m/s at ground
floor. Each floor has a thickness of 3 m and density .
3
325kg/m
oor
=
fl
ρ
2. The second model has 9 soft layers, plus 9 stiff and heavy slabs. The soft layers have the
same height and shear wave velocity as our first model, but reduced density of
.such that the total mass of both models remain the same. Fig.
3
200kg/m
floor
ρ = 7.8 illustrates
our second analytical model with slabs. As it can be seen, the slabs are deployed between the soft
layers and have density and shear wave velocity
3
1500kg/m
slab
ρ = 1200m/s
slab
β = , and
thickness of 0.25 m.

(a) (b)
Roof
a) The 9-layer shear beam model, b) shear beam model with 9 added slabs at the
layers’ boundaries
Figure 7.8
29.25m
Ground floor
2
nd
floor
3
rd
floor
9
th
floor
0.25 m
3 m
Roof
9@3m
GF
190

7.2.2 Identification Results
Table 7.3 presents model properties and observed pulse arrival times for each model. The results
for IRFs and TFs are plotted in figures 7.9 through 7.15. Figure 7.9 shows that the IRF of the
second model is significantly distorted due to the inserted slabs, and the pulse arrival time is
noticeably larger (i.e. 0.39 s) as opposed to the travel time of model with no slab (i.e. 0.28 s).
Fig. 7.10 compares the TFs of the two models. As it can be seen, the TF of model with slabs has
smaller modal frequencies in comparison to the model with no slab, presenting a softer structure.
Also, the TF levels off to zero after its eighth mode at 9.2 Hz. This is due to the significant
contrast of the layers and the slabs, which turn the TF resemble that of a discrete lumped mass
shear-building model.

Illustration of significant scattering phenomena in models with inserted slabs.
Comparison of shear wave velocities and pulse arrival times of model with no slab, model with
inserted slabs, and two models with Adjusted velocity profiles. AF denotes the adjusting factor
for each layer. (0-9 Hz, 2% ζ = )
Table 7.3
Floor
i
h

[m]
9-layer model
with no slab
9-layer
model with
9 inserted
slabs
Adjusted model
with slabs
Case 1: Uniform adj.
Adjusted model
with slabs
Case 2: Local adj.
obs
i
t
[s]
obs
i
t
[s]
i
AF

obs
i
t
[s]
i
AF

i
β

[m/s]
i
β

[m/s]

i
β

[m/s]

obs
i
t
[s]
8+9 6 80 0.0715 0.1165 1.32 105.6 0.076 1.63 130.4 0.067
7 3 90 0.112 0.1675 1.32 118.8 0.116 1.26 113.4 0.103
6 3 90 0.138 0.2095 1.32 118.80.1451.615 145.40.123
5 3 100 0.1725 0.2405 1.32 132 0.176 0.9 90 0.172
4 3 100 0.199 0.284 1.32 132 0.2051.642 164.2 0.18
3 3 110 0.2285 0.324 1.32 145.20.2311.356 149.20.229
2 3 110 0.254 0.350 1.32 145.2 0.258 1.02 112.2 0.25
Gnd 3 120 0.2805 0.390 1.32 158.4 0.282 1.509 181.1 0.271

191

Figure 7.9 IRFs of models with and without slabs; Pulse arrival time elongation illustration

Comparison of TFs of models with and without slabs (Fig. 7.8) Figure 7.10
192

7.2.3 Analytical Dispersion for a Periodic Structure
Fukuwa and Matsushima (1994) studied wave propagation in 1D periodic structures in
which each cell consists of a shear beam segment with rigid mass at both ends. For a building
model with slabs, the mass of the slabs is divided between the cells above and below. In this
section, the wave velocities identified by the wave method presented in this dissertation are
compared against the analytical values derived for the periodic structure. The two limiting cases,
when the soft segment has no mass and when the lumped mass is zero correspond to a lumped
mass and uniform shear beam models.
Fukuwa and Matsushima (1994) derived the dispersion relation for the cell, which gave
analytical expressions for the phase and group velocities in the periodic structure. The expression
of the phase velocity is
phase
1
cos sin cos
2
VV
β
αβ
β β
−
=
⎛⎞
−+
⎜⎟
⎝⎠

(7.1)
where V is the shear wave velocity of the soft segment, and β and α are dimensionless
frequency and mass ratio defines as
2
,
f m
l
Vl
π
βα
ρ
==
(7.2)
In Eqn (7.2), l is the length of the soft segment (i.e. the floor height), and ρ is the mass density
of the soft layer, and m is the mass of the rigid part per unit cross-sectional area. Then 0 α =
corresponds to a uniform shear beam and α = ∞ corresponds to a lumped mass model. Eqn (7.1)
implies that
phase
V is function of frequency and can be real or complex. The bands where it is
real are called pass-bands, and where it is imaginary are called stop-bands. The latter have the
effect of large damping.
193

Let us divide the wave propagation in regions such that
( 1) , 0,1, 2,... jj j πβπ ≤< + = (7.3)
Each region starts with a pass-band and ends with a stop-band. Let’s rewrite eqn. (7.3) in terms of
frequency for the first region only ( 0 β π ≤ < ):
01/0
22
f
VV
fN
lN
N
≤< ÷ ≤ <
l
(7.4)
where N is the number of stories in the building. Recalling the relation between the fundamental
frequency of a uniform shear beam and total wave travel time (
1
1/ 4 f τ = ), eqn. (7.4) can be
written as:
1
02 f Nf ≤< (7.5)
Of interest for the identification method presented in this dissertation is the first pass-band.
Let us assume 1 α = . Then, using eqn. (7.5), a rule of thumb value for the pass-band region is:

1
0, for
pass
fNf 1 α ≤< =
(7.5)
The pass-bands become wider as α decreases. For the model of Millikan library NS response,
with 0.4, 400m/s V α== , the first stop-band starts at about 35 Hz, and the dispersion over 0-
15 Hz is very small.

194

7.2.4 Comparison of Identification Results with Analytical Predictions
It is of interest to compare identification results using the algorithms presented in this paper with
the analytical results predicted by the model of Fukuwa and Matsushima (1994). For a more
meaningful comparison, we analyze a model of a 9-story building that is uniform along the
height. The model has same dimensions as the model in Fig. 7.8b, and the shear wave velocity of
the soft layers is . The mass ratio of the model is 100m/s V = 0.6 α = .
Fig. 7.11 and 7.12 show the dispersion curves predicted by eqn. 7.1 in the first pass-band for
three mass ratios (the red lines). It can be seen that the phase velocity decreases with frequency,
but very gradually for lower frequencies, approaching a value for that is lower than the
shear wave velocity, by a factor that increases with increasing mass ratio
0 f →
α . For 0.6 α = ,
0.8, and the pass-bands ends at about 11 Hz. /
ph
VV ≈
The horizontal lines in Fig. 7.11 represent empirically measured as identified from
pulse time shifts for cut-off frequencies 4 Hz and 8 Hz (i.e. by the Direct algorithm). The lines
extend over the respective bands used for the identification. It can be seen that the measured
phase velocities in the two bands correctly indicate decreasing phase velocity with increasing
frequency.
ph
V
The dots in Fig. 7.12 represent empirically measured as identified from the resonant
frequencies of the beam (read from the transfer function) assuming that the model with slabs, for
lower frequencies, deforms approximately as a uniform shear beam. The relation between the
phase velocity and the frequency of the n-th mode (
ph
V
n
f ) is:
4
() , 1,2,
21
n
ph n
Hf
Vf n
n
≈=
−
… (7.6)
195

where
4
21
n
Hf
n −
is the velocity of a uniform fixed-base shear beam that has the same n-th mode
frequency as the dispersive slab model, and H is total height of the model. It can be seen that the
dots practically coincide with the analytical dispersion curve.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.4
0.5
0.6
0.7
0.8
0.9
1
Dispersion Curves for uniform shear beam with slabs
f (Hz)
V
phase
/ V

α=0.3
α=1
α=0.6
Fukuwa et al. 1994
Lowpass filtered models at
4 Hz and 8 Hz - This study
Dispersion curves for uniform shear beam with added lumped mass. Comparison
of analytical solution with a slab model. Velocities are estimated using low-pass filtered IRFs.
Figure 7.11
196

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.4
0.5
0.6
0.7
0.8
0.9
1
f (Hz)
V
phase
/ V
α=0.3
α=0.6
α=1
Wave velocities at
resonant frequencies of
model - This study
Fukuwa et al. 1994
The same as Fig. 7.11, but the wave velocities are estimated using resonance
frequencies of the slab model.
Figure 7.12

7.2.5 Effect of the Stop Band on the Model Impulse Responses
It is of interest how approaching the stop band or including part of it affects the impulse response
functions of a building. To investigate this, we analyze low-pass IRFs for the model with slabs
(see Section 7.2.1) in two bands, one of which has the cut-off frequency entirely within the pass
band (
max
f = 8 Hz), while the other one has cut-off frequency near the beginning of the first stop
band (
max
f = 10 Hz), where the dispersion is significant (see Fig. 7.11) and the transfer-function
amplitudes are small. Figure 7.13 shows the corresponding impulse response functions. It can
be seen from Fig. 7.13 that the source pulse is propagating practically undispersed in the first
band, while it becomes significantly dispersed in the second band (0-10 Hz) as the
max
f gets
very close to the stop-band edge (see Fig. 7.13).

197

IRFs of model with slabs for different cut-off frequencies,
max
f Figure 7.13

7.2.6 Assessment of the Goodness of the Adjusted Model
In the first section, for the case of Millikan library, it was shown that an adjusted model with
slabs can closely mimic the IRFs and TFs of a model with no slabs. Here, we again adjust our
slab model shear wave velocities in two different ways. In the first approach, we uniformly
increase the shear wave velocities of all layers by an adjusting factor (AF). This factor (i.e. 1.32)
is found by trial and error such that the pulse arrival time at the base of the model with slab is
identical to the non-slab model. In the second approach, the layers are adjusted locally using their
local factors. These local adjusting factors are determined by dividing the layers’ domain travel
times observed from IRFs of the two models. The results are presented in Fig. 7.14 and 7.15.
198

Comparison of IRFs of model with no slab and the adjusted models with slabs Figure 7.14

Comparison of TFs of model with no slab and the adjusted models with slabs
Figure 7.15

199

Figure 7.14 illustrates the IRFs of the two adjusted models and the model with no slab. It
can be seen that the uniformly adjusted model with slab is almost identical to the model with no
slab. Note that, the slab model and the model with no slab are slightly different in their total mass
and height (see Fig. 7.8), so that small mismatch in IRFs and TFs is expected. In contrast to the
uniformly adjusted model, IRFs of locally adjusted model do not match that of non-slab model.
Herein, we conclude that the local adjustment is not an acceptable approach to modify a model
with slabs, so that it can mimic the non-slab model’s response. Fig. 7.15 illustrates the TFs of the
adjusted models and the non-slab model. The results show that the TF of uniformly adjusted
model is coincident with no-slab model up our cut-off frequency, 9 Hz. These results again
confirm that this simple adjusting is capable enough to identify rapidly the shear wave velocities
for a model with slabs.
In summary, we have assessed the appropriateness of ray theory in identifying the building
shear wave velocities from observed response by including the slabs in our models. Considering
the case of Millikan library, it is found that ray theory is not appropriate for the identification of
the model with 9 slabs. For the model with slabs, the actual total pulse travel time to the base is
larger than the sum of the domain travel times, which is attributed to additional phase delays due
to wave scattering from the slabs. It is concluded that the identification of the shear wave
velocity distribution for higher resolution models requires rigorous model fitting (i.e. waveform
inversion of model including the slabs). An alternative identification approach to the LSQ model
fitting with slabs is the model adjusting. In this approach, and approximation can be quickly
obtained by scaling (increasing) the wave velocities in the soft layers uniformly by an adjusting
factor (AF). The results show that the adjusted model with slabs predicted very closely the pulse
arrival time at the individual floors and also the total travel time to the base.
The transfer-functions of the identified models are also analyzed and compared. It was
found that the TF of the fitted LSQ model with slab coincides with the non-slab fitted model. It is
200

also shown that the simple adjusted model with slabs can closely mimic the TF of the fitted
model with and without slabs. Such an agreement in the effective frequency band (
max
0 f − )
implies that the non-slab model is able to accurately estimate the velocity profile of our model.
Although the estimated velocities are uniformly smaller (by a constant factor) than the velocities
in a more realistic model with slabs, the relative change in the floors’ velocities, which is our
damage/again indicator, will remain the same in both scenarios. Hence, in this dissertation, we
chose to use layered shear beam with no slab for the identification purposes, provided that the
chosen effective frequency band is such that it only covers the first few dominant modes of
vibration (e.g. three modes for Millikan library, N-S response).

201

7.3 Summary and Conclusion
The effect of modeling slabs as separate layers was examined and discussed in this chapter.
First, we studied this effect on the identification of Millikan library NS response during Yorba
Linda, 2002 earthquake. It is concluded that (1) the Ray theory interpretation of the wave
propagation in the model is not appropriate for a model with slabs. For the model with slabs, the
actual total pulse travel time to the base is larger than the sum of the domain travel times, which
is attributed to additional phase delays due to wave scattering from the slabs. (2) Also, it is
concluded that the model with added slabs is dispersive. The level of dispersion is examined
through a numerical model and compared with analytical derivations by Fukuwa and Mitsushima
(1994).
An alternative identification approach to the LSQ model fitting is the model adjusting. The
results show that (3) the adjusted model with slabs predicted very closely the pulse arrival time at
the individual floors and also the total travel time to the base. (4) It was found that the TF of the
fitted LSQ model with slab coincides with the non-slab fitted model. It is also shown that the
simple adjusted model with slabs can closely mimic the TF of the fitted model with and without
slabs. (5) Such an agreement in the effective frequency band (
max
0 f − ) implies that the non-
slab model is able to accurately estimate the velocity profile of our model. Although the estimated
velocities are uniformly smaller (by a constant factor) than the velocities in a more realistic model
with slabs, the relative change in the floors’ velocities, which is our damage/again indicator, will
remain the same in both scenarios.
202

Chapter 8
Dispersion due to Soil-Structure Interaction and
Why the Method is not Sensitive to the SSI
Effects
In this chapter, nonparametric techniques for estimation of wave dispersion in buildings by
seismic interferometry are applied to a simple model of a soil-structure interaction (SSI) system
with coupled horizontal and rocking response. The system consists of a viscously damped shear-
beam, representing a building, on a rigid foundation embedded in a half-space. The analysis
shows that (1) wave propagation through the system is dispersive. The dispersion is characterized
by lower phase velocity (softening) in the band containing the fundamental system mode of
vibration, and little change in the higher frequency bands, relative to the building shear wave
velocity. This mirrors its well known effect on the frequencies of vibration, i.e. reduction for the
fundamental mode (softening) and no significant change for the higher modes of vibration, in
agreement with the duality of the wave and vibrational nature of structural response.
Nevertheless, the phase velocity identified from broader band IRFs is very close to the
superstructure shear wave velocity, as found by the earlier study. The analysis reveals that (2) the
reason for this apparent paradox is that the latter estimates are biased towards the higher values,
representative of the higher frequencies in the band, where the response is less affected by SSI. It
is also discussed that (3) bending flexibility and soil flexibility produce similar effects on the
phase velocities and frequencies of vibration of a building.
203

This Chapter is based on the following technical note:
Rahmani M, Ebrahimian M, Todorovska MI (2014). Wave dispersion in high-rise buildings due
to soil-structure interaction, Earthq. Eng. Struct. Dyn., DOI: 10.1002/eqe.2454.

8.1 Interferometric Identification of a Soil-Structure
Interaction Model
In this section, we revisit the problem of interferometric identification of a soil structure
interaction (SSI) model, studied earlier by Todorovska (2009a), to explain why the identified
wave velocity is not affected by the SSI. We also discuss the feasibility of estimation of the true
fixed-base frequency of the superstructure, from recorded horizontal motions only, when the
response is affected simultaneously by bending and soil-structure interaction.
Todorovska (2009a) analyzed pulse propagation in simple linear SSI model, with shear beam
as a structure and with coupled horizontal and rocking response, by following pulses in low-pass
filtered impulse response functions at different locations in the structure. The study demonstrated
that, if the IRFs are broader-band, the pulse wave travel time through the structure, τ , and,
consequently, the wave velocity / cH τ = , where H is the height, are not sensitive to the effects
of SSI. In contrast, the TF are affected (Luco et al. 1988). A subsequent analysis (Todorovska,
2009b) of 1/ (4 ) τ , used as proxy for the building fixed-base frequency of a building, explained to
what degree the observed wandering of the NS apparent frequency of Millikan library was caused
by changes in the structure as opposed to changes in the soil.
In this section, we explain why the pulse time shift in broader-band impulse response
functions is not sensitive to the SSI effects, through analysis of impulse response functions of the
same SSI model but in subbands, and fitting with the waveform inversion of IRFs. The previous
explanations have been only speculative. We also discuss the feasibility to measure the true fixed-
base frequency when both SSI and bending are present.
204

8.1.1 Models and Parameters
Without loss of generality, the SSI model consists of a shear beam supported by a semi-circular
foundation embedded in uniform elastic half-space and excited by incident wave (see Fig. 8.1a,
left). The foundation has three degrees of freedom, horizontal and vertical translations and
rotation in the x-z plane. In the linear solution, the horizontal and rocking motions are coupled,
while the vertical motion is uncoupled. The model impedances and derivation of the solution can
be found in (Todorovska, 2009a). The model dimensions and material properties were chosen to
match approximately the NS response of Millikan Library to the Yorba Linda, 2002 earthquake.
Of interest for this study are the height of the shear beam, H = 44 m, and its shear wave velocity,
=457.6 m/s. The latter was obtained based on fixed-base frequency
S
c
1
f = 2.6 Hz obtained by
trial and error to that the model apparent frequency matches approximately the observed apparent
frequency, for shear wave velocity in the soil of 300 m/s.
The SSI model is used to produce “observed” absolute response of the building, (; ) uz ω ,
where corresponds to the top, and transfer-function (TF) 0 z =
(; )
ˆ
uz
(,0; )
(0; )
hz
u
ω
ω
ω
=
(8.1)
from which “observed” band-pass filtered and regularized IRFs are computed, for virtual source
at the roof. The IRFs of the fitted model, which is a fixed base uniform shear beam (see Fig.
8.1a, right), are calculated in a similar fashion.
Fig. 8.1b shows comparison of the SSI model and fixed-base superstructure TFs, and of the
corresponding IRFs at roof and ground floor levels, low-pass filtered at 15 Hz. The filtered IRFs
look very similar to the observed ones during the Yorba Linda earthquake of 2002 over the entire
band of the records (0-25 Hz), because the building, which is very stiff in the NS direction, itself
acts as a filter of the higher frequencies. It can be seen that the SSI model and fixed base
205

superstructure TFs differ significantly in the frequency and damping of the first mode, which is
shifted towards lower frequencies and is more damped in the SSI model response, due to soil-
structure interaction through foundation rocking. In contrast, the time shifts of the pulses are very
close.
Wave velocity is identified by the waveform inversion algorithm, which minimizes the
least squares error between the observed and fitted model IRFs at ground level within chosen
time windows (details of fitting windows length is discussed in chapter 2). In addition, wave
velocity was identified from readings of the observed modal frequencies, based on the
relationship between the resonant frequencies of a fixed-base uniform shear beam and its shear
wave velocity. This relationship gives
LSQ
eq
c
res
c
11
() 4 cf Hf = ,
22
() 4 /3 cf Hf = , etc.,
where
33
()4 /5 cf Hf =
i
f is the modal frequency.
8.1.2 Analysis and Results
Table 8.1 summarizes the results of fitting a fixed-base uniform shear beam in the SSI model
response. Identified shear wave velocities of models, , are found by least squares fit of
low-pass filtered IRFs in three bands containing the first one, two and three modes of vibration
(left; 0-5 Hz, 4.5-10.4 Hz and 10.5-16.5 Hz), and by least squares fit of band-pass filtered IRFs
for three subbands (right; 0-5 Hz, 0-11 Hz and 0-15 Hz), and their comparison with the
theoretical velocity of the superstructure, . In this table, is the initial value used for fitting
that is an estimate found directly from IRFs by measuring pulse travel time from peaks of the
IRFs at ground level. Similarly, Table 8.2 shows the identified values from the apparent
frequencies, , for the first three modes and a comparison with . The results are shown
graphically in Figs
LSQ
eq
c
D
eq
c
S
c
()
res
i
cf
S
c
8.2 and 8.3.

206

a) Sketches of the soil-structure interaction model used to generate observed
response (left) and of the model that is fitted (right). b) Impulse response functions, for virtual
source at roof (left), and transfer functions, w.r.t. base motion (right) of the SSI model and of its
superstructure if it were fixed-base.
Figure 8.1

207

Fig. 8.2 shows the results of fitting IRFs in low-pass bands. Part a) shows comparison of the
SSI model and fitted model IRFs for the broadest band (0-15 Hz) and the corresponding TFs. The
time windows as well as the frequency band for the fit are shown. It can be seen that, for this
band, the fitted IRFs and TFs are very close to the fixed base superstructure (see Fig. 8.1b). In
part b), the identified velocities over the low-pass bands, , are compared with the
theoretical velocity of the superstructure, (the bars indicate the extent of the bands). It can be
seen that, over the band that contains only the first mode, which is most affected by SSI, the
identified velocity is much smaller than , by 28%, while, over the broader bands, which
contain also additional mode(s), which are less affected by SSI, the identified velocities are very
close to , within 7%.
LSQ
eq
c
S
c
c
S
S
c
Similarly, Fig. 8.3 shows the results of the identification in three narrower subbands. It can
be seen that, for all three subbands, the IRFs of the fitted models are very close to the observed
IRFs. The TFs are also close, in terms of agreement of the modal frequencies within the subband
over which the IRFs were fitted, but not necessarily elsewhere. It can be seen that, in the higher
subband, is within 1% from , while, in the first subband, it is 28% smaller than .
Further, the estimates are shown, plotted as dots at
LSQ
eq
c
S
c
S
c
()
res
i
cf
i
f f = (Table 8.2). It can be seen
that is 36% smaller than .
1
() f
res
c
S
c
The plot of identified velocities in subbands in Fig. 8.3b constitutes experimentally
measured dispersion of the SSI model. It is seen that wave propagation in the SSI model is
dispersive, and that the system softening is reflected also in the reduced pulse velocity near the
fundamental mode frequency, which is the modal frequency that is most affected by the SSI. For
that reason, the pulse travel time in a band that contains only the fundamental mode is affected by
the SSI, as well as corresponding estimates of wave velocity, as reported earlier (Todorovska,
2009a).
208

The pale curve in the background in Figs 8.2b and 8.3b represents the phase velocity
of the first wave mode of a fixed-base Timoshenko beam model of the NS response of Millikan
library (Ebrahimian and Todorovska, 2013; Ebrahimian et al. 2014). In contrast to the shear
beam model, Timoshenko beam model accounts for deformation of a building due to both shear
and bending flexibility (Boutin et al. 2005). Comparison with the wave velocities of the SSI
system and its shear beam superstructure shows that flexibility in flexure, in addition to shear,
and SSI affect similarly the phase velocity, both causing softening in the band around the
fundamental mode. In terms of phase velocity, the softening is manifested by , and,
in terms of the modal frequencies, it is manifested by increase in the ratio
,
1
TB ph
c
2
)
1
() ( cf cf <
21
/ f f , which for shear
beam superstructure becomes greater than 3. Because of this property, neither the nature nor the
severity of the SSI effects for a full-scale building can be assessed solely from observed ratio
2
/
1
f f or estimated ratio .
21
()/ ( ) cf cf

Identified wave velocity in the SSI system from low-pass and band-pass filtered
impulse response functions in different bands ( are estimates obtained directly from pulse
travel time and and
D
eq
c
LSQ
eq
c σ are waveform inversion estimates and the corresponding standard
deviation). Comparison with the true velocity of the superstructure is also shown.
S
c
Table 8.1
Low pass filtered IRFs, = 457.6 m/s
S
c
S
c Band-pass filtered IRFs, = 457.6 m/s
Band
[Hz]
D
eq
c
[m/s]
LSQ
eq
c
[m/s]
LSQ
eq
c
σ

LSQ
eq
S
c
c

Band
[Hz]
D
eq
c
[m/s]
LSQ
eq
c
[m/s]
LSQ
eq
c
σ

LSQ
eq
S
c
c

0-5 345.1 328.5 0.7% 0.72 0-5 345.1 328.5 0.7% 0.72
0-11 440 424.3 1.0% 0.93 4.5-10.4 440 454 0.5% 0.99
0-15 440 443 0.9% 0.97 10.5-16.5 463.2 459.4 0.9% 1.00

Identified wave velocities, , of the SSI system from the system
frequencies,
()
res
i
cf
i
f , and comparison with the true shear wave velocity of the superstructure, .
S
c
Table 8.2

209

Mode
No.
i
f
[Hz]
()
res
i
cf
[m/s]
s
c
[m/s]

()
res
i
S
cf
c

1 1.66 292.2 457.6 0.64
2 7.7 451.7 457.6 0.99
3 13.1 461.1 457.6 1.01

210

Results of identification of the SSI model by fitting low-pass filtered impulse
response functions. a) Comparison of the observed and fitted model impulse response functions
(left) and transfer-functions (right) for fit in the band 0 - 15 Hz. b) Comparison of the identified
velocities, , in three low-pass bands with the true superstructure velocity, . The phase
velocity of an approximate Timoshenko beam model of the same building is shown in
the background (redrawn from Ebrahimian et al. (2014)).
LSQ
eq
c
S
c
ph ,
1
TB
c
Figure 8.2

211

Same as Fig. 8.2 but for identification by fitting band-pass filtered impulse
response functions. The dots in part b) show estimates of wave velocity from the SSI model
apparent frequencies.
Figure 8.3
212

8.2 Discussion and Conclusion
The presented analysis of the SSI model in Fig. 8.1 showed that (1) wave propagation in a shear-
beam building model on flexible soil is dispersive when foundation rocking is present, coupled
with the foundation horizontal motion. (2) The identified wave velocity from broader-band IRFs
of recorded horizontal response is less affected by SSI because it is biased towards the values
representative of the higher frequencies of the band, which are less affected by SSI. For the SSI
model in Fig. 8.1, in which the superstructure is a shear beam, the identified phase velocity is
close to the beam shear wave velocity. Further, (3) because the shear beam is lightly damped,
and, therefore, the wave dispersion due to material damping is small, extrapolation of the fitted
model to lower frequencies predicts closely the actual fixed-base properties in that band,
including the fundamental fixed-base frequency.
The results confirm that, if a building behaves closely to a uniform shear beam, its
fundamental fixed-base frequency can be obtained only from a pair of recorded horizontal
response, at base and roof. Otherwise, only a proxy is obtained by fitting a uniform shear beam.
It is of interest if a closer proxy to the true value can be obtained by applying the same concept
(i.e. fitting a model on a band in which the response is less sensitive to SSI and then extrapolating
the model outside that band) with a more realistic model. Exploratory analyses, involving fitting
uniform and layered Timoshenko beam models, can be found in recent conference papers
(Ebrahimian and Todorovska, 2014).

213

214

Chapter 9
Modeling Error Due to Bending Deformation
The identification method presented in this dissertation was based on the assumption that the
building deforms in shear only, and that the wave propagation is predominantly one-dimensional,
vertically through the structure. While these assumptions may be appropriate for structures that
behave like shear beam (e.g. moment frame buildings), and for exploratory data analyses and
initial method development, in general, their violation leads to modeling error. In this chapter,
the modeling error incurred by ignoring the bending deformation is investigated. For this
purpose, uniform Timoshenko beam models of Millikan Library NS and EW response are used to
generate “observed” response, which is used as input in identification by fitting a layered shear
beam. The least squares fit algorithm is used for the fit (see Chapter 2.2.2). The distortions in the
identified velocity profiles are analyzed and the modeling error is measured. The modeling error
may be relevant for inferences on the spatial distribution of damage from detected changes in
such velocity profiles. Wave propagation in Timoshenko beam building models have been
analyzed by Ebrahimian and Todorovska (2014).
This chapter is based on materials presented in:
Ebrahimian M, Rahmani M, Todorovska MI (2014). Nonparametric identification of wave
dispersion in high-rise buildings by seismic interferometry, Earthq. Eng. Struct. Dyn., DOI:
10.1002/eqe.2453.

215

9.1 Benchmark Timoshenko Beam Models
Lateral force resisting system of the Millikan library is a combination of a moment resisting
frames and shear walls. Because of the added shear stiffness by these walls, the relative
contribution of bending deformation to the total response is larger than what it would have been
otherwise.
Table 9.1 shows the parameters of approximate, fixed-base, uniform Timoshenko beam
models of the NS and EW responses, obtained based on the building geometry and acceleration
records of Yorba Linda earthquake of 2002 (M=4.8, R=40 km) (Ebrahimian et al. 2014). The
beam height, H , was chosen based on the roof height above ground level, where the highest
sensor is located. The radius of gyration
g
r was estimated based on the dimensions of the
building cross-section (W ). For the damping constant, μ , small reasonable values were assumed
(Ebrahimian et al. 2014). The beam’s material is characterized by mass density ρ, Young’s
modulus and shear modulus . The shear wave velocity, E G /
S
cG ρ = , and ratio / RG E = ,
were determined by matching the frequencies of the first two modes, after correcting the observed
(apparent) frequency of the first mode for SSI effects, based on findings of Luco et al. (1988).
The target fixed-base fundamental mode frequency of the TB model was assigned value that is
higher than the observed apparent frequency by 22% for the NS and 12% for the EW responses.
No adjustment was made to the observed second mode frequency.
Note that the shear stress on the section of the beam, which must be zero on the lateral
boundaries, is non-uniform. Therefore, the shear wave velocity of the Timoshenko beam has to be
adjusted by a shear factor k (Cowper, 1966; Ebrahimian and Todorovska, 2014). Table 9.1 also
shows the shear wave velocity in the beam,
G
TB
S
c
SG
ck = , where = 5/6, assuming uniform
G
k
216

distribution of rigidity over the rectangular cross-section (Mead, 1985). We refer to these models
(table 9.1) as TB model NS and TB model EW.

Table 9.1 Properties of an approximate uniform TB model of Millikan library NS and EW fix-
base responses (Ebrahimian et al. 2014).
S
c
TB
S
c

[m/s]

[m/s]
μ
H
[m]
W
[m]
TB
Model
G
R
E
=

[s]
NS 21 39 716 653 0.42 0.002
EW 23 39 526 480 0.82 0.003

9.2 Modeling Error in Fitting a Layered Shear-Beam
In this numerical experiment, a 3-layer shear-beam model is fitted in the response of the TB by
matching the causal and acausal pulses in broader-band IRFs at several levels. The phase
velocities were estimated directly from the pulse time shifts, and also by the waveform inversion
algorithm, which fits the pulses as waveforms over selected time intervals. The corresponding
estimates are indicated as
D
j
c and
LSQ
j
c , where j is the order of the layer measured from the
top. Quality factor was assumed (equivalent to 2% damping ratio). The results are
shown in Table 9.2 (
50 Q =
j
σ in this table is the standard deviation of the fit in layer j ). It is seen that
the profile of the fitted model shows decreasing shear wave velocity towards the top, even though
the structure that is identified has uniform properties. The distortion is the largest in the top
layer, where it is a factor of 2 (table 9.2). This distortion occurs because the layered shear beam
is compensating for the progressively larger deflection of the TB towards the top by the
progressively lower shear wave velocity.
217

Table 9.2 Identified shear wave velocities,
D
j
c and
LSQ
j
c , of an equivalent 3-layer shear beam
fitted in the response of uniform TB model NS (top) and EW (bottom) by the direct and least
squares fit algorithms.
TB
S
c ζ =2%; TB Model: NS, 0-15 Hz, = 653 m/s

Layer
No.
Floors
D L
.

Fig. 9.1 shows a comparison of the observed (TB) and fitted model (layered shear beam)
IRFs at four levels (left), transfer-functions between roof and ground floor motions (center), and
mode shapes of the first and second mode of vibration (right). The plot of the TFs shows that
there is a mismatch of the frequencies of vibration, even though the pulse time shifts match
closely. This demonstrates that, if the bending behavior of the structure that is identified is
significant, the fitted layered shear beam cannot produce a good match of both IRFs and TFs.
Then, the agreement of the TFs can be used as an indicator of the significance of bending for the
structure that is identified.
j
h
[m]
j
τ
[s]

D
j
c
[m/s]
j
TB
S
c
c

LSQ
j
c
[m/s]
j
σ

[m/s]
/
LSQ
j j
c σ
[%]
SQ
j
TB
S
c
c
1 Roof-7 12.8 0.0425 301 0.5 250 7.8 3.1 0.4
2 7-4 12.8 0.0250 512 0.8 578 46.3 8 0.9
3 4-GF 13.4 0.0175 766 1.2 669 67.1 10 1.0
TB Model: EW, 0-7.5 Hz,
TB
S
c ζ =2%; = 480 m/s

Layer Floors
D L
j
h
[m]
j
τ
[s]

D
j
c
[m/s]
j
TB
S
c
c

LSQ
j
c
[m/s]
j
σ

[m/s]
/
LSQ
j j
c σ
[%]
SQ
j
TB
S
c
c
1 Roof-7 12.8 0.083 154 0.3 129 4 3.1 0.3
2 7-4 12.8 0.033 386 0.8 344 30 8.7 0.7
3 4-GF 13.4 0.033 404 0.85 422 40.8 9.7 0.9
218

Figure 9.1 Assessment of modeling error in identified equivalent 3-layer shear-beam velocity
profile fitted in low-pass filtered IRFs of TB model NS (top) and TB model EW (bottom) of
Millikan Library (Table 4). Comparison of fitted and observed impulse responses (left), transfer-
function (center) and mode-shapes (right).

219

220

9.3 Conclusion
The main finding of this study is that ignoring bending in interferometric identification of
buildings from propagating pulse produces artificial softening in the identified shear velocity
profile near the top. This is because the layered shear beam is compensating for the progressively
larger deflection of the TB towards the top by the relatively lower shear wave velocity. Despite
the distortion of the identified velocity profile, the layering structure of a shear beam can still be
used to monitor change in stiffness in different parts of the structure. However, further
investigation is needed to find the possible effects of the distortion on the accuracy of the result.

Chapter 10
Summary and Conclusion
This chapter presents a brief summary of the important findings of this dissertation research and
presents the general conclusions of the study. It also presents suggestions for future work on the
wave method for structural system identification and health monitoring. Detailed discussions of
the findings and conclusions of the individual chapter are presented at the end of each chapter.
This chapter is intended to briefly highlight the most important findings along with our
recommendations for future studies.

221

10.1 Summary of Major Findings and Conclusions
This dissertation presented a methodology for structural system identification and health
monitoring of high rise buildings based on a layered shear beam model. The layers comprise in
general of a group of floors, but they can also represent individual floors and the floor slabs
(modeled as thin, heavy and stiff layers). The model is fitted by matching traveling pulses in low-
pass filtered the impulse response functions obtained from recorded acceleration response during
an earthquake. It identifies an equivalent shear beam velocity profile and monitors its changes in
time, during a stronger earthquake or between smaller events.
In the methodology section (Chapter 2), analytical recursive expressions for the model
transfer-functions and low pass filtered impulse response functions are presented, three fitting
algorithms and an automated moving window algorithm, which produces time-velocity graphs.
Two of the algorithms (ray) match only the pulse time shifts, while the third one (wave) matches
the pulse amplitudes as waveforms, i.e. their shape and amplitudes, in the least squares (LSQ)
sense, simultaneously at all levels where motion was recorded. The Levenberg-Marquardt method
for nonlinear least squares estimation was used for the fit, with initial condition that is the fit by
the simpler ray algorithm.
Results are presented for three full-scale buildings (Chapters 4, 5 and 6), which were
selected for detailed analysis out of about 15 buildings, for which preliminary identification
analysis was performed by the simpler ray algorithm. One of the case studies is a tall steel frame
building (Los Angeles 54-story Office Building), for which records of six earthquakes were used,
none of which produced observed damage. The second case study is a densely instrumented 9-
story RC building with shear walls and central core (Millikan Library), for which small amplitude
records of one earthquake were used. The third case study is a 12 story RC building (Sherman
Oaks 12-story Office Building) damaged by the San Fernando earthquake of 1971. In addition to
the full-scale buildings, models of a building were also analyzed (Chapters 3, 7, 8 and 9). The
222

models are: a layered shear beam in which the floor slabs are part of the softer layers, a shear
beam with the floor slabs considered explicitly as separate thin, stiff and heavy layers, a uniform
shear beam as part of a soil-structure interaction (SSI) model with coupled horizontal and rocking
response, and a Timoshenko beam model identified by fitting a layered shear beam model.
The analyses of models and full-scale buildings provided valuable insight into the nature of
wave propagation in real buildings, and the capabilities and limitations of the method. The major
findings are summarized as follows.
1. The layered shear beam model, which views the building as continuum, is meaningful only
for longer wavelengths, which are less sensitive to the inhomogeneities in the structure.
Therefore, it is necessary to low-pass filter both the observed and model impulse response
functions before fitting. The choice of the filter is perhaps the most important parameter for
the fit, which should be chosen carefully. Lowering the cut-off frequency
max
f helps control
the effects of dispersion, which is not accounted for by the model, except for dispersion due
to material damping, which is relatively small). The cut-off frequency also controls the spatial
resolution and accuracy of the identification, which are higher for higher cut-off frequency
(i.e. larger bandwidth of the signals). Therefore, some optimal value of
max
f should be
chosen for a particular building.
2. The ray theory interpretation of wave propagation, which has been used successfully to solve
many problems in seismology and geophysics, has to be used with care for buildings. As
shown in Chapter 7, it is not appropriate for a building model in which the slabs are
considered explicitly. According to this simplified theory of wave propagation, which ignores
scattering, the total wave travel time from base to roof would be essentially the sum of travel
times through the soft layers. The actual travel time however is longer, because of delays due
to scattering from the slabs. Constructing models with slabs for forward analysis, and
identification based on such models must consider wave scattering from the slabs.
223

3. The ray theory interpretation of wave propagation in buildings is appropriate when the model
consists of soft layers only (representing groups of floors) and the impedance contrast
between the layers is not large, as in the case of a model with slabs. Therefore, the building
can be identified directly from readings of the pulse time shifts, i.e. the travel times, and the
distances travelled (referred to in this dissertation as “direct algorithm” or “simple ray
algorithm”). However, the accuracy of the identification by this simple ray algorithm may
not be sufficient for buildings, in which the distances traveled are relatively short compared
to distances in seismology and geophysics, and there is a need for high accuracy of the
identification. As shown in Chapter 3, there is a trade-off between accuracy and resolution of
the ray algorithm, for a given cut-off frequency, and the accuracy is poor for models with
thinner layers, which may be the choice for a densely instrumented structure, and/or stiffer
structures. This phenomenon was explained by the Heisenberg-Gabor uncertainty principle
for signals, and is contrary to the common belief that the accuracy of the identification by
picking the time of the pulses is controlled by the sampling rate only.
4. The least squares fitting algorithm was shown to have superior accuracy, in comparison with
the simpler ray algorithms, and should be used for SHM of buildings. The Levenberg-
Marquardt method for least squares estimation, which is fast but requires good initial
estimates in order to converge, was shown to converge for all three full-scale case studies,
with the initial estimates obtained by the simplest ray algorithm
5. As shown in Chapter 8 on an SSI model, wave propagation through the SSI system is
dispersive. However, if the band of the fit is sufficiently wide (includes at least the second
mode of vibration), the identified wave velocity profile of the building by this method is not
sensitive to the effects of soil-structure interaction even in the more general case when
foundation rocking is present. Therefore, the detected changes will not be sensitive to
nonlinear soil-response. The estimated damping, however, is sensitive to soil-structure
224

interaction, and depends on the damping in the structure and the radiation damping associated
with foundation rocking. Therefore, the structural damping cannot be estimated by this
method, unless the radiation damping due to rocking can be proven to be negligible.
6. The results for Los Angeles 54-story building (Chapter 4) showed that the layered shear beam
is a very good physical model for tall Moment Resistant Frame (MRF) buildings. The
analysis revealed permanent reduction of the structural stiffness of about 10% even though
the building had shown no signs of damage, demonstrating high accuracy and sensitivity of
the identification.
7. The analysis of Sherman Oaks 12-story Office Building (Chapter 6) showed that the
automated time velocity algorithm is robust even when applied to a damaged building, and is
sensitive even to light damage of RC buildings.
8. The analysis of Millikan library (Chapter 5) showed that wave propagation in this building is
dispersive, in particular for the EW response. Nevertheless, a layered shear beam could be
fitted by an appropriate choice of frequency band. Further, aliasing errors were discussed
which occur when the layer thickness is small compared to the data bandwidth.
9. Finally, the analysis of modeling error (Chapter 9) showed that, in structures with shear walls,
which are affected by bending deformation, the identified velocity profile is distorted in the
sense that is suggests lower shear wave velocity (softer structure) towards the top.

It is concluded that the layered shear beam model and the interferometric identification
provided valuable insight into the nature of wave propagation in real buildings useful for further
development of the wave SHM method. Within carefully chosen frequency bands, it can be used
for SHM of many high rise buildings. The simplicity of the equations of motion and the minimum
number of stiffness related parameters that need to be identified makes it a natural first choice of
model to fit.
225

226

10.2 Recommendations for Future Work
The method can be further improved by extending it to fitting more realistic models, i.e.
such that account for wave dispersion, for coupling of the different motions and for modeling the
details of the structure, which will account for the wave dispersion at higher frequencies. Fitting
such models would make it possible to extend the band of the fit, which would improve the
spatial resolution of the identification and the capability to detect more localized and light
damage. Eventually, Finite Element model of the structure can be fitted in a similar fashion and
updated by iteratively matching impulse response functions. Further, the method should be
extended to include a probabilistic decision system for issuing an alarm. Some other suggestions
for future studies are as follows.

Effect of Response Coupling
It is also important to understand the effect of coupling between different components of
motion on our identification results. For example, the transverse direction of buildings with
rectangle plan (e.g. Torre Central Building, Chile) significant torsional response is seen in.
The Effect of Sensor Placement
There are stations where the vibrational sensors are located at various locations at different
levels, and not vertically lined up. It is important to examine the effect of sensor placement on the
identification results and their wandering.
Extending the Application of Method to Other Infrastructures
The wave-based methodology can be extended for identification and possibly damage/aging
detection of infrastructures, such bridges, offshore platforms, dams, etc.

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About the Author

Mehran Rahmani was born and raised in Mahmood-Abad, a coastal
city by the Caspian Sea in the northern province of Mazandaran, Iran.
Passing the national universities entrance exam, he entered University
of Mazandaran (Noushirvani) in 2002 as a civil engineering major with
an emphasis in structural engineering. Graduating as an honor student
with the highest GPA in the class of 2006, Mehran succeeded to enter
Master’s program in structural engineering in the Sharif University of
Technology, Tehran, Iran. His Master’s thesis was on the assessment
of soil-structure interaction (SSI) effect in buildings with embedded foundation. The research
provided alternative solutions for the Inertial and Kinematic interaction due to SSI effect and
presented critical assessment of FEMA 440 provision.
Receiving a Research Assistantship, Mehran started his PhD studies in the civil engineering
department at USC in Fall 2009. He joined the Strong Motion Research Group to work on a
project on Structural Health Monitoring and Damage Detection in buildings funded by the U.S.
National Science Foundation (NSF) under the direction of Professor Maria Todorovska. During
his graduate studies, Mehran acquired expertise in several areas in earthquake engineering such as
wave propagation, nondestructive testing, performance-based design, site response, and
engineering seismology. Through his second Master’s degree in Electrical Engineering at USC,
he acquired knowledge in Signal Processing and Estimation Methods.
Mehran has authored and co-authored more than 10 peer-reviewed technical papers. He is an
active member of professional societies including ASCE, EERI, AISC, and ACI, and also a
technical reviewer for several earthquake engineering journals.
237

Abstract (if available)

Abstract Monitoring the integrity of a building based on instrumental data and detecting damage during or soon after an earthquake or some other natural or manmade disaster may significantly reduce loss of life and injuries caused by potential collapse of a weakened structure during shaking from aftershocks and facilitate emergency response in large cities. Likewise, it would help prevent or reduce loss of function of critical facilities and monetary losses caused by needless evacuation of a structure that is safe. This dissertation presents a wave method for structural system identification and structural health monitoring (SHM) of high‐rise buildings. The method is robust when applied to real structures and large amplitude response, and is not sensitive to the properties of the underlying soil and its changes. It is intended for use in seismic alert systems as well as for general condition monitoring. ❧ The method uses data from an array of accelerometers, identifying the velocities of waves propagating vertically through the structure, and detects changes in these velocities, possibly caused by damage. The identification is based on fitting a layered shear beam model of the buildings, such that the layers correspond, in general, to a group of floors. The fitting involves matching propagating pulses in impulse response functions computed at different levels in the building. Three identification algorithms are presented: 1) the direct (ray) algorithm, which is based on reading the pulse arrival time, 2) the time shift matching algorithm, which is an iterative version of the direct algorithm, and 3) the waveform inversion algorithm, which involves nonlinear least squares (LSQ) fit of the pulses as waveforms over a selected time interval. The latter is extended to an automated moving window analysis, i.e. time-velocity analysis, for detecting changes during an earthquake. ❧ Detailed applications of the method are presented for three full‐scale buildings: (1) a typically instrumented tall steel building during six earthquakes over a period of 19 years, none of which caused damage (Los Angeles 54‐story Office Building), (2) a densely instrumented high‐rise reinforced concrete (RC) building during one smaller local earthquake (Millikan library in Pasadena), and (3) a damaged high‐rise RC building (Sherman Oaks 12‐story Office Building, during the San Fernando earthquake of 1971). Also, analyses of models are presented, which demonstrate the insensitivity of the method to the effects of soil‐structure interaction, the effects of the slabs on the wave propagation, and the modeling error when the method is applied to buildings with significant bending deformation (e.g. buildings with shear walls). ❧ It is concluded that the presented method is robust, accurate and sensitive to damage, being able to detect considerable changes in an RC building that has been only lightly damaged, and permanent changes over time in a tall steel building that has shown no signs of damage. With careful consideration of the frequency bands for the fit, it can be applied to many high rise buildings typical of metropolitan areas, such as, e.g. Los Angeles and San Francisco in California, and urban areas in general. Directions for research on further development of the method are identified.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Rahmani, Mohammadtaghi (author)

Core Title Wave method for structural system identification and health monitoring of buildings based on layered shear beam model

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering (Structural Engineering)

Publication Date 06/17/2014

Defense Date 05/19/2014

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

Tag Impulse response function,Los Angeles 54-story office building,Millikan Library building,OAI-PMH Harvest,seismic interferometry,shear beam model,Sherman Oaks 12-story Office Building,soil‐structure interaction,structural damage detection,Structural health monitoring,structural system identification,Tall buildings,wave dispersion,wave propagation in buildings,wave travel time

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Todorovska, Maria I. (committee chair), Lee, Vincent W. (committee member), Trifunac, Mihailo D. (committee member), Udwadia, Firdaus E. (committee member), Wong, Hung Leung (committee member)

Creator Email mrahmani@usc.edu,mt.rahmani@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-421015

Unique identifier UC11285876

Identifier etd-RahmaniMoh-2557.pdf (filename),usctheses-c3-421015 (legacy record id)

Legacy Identifier etd-RahmaniMoh-2557.pdf

Dmrecord 421015

Document Type Dissertation

Format application/pdf (imt)

Rights Rahmani, Mohammadtaghi

Type texts

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

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

Repository Name University of Southern California Digital Library

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