VERIFICATION OF COMPUTER ANALYSIS
MODELS FOR SUSPENSION BRIDGES
by
Masanobu Shinozuka, Distinguished Professor and Chair
and
Debasis Karmakar, Graduate Student
Samit Ray Chaudhuri, Postdoctoral Scholar
Ho Lee, Assistant Researcher
Department of Civil and Environmental Engineering
University of California, Irvine
Report No: CA/ UCI- VTB- 2009
August 2009
Final Report Submitted to
the California Department of Transportation
under Contract No: RTA- 59A0496
VERIFICATION OF COMPUTER ANALYSIS
MODELS FOR SUSPENSION BRIDGES
Final Report Submitted to the Caltrans under
Contract No: RTA- 59A0496
by
Masanobu Shinozuka, Distinguished Professor and Chair
and
Debasis Karmakar, Graduate Student
Samit Ray Chaudhuri, Postdoctoral Scholar
Ho Lee, Assistant Researcher
Department of Civil and Environmental Engineering
University of California, Irvine
Report No: CA/ UCI- VTB- 2009
August 2009
ii
STATE OF CALIFORNIA × DEPARTMENT OF TRASPORTATION
TECHNICAL REPORT DOCUMENTAION PAGE
TR0003 ( REV. 9/ 99)
1. REPORT NUMBER
CA/ UCI- VTB- 2009
2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER
5. REPORT DATE
August 2009
4. TITLE AND SUBTITLE
VERIFICATION OF COMPUTER ANALYSIS MODELS FOR
SUSPENSION BRIDGES
6. PERFORMING ORGANIZATION CODE
UC Irvine
7. AUTHOR
Masanobu Shinozuka, Debasis Karmakar, Samit Ray Chaudhuri,
and Ho Lee
8. PERFORMING ORGANIZATION REPORT
NO.
10. WORK UNIT NUMBER
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Department of Civil and Environmental Engineering
The Henry Samueli School of Engineering
University of California, Irvine
Irvine, CA 92697- 2175
11. CONTRACT OR GRANT NUMBER
RTA- 59A0496
13. TYPE OF REPORT AND PERIOD
COVERED
Final Report
12. SPONSORING AGENCY AND ADDRESS
California Department of Transportation ( Caltrans)
Division of Research and Innovation
1227 O Street, MS- 83
Sacramento, CA 95814
14. SPONSORING AGENT CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles
and Long Beach ports, two busiest ports in the west coast of USA. The bridge carries an
overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 45,500,
many of which are cargo trucks. Based on the recent finding that the main span of the Vincent
Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a
devastating earthquake, the bridge underwent a major retrofit in spring 2000, mainly using visco-elastic
dampers. This study focuses on performance evaluation of the retrofitted bridge under
seismic, wind and traffic loads. A member- based detailed three- dimensional Finite Element ( FE) as
well as panel- based simplified models of the bridge are developed. In order to show the
appropriateness of these models, eigenproperties of the bridge models are evaluated and compared
with the system identification results obtained using ambient vibration. In addition, model validation
is also performed by simulating and comparing with the measured dynamic response during two
recent earthquakes. FE model is also updated using a sensitivity- based parameter updating method.
Effect of spatial variability of ground motions on seismic displacement and force demands is
investigated. To record actual wind velocity and direction, three anemometers are installed at three
different locations of the bridge. Response of the bridge is computed under wind velocity. Finally,
analysis of the bridge under traffic load is also carried out.
17. KEYWORDS
Suspension Bridge, System Identification, Retrofit,
Fragility Curve, Earthquake, Wind, Traffic
18. DISTRIBUTION STATEMENT
No restrictions.
19. SECURITY CLASSIFICATION ( of this report)
Unclassified
20. NUMBER OF PAGES
178
21. COST OF REPORT CHARGED
ii i
DISCLAIMER: The contents of this report reflect the views of the
authors who are responsible for the facts and the
accuracy of the data presented herein. The contents
do not necessarily reflect the official views or
policies of the STATE OF CALIFORNIA or the
Federal Highway Administration. This report does
not constitute a standard, specification or regulation.
The United States Government does not endorse
products or manufacturers. Trade and
manufacturers’ names appear in this report only
because they are considered essential to the object
of the document.
iv
SUMMARY
The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los
Angeles and Long Beach ports, two busiest ports in the west coast of USA. The bridge
carries an overwhelming number of traffic with an Annual Average Daily Traffic
( AADT) volume of 45,500, many of which are cargo trucks. Based on the recent finding
that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes
fault, which has the capacity to produce a devastating earthquake, the bridge underwent a
major retrofit in spring 2000, mainly using visco- elastic dampers.
This study focuses on performance evaluation of the retrofitted bridge under seismic,
wind and traffic loads. A member- based detailed three- dimensional Finite Element ( FE)
as well as panel- based simplified models of the bridge are developed. In order to show
the appropriateness of these models, eigenproperties of the bridge models are evaluated
and compared with the system identification results obtained using ambient vibration. In
addition, model validation is also performed by simulating and comparing with the
measured dynamic response during two recent earthquakes. Tornado diagram and first
order second moment ( FOSM) methods are applied for evaluating the sensitivity of
different parameters on the eigenproperties of the FE models. The study indicates that the
mass density of deck slab and elastic modulus of bottom chord are very important
parameters to control eigenproperties of the models. FE model is also updated using a
sensitivity- based parameter updating method.
Considering a set of strong ground motions in the Los Angeles area, nonlinear time
history analyses are performed using the FE models developed and seismic fragility
curves are derived comparing the ductility demand with the ductility capacity at critical
v
tower sections. Effect of spatial variability of ground motions on seismic displacement
and force demands is also investigated. To generate spatially correlated nonstationary
acceleration time histories compatible with design spectrum at each location. A new
algorithm is developed involving evolutionary power spectral density function ( PSDF)
and with the aid of spectral representation method. It has been found that, in some
locations on the bridge deck, the response is higher when the spatially variable ground
motion is considered as opposed to the uniform ground motion time histories having the
highest ground displacement.
To record actual wind velocity and direction, three anemometers are installed at three
different locations of the bridge. The fluctuating component of the wind velocity
measured at these three locations are found to be non- Gaussian. They are used for
simulation of fluctuating component of wind velocity throughout the span and along the
tower on the basis of three different simulation methods ( i) newly developed non-
Gaussian conditional method, ( ii) Gaussian conditional method, and ( iii) Gaussian
unconditional method. Response of the bridge is computed under wind velocity using
these three different methods. It is observed that the non- Gaussian conditional simulation
technique yields higher response than both Gaussian conditional and Gaussian non-conditional
techniques. Finally, analysis of the bridge under traffic load is also carried out
and a critical evaluation of shear force in deck shear connectors is performed.
vi
ACKNOWLEDGEMENT
The research presented in this report was sponsored by the California Department of
Transportation ( Caltrans) with Dr. Li- Hong Sheng as the project manager. The authors
are indebted to Caltrans for its support of this project and to Dr. Li- Hong Sheng for his
helpful comments and suggestions.
viii
TABLE OF CONTENTS
Page
LIST OF FIGURES vi
LIST OF TABLES xiii
ABSTRACT xv
CHAPTER 1 Introduction 1
1.1 Background 1
1.2 Literature Survey 3
1.3 Objective and Scope 10
1.4 Dissertation Outline 12
CHAPTER 2 Finite Element Modeling of Vincent Thomas Bridge 13
2.1 Background 13
2.2 Calculation of Dead Weight 13
2.3 Calculation of the Initial Shape of the Cable 14
2.4 Panel- Based Simple Model 14
2.4.1 Moment of Inertia ( Iz) 18
2.4.2 Torsional Constant ( J) 19
2.5 Member- Based Detail Model 24
2.5.1 Cable Bent 27
2.5.2 Deck Shear Connector 27
2.5.3 Dampers 28
2.5.4 Suspended Truss 28
2.5.5 Suspenders 28
2.6 Eigen Value Analysis 29
ix
2.7 Closure 31
CHAPTER 3. System Identification and Model Verification 33
3.1 Background 33
3.2 Evaluation of Eigenproperties using Ambient Vibration Data 33
3.3 Comparison of System ID Result with Analytical
Eigen Properties 37
3.4 Modal Parameter Identification from Chino Hills
Earthquake Response 38
3.5 Effect of Parameter Uncertainty on Modal Frequency 40
3.5.1 Soil Spring Modeling 40
3.5.2 Uncertain Parameters Considered 41
3.5.3 Analysis methods 43
3.5.4 Sensitivity of Modal Frequencies 46
3.6 Finite Element Model Updating 55
3.6.1 Sensitivity Based Model Updating 56
3.6.2 Selection of Modes and Parameters 59
3.6.2.1 Selection of Modes 59
3.6.2.2 Selection of Parameters 60
3.6.3 Updated Results 61
3.7 Closure 64
CHAPTER 4 Seismic Analysis 65
4.1 Background 65
4.2 Scope 67
4.3 Response Analysis under Northridge Earthquake 68
4.4 Response Analysis under Chino Hills Earthquake 70
4.5 Generation of Fragility Curves 73
4.6 Simulation of Ground Motion Considering Spatial Variability 78
4.6.1 Generation of Evolutionary PSDF from Given
Ground Motion using STFT 78
4.6.2 Generation of Evolutionary PSDF from Given
x
Ground Motion using Wavelet Transform 79
4.6.3 Simulation of One- Dimensional Multi- Variate
( 1D- mV), Nonstationary Gaussian Stochastic Process 82
4.6.4 Simulation of Seismic Spectrum Compatible
Accelrograms 85
4.6.5 Examples of Generated Seismic Ground Motion 90
4.7 Results 99
4.8 Closure 105
CHAPTER 5 Wind Sensor Installation and Wind Speed Measurement 106
5.1 Background 106
5.2 Anemometer and Data Acquisition System 107
5.2.1 Anemometer for Vantage Pro2 107
5.2.2 Anemometer Transmitter with Solar Power 107
5.2.3 Wireless Repeater with Solar Power 107
5.2.4 Wireless Weather Envoy ( Wireless Receiver) 110
5.2.5 WeatherLink Software for Data Collection 110
5.2.6 Data Acquisition Software Developed 110
5.2.7 Experimental Setup 111
5.2.8 Anemometer Installation and Data
Acquisition System 112
5.3 WeatherLink Software for Data Collection 114
5.4 Recorded Wind Velocities 115
5.5 Closure 117
CHAPTER 6 Wind Buffeting Analysis 118
6.1 Background 118
6.2 Scope 121
6.3 Conditional Simulation of Gaussian Random Processes 122
6.3.1 Conditional Simulation in Frequency Domain 123
6.4 Conditional Simulation of Non- Gaussian Random Processes 124
xi
6.5 Simulation of Spatially Correlated Gaussian
Wind Velocity Fluctuations 128
6.6 Conditional Simulation of Gaussian Wind
Velocity Fluctuations 135
6.7 Conditional Simulation of non- Gaussian
Wind Velocity Fluctuations 138
6.8 Buffeting Force Calculation 152
6.9 Buffeting Response of Vincent Thomas Bridge 153
6.10 Closure 157
CHAPTER 7 Traffic Load Analysis 158
7.1 Background 158
7.2 Moving Load Analysis 158
7.3 Closure 165
CHAPTER 8 Conclusions and Future Work 166
8.1 Summary and Conclusions 166
8.2 Future Work 169
REFERENCES 171
xii
LIST OF FIGURES
Page
Figure 2.1 The shape of the initial cable profile under dead load 18
Figure 2.2 Cross section of deck 19
Figure 2.3 Location of stringers in one side of the deck 20
Figure 2.4 Commonly used lateral bracing systems and stiffening girders 21
Figure 2.5 Horizontal system ( K- type) 21
Figure 2.6 Vertical web system ( Worren type) 22
Figure 2.7 Different sections of the tower 26
Figure 2.8 Typical tower cross- section 26
Figure 2.9 The detailed model of one panel 28
Figure 2.10 Deck shear connector ( before retrofit) 29
Figure 2.11 Deck shear connector ( after retrofit) 29
Figure 2.12 K- truss modifications after retrofit 30
Figure 2.13 Suspender modifications after retrofit 30
Figure 2.14 First three mode shapes of the simple model 31
Figure 3.1 Location and direction of sensors installed in the bridge 35
Figure 3.2 Vertical accelerometer data used in the study 36
Figure 3.3 Lateral accelerometer data used in the study 36
Figure 3.4 Plot of SV vs. Frequency 37
Figure 3.5 Detailed model in SAP 2000 with foundation springs 41
Figure 3.6 Tornado diagram considering 19 parameters 51
xiii
Figure 3.7 Relative variance contribution ( neglecting correlation
terms) from FOSM analysis 54
Figure 3.8 Three- dimensional finite element model of
Vincent Thomas Bridge 55
Figure 3.9 Procedure for the sensitivity- based model updating 58
Figure 3.10 Comparison of frequency differences using the
initial and updated FE models 62
Figure 4.1 Location and direction of sensors 69
Figure 4.2 Comparison of measured and calculated longitudinal
displacement at channel # 10 location 69
Figure 4.3 Comparison of analytical lateral response at channel 5
due to ground motions at east anchorage, east tower and
west tower with field measured response 71
Figure 4.4 Comparison of analytical lateral response at channel 3
due to ground motions at east tower with field measured response 72
Figure 4.5 Comparison of analytical vertical response at channel 17 due
to ground motions at east tower with field measured response 72
Figure 4.6 Comparison of analytical longitudinal response at channel 10
due to ground motions at east tower with field measured response 73
Figure 4.7 Before and after retrofit Fragility curves for different damage levels 77
Figure 4.8 Evolutionary PSDF of LA21 earthquake record using STFT method 79
Figure 4.9 Evolutionary PSDF of LA21 earthquake record using
wavelet transform 82
Figure 4.10 Iterative scheme to simulate spectrum compatible
acceleration time histories 89
Figure 4.11 Different support locations of the bridge 91
Figure 4.12 Acceleration time history of LA 21 scenario earthquake 92
Figure 4.13 Acceleration time history at location 1 93
Figure 4.14 Acceleration time history at location 2 94
xiv
Figure 4.15 Acceleration time history at location 3 94
Figure 4.16 Acceleration time history at location 4 95
Figure 4.17 Acceleration time history at location 5 95
Figure 4.18 Acceleration time history at location 6 96
Figure 4.19 Displacement time history at location 3 96
Figure 4.20 Displacement time history at location 6 97
Figure 4.21 Comparison between simulated and design spectra at
location 1 using STFT 97
Figure 4.22 Comparison between simulated and design spectra at
location 3 using STFT 98
Figure 4.23 Comparison between simulated and design spectra at
location 1 using Wavelet 98
Figure 4.24 Comparison between simulated and design spectra at
location 3 using Wavelet 99
Figure 4.25 Absolute axial force demand envelope for the bridge girder 101
Figure 4.26 Absolute shear force demand envelope for the bridge girder 101
Figure 4.27 Absolute moment demand envelope for the bridge girder 102
Figure 4.28 Absolute torsional force demand envelope for the bridge girder 102
Figure 4.29 Absolute axial force demand envelope for the east tower of the bridge 103
Figure 4.30 Absolute shear force demand envelope for the east tower of the bridge 103
Figure 4.31 Absolute moment demand envelope for the east tower of the bridge 104
Figure 4.32 Absolute torsional force demand envelope for the east tower
of the bridge 104
Figure 5.1 Anemometer 108
Figure 5.2 Anemometer transmitter with solar power 109
xv
Figure 5.3 Wireless repeater with solar power 109
Figure 5.4 Wireless Weather Envoy ( Wireless Receiver) 110
Figure 5.5 Layout of the data acquisition system 111
Figure 5.6 Locations of anemometers, transmitters, repeaters and receivers 113
Figure 5.7 Distance between different components 113
Figure 5.8 Installation of anemometers, transmitters and repeaters
( a) Top of the east tower
( b) Vertical post on deck
( c) East tower platform
( d) Anchorage house wall 115
Figure 5.9 Screen shots from Weather Link and data acquisition system
( a) Anemometer # 1
( b) Anemometer # 2
( c) Anemometer # 3
( d) Data acquisition system 116
Figure 5.10 Wind velocity recorded for 24 hrs on
April 8, 2009 ( 1 sample/ min) 116
Figure 5.11 Wind velocity recorded for 30 minutes on
April 15, 2009 ( 1 sample/ 3s) 117
Figure 6.1 Flow chart of conditional simulation of
non- Gaussian random processes 127
Figure 6.2 Installed anemometer locations on VTB 133
Figure 6.3 Locations of “ aerodynamic” nodes along the bridge deck 133
Figure 6.4 Horizontal wind velocity fluctuations at different locations
along the deck ( around anemometer # 2) in m/ s
from Gaussian unconditional simulation 134
Figure 6.5 Horizontal wind velocity fluctuations at different locations
along the deck ( around anemometer # 1) in m/ s
from Gaussian unconditional simulation 134
Figure 6.6 Horizontal wind velocity fluctuations at two different
locations from Gaussian unconditional simulation 135
xvi
Figure 6.7 Measured wind velocity fluctuation at anemometer # 1 location 136
Figure 6.8 Measured wind velocity fluctuation at anemometer # 2 location 136
Figure 6.9 Measured wind velocity fluctuation at anemometer # 3 location 136
Figure 6.10 Horizontal wind velocity fluctuations at different locations along
the deck ( around anemometer # 2) in m/ s from Gaussian
conditional simulation 137
Figure 6.11 Horizontal wind velocity fluctuations at different locations along
the deck ( around anemometer # 1) in m/ s from Gaussian
conditional simulation 137
Figure 6.12 Horizontal wind velocity fluctuations at two different
locations from Gaussian unconditional simulation 138
Figure 6.13 Actual and analytical PDF of wind velocity
fluctuation measured at anemometer # 1 location 140
Figure 6.14 Actual and analytical PDF of wind velocity
fluctuation measured at anemometer # 2 location 140
Figure 6.15 Actual and analytical PDF of wind velocity
fluctuation measured at anemometer # 3 location 141
Figure 6.16 Actual and analytical CDF of wind velocity fluctuation
measured at anemometer # 1 location 141
Figure 6.17 Actual and analytical CDF of wind velocity fluctuation
measured at anemometer # 2 location 142
Figure 6.18 Actual and analytical CDF of wind velocity fluctuation
measured at anemometer # 3 location 142
Figure 6.19 Horizontal wind velocity fluctuations at two different
locations from non- Gaussian conditional simulation 144
Figure 6.20 Simulated and target CDF of wind velocity fluctuation
at point # 10 144
Figure 6.21 Simulated and target CDF of wind velocity fluctuation
at point # 16 145
xvii
Figure 6.22 Comparison of PSDF from simulated wind velocity
fluctuation and target PSDF at point # 10 115
Figure 6.23 Comparison of PSDF from simulated wind velocity
fluctuation and target PSDF at point # 16 146
Figure 6.24 Comparison of PSDF from measured velocity fluctuation at
anemometer # 1 and assumed analytical PSDF 147
Figure 6.25 Comparison of PSDF from measured velocity fluctuation at
anemometer # 2 and assumed analytical PSDF 147
Figure 6.26 Comparison of PSDF from measured velocity fluctuation at
anemometer # 3 and assumed analytical PSDF 148
Figure 6.27 Horizontal wind velocity fluctuations at different locations
along the deck ( around anemometer # 2) in m/ s
from non- Gaussian conditional simulation 148
Figure 6.28 Horizontal wind velocity fluctuations at different locations
along the deck ( around anemometer # 1) in m/ s
from non- Gaussian conditional simulation 149
Figure 6.29 Simulated wind velocity fluctuations at location # 10
with three different simulation techniques 149
Figure 6.30 Simulated wind velocity fluctuations at location # 16
with three different simulation techniques 150
Figure 6.31 Horizontal wind velocity fluctuations at different locations
along the tower ( around anemometer # 2) in m/ s
from Gaussian unconditional simulation 150
Figure 6.32 Horizontal wind velocity fluctuations at different locations
along the tower ( around anemometer # 2) in m/ s
from Gaussian conditional simulation 151
Figure 6.33 Horizontal wind velocity fluctuations at different locations
along the tower ( around anemometer # 2) in m/ s from
non- Gaussian conditional simulation 151
Figure 6.34 Schematic diagram for aerodynamic forces on bridge deck 155
Figure 6.35 Simulated lateral deck displacements at the center of the mid span 156
Figure 6.36 Simulated vertical deck displacement at the center of the mid span 156
xviii
Figure 7.1 Plan view of deck shear connectors before and after retrofit 159
Figure 7.2 Deck shear connector 159
Figure 7.3 Deck shear connector design drawing 160
Figure 7.4 HS20- 44 AASTHO traffic loading 161
Figure 7.5 Different traffic load cases 162
Figure 7.6 Axial force in shear connector due to
traffic load ( before and after retrofit) 163
Figure 7.7 Vertical shear force in shear connector due to
traffic load ( before and after retrofit) 163
Figure 7.8 Longitudinal shear force in shear connector due to
traffic load ( before and after retrofit) 164
Figure 7.9 Shear key in east side span 164
xix
LIST OF TABLES
Page
Table 2.1 Calculated dead load ( kip/ ft) along the length of bridge 16
Table 2.2 Calculated nodal coordinates of the cable- only system 17
Table 2.3 Calculated sectional properties of panels 23
Table 2.4 Calculated sectional properties of the tower
sections ( before retrofit) 27
Table 2.5 Calculated sectional properties of the tower
sections ( after retrofit) 27
Table 2.6 Comparison of modal frequencies in Hz ( before retrofit) 31
Table 2.7 Comparison of modal frequencies in Hz ( after retrofit) 32
Table 3.1 Location and direction of accelerometers 35
Table 3.2 Comparison of modal frequencies in Hz ( before retrofit) 39
Table 3.3 Comparison of modal frequencies in Hz ( after retrofit) 39
Table 3.5 Location and number of piles considered 41
Table 3.6 Parameters considered for sensitivity analysis 43
Table 3.7 Comparison of natural frequencies 59
Table 3.8 Parameters selected for adjustment 60
Table 3.9 MAC matrix of updated FE model 62
Table 3.10 Comparison of natural frequencies between
baseline and updated FE model 63
Table 3.11 Updated design parameters 63
Table 4.1 Different support motions considered with channel numbers 70
xx
Table 4.2 Details of the motions considered in this study for fragility
Development 77
Table 4.3 Site coefficient parameters to calculate design
spectra at different supports 92
Table 4.4 Displacement demand comparison 100
Table 5.1 Settings of different repeaters 114
Table 5.2 Settings of different receivers 114
Table 6.1 Properties for assumed generalized extreme value distribution 141
Table 7.1 Shear stress developed in shear key bolts 165
1
CHAPTER 1
INTRODUCTION
1.1 Background
Throughout the history of suspension bridges, their tendency to vibrate under different
dynamic loadings such as wind, earthquake, and traffic loads has been a matter of
concern. The failure of the Tacoma Narrows bridge in 1940 has pointed out that the
suspension bridges are vulnerable to wind loading ( Rannie 1941). It is now widely
accepted that the wind- induced vibration of suspension bridges may be significant and
should be taken into consideration. Similar conclusions have also been drawn for other
dynamic loadings. As a prerequisite to the investigation of aerodynamic stability, traffic
impact, soil- structure interaction and earthquake resistant design of suspension bridges, it
is necessary to know certain dynamic characteristics such as the natural frequencies and
the possible modes of vibration. Several investigations have been taken place in recent
years to determine the vibrational properties of suspension bridges. However, the
complexity of a suspension bridge structure makes the determination of vibrational
characteristics difficult.
With the advent of computers, non- conventional structures like suspension
bridges are analyzed with the finite element ( FE) analysis technique. There are several
commercially available finite element software packages that are used by practicing
2
engineers as well as researchers, which can evaluate the response of a suspension bridge
from operational traffic, wind and earthquake loads taking into account both material and
geometric non- linear behavior.
In addition to analytical modeling and response analysis of suspension bridges,
field tests are also very important from the analysis and design point of view. Field test
results not only give experimental data but also help us to understand the behavior of the
structure and to calibrate the analytical model. To perform field tests, it is necessary to
measure, input loadings such as wind velocity at different pints and earthquake ground
acceleration at different support locations, and output responses such as acceleration,
velocity and displacement time history at different points of the bridge.
For predicting response of long span suspension bridges under random wind, the
most widely used method is the frequency domain analysis. In theory, the frequency
domain solution is accurate, when the load- response relationship is linear. Although the
structural elements in a suspension bridge generally behave in a linear elastic fashion
under normal loading, the overall load- displacement relationship exhibits geometrical
nonlinearity, particularly when it is subjected to high wind. Therefore, in this case, a
frequency domain analysis may not be appropriate. One way in which the limitation of
the frequency domain analysis can be overcome is the use of Monte Carlo simulation
technique. One of the most important components of the Monte Carlo simulation method
is the generation of sample functions of stochastic processes, fields, or waves those are
involved in the problem. For buffeting analysis, wind velocity fluctuation in the
horizontal and vertical directions needed to be digitally simulated and fed into the
equation of motion. Since the length of a modern suspension bridge generally exceeds 1
3
km, the simulated sample functions must accurately describe the probabilistic
characteristics not only in terms of temporal variation but also in spatial distribution.
Similarly for seismic response, critical members of the bridge may undergo significant
nonlinear deformation and a simple response spectrum method for analyzing such
response may not be adequate. In addition, there may be significant variation of ground
motion from one support of the bridge to the other.
1.2 Literature Survey
Theoretical and practical treatises on the vibrational characteristics and the dynamic
analysis of suspension bridges, have been developed by many authors, especially after the
disastrous collapse of the Tacoma Narrows Bridge in 1940 ( Rennie 1941). Bleich et al.,
1950 studied the free vertical and torsional vibration by solving a forth order linearized
differential equation. In addition, an approximate method of the Rayleigh- Ritz type
solution was suggested. However, the procedure is applicable only for calculating the
lowest few modes due to the great level of complexity and redundancy of higher modes
of suspension bridges. Steinman, 1959 introduced a number of simplified formulas for
estimating the natural frequencies and the associated mode shapes of vibration, both
vertical and torsional, of suspension bridges. Japanese researchers ( Konishi et al 1965;
Konishi and Yamada 1969; Yamada and Takemiya 1969, 1970; Yamada and Goto 1972;
and Yamada et al. 1979) performed extensive studies to investigate the vertical and
lateral vibration as well as the tower- pier system of a three- span suspension bridge by
using a lumped- mass system interconnected by spring elements. In their analysis for the
4
suspended structure, they assumed simple harmonic excitations and applied it separately
to each supporting point. They reported that there was a fairly significant contribution
from the higher modes to the bending response and a large number of modes should be
included to accurately determine the dynamic response of suspension bridges.
The geometrically nonlinear behavior of suspension bridges was considered
( Tezcan and Cherry 1969) due to large deflection and presented an iterative technique for
the nonlinear static analysis by using tangent stiffness matrices. These matrices are
incorporated in obtaining the free- vibrational modes of the structure. In their analysis, the
bridge was modeled as a three- dimensional lumped mass system. They calculated the
response of the bridge considering three orthogonal components of uniform ground
motion and pointed out that the longitudinal motion of the deck as well as the vertical
motion of the tower were small and therefore could be neglected.
Major advances in studying the dynamic characteristics of suspension bridges
have been achieved through the use of finite element method and linearized deflection
theory ( Abdel- Ghaffar 1976, 1977, 1978a, 1978b, 1979, 1980 and 1982). Natural
frequencies, mode shapes, and energy capacities of the different structural components
for vertical, torsional, and lateral vibrations were investigated. Several examples were
presented and the applicability of the proposed methods was illustrated by comparing the
results obtained from analyzing the Vincent- Thomas bridge ( Los Angeles Harbor) with
the results of full- scale ambient vibration tests ( Abdel- Ghaffar 1976, 1978 and Abdel-
Ghaffar and Housner 1977). Some researchers ( Abdel- Ghaffar and Rubin 1983a and
Abdel- Ghaffar and Rubin 1983b) studied the effect of large amplitude nonlinear free
coupled vertical- torsional vibrations of suspension bridges using a continuum approach
5
where approximate solutions of the nonlinear coupled equations were conducted.
Nonlinearities due to large deflections of cables, the axial stretching of stiffening
structure, and the nonlinear curvature of the stiffening structure were considered. It was
mentioned that the importance of geometric nonlinearities arises only for very high
amplitude vibration. Also, they studied using two- dimensional models the directional
vertical, torsional, and lateral earthquake response, in both time and frequency domains,
of long- span suspension bridges subjected to multiple- input excitations ( Abdel- Ghaffar
and Rubin 1982; Abdel- Ghaffar and Rubin 1983c and Abdel- Ghaffar et al. 1983). In
addition, they considered a simplified model for the tower- pier system and investigated
the longitudinal vibration response taking into account the flexibility and damping
characteristics of the underlying and surrounding soil. They applied their procedure to the
tower- pier system of the Golden Gate bridge ( San Francisco) and different soil conditions
were used.
The vertical response of suspension bridges has been studied to seismic
excitations using a stochastic approach ( Dumanoglu and Severn 1990). They applied their
method to three suspension bridges using one set of earthquake records and a filtered
white noise as well. They pointed out that the accuracy of that approach, in comparison to
the time- history approach, depends upon the magnitudes of the fundamental period of the
bridge under consideration. They reported that, for long- span suspension bridges like the
Bosporus ( in Turkey) and Humber ( in England) bridges, the response results of the
stochastic approach should be cautiously assessed, especially when the earthquake
records are not zero- padded.
6
Some researchers ( Lin and Imbsen 1990; Ketchum and Seim 1991 and Ketchum
and Heledermon 1991) carried out an investigation on the Golden Gate bridge by
developing an elaborate 3- D finite element model. The lower wind- bracing system of the
bridge was considered to carry a light train. They incorporated different elements types
and performed a nonlinear static analysis to determine the stiffness of the bridge in its
dead- load state and used this matrix in the solution for the natural frequencies and mode
shapes. Their model is verified by comparing its results with those obtained from
previous studies ( Abdel- Ghaffer and Scanlan 1985a and Abdel- Ghaffer and Scanlan
1985b). They reported that most of the lowest modes involving vibration of cables and
torsional motion of the deck are not relevant to the earthquake performance of the bridge.
A 3D finite element model was proposed for the Vincent- Thomas bridge ( Niazy
et al. 1991). They considered geometrical nonlinearities in suspension bridges, and an
iterative nonlinear static analysis technique was adopted. The stiffening truss, tower and
cable- bent elements, were modeled as 3- D frame elements and cable elements were
modeled as 3- D truss elements. In their study, 50 lowest natural frequencies and the
corresponding mode shapes of the bridge model were determined in its dead- load
configuration. However, in their modeling they did not consider the actual mass
distribution over the length of the bridge. They considered uniform mass distribution over
the center span and the side spans. Initial shape of the cable is one of the important
parameters in the analysis of suspension bridges. A non- linear shape- finding analysis was
used for a self- anchored suspension bridge named Yongjong Grand Bridge ( Kim et al.,
2002). The shape- finding analysis determines the coordinates of the main cable and
7
initial tension of main cable and hangers, which satisfies the design parameters at the
initial equilibrium state under full dead loads.
Several models and expressions have been proposed ( Davenport 1968) in relation
to spatial variation of wind velocity fluctuation. For a more complete bibliography, the
reader is referred to Simiu and Scanlan ( 1996). The analytical work by Beliveau et al.,
1977 combined the effect of buffeting and self- excited forces. They used a two degrees of
freedom mathematical model. Even though simulation techniques have been reported
since 1970 ( Shinozuka and Jan 1972), some earlier studies assumed uniformly distributed
wind velocity fluctuations for the nonlinear time history analysis of cable- supported
bridges ( Arzoumanidis 1980). In past decades, a number of researchers reported on
efficient methods for generating spatially correlated wind velocity fluctuations ( Li and
Kareem 1993; Shinozuka and Deodatis 1996; Deodatis 1996; Facchini 1996; Yang et al.
1997; Paola 1998; Paola and Gullo 2001). As a result of improvements in simulation
techniques as well as computational speed, the time- domain approach has been utilized
more frequently in recent buffeting analyses of long- span cable- supported bridges to take
aerodynamic and/ or geometric nonlinearity into consideration ( Aas- Jakobsen and
Strømmen 1998, 2001; Minh et al. 1999; Ding and Lee 2000; Chen et al. 2000; Chen and
Kareem 2001; Lin et al. 2001). Kareem’s group, in particular, has reported extensively on
the line of time domain analysis framework for use in predicting aerodynamic nonlinear
responses by incorporating frequency dependent parameters of unsteady aerodynamic
forces by utilizing a rational function approximation technique ( Chen and Kareem 2001).
This technique is also readily available for the structure- originated nonlinearity in
buffeting analysis. However, only a few studies utilized a nonlinear analysis procedure
8
for estimating buffeting response using structural nonlinearity, which is potentially
involved in long- span cable- supported bridges, has been taken into consideration ( Ding
and Lee 2000; Lin et al. 2001).
The spatial variation of earthquake ground motions may have significant effect on
the response of long span suspension bridges. Abdel- Ghaffar and Rubin ( 1982) and
Abdel- Ghaffar and Nazmy ( 1988) studied response of suspension and cable- stayed
bridges under multiple support excitations. Zerva ( 1990) and Harichandran and Wang
( 1990) examined the effect of spatial variable ground motions on different types of bridge
models. Harichandran et al. ( 1996) studied the response of long- span bridges to spatially
varying ground motion. Deodatis et al. ( 2000) and Kim and Feng ( 2003) investigated the
effect of spatial variability of ground motions on fragility curves for bridges. Lou and
Zerva ( 2005) analyzed the effects of spatially variable ground motions on the seismic
response of a skewed, multi- span, RC highway bridge. Most of the aforementioned
studies dealt with simple FE models of the bridge, as a result response of critical
members could not be evaluated. In the present analysis a panel based detailed 3D FE
model of a long span suspension bridge is utilized.
In this study, an iterative algorithm is proposed to generate spatially variable,
design spectrum compatible acceleration time histories at different support locations of
the bridge. The proposed algorithm is used to generate synthetic ground motions at six
different points on the ground surface. For generating non- stationary accelerograms,
previously researchers used time dependent envelope function on top of simulated
stationary ground motions ( Deodatis 1996). In this study by using evolutionary power
spectral density function from the mother accelerogram, a new algorithm has been
9
proposed to simulate spatial variable ground motions. In the simulated acceleration time
histories the temporal variations of the frequency content are same as the mother
accelerogram. Mukherjee and Gupta ( 2002) proposed a new wavelet based approach to
simulate spectrum compatible time histories. But they only considered one design
spectrum and simulated one accelrogram from a single mother acceleration time history.
Sarkar and Gupta ( 2006) developed a wavelet based approach to simulate spatially
correlated and spectrum compatible accelerogram.
So far in a broad sense two approaches have been introduced by researchers
regarding conditional simulation. The two approaches are based on “ kriging” ( Krige,
1966) ( linear estimation theory applied to random functions) and conditional probability
density function. Vanmarcke and Fenton ( 1991) applied conditional simulation of to
simulate Fourier coefficients using kriging technique. Kameda and Morikawa ( 1992 and
1994), used an analytical framework based on spectral representation method, derived
joint probability density functions of Fourier coefficients obtained from the expansion of
conditioned random processes into Fourier series. They calculated conditional
expectations and variances of the conditioned random processes and considered their
first- passage probabilities. Hoshiya ( 1994) considered a conditional random field as a
sum of its kriging estimate and the error. He simulated the kriging estimate and the error
separately and combined them to get the Gaussian conditionally simulated field. In all the
above studies the investigators considered Gaussian processes and Gaussian random
fields.
Sometimes the assumption of Gaussian wind loading is not correct. In those cases,
conditional simulation of non- Gaussian wind velocity field should be used. Elishakoff et
10
al. ( 1994) combined the conditional simulation technique of Gaussian random fields by
Hoshiya ( 1994) and the iterative procedure for unconditional simulation of non- Gaussian
random fields by Yamazaki and Shinozuka ( 1988), to conditionally simulate time-independent
non- Gaussian random fields. Gurley and Kareem ( 1998) developed a
procedure for conditional simulation of multivariate non- Gaussian velocity/ pressure
fields. For mapping the Gaussian process to non- Gaussian process and vice versa, they
used modified Hermite transformation using Hermite polynomial function.
For buffeting analysis of long span cable supported bridges Chen ( 2001), Kim
( 2004) used time domain analysis to consider the effect of non- linearity in the structure.
Also they only considered the wind forces on the deck only. They neglected the coupling
effect of wind forces on tower and cable. Sun ( 1999) considered the coupling effect of the
aeroelastic forces on the bridge deck, towers and cables. But they did not consider a 3D
detailed finite element ( FE) model of the bridge. Recently, He ( 2008) considered a
detailed 3D model for buffeting analysis.
1.3 Objectives and Scope
The main purpose of this research is to evaluate the performance of a long span
suspension bridge under seismic, wind, and traffic loads. A member- based detailed three-dimensional
Finite Element ( FE) as well as panel- based simplified models of the bridge
are developed. In order to show the appropriateness of these models, eigenproperties of
the bridge models are evaluated and compared with the system identification results
obtained using ambient vibration. In addition, model validation is also performed by
11
simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino
Hills earthquake and comparing with the measured response. Tornado diagram and first
order second moment ( FOSM) methods are applied for evaluating the sensitivity of
different parameters on the eigenproperties of the FE models. This kind of study will be
very helpful in selecting parameters and their variability ranges for FE model updating of
suspension bridges.
Considering a set of strong ground motions in the Los Angeles area, nonlinear
time history analyses are performed and the ductility demands of critical sections of the
tower are presented in terms of fragility curves. Effect of spatial variability of ground
motions on seismic displacement demand and seismic force demand is investigated. To
generate spatially correlated design spectrum compatible nonstationary acceleration time
histories, a newly developed algorithm using evolutionary power spectral density
function ( PSDF) and spectral representation method is used.
To simulate the wind velocity field accurately for the bridge site, measurement of
the wind velocity is needed at the bridge location. For colleting actual wind data i. e. wind
velocity and direction, three anemometers have been installed at three different locations
of the bridge, so that the wind velocity field can be simulated in both horizontal and
vertical directions. The measured wind velocity fluctuation data have been used for
conditional simulation of wind velocity fluctuation field.
Finally, response of Vincent Thomas Bridge under conditionally simulated wind
velocity field is also presented in this study. A new simulation technique for conditional
simulation of non- Gaussian wind velocity fluctuation field is proposed and used for
12
buffeting analysis of the bridge under simulated wind load. Analysis of the Vincent
Thomas bridge under traffic load is also carried out in this study.
1.4 Dissertation Outline
The dissertation contains the following chapters
Chapter 2 summarizes the finite element ( FE) modeling of the before and after
retrofitting of the bridge.
Chapter 3 presents the system Identification results obtained from response of the
bridge and compared with modal parameters obtained from analytical model. A
sensitivity analysis is also carried out.
Chapter 4 proposes a new methodology to simulate spectrum compatible spatial
variable ground motions. Response variability due to spatial variation in ground motion is
also assessed.
Chapter 5 describes the wind sensors installation in the bridge and data collection.
Chapter 6 proposes a new methodology to conditionally simulate non- Gaussian
wind velocity fluctuation profiles using the data collected by anemometers at the bridge
site. Wind buffeting analysis also carried out using the simulated wind velocity
fluctuation profile.
Chapter 7 describes the traffic load analysis.
13
CHAPTER 2
FINITE ELEMENT MODELING OF VINCENT THOMAS BRIDGE
2.1 Background
With the advent of high- speed computer, major advances in studying the dynamic
characteristics of suspension bridges have been achieved through the use of finite element
method. In addition, effort has also been given for developing simplified models that can
predict response consistent with detailed model. In recent years, several commercially
available finite element software packages have been used by practicing engineers as well
as researchers to evaluate the response of a suspension bridge from operational traffic,
wind and earthquake loads taking into account both material and geometric non- linear
behavior. This chapter focuses on numerical modeling of the Vincent Thomas Bridge. A
member- based detailed three- dimensional Finite Element ( FE) as well as a panel- based
simplified model of the Vincent Thomas bridge have been developed for before and after
retrofit of the bridge.
2.2 Calculation of Dead Weight
The dead load along the length of the bridge has been calculated. Table 1 shows the
calculated dead load of the different components of the bridge. It has been found that the
weight per unit length of the bridge in the center span is very close to the design value of
14
7.2 kip/ ft indicated in the design drawing. The dead load calculation is also compared
with the values reported by Abdel- Ghaffer, 1976 shown in Table 2.1.
2.3 Calculation of the Initial Shape of the Cable
Initial shape of the cable is one of the important parameters in the modeling of
suspension bridges. Initial shapes of the cables of Vincent Thomas Bridge have been
calculated using non- linear shape- finding analysis and subsequently used in the FE
model. The shape- finding analysis determines the coordinates of the main cable and
initial tension of main cable and hangers, which satisfies the design parameters at the
initial equilibrium state under full dead loads. Details of the analysis methodology and
software are described in Kim et al., 2002. The shape of the initial cable profile in the
form of preliminary and final configurations are tabulated in Table 2.2 and the initial
cable profile is plotted in Figure 2.1.
2.4 Panel- Based Simple Model
For simplified panel- based modeling, the girders and diaphragms are considered as
equivalent 3D frame elements. The cable and suspender are modeled as 3D truss element.
Also as in the case of detailed model, truss and cable bent were modeled with frame
elements. Dampers are also included in the simplified model only at the tower and girder
connections. . FE modeling is done with SUCOT ( Kim, 1993) and SAP 2000 V10
( Computer and Structures, 2002). Area of the stiffening girder is set equal to the sum of
the area of top chord, bottom chord and web.
15
Table 2.1 Calculated dead load ( kip/ ft) along the length of bridge
Present study
Different components Center
span
Side
span
Abdel-
Ghaffer, 1976
Curb 0.066 0.066
Bracket 0.019 0.019
Crash barrier 0.413 0.413
Sub total 0.498 0.498 0.203
Grating 0.036 0.036
Railing 0.0414 0.0414
Fence 0.131 0.131
Sub total 0.208 0.208 0.199
Lightweight concrete 2.521 2.521 2.592
Reinforcement steel 0.173 0.173 0.173
Stringers 0.544 0.544 0.682
Bracings 0.154 0.154
Sub total 3.392 3.392 3.447
Floor Truss 0.41 0.41
Inspection walkway 0.098 0.098
Inspection rail 0.052 0.052
Wind shoe 0.008 0.008
Bridge floor
Sub total 0.568 0.568 0.613
Top chord 0.313 0.313 0.315
Bottom chord 0.307 0.291 0.302
Gusset plate, splice 0.234 0.234 0.124
Web ( diagonal) 0.162 0.166 0.142
Post ( vertical) 0.055 0.055 0.053
Strut, rivet, bolt etc 0.007 0.007 0.007
Stiffening truss
Sub total 1.078 1.066 0.943
K- truss 0.161 0.154 0.159
Lateral system
Sub total 0.161 0.154 0.159
Cable 0.971 0.971 1.025
Suspenders 0.066 0.065 0.054
Cable
Sub total 1.037 1.036 1.079
Cable and suspender weight 1.037 1.036 1.079
Suspended structure weight 5.905 5.886 5.564
Total weight 6.942 6.922 7.170
For SI: 1 kip/ ft = 14.593 kN/ m
16
Table 2.2 Calculated nodal coordinates of the cable- only system
Y ( ft) Z ( ft)
Preliminary Final X ( ft) Preliminary Final
configuration configuration configuration configuration
Remark
1256.500 29.5833 29.5833 163.1400 163.1400
Cable
bent
1221.840 29.7626 29.7628 172.5984 172.6094
1190.780 29.9295 29.9297 181.5906 181.6075
1159.720 30.1039 30.1042 191.0963 191.1143
1128.660 30.2867 30.2869 201.1168 201.1324
1097.600 30.4782 30.4784 211.6536 211.6664
1066.540 30.6787 30.6787 222.7081 222.7190
1035.480 30.8882 30.8882 234.2820 234.2919
1004.420 31.1069 31.1069 246.3771 246.3866
973.360 31.3349 31.3349 258.9949 259.0050
942.300 31.5722 31.5723 272.1375 272.1489
911.240 31.8188 31.8190 285.8068 285.8202
880.180 32.0749 32.0751 300.0048 300.0211
849.120 32.3404 32.3407 314.7336 314.7525
818.060 32.6154 32.6157 329.9955 330.0132
787.000 32.8998 32.9000 345.7927 345.8045
750.000 33.2500 33.2500 365.2600 365.2600 Tower
714.380 32.9474 32.9482 351.2700 351.3089
683.320 32.6931 32.6946 339.6119 339.6821
652.260 32.4485 32.4505 328.4872 328.5843
621.200 32.2136 32.2161 317.8940 318.0124
590.140 31.9885 31.9913 307.8302 307.9643
559.080 31.7732 31.7762 298.2938 298.4394
528.020 31.5678 31.5710 289.2832 289.4360
496.960 31.3723 31.3756 280.7965 280.9516
465.900 31.1870 31.1901 272.8323 272.9846
434.840 31.0117 31.0147 265.3889 265.5335
403.780 30.8468 30.8495 258.4649 258.5980
372.720 30.6922 30.6946 252.0591 252.1778
341.660 30.5482 30.5502 246.1702 246.2718
310.600 30.4149 30.4165 240.7971 240.8811
279.540 30.2925 30.2937 235.9389 236.0069
248.480 30.1812 30.1821 231.5945 231.6482
217.420 30.0812 30.0819 227.7631 227.8042
17
Table 2.2 Calculated nodal coordinates of the cable- only system ( contd.)
Y ( ft) Z ( ft)
Preliminary Final X ( ft) Preliminary Final
configuration configuration configuration configuration
Remark
186.360 29.9930 29.9934 224.4441 224.4743
155.300 29.9168 29.9170 221.6367 221.6577
124.240 29.8531 29.8531 219.3405 219.3539
93.180 29.8024 29.8023 217.5550 217.5626
62.120 29.7655 29.7653 216.2799 216.2833
31.060 29.7429 29.7427 215.5150 215.5158
0.000 29.7353 29.7350 215.2600 215.2600 Center
For SI: 1 ft = 0.3048 m
0
50
100
150
200
250
300
350
400
- 1500 - 1000 - 500 0 500 1000 1500
Length ( ft)
Height ( ft)
Figure 2.1 The shape of the initial cable profile under dead load
Calculations of other cross- sectional properties of girder ( moment of inertia and torsional
constant) are given as follows:
z
x
y
18
2.4.1 Moment of Inertia ( Iz)
Moment of inertia of various members is computed from the equations in the table below
and their values are given following the table.
Chord Slab Stringer
( / 2) 2 2
I = A× e × y
12
3 bh
Iy = 
=
=
4
1
2
i
i i Iy A d
Chord:
side span = 2 2 2 2 55.56in × 29.585 × 2ea× 2( both) = 194,520in ft
center span = 2 2 2 2 53.78in × 29.585 × 2ea× 2( both) = 188,288in ft
Slab:
Figure 2.2 shows the cross section of the deck.
Figure 2.2 Cross section of deck
4
3 3
6744.9
12
54.5 0.5
12
ft
bh
Iy =
×
= =
For equilibrant steel section: 4 449.7
15
6744.9
Iy = = ft
27.25  27.25 
CL
0.5 
19
Stringer:
Figure 2.3 shows the location of stingers in one side of the deck.
Figure 2.3 Location of stringers in one side of the deck
For one side:
2 2 2 2 4
4
1
2 Iz A d 0.1389( 3.5 10.5 17.5 24.5 ) 142.93 ft
i
i i = = + + + = 
=
So, for one stiffening girder 4 142.93 367.78
2
449.7
Iz = + = ft
From ( Abdel- Ghaffer, 1976),
slab + stringers : ( 105,000+ 290) sq. in. sq. ft./ 144/ 2= 4 365.59 ft
2.4.2 Torsional Constant ( J)
Figure 2.4 shows commonly used lateral bracing systems and stiffening girders for
suspension bridges.
i i i i J = 2 b b d ;
i vi i hi
i i vi hi
i
b d
b d
μ μ
μ μ
b
× + ×
=
2 2
2 2 2 web : A 0.117 ft ; k truss : A 0.115 ft , Av 0.132 ft d d
= − = =
b = 59.17 ´ , d= 15 ´ 0.25 2.5
2( 1 )
= \ = 
+
=
G
E E
G μ
μ
CL
0.5 
3.5  7  7  7 
20
Figure 2.4 Commonly used lateral bracing systems and stiffening girders
( Abdel- Ghaffer 1976)
To calculate the torsional constant of the suspension bridge girder two coefficients are
used. Here, h
μ is the coefficient for horizontal K- type system and V
μ is for vertical
Worren type web system. Figure 2.5 shows the horizontal K- type system and Figure 2.6
shows the for vertical Worren type web system. The procedure to calculate those two
coefficients are shown below.
Figure 2.5 Horizontal system ( K- type)
 2
59.17 
31.08 
Ad
Av
21
= = °
− ) 43.6
31.08
59.17 / 2
tan ( 1
2
a
0.134
)
0.082 0.114 sin 43.6
2 0.114 0.082 sin 43.6 cos 43.6
2.5 (
)
sin
2 sin cos
(
3
2
2
3
2 2
2
=
+ × °
× × × ° × °
= ×
+ ×
× × × ×
=
a
a a
μ
v d
d v
A A
A A
G
E
h
Figure 2.6 Vertical web system ( Worren type)
= = °
− ) 44
31.08/ 2
15
tan ( 1
1 a
sin cos 2.5 0.137 sin 44 cos44 0.119 2
1 1
2
μ = × × a × a = × × ° × ° = v Ad
G
E
0.032
59.17 0.119 15 0.134
59.17 15 0.119 0.134
2 2 2 2
=
× + ×
× × ×
=
× + ×
=
i vi i hi
i i vi hi
i
b d
b d
μ μ
μ μ
b
4 Ji 2 ibidi 2 0.032 59.17 15 56.416 ft = b = × × × =
The sectional properties computed in this section ( Section 1) are summarized in Table 2.3
below for each panel.
15'
31.08'
 1
Ad
22
Table 2.3 Calculated sectional properties of panels
Area Torsional Constant Iz Iy
Panel No.
ft 2 ft 4 ft 4 ft 4
1 0.958 24.342 369.010 39.367
2 0.931 23.021 369.010 39.367
3 0.911 22.944 369.010 38.234
4 0.941 21.304 369.010 41.676
5 0.972 21.304 369.010 43.398
6 0.902 16.515 369.010 43.398
7 0.972 21.304 369.010 43.398
8 0.938 19.141 369.010 43.398
9 0.938 19.141 369.010 43.398
10 0.938 19.141 369.010 43.398
11 0.972 21.304 369.010 43.398
12 1.010 23.308 369.010 43.398
13 0.979 23.308 369.010 41.676
14 0.948 24.704 369.010 38.234
15 0.968 24.797 369.010 39.367
16 0.968 24.797 369.010 39.367
17 0.866 19.188 369.010 39.367
18 0.866 19.188 369.010 39.367
19 0.846 19.141 369.010 38.234
20 0.811 16.523 369.010 38.234
21 0.811 16.523 369.010 38.234
22 0.841 16.523 369.010 39.957
23 0.824 15.034 369.010 39.957
24 0.824 15.034 369.010 39.957
25 0.824 15.034 369.010 39.957
26 0.824 15.034 369.010 39.957
27 0.855 15.034 369.010 41.680
28 0.907 19.141 369.010 41.680
29 0.907 19.141 369.010 41.680
30 0.968 19.141 369.010 45.117
31 0.968 19.141 369.010 45.117
32 0.968 19.141 369.010 45.117
33 0.968 19.141 369.010 45.117
34 0.916 15.034 369.010 45.117
35 0.916 15.034 369.010 45.117
23
Table 2.3 Calculated sectional properties of panels ( contd.)
Area Torsional Constant Iz Iy
Panel No.
ft 2 ft 4 ft 4 ft 4
36 0.916 15.034 369.010 45.117
37 0.916 15.034 369.010 45.117
38 0.916 15.034 369.010 45.117
39 0.916 15.034 369.010 45.117
40 0.916 15.034 369.010 45.117
37 0.916 15.034 369.010 45.117
41 0.916 15.034 369.010 45.117
42 0.916 15.034 369.010 45.117
43 0.916 15.034 369.010 45.117
44 0.916 15.034 369.010 45.117
45 0.916 15.034 369.010 45.117
46 0.916 15.034 369.010 45.117
47 0.916 15.034 369.010 45.117
48 0.968 19.141 369.010 45.117
49 0.968 19.141 369.010 45.117
50 0.968 19.141 369.010 45.117
51 0.907 19.141 369.010 41.680
52 0.907 19.141 369.010 41.680
53 0.907 19.141 369.010 41.680
54 0.824 15.034 369.010 39.957
55 0.824 15.034 369.010 39.957
56 0.824 15.034 369.010 39.957
57 0.824 15.034 369.010 39.957
58 0.841 16.523 369.010 39.957
59 0.811 16.523 369.010 38.234
60 0.811 16.523 369.010 38.234
61 0.846 19.141 369.010 38.234
62 0.866 19.188 369.010 39.367
63 0.866 19.188 369.010 39.367
64 0.968 24.797 369.010 39.367
65 0.968 24.797 369.010 39.367
66 0.948 24.704 369.010 38.234
67 0.979 23.308 369.010 41.676
68 1.010 23.308 369.010 43.398
69 0.972 21.304 369.010 43.398
24
Table 2.3 Calculated sectional properties of panels ( contd.)
Area Torsional Constant Iz Iy
Panel No.
ft 2 ft 4 ft 4 ft 4
70 0.938 19.141 369.010 43.398
71 0.938 19.141 369.010 43.398
72 0.938 19.141 369.010 43.398
73 0.972 21.304 369.010 43.398
74 0.902 16.515 369.010 43.398
75 0.972 21.304 369.010 43.398
76 0.941 21.304 369.010 41.676
77 0.911 22.944 369.010 38.234
78 0.931 23.021 369.010 39.367
79 0.958 24.342 369.010 39.367
80 0.958 24.342 369.010 39.367
For SI: 1 ft = 0.3048 m
Calculation of tower cross- sectional properties:
For thin walled closed sections the torsional constant is given by the following formula
( Bredt’s formula):

=
t
ds
A
J
2 4
Different sections of the tower is shown in Figure 2.7 and a typical plan view of the tower
section is shown in Figure 2.8. Table 2.4 and 2.5 show the calculated sectional properties
of the tower section at different heights for before and after retrofit models respectively.
2.5 Member- Based Detail Model
Finite Element modeling of the detailed structure is done with the help of SAP 2000 V10
( Computer and Structures, 2002). The cables and suspenders are modeled as 3D elastic
truss elements. The chords, vertical members and the diagonal members in the stiffening
25
Figure 2.7 Different sections of the tower
Figure 2.8 Typical tower cross- section
X
Y
1
2
3
4
42.89  5
52.58 
52.58 
85.50 
85.66 
26
Table 2.4 Calculated sectional properties of the tower sections ( before retrofit)
Area Ix Iy Torsional Constant
Section No.
ft 2 ft 4 ft 4 ft 4
1 3.18 20.42 21.07 17.81
2 4.35 42.75 48.25 25.69
3 4.92 57.64 65.06 26.89
4 4.93 60.32 65.86 29.14
5 5.47 76.11 90.37 34.20
For SI: 1 ft = 0.3048 m
Table 2.5 Calculated sectional properties of the tower sections ( after retrofit)
Area Ix Iy Torsional Constant
Section No.
ft 2 ft 4 ft 4 ft 4
1 3.66 23.48 24.23 23.55
2 5.00 49.16 55.49 33.98
3 5.66 66.29 74.82 35.57
4 5.67 69.37 75.74 38.54
5 6.29 87.53 103.93 45.23
For SI: 1 ft = 0.3048 m
girder are modeled as 3D truss elements. Also members in the diaphragm are modeled as
truss elements. The tower, the cable bent leg, and strut members are modeled as frame
elements. The reinforced- concrete deck is modeled as shell element and the supporting
stringers are modeled as beam elements.
Hydraulic, viscous dampers between tower and the suspended structure are also
modeled according to their properties mentioned in the design drawing. Mass is taken
distributed over each and every member. To consider the mass of non- structural
components, equivalent point mass and mass moment of inertia are distributed at joints in
the diaphragm.
The most important structural components that are considered for post- retrofit
modeling are suspended truss system, deck shear connectors, cable bent cross- sections,
27
suspenders and dampers installed. Figure 2.9 shows detailed model of one panel and
construction drawing.
Figure 2.9 The detailed model of one panel
2.5.1 Cable Bent
Four feet of stiffening truss in the cable bent was removed to allow free oscillations of the
side spans of the bridge. Also, the cable bent cross section was changed. This change in
the cross section is considered in the post- retrofit modeling of the bridge. Cross sectional
properties of the modified sections are calculated and used in the post- retrofit analysis.
2.5.2 Deck Shear Connector
Deck shear connectors were replaced with new types. Deck shear connectors of the
original structure were removed and then a new set was introduced. Figures 2.10 and 2.11
( taken from Design Drawing) shows the comparison between the shapes of the deck
shear connectors before and after retrofit. The FE modeling is done according to this
design drawing.
ECLAEBMLEENT TSRTIUFSFSENING
CABEHLLEAE NNMGOEEDNRETS
28
Figure 2.10 Deck shear connector ( before
retrofit)
Figure 2.11 Deck shear connector ( after
retrofit)
2.5.3 Dampers
Total of 48 dampers were installed in the bridge as a retrofit measure with 16 dampers
installed in each tower, at the junction between tower and girder connection. In each
cable bent, 4 dampers were installed. In the middle of each side span a new diaphragm
was inserted. At the location of the inserted diaphragm, 4 more dampers were installed in
each side span. These 8 dampers in the side spans were non- linear dampers having the
form of F = cvn where n = 0.5. For all other dampers, n = 1.0 is used.
2.5.4 Suspended Truss
The suspended truss structure was modified by inserting new members and also replacing
some members in the K- truss in the middle span as well as in the side spans. Figure 2.12
( taken from Design Drawing) shows the modifications made in the K- truss.
2.5.5 Suspenders
Some suspenders in the middle span were replaced with new suspenders. Figure 2.13
( taken from Design Drawing) describes the modified suspenders in the middle span.
29
Figure 2.12 K- truss modifications after retrofit
Figure 2.13 Suspender modifications after retrofit
2.6 Eigen Value Analysis
First 100 eigen vectors were calculated with a convergence tolerance of 1.0 10 5 −
× . Table
2.6 shows the comparison of modal frequencies obtained from before retrofit panel- based
simple model and member- based detailed model with analytical eigen properties of the
bridge obtained by previous researchers. Table 2.7 shows the aforementioned comparison
of results obtained from after retrofit model of the bridge, In the tables the computed
modal frequencies were obtained from the FE models by using SUCOT ( Kim, 1993) and
30
SAP 2000 V 10 ( Computers and Structures, 2002). It can be seen from the results that
the computed modal frequencies obtained from the SUCOT and the SAP 2000 panel
based model are having a good match with the calculated frequencies from finite element
models developed by previous researchers. First three modes obtained from the SAP
2000 model are shown in Figure 2.14.
Figure 2.14 First three mode shapes of the simple model
Table 2.6 Comparison of modal frequencies in Hz ( before retrofit)
Present study
Panel- based
simple
Member-based
detailed
Dominant
Motion
Abdel-
Ghaffar,
1976
Niazy et
al., 1991
SUCOT
SAP
2000
SAP
2000
L *- S * 0.173 0.169 0.159 0.152 0.161
V *- AS * 0.197 0.201 0.210 0.223 0.221
V- S 0.221 0.224 0.232 0.239 0.226
V- S 0.348 0.336 0.460 0.384 0.363
V- AS 0.346 0.344 0.456 0.495 0.369
L- AS 0.565 0.432 0.472 0.448 0.503
T *- S 0.449 0.438 0.483 0.482 0.477
V- S 0.459 0.442 0.500 0.538 0.479
* L: Lateral, S: Symmetric, V: Vertical, AS: Asymmetric and T: Torsional
31
Table 2.7 Comparison of modal frequencies in Hz ( after retrofit)
Ingham Present Study
et al., 1997
( ADINA)
Fraser,
2003
( ADINA)
Panel- based Simple
Member-based
Detailed
Dominant
motion
Simple Detailed Detailed SUCOT
SAP
2000
SAP 2000
L- S 0.162 0.135 0.130 0.161 0.152 0.160
V- AS 0.197 0.171 0.182 0.210 0.218 0.220
V- S 0.232 0.229 0.226 0.232 0.235 0.226
V- S - - - 0.360 0.369 0.362
V- AS - - - 0.453 0.469 0.372
L- AS 0.535 0.420 0.409 0.473 0.447 0.494
T- S 0.588 0.510 0.511 0.490 0.484 0.482
V- S - - - 0.498 0.513 0.486
* L: Lateral, S: Symmetric, V: Vertical, AS: Asymmetric and T: Torsional
2.7 Closure
In this chapter numerical modeling has been achieved for Vincent Thomas Bridge. A
member- based detailed three- dimensional Finite Element ( FE) as well as a panel- based
simplified model of the Vincent Thomas Bridge have been developed for the bridge
before and after retrofit. First eight modal frequencies obtained from FE models
developed using different commercially available softwares have been compared. The
results obtained from this study are also compared with previous results obtained for the
bridge. It has been observed that the first lateral modal frequency for the member- based
detailed model is 20% higher than those presented in previous studies. It is also found
32
that results of panel- based simple models are in good agreement with those obtained from
the detailed model and those reported in previous similar studies.
33
CHAPTER 3
SYSTEM IDENTIFICATION AND MODEL VERIFICATION
3.1 Background
To ensure the validity of the analytical finite element model of a massive structure like a
suspension bridge, the eigenproperties obtained from the analytical model should be
compared with the identified modal frequencies. Modal identification can be done by
using measured ambient vibration response or strong motion earthquake response of the
bridge. The response of the bridge, under external excitations, is measured with the help
of acceleration measuring sensors installed at different locations of the bridge.
3.2 Evaluation of Eigenproperties using Ambient Vibration Data
Experimental modal analysis has drawn significant attention from structural engineers for
updating the analysis model and estimating the present state of structural integrity.
Forced vibration tests such as impact tests can be carried out to this end. However, it is
usually restricted to small- size structures or to their components. For large structures such
as dams, and long- span bridges, ambient vibration tests under wind, wave, or traffic
loadings are the effective alternatives. In this study, modal parameters were obtained
using the frequency domain decomposition technique ( Otte et al, 1990 and Brincker et al.,
2000) which is one of the frequency domain methods without using input information.
It is very difficult, if not impossible, to identify closely spaced modes using the
34
peak- picking ( PP) method. In this case, the frequency domain decomposition ( FDD)
method that utilizes the singular value decomposition of the PSD matrix may be used to
separate close modes ( Brincker et al., 2000). The method was originally used to extract
the operational deflection shapes in mechanical vibrating systems ( Otte et al, 1990). The
natural frequencies are estimated from the peaks of the PSD functions in the PP method.
On the other hand, they are evaluated from singular value ( SV) functions of the PSD
matrix in the FDD method.
( ) ( ) ( ) ( )
T
S yy w = U w s w V w ( 3.1)
where ( ) m m N N
yy S R ×
w Î is the PSD matrix for output responses ( ) m N y t Î R , ( ) m m N N s R ×
w Î
is a diagonal matrix containing the singular values of its PSD matrix, and, U ( w ) ,
( ) m m N N V R ×
w Î are corresponding unitary matrices. m N is the number of measuring points.
The general multi- DOF system can be transformed to the single DOF system nearby its
natural frequencies by singular value decomposition. The mode shape can be estimated as
the first column vector of the unitary matrix of U since the first singular value may
include the structural mode nearby its natural frequencies. However in the closely spaced
modes, the peak of largest singular values at one natural frequency indicates the structural
mode and adjacent second singular value may indicate the close mode.
Figure 3.1 shows the layout of the acceleration sensors installed in the bridge site.
Table 3.1 describes the location and direction of all the accelerometers present in the
bridge site. Figure 3.2 shows the vertical accelerometers and Figure 3.3 shows the lateral
35
accelerometers used in the modal identification of the bridge structure. Figure 3.4 shows
the plot of SV vs. frequency for the acceleration data obtained from vertical channels.
Figure 3.1 Location and direction of sensors installed in the bridge
Table 3.1 Location and direction of accelerometers
Sensor Number Sensor Location Sensor Direction
22, 15, 16, 17, 18, 21 Truss top/ Deck Vertical
2, 4, 5, 6, 7 Truss top/ Deck Lateral
12 Truss top/ Deck Longitudinal
3 Truss bottom Lateral
8 Tower Lateral
10, 11 Tower Longitudinal
14, 19, 20 Tower base Vertical
1, 9 Tower base Lateral
13, 23 Tower base Longitudinal
26 Anchorage Vertical
24 Anchorage Lateral
25 Anchorage Longitudinal
36
Figure 3.2 Vertical accelerometer data used in the study
Figure 3.3 Lateral accelerometer data used in the study
37
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
x 10
4
Frequency ( Hz)
SV
SV of vertical acc.( CH- 15- 22)
Figure 3.4 Plot of SV vs. Frequency
3.3 Comparison of System ID Result with Analytical Eigen Properties
In this study, modal parameters have been obtained using the frequency domain
decomposition ( FDD) technique ( Brincker et al. 2000) which is one of the frequency
domain methods without using in- put information. The method utilizes the singular value
decomposition of the PSD matrix and may be used to separate close modes. Total 15
ambient vibration recording has been used for this purpose from the installed sensors.
The data were recorded from April, 2003 to October, 2004, over 1 year 6 months record
has been considered for system identification analysis. Average identified modal
frequencies obtained from 15 data- set are considered as final identified modal
frequencies from the ambient vibration data. Figure 1 shows previously installed sensor
locations on the bridge. For system ID from ambient vibration data, vertical sensors 15,
16, 17, 18, 21, 22 and lateral sensors 4, 5, 6, 7 are used. Sensor # 3 in the lateral direction
is excluded because it provided some noisy data.
Table 3.2 and 3.3 below shows the comparison of modal frequencies before and after
retrofitting of the bridge. Modal identification results from ambient vibration data are
also tabulated in Table 3.3. It can be seen from Table 3.3 that in the first mode of
38
vibration, the structure is a little bit stiffer in the simple model rather than detailed model.
In case of first mode of vibration the system ID result matches with the frequency
obtained from the detailed model. Also, from the second mode and above both the
analytical and system ID results shows pretty good match. On an average sense, it can be
seen from Table 3.3 that system ID results show pretty good match with detailed model.
3.4 Modal Parameter Identification from Chino Hills Earthquake Response
Chino Hills earthquake data recorded at the bridge site are also used in the modal
identification. Chino Hills earthquake occurred on July 29, 2008, in Southern California.
The epicenter of the magnitude 5.4 earthquake was in Chino Hills, approximately 45 km
east- southeast of downtown Los Angeles. Table 3.4 compares the modal frequencies of
the bridge obtained from ambient vibration and Chino Hills earthquake data. These two
identified frequencies matches very well. Note also that the two other previous studies
( Ingham et al. 1997 and Fraser 2003) involving detailed models under predict modal
frequencies significantly for the first two modes. Results from these two studies are also
tabulated in Table 3.4.
39
Table 3.2 Comparison of modal frequencies in Hz ( before retrofit)
Identified
( System ID)
Computed
Present Study
Panel- based
Simple
Member-based
Detailed
Dominant
Motion
Abdel-
Ghaffar
and
Housner,
1977
( Ambient)
Niazy
et al.,
1991
( Whittier)
Ingham
et al., 1997
( Northridge)
Abdel-
Ghaffar,
1976
Niazy
et al.,
1991
SUCOT
SAP
2000
SAP
2000
L *- S * 0.168 0.149 0.145 0.173 0.169 0.159 0.152 0.161
V *- AS * 0.216 0.209 - 0.197 0.201 0.210 0.223 0.221
V- S 0.234 0.224 0.222 0.221 0.224 0.232 0.239 0.226
V- S 0.366 0.363 0.370 0.348 0.336 0.460 0.384 0.363
V- AS - 0.373 - 0.346 0.344 0.456 0.495 0.369
L- AS 0.623 0.459 0.417 0.565 0.432 0.472 0.448 0.503
T *- S 0.494 0.513 0.556 0.449 0.438 0.483 0.482 0.477
V- S 0.487 0.448 - 0.459 0.442 0.500 0.538 0.479
* L: Lateral, S: Symmetric, V: Vertical, AS: Anti- Symmetric, T: Torsional
Table 3.3 Comparison of modal frequencies in Hz ( after retrofit)
Identified
( System ID)
Computed
Ingham Present Study
et al., 1997
( ADINA)
Fraser,
2003
( ADINA)
Panel- based
Simple
Member-based
Detailed
Dominant
Motion Fraser,
2003
He
et al.,
2008
Present
Study
( Ambient)
Simple Detailed Detailed SUCOT
SAP
2000
SAP
2000
L- S 0.150 - 0.162 0.162 0.135 0.130 0.161 0.152 0.160
V- AS - 0.168 0.219 0.197 0.171 0.182 0.210 0.218 0.220
V- S 0.233 0.224 0.229 0.232 0.229 0.226 0.232 0.235 0.226
V- S 0.367 0.356 0.369 - - - 0.360 0.369 0.362
V- AS - - - - - - 0.453 0.469 0.372
L- AS - - 0.534 0.535 0.420 0.409 0.473 0.447 0.494
T- S - 0.483 0.471 0.588 0.510 0.511 0.490 0.484 0.482
V- S - - - - - - 0.498 0.513 0.486
* L: Lateral, S: Symmetric, V: Vertical, AS: Anti- Symmetric, T: Torsional
40
Table 3.4 Comparison of modal frequencies ( in Hz) of the Vincent Thomas Bridge
Identified
( System ID)
Computed
Mode
Number
Dominant
Motion Ambient
Vibration
Chino
Hills
Earthquake
SAP 2000
( Present
Study)
Ingham
et al.,
1997
Fraser,
2003
1 L *- S * 1 0.162 0.168 0.160 0.135 0.130
2 V *- AS * 1 0.219 - 0.220 0.171 0.182
3 V- S1 0.229 0.228 0.222 0.229 0.226
4 V- S2 0.369 0.362 0.362 - -
5 V- AS2 - 0.467 0.372 - -
6 T *- S1 0.471 0.491 0.478 0.510 0.511
7 V- S3 - - 0.483 - -
8 L- AS1 0.534 - 0.491 0.420 0.409
* L: Lateral, S: Symmetric, V: Vertical, AS: Anti- Symmetric, T: Torsional
3.5 Effect of Parameter Uncertainty on Modal Frequency
3.5.1 Soil Spring Modeling
To consider the effect of soil structure interaction kinematic three translational and three
rotational soil springs with their coupling effects are considered at the foundations of east
tower, west tower, east cable bent, west cable bent, east anchorage and west anchorage.
The stiffness of the soil springs are calculated from the equivalent pile group stiffness at
the foundations discussed earlier. Table 3.5 gives the number of piles at different
foundations considered for the FE model of the bridge. Figure 3.5 shows the finite
element model of the bridge with foundation springs.
41
Table 3.5 Location and number of piles considered
Location Number of piles
East tower 167
West tower 167
East cable bent 48
Wast cable bent 48
East anchorage 188
West anchorage 188
Figure 3.5 Detailed model in SAP 2000 with foundation springs
3.5.2 Uncertain Parameters Considered
For model updating purpose, in ideal case, all parameters related to elastic, inertial
properties and boundary conditions should be considered. However, if too many
parameters are considered for model updating then chances of obtaining unreliable model
increases ( Zhang et al., 2001). For this reason, parameter selection is a very important
East tower
West tower
East cable bent
East anchorage
42
task in model updating process. Practically if the parameters considered do not have
much effect on the modal frequencies and mode shapes, then they should be excluded
from the list. Therefore a comprehensive eigenvalue sensitivity study is performed to
figure out the most sensitive parameters to be considered for suspension bridge finite
element model calibration.
Total 19 parameters are considered for the sensitivity analysis. The selection of
these parameters is based on the outcome of previous research ( Zhang et al., 2001) and
engineering judgments. Elastic modulus and mass density of different set of structural
members, boundary conditions ( deck and tower connection and deck and cable bent
connection) and stiffness of the soil springs are considered as variable parameters.
However the cable and the concrete deck have homogeneous properties, but due to
corrosion the structural strength may get decreased over the service life of the bridge. To
capture that effect, elastic modulus and mass density of cable and concrete deck is
considered as variable parameters in the analysis. Also, for the generality of the analysis
kinematic spring stiffnesses ( soil spring stiffness) are also considered as variable
parameters in the analysis. Since there was no tower dominant mode in the considered
first 8 mode shapes, therefore, the stiffness and inertial properties of the tower is not
considered as a variable parameter in the present study.
For evaluating the effect of uncertainty in the modal parameters of Vincent Thomas
Bridge, uncertainty associated with elastic and inertial property of different members is
represented by assigning a mean and standard deviation in terms of coefficient of
variation for each parameter. The mean values considered here are calculated based on
the design drawing of the bridge. Table 3.6 lists these parameters with their mean values.
43
To asses the sensitivity, coefficients of variation ( COV) of all the parameters are
considered as 10%. In the analysis, all the 36 values of the spring stiffness matrices are
varied by 10% for the case of east tower, west tower, east cable bent and west cable bent.
For the first- order second- moment ( FOSM) analysis only lateral translational stiffness of
each foundation spring is considered.
Table 3.6 Parameters considered for sensitivity analysis
Serial
Number
Parameters Mean Value
1 Side link elastic modulus 2.00 × 10 8 kPa
2 Cable bent and girder connection elastic modulus 2.00 × 10 8 kPa
3 Top Chord Elastic Modulus 2.00 × 10 8 kPa
4 Top Chord Mass Density 7.85 kg/ m 3
5 Bottom Chord Elastic Modulus 2.00 × 10 8 kPa
6 Bottom Chord Mass Density 7.85 kg/ m 3
7 Stringer Elastic Modulus 2.00 × 10 8 kPa
8 Stringer Mass Density 7.85 kg/ m 3
9 Deck Slab Elastic Modulus 2.48 × 10 7 kPa
10 Deck Slab Mass Density 1.48 kg/ m 3
11 Main Cable Elastic Modulus 1.66 × 10 8 kPa
12 Main Cable Mass Density 8.37 kg/ m 3
13 Suspender Elastic Modulus 1.38 × 10 8 kPa
14 Suspender Mass Density 7.85 kg/ m 3
15 Wind Shoe Elastic Modulus 2.00 × 10 8 kPa
16 East Tower Spring 1.30 × 10 6 kPa
17 East Cable Bent Spring 7.35 × 10 6 kPa
18 West Tower Spring 1.19× 10 6 kPa
19 West Cable Bent Spring 4.65× 10 6 kPa
3.5.3 Analysis methods
Reduction of the number of uncertain parameters cuts down the computational effort and
cost. One way of doing this is to identify those parameters with associated ranges of
uncertainty that lead to relatively insignificant variability in response and then treating
these as deterministic parameters by fixing their values at their best estimate, such as the
mean. For ranking uncertain parameters according to their sensitivity to desired response
44
parameters, there are various methods such as tornado diagram analysis, first order
second moment ( FOSM) analysis, and Monte Carlo simulation ( Porter et al. 2002, Lee
and Mosalam 2006). Monte Carlo simulation, which is computationally demanding due
to the requirement of a large number of simulations, especially for a model consisting of
a large number of degrees of freedom as in the case here, is not used in this study because
of these practical considerations. Instead, the tornado diagram analysis and the FOSM
analysis have been used here due to their simplicity and efficiency to identify sensitivity
of uncertain parameters.
For the tornado diagram analysis, all uncertain parameters are assumed as random
variables, and for each of these random variables, two extreme values the 84 th percentile
and 16 th percentile corresponding to assumed upper and lower bounds, respectively, of its
probability distribution have been selected. One can observe that these extreme values
come from the normal distribution assumption, mean + standard deviation and mean –
standard deviation, respectively representing their upper and lower bounds. Using these
two extreme values for a certain selected random variable, the modal frequencies of the
model has been evaluated for both cases, while all other random variables have been
assumed to be deterministic parameter with values equal to their mean value. The
absolute difference of these two modal frequency values corresponding to the two
extreme values of that random variable, which is termed as swing of the modal frequency
corresponding to the selected random variable, is calculated.
This calculation procedure has then been repeated for all random variables in question.
Finally, these swings have been plotted in a figure from the top to the bottom in a
descending order according to their size to demonstrate the relative contribution of each
45
variable to the specific mode under question. It is noteworthy that longer swing implies
that the corresponding variable has larger effect on the modal frequency than those with
shorter swing.
For the FOSM analysis, the modal frequency has been considered as a random
variable Y, which has been expressed as the function of random variables, Xi ( for i = 1 to
N) denoting uncertain parameters and Y is given by
( , ,..., ) 1 2 N Y = g X X X ( 3.2)
Let Xi has been characterized by mean μX and variance s X
2 . Now, the derivatives of g( X)
with respect to Xi , one can express Y by expanding Eq. ( 3.2) in Taylor series as
LLL
K
+ − − +
= + −
 

= =
=
i j
N
j
i X j X
N
i
i
N
i
X X X i X
X X
g
X X
X
g
Y g X
i j
N i
d d
d
μ μ
d
d
μ μ μ μ
2
1 1
1
( )( )
2!
1
( )
1!
1
( , , , )
1 2
( 3.3)
Considering only the first order terms of Eq. ( 3.3) and ignoring higher order terms Y can
be approximated as
i
N
i
X X X i X X
g
Y g X
N i
d
d
μ μ μ μ 
=
» + −
1
( )
1!
1
( , , , )
1 2
K ( 3.4)
Taking expectation of both sides, the mean of Y,   μY can be expressed as
( , , , )
Y X1 X2 X N μ g μ μ μ K » ( 3.5)
Utilizing the second moment of Y as expressed in Eq. ( 3.4) and simplifying, the variance
of Y, s Y
2 can be derived as
 
= =
»
N
i
N
j j
N
i
N
Y i j X
g X X X
X
g X X X
X X
1 1
2 1 2 1 2 ( , ,..., ) ( , ,..., )
cov( , )
d
d
d
d
s
46
j
N
i
N
N
i
N
j i
X X
i
N
X
N
i X
g X X X
X
g X X X
X
g X X X
i i j
d
d
d
d
r
d
d
s
( , ,..., ) ( , ,..., ) ( , ,..., ) 1 2 1 2
1
2
2 1 2
1
  
= = ¹
+  


 


» ( 3.6)
where
i j X X
r denotes correlation coefficient for random values Xi and Xj ( i. e., coefficient
defining the degree to which one variable is related to another). The partial derivative of
( , ,..., ) 1 2 N g X X X with respect to Xi has been calculated numerically using the finite
difference equation given below
i
i i N i i N
i
N
x
g x x x x g x x x x
X
g X X X
D
+ D − − D
=
2
( , ,..., ) ( , ,..., , ) ( , ,..., , ) 1 2 1 2 1 2
μ μ
d
d
( 3.7)
In this case, a large number of simulations were performed varying each input parameter
individually to approximate the partial derivatives as given in Eq. ( 3.7). For these
calculations, the mean and the standard deviation values given in Table 3.6 are used.
For these sensitivity analyses, at first, the reference model with mean parameters of
each 19 random variable considered in this study is analyzed. Then the analyses have
been carried out using their lower and then upper bounds. Altogether 39 cases of modal
analysis are performed for each set of parameters, modal frequencies expressed
as ( , ,..., ) 1 2 N Y = g X X X is observed.
3.5.4 Sensitivity of Modal Frequencies
For tornado diagram analysis, all the 19 parameters shown in Table 3.6 are used for total
8 mode shapes. Figures 3.6 ( a- h) show tornado diagrams for 8 modes developed
according to the procedure in section 3.5.3. The vertical line in the middle of tornado
47
diagrams indicates modal frequency value calculated for a certain mode considering only
the mean values of all random variables and the length of each swing ( horizontal bar)
represents the variation in the modal frequency due to the variation in the respective
random variable.
It is clear from Figures 3.6 ( a- e) that, up to mode # 5 deck slab mass density and
bottom chord elastic modulus have almost the largest contribution in response variability.
In mode numbers 2, 3, 6, and 7, mostly vertical and torsional modes, main cable elastic
modulus is significant contributor of the response variability. One can also notice from
Figures 3.6 ( a- h) that couple of swings are asymmetric about the vertical line. This skew
of the modal frequency distributions implies that the problem is highly nonlinear. In other
words, the same amount of a positive and a negative change in these parameters does not
produce the same amount of variation in modal frequency. This skewness is very clear
for 2 nd mode in case of main cable elastic modulus variation. Since the 2 nd mode is
vertical antisymmetric, increase is main cable elastic modulus does not have much effect
on increase in frequency but decrease in the stiffness of main cable decreases the
frequency by 8% from the base model frequency.
Interestingly, deck slab stiffness has most contribution in the 1 st mode, but it does
not have any contribution in rest of the modes except the 8 th mode. Most of the boundary
condition ( P1, P2, and P15) and soil spring ( P16, P17, P18, and P19) related parameters
have very insignificant effect on response variability.
For FOSM method, analyses have been carried out to determine the sensitivity of
modal frequencies to the uncertainty in each random variable. Focus has been placed on
the variance of modal frequency when considering uncertainties of 19 input parameters.
48
Figures 3.7 ( a- h) show relative variance contributions of each parameter to the modal
frequency when the correlation, as given in the second term of Eq. ( 3.6), is neglected.
From this figure, it can be observed that the uncertainties in the deck slab mass density
and bottom chord elastic modulus contribute mostly to the variance of modal frequencies.
This is the same trend as observed from the tornado diagram analysis for all the 8 modes
considered.
49
0.156 0.157 0.158 0.159 0.160 0.161 0.162 0.163 0.164
Modal Frequency ( Hz) : Mode 1 ( LS- 1)
P 10
P 9
P 5
P 3
P 12
P 7
P 8
P 4
P 6
P 11
P 14
P 18
P 13
P 1
P 2
P 15
P 16
P 17
P 19
0.216 0.217 0.218 0.219 0.220 0.221 0.222 0.223 0.224 0.225
Modal Frequency ( Hz) : Mode 2 ( VAS- 1)
P 1 0
P 5
P 1 2
P 1 1
P 8
P 4
P 6
P 3
P 9
P 1 4
P 7
P 1 3
P 1 6
P 1
P 2
P 1 5
P 1 7
P 1 8
P 1 9
( a)
( b)
P10
P5
P12
P11
P8
P4
P6
P3
P9
P14
P7
P13
P16
P1
P2
P15
P17
P18
P19
50
0.218 0.220 0.222 0.224 0.226 0.228
Modal Frequency ( Hz) : Mode 3 ( VS- 1)
P 1 0
P 1 1
P 5
P 1 2
P 8
P 4
P 6
P 3
P 9
P 1 4
P 7
P 1 3
P 1
P 2
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
0.354 0.356 0.358 0.360 0.362 0.364 0.366 0.368 0.370
Modal Frequency ( Hz) : Mode 4 ( VS- 2)
P 1 0
P 5
P 1 2
P 3
P 8
P 4
P 6
P 9
P 1 4
P 7
P 1 3
P 1 6
P 1
P 1 1
P 1 8
P 2
P 1 5
P 1 7
P 1 9
0.364 0.366 0.368 0.370 0.372 0.374 0.376 0.378 0.380
Modal Frequency ( Hz) : Mode 5 ( VAS- 2)
P 1 0
P 5
P 1 2
P 3
P 8
P 4
P 6
P 9
P 1 4
P 7
P 1 1
P 1 3
P 1 8
P 1 6
P 1
P 2
P 1 5
P 1 7
P 1 9
P10
P11
P5
P12
P8
P4
P6
P3
P9
P14
P7
P13
P1
P2
P15
P16
P17
P18
P19
P10
P5
P12
P3
P8
P4
P6
P9
P14
P7
P13
P16
P1
P11
P18
P2
P15
P17
P19
P10
P5
P12
P3
P8
P4
P6
P9
P14
P7
P11
P13
P18
P16
P1
P2
P15
P17
P19
( c)
( e)
( d)
51
0.468 0.470 0.472 0.474 0.476 0.478 0.480 0.482 0.484 0.486 0.488
Modal Frequency ( Hz) : Mode 6 ( TS- 1)
P 1 1
P 1 2
P 1 0
P 6
P 4
P 9
P 3
P 8
P 5
P 7
P 1 4
P 1 3
P 1 6
P 1
P 1 8
P 1 5
P 1 7
P 1 9
0.474 0.476 0.478 0.480 0.482 0.484 0.486 0.488 0.490 0.492
Modal Frequency ( Hz) : Mode 7 ( VS- 3)
P 1 0
P 1 1
P 1 2
P 5
P 8
P 3
P 4
P 6
P 9
P 1 4
P 7
P 1 3
P 1 5
P 1 6
P 1 8
P 1
P 2
P 1 7
P 1 9
0.479 0.482 0.485 0.488 0.491 0.494 0.497 0.500 0.503
Modal Frequency ( Hz) : Mode 8 ( LAS- 1)
P 1 2
P 1 0
P 9
P 5
P 7
P 3
P 8
P 1 4
P 4
P 1 1
P 6
P 1 8
P 1 6
P 1
P 2
P 1 3
P 1 5
P 1 7
P 1 9
P11
P12
P10
P6
P4
P9
P3
P8
P5
P7
P14
P13
P16
P1
P2
P18
P15
P17
P19
P10
P11
P12
P5
P8
P3
P4
P6
P9
P14
P7
P13
P15
P16
P18
P1
P2
P17
P19
P12
P10
P9
P5
P7
P3
P8
P14
P4
P11
P6
P18
P16
P1
P2
P13
P15
P17
P19
Figure 3.6 Tornado diagram considering 19 parameters
( f)
( g)
( h)
52
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.050
0.002
0.475
0.236
0.021
0.029
0.006
0.096
0.009
0.075
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Relative variance Modal Frequency ( Hz) : Mode 1 ( LS- 1)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.091
0.055
0.601
0.002
0.027
0.000
0.009
0.197
0.009
0.009
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Relative variance Modal Frequency ( Hz) : Mode 2 ( VAS- 1)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.068
0.259
0.528
0.001
0.023
0.000
0.009
0.099
0.009
0.005
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Relative variance Modal Frequency ( Hz) : Mode 3 ( VS- 1)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
( a)
( b)
( c)
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
53
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.064
0.000
0.467
0.003
0.020
0.000
0.006
0.406
0.007
0.025
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Relative variance Modal Frequency ( Hz) : Mode 4 ( VS- 2)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
0.000
0.000
0.000
0.000
0.000
0.001
0.000
0.088
0.000
0.523
0.004
0.020
0.000
0.007
0.322
0.008
0.027
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Relative variance Modal Frequency ( Hz) : Mode 5 ( VAS- 2)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.237
0.480
0.155
0.027
0.007
0.003
0.040
0.003
0.029
0.018
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Relative variance Modal Frequency ( Hz) : Mode 6 ( TS- 1)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
( d)
( e)
( f)
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
54
0.000
0.000
0.000
0.000
0.000
0.001
0.000
0.098
0.372
0.423
0.004
0.017
0.000
0.006
0.059
0.007
0.014
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Relative variance Modal Frequency ( Hz) : Mode 7 ( VS- 1)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
0.000
0.000
0.000
0.000
0.000
0.003
0.000
0.774
0.001
0.099
0.076
0.004
0.013
0.001
0.020
0.002
0.006
0.000
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Relative variance Modal Frequency ( Hz) : Mode 8 ( LAS- 1)
P 1
P 2
P 3
P 4
P 5
P 6
P 7
P 8
P 9
P 1 0
P 1 1
P 1 2
P 1 3
P 1 4
P 1 5
P 1 6
P 1 7
P 1 8
P 1 9
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
( h)
( g)
Figure 3.7 Relative variance contribution ( neglecting correlation terms) from FOSM analysis
55
3.6 Finite Element Model Updating
A detailed three- dimensional finite element ( FE) model of Vincent Thomas Bridge was
developed using the finite element analysis code ADINA 8.3. This finite element model
is composed of 3D elastic truss elements to represent the main cables and suspenders, 2D
shell elements to model the bridge deck and beam elements to model the stiffening
trusses and tower shafts. The ADINA bridge model is shown in Figure 3.8.
Figure 3.8 Three- dimensional finite element model of Vincent Thomas Bridge
For updating the original ADINA model an improved sensitivity- based parameter
updating method is employed ( Zhang et al., 2001). The method is based on the eigen
value sensitivity to some selected structural parameters that are assumed to be bounded
within some prescribed regions according to the degrees of uncertainty and variation
existing in the parameters, together with engineering judgment. The changes of these
parameters are found by solving a quadratic programming problem.
56
3.6.1 Sensitivity Based Model Updating
The structural parameters affecting the natural frequencies are selected to construct the
design parameter vector a P . The eigenvalue vector based on the designed parameters is
denoted as a
l , while the measured eigenvalue vector as m
l . The error vector is defined
as m a
d l = l − l . The updating process minimizes the error vector by changing the design
parameter vector a P . The variation of design parameter vector d p can be determined by
d l = S d p ( 3.8)
where S is the sensitivity matrix that represents the variation of natural frequencies of
the model due to the variation of design parameter vector.
The solution of Eq. ( 3.8) can be solved by the following iterative updating procedures.
p p p k k
= + d
+ 1 ( 3.9)
l = l + d l
+ k
a
k
a
1 ( 3.10)
where k
a
k p , l are the parameter vector and eigenvalue vector of FE model, respectively,
at the k- th updating step. The iterative updating is repeated until the updated eigenvalue
vector k
a
l converges to the measured eigenvalue vector m
l .
57
The criteria of convergence are used as
tolerance
f
f f
m i
m i
k
a i
i
£
−
,
, , max ( 3.11)
tolerance
k
a i
k
a i
k
a i
i
£
−
−
−
1
,
1
, , max
l
l l
( 3.12)
where k
a i f , and k
a, i
l are the i- th natural frequency and corresponding eigenvalue at k- th
update, and m i f , the measured i- th natural frequency. The following optimization
problem is applied to determine d p in Eq. ( 3.8) ( Friswell and Mottershead, 1994).
J J ( S p ) W ( S p ) p W p p
T
e
T
d l d d l d d d
2
1
2
1
min 1 2
+ = − − +
F
( 3.13)
subject to l u b £ d p £ b
The first term in right hand side of Eq. ( 3.13) represents the objective function to
minimize the error vector, while the second term to minimize the variation of design
parameter vector. e W and p W are weighting functions. The constrained optimization
solutions as outlined in Eq. ( 3.13) are incorporated into an iterative procedure as shown
in Figure 3.9 for the model updating Vincent Thomas Bridge.
58
Input:
u l b , b a
k p p
k
=
= 0,
Yes
Convergence
Criterion
Satisfy?
Compute:
u
Constrained Optimization
Determine d p
1
1
= +
= +
+
k k
p p p k k k
d
STOP
No
a p e p , W , W
k k
a
k
S
p
,
FE model
® l
®
k
l
k
l
k
u
k
b b p
b b p = −
= −
k Figure 3.9 Procedure for the sensitivity- based model updating
59
3.6.2 Selection of Modes and Parameters
3.6.2.1 Selection of Modes
Average values of the identified modal frequencies obtained from 14 different ambient
vibration data recorded at the bridge site are considered as target frequencies for further
ADINA model updating. Those 14 ambient vibration data were recorded from April,
2003 to October 2004. In the study, it is decided to select 8 modes to be matched between
the updated FE analysis and the measured results. These include five vertical- dominant;
two lateral- dominant; one torsional- dominant modes of the deck. Table 3.7 shows the
modal frequencies and percentage error in modal frequencies of Initial ( original) and
Baseline ADINA model results with respect to identified frequencies obtained from the
ambient vibration measurement data.
Table 3.7 Comparison of natural frequencies
Mode Measured frequency
Updated FE model Type ( Hz) Initial Baseline Initial Baseline
1 L- S 0.161 0.131 0.148 - 18.63 - 7.83
2 V- AS1 0.221 0.206 0.210 - 6.79 - 5.02
3 V- S1 0.233 0.226 0.227 - 3.00 - 2.66
4 V- S2 0.374 0.363 0.371 - 2.94 - 0.86
5 V- S3 0.474 0.460 0.470 - 2.95 - 0.78
6 L- AS 0.476 0.411 0.462 - 13.66 - 2.90
7 T- S 0.538 0.500 0.506 - 7.06 - 6.02
8 V- AS2 0.568 0.568 0.583 0.00 2.66
Mode no.
Finite element analyzed frequencies err.(%)
60
3.6.2.2 Selection of Parameters
All possible parameters relating to the geometric, structural properties as well as the
boundary conditions should be considered for adjustment in the updating procedure.
However, if the parameters are found to have little or no effect on the targeted vibration
modes, then they can be excluded from parameters list. After removing those parameters
with very small sensitivities, total 17 different parameters are considered for this analysis.
For this purpose, a sensitivity study is done and is explained in Section 3.5. They are
summarized in Table 3.8 together with their initial estimates.
Table 3.8 Parameters selected for adjustment
Structure parameters Variations in %
Stiffening truss Top chord Elastic modulus 29000 kip/ in 2 15
Mass density 8.71E- 07 kip/ in 3 15
Bottom chord Elastic modulus 29000 kip/ in 2 20
Mass density 8.71E- 07 kip/ in 3 15
Diagonal Mass density 8.71E- 07 kip/ in 3 20
lateral brace
( k- truss)
Elastic modulus 29000 kip/ in 2 10
Mass density 1.35E- 06 kip/ in 3 20
Stringers Elastic modulus 29000 kip/ in 2 20
Mass density 9.02E- 07 kip/ in 3 10
Deck Slab Elastic modulus 2825 kip/ in 2 30
Mass density 2.01E- 07 kip/ in 3 5
Cable Main cable Initial strain 1 20
Elastic modulus 29000 kip/ in 2 20
Mass density 7.71E- 07 kip/ in 3 15
Suspender Mass density 7.65E- 07 kip/ in 3 15
Tower Elastic modulus 29000 kip/ in 2 15
Mass density 7.62E- 07 kip/ in 3 15
Initial estimation
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
61
3.6.3 Updated Results
The allowable errors permitted for the check of natural frequency convergence was
applied 6% for the general modes, while 3% for the first and second modes. If the ratio of
variation for the eigenvalue is lower than 0.1%, then the iteration is also ended. For the
cable supported bridge of which modes are closely spaced, the disorder between adjacent
modes should be critically checked. The following MAC ( Modal Assurance Criteria) is
applied to the each set of two updated natural modes ( Friswell and Mottershead, 1994).
( ) ( )
0 1
1 , , 1 , ,
2
, ,
= £ £  

= =
= MAC MAC p
l
a
l j
a
l j
p
l
e
l i
e
l i
p
t l
a
l j
e
l i
f f f f
f f
( 3.14)
If the two shape vectors a e
f , f to be compared are identical, then MAC becomes 1,
while if the two shape vectors are orthogonal, MAC becomes 0. Therefore, MAC can be
utilized to prevent disorder between the calculated and measured frequency. MAC also
provides the criteria for the reliability of the developed model after model updating. The
MACs are listed in Table 3.9. The differences between the measured and the calculated
frequencies for the initial and the final updated FE modes are showed in Figure 3.10.
Table 3.10 shows the natural frequencies of the baseline model and updated model. For
most of the modes, the discrepancies between measured frequencies and updated
frequencies decreased less than 3%, while a few modes such as the first lateral frequency
shows about 4% discrepancy. However, the discrepancy between measured and baseline
model was about 19% and the current updating decreases the error in amount of 4%.
62
Table 3.9 MAC matrix of updated FE model
#
1 2 3 4 5 6 7 8 1 0.542 0.015 0.007 0.000 0.000 0.000 0.000 0.000
2 0.001 0.531 0.000 0.000 0.000 0.000 0.000 0.000
3 0.003 0.009 0.490 0.000 0.000 0.000 0.000 0.000
4 0.000 0.000 0.000 0.500 0.000 0.000 0.000 0.000
5 0.000 0.000 0.000 0.000 0.500 0.000 0.000 0.000
6 0.000 0.000 0.000 0.000 0.000 0.538 0.000 0.000
7 0.090 0.005 0.001 0.003 0.006 0.094 0.541 0.000
8 0.001 0.003 0.008 0.113 0.002 0.000 0.000 0.486
- 20
- 18
- 15
- 13
- 10
- 8
- 5
- 3
0
3
5
L- S1 V- AS1 V- S1 V- S2 V- S3 L- AS1 T- S1 V- AS2
Frequency differences(%)
Modes
Initial FE model
Baseline FE model
Updated FE model
Figure 3.10 Comparison of frequency differences using the initial and updated
FE models
63
Table 3.10 Comparison of natural frequencies between baseline and updated FE model
Frequency ( Hz) Frequency ( Hz) Err.(%) Frequency ( Hz) Err.(%) Frequency ( Hz) Err.(%)
1 L- S 0.161 0.131 - 18.63 0.148 - 7.83 0.155 - 4.04
2 V- AS1 0.221 0.206 - 6.79 0.210 - 5.02 0.215 - 2.90
3 V- S1 0.233 0.226 - 3.00 0.227 - 2.66 0.233 - 0.09
4 V- S2 0.374 0.363 - 2.94 0.371 - 0.86 0.373 - 0.19
5 V- S3 0.474 0.460 - 2.95 0.470 - 0.78 0.478 0.80
6 L- AS 0.476 0.411 - 13.66 0.462 - 2.90 0.487 2.25
7 T- S 0.538 0.500 - 7.06 0.506 - 6.02 0.538 - 0.04
8 V- AS2 0.568 0.568 0.00 0.583 2.66 0.587 3.31
Mode no.
Finite element analyzed frequencies
Mode
Type
Initial Identified Baseline Updated
The variations of design parameters are also important to estimate reliability and
effectiveness of updating results. The variations of design parameters are well limited in
permitted arrange that can be regarded as reasonable as shown in Table 3.11.
Table 3.11 Updated design parameters
Structure parameters Initial estimation Updated value Percent changes
Stiffening truss Top chord Elastic modulus( kip/ in 2 ) 29000 30815 6.3
Mass density( kip/ in 3 ) 8.71E- 07 7.85E- 07 - 9.8
Bottom chord Elastic modulus( kip/ in 2 ) 29000 33350 15.0
Mass density( kip/ in 3 ) 8.71E- 07 7.58E- 07 - 13.0
Diagonal Mass density( kip/ in 3 ) 8.71E- 07 7.49E- 07 - 14.0
lateral brace
( k- truss) Elastic modulus( kip/ in 2 ) 29000 29442 1.5
Mass density( kip/ in 3 ) 1.35E- 06 1.14E- 06 - 15.0
Stringers Elastic modulus( kip/ in 2 ) 29000 24650 - 15.0
Mass density( kip/ in 3 ) 9.02E- 07 8.16E- 07 - 9.5
Deck Slab Elastic modulus( kip/ in 2 ) 2825 3390 20.0
Mass density( kip/ in 3 ) 2.01E- 07 1.82E- 07 - 9.2
Cable Main cable Initial strain 1.00 1.15 15.0
Elastic modulus( kip/ in 2 ) 29000 24650 - 15.0
Mass density( kip/ in 3 ) 7.71E- 07 7.45E- 07 - 3.3
Suspender Mass density( kip/ in3) 7.65E- 07 8.41E- 07 10.0
Tower Elastic modulus( kip/ in 2 ) 29000 27931 - 3.7
Mass density( kip/ in 3 ) 7.62454E- 07 7.87E- 07 3.3
64
3.7 Closure
To demonstrate the appropriateness of the bridge models developed in the previous
chapter, eigen properties of the models are evaluated in this chapter and compared with
those of the system identification results obtained using frequency domain decomposition
technique on ambient vibration and recorded earthquake response data. After that, a
comprehensive sensitivity analysis is performed considering 19 different structural and
soil spring parameters. First eight modal frequencies are considered for the sensitivity
study. Tornado diagram and FOSM methods are applied for the sensitivity study. It is
observed that the mass density of deck slab and elastic modulus of bottom chord
contributes most to the modal frequencies of the bridge. This kind of study will be very
helpful in selecting parameters and their variability ranges for FE model updating of
suspension bridges.
In this study, a sensitivity based automatic model updating procedure is presented,
which solves an optimization problem for model error minimization. Four vertical
vibration modes, two lateral modes, one torsional mode and 17 design parameters are
selected for the problem. Updated results show that the model error could be reduced
from 0~ 18% to 0~ 4% in terms of modal frequency ratio. During the optimization
procedure, the target error bounds were 3% for the lower vertical modes and 6% for the
horizontal modes. In order to prevent mode interchange due to the closely spaced
frequencies of the three dimensional FE model, MACs are introduced to verify the
updated results through the optimization procedure.
65
CHAPTER 4
SEISMIC ANALYSIS
4.1 Background
The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both
Los Angeles and Long Beach ports, two of the busiest ports in the west coast of USA.
Thus, the bridge carries an overwhelming number of traffic with an Annual Average
Daily Traffic ( AADT) volume of 100,000, many of which are cargo trucks. Based on the
recent finding that the main span of the Vincent Thomas Bridge crosses directly over the
Palos Verdes fault, which has the capacity to produce a devastating earthquake, in spring
2000, the bridge underwent a major retrofit using visco- elastic dampers. This study
focuses on seismic vulnerability of the retrofitted bridge. A member- based detailed three-dimensional
Finite Element ( FE) as well as panel- based simplified models of the bridge
are developed. In order to show the appropriateness of these models, eigenproperties of
the bridge models are evaluated and compared with the system identification results
obtained using ambient vibration. In addition, model validation is also performed by
simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino
Hills earthquake and comparing with the measured response. Finally, considering a set of
strong ground motions in the Los Angeles area, nonlinear time history analyses are
performed and the ductility demands of critical sections are presented in terms of fragility
curves. The study shows that a ground motion with PGA of 0.9g or greater will result in
plastic hinge formation at one or more locations with a probability of exceedance of 50%.
66
Also, it is found that the effect of damper is minimal for low to moderate earthquakes and
high for strong earthquakes.
The spatial variation of earthquake ground motions may have significant effect on
the response of long span suspension bridges. Abdel- Ghaffar and Rubin ( 1982) and
Abdel- Ghaffar and Nazmy ( 1988) studied response of suspension and cable- stayed
bridges under multiple support excitations. Zerva ( 1990) and Harichandran and Wang
( 1990) examined the effect of spatial variable ground motions on different types of bridge
models. Harichandran et al. ( 1996) studied the response of long- span bridges to spatially
varying ground motion. Deodatis et al. ( 2000) and Kim and Feng ( 2003) investigated the
effect of spatial variability of ground motions on fragility curves for bridges. Lou and
Zerva ( 2005) analyzed the effects of spatially variable ground motions on the seismic
response of a skewed, multi- span, RC highway bridge. Most of the aforementioned
studies dealt with simple FE models of the bridge, as a result response of critical
members could not be evaluated. In the present analysis a panel based detailed 3D FE
model of a long span suspension bridge is utilized.
For design purpose of important structures in a site, U. S. Geological Survey
( USGS) provides a set of scenario earthquakes specified for a site. To consider spatial
variability of ground motions one needs to know the ground excitations at different
supports of a long span suspension bridge. For generating spatial variable ground motions
from a scenario earthquake compatible to different design spectra for different supports
( as the local soil conditions will be different for different supports) a new algorithm is
proposed using evolutionary power spectral density function ( PSDF) of the scenario
earthquake specified for the site. Evolutionary PSDF of LA21 scenario earthquake is
67
estimated by using short- time Fourier transform ( STFT) and wavelet transform ( WT)
methods. Two evolutionary PSDFs thus developed maintain the same total energy
possessed by the time history data. Using the evolutionary 20 sets of simulated ground
motions for six different spatially correlated supports are generated. Ensemble average of
5% damped spectral acceleration response spectra obtained from simulated earthquake
time histories are compared with the design response spectra for all the support locations.
Good match has been found with the target design acceleration response spectra with the
simulated one.
Simulated spatially variable ground motions are used in calculating the response
of the bridge. In addition to spatial variable seismic ground motions, two uniform ground
motions are also considered for comparison purpose. The seismic responses of the bridge
deck and the east tower are calculated using those three different cases and compared in
both seismic displacement demand and seismic force demand.
4.2 Scope
FE model validation of the bridge is also performed by simulating the dynamic response
during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing
with the measured response from installed acceleration sensors. Considering a set of
strong ground motions in the Los Angeles area, nonlinear time history analyses are
performed and ductility demands of critical tower section are presented in terms of
seismic fragility curves. Effect of spatial variability of ground motions on seismic
displacement demand and seismic force demand is investigated. To generate spatially
68
correlated spectrum compatible nonstationary acceleration time histories, a newly
developed algorithm using evolutionary PSDF and spectral representation method is
used.
4.3 Response Analysis under Northridge Earthquake
To validate the developed numerical models ( discussed in Chapter 2), time
history analysis is performed using the 1994 Northridge earthquake ( Mw = 6.7) ground
motions recorded at the bridge sites. Newmark- Beta method is used with g = 0.5 and b =
0.25 for this purpose. The ground motions and the bridge response during the Northridge
earthquake are collected from the sensors installed at the bridge site ( California Strong
Motion Instrumentation Program ( CSMIP) [ http:// www. strongmotioncenter. org/]). Since
the earthquake occurred before the retrofit, detailed model before the retrofit is used here.
To consider the effect of spatial variation, different ground motions are
considered at different support locations, wherever possible. In some cases, due to the
unavailability of recorded support motions, ground motions recorded at the nearest
support is considered. Figure 4.1 shows the location of sensors and Table 4.1 illustrates
the list of supports on which ground motions are applied for this analysis. Figure 4.2
shows comparison of measured and calculated longitudinal displacement at the top of the
east tower location ( channel # 10) of the bridge. The plot shows good match between the
calculated and field measured responses.
69
Figure 4.1 Location and direction of sensors
- 8
- 4
0
4
8
0 20 40 60 80 100 120
Time ( Sec)
Displacement ( cm)
Measured Calculated
Figure 4.2 Comparison of measured and calculated longitudinal displacement at channel
# 10 location
Vertical
Lateral
Longitudinal
East Anchorage
West Tower
70
Table 4.1 Different support motions considered with channel numbers
Location Longitudinal Lateral Vertical
East Anchorage Ch. 25 Ch. 24 Ch. 26
East Cable Bent* Ch. 13 Ch. 9 Ch. 19
East Tower Ch. 13 Ch. 9 Ch. 19
West Anchorage* Ch. 23 Ch. 1 Ch. 14
West Cable Bent* Ch. 23 Ch. 1 Ch. 14
West Tower Ch. 23 Ch. 1 Ch. 14
* No recording at these locations
4.4 Response Analysis under Chino Hills Earthquake
To study the developed numerical model, time history analysis is performed using the
2008 Chino Hills earthquake ( Mw = 5.4) ground motions recorded at the bridge sites.
Newmark- Beta method is used with g = 0.5 and b = 0.25 for this purpose with time step
equal to 0.01 sec. The ground motions and the bridge response during the Chino Hills
earthquake are collected from the sensors installed at the bridge site ( California Strong
Motion Instrumentation Program ( CSMIP) [ http:// www. strongmotioncenter. org/]). Figure
1 shows the location of sensors already installed in the bridge. Since the earthquake
occurred after the retrofit, detailed model after the retrofit is used here. Three directional
components of ground motions recorded at east anchorage, east tower and west tower are
applied uniformly over all the supports to study which set of ground motions will give
much more accurate results. Figure 4.3 shows the comparison of analytical lateral
response at channel 5 due to ground motions at east anchorage, east tower, west tower
and considering spatial variation in ground motion with field measured response. It can
be seen from figure 4.3 that the analytical response due the ground motion recorded at
east tower matches well with the measured response. Figures 4.4, 4.5 and 4.6 show
71
comparison of analytical lateral, vertical and longitudinal responses at different channels
due to ground motions at east tower with field measured response. These plots shows
good match between the analytical and field measured responses.
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
0 20 40 60 80 100 120
Time ( sec)
Displacement ( cm)
Measured
Computed_ East Anchorage
Computed_ East Tower
Computed_ West Tower
Computed_ Spatial
Figure 4.3 Comparison of analytical lateral response at channel 5 due to ground motions
at east anchorage, east tower and west tower with field measured response
72
- 1.2
- 1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120
Time ( sec)
Displacement ( cm)
Measured
Computed based on east tower data
Figure 4.4 Comparison of analytical lateral response at channel 3 due to ground motions
at east tower with field measured response
- 1.5
- 1
- 0.5
0
0.5
1
1.5
0 20 40 60 80 100 120
Time ( sec)
Displacement ( cm)
Measured
Computed based on east tower data
Figure 4.5 Comparison of analytical vertical response at channel 17 due to ground
motions at east tower with field measured response
73
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
0 20 40 60 80 100 120
Time ( sec)
Displacement ( cm)
Measured
Computed based on east tower data
Figure 4.6 Comparison of analytical longitudinal response at channel 10 due to ground
motions at east tower with field measured response
4.5 Generation of Fragility Curves
It is clear from the previous literature, especially those studies in the aftermath of 1995
Kobe ( Hyogo- ken Nanbu) earthquake at Japan that the bridge deck and cables of
suspension bridges are less vulnerable under strong earthquake ground motion ( remain
elastic) while the tower is the most vulnerable part. In order to simplify analysis, in this
study, only the towers are modeled as nonlinear elements. Remaining elements of the
bridge are considered as linear. Each tower leg is constructed with members of 5 different
cross sections. A total of 40 plastic hinges are introduced at all four tower legs. An elasto-plastic
behavior with 3% strain hardening is considered for the material models of these
plastic hinges. 2% Raleigh damping is used for the first and tenth modes. Forty ground
motions representing 2% in 50 years and 10% in 50 years of hazard level as specified by
74
FEMA/ SAC are used for evaluating seismic vulnerability of the retrofitted bridge. The
motions cover wide range peak characteristics with Peak Ground Acceleration ( PGA)
ranging from 0.42 to 1.30g. Note also that these motions include expected motions from
Palos Verdes fault, the fault crossing the main span of this bridge. For nonlinear time
history analysis, direct time integration is used in the framework of SAP 2000. Motions
are applied in the lateral direction of the bridge and no spatial variation is considered.
After performing the nonlinear time history analysis, the ductility demands of all the
critical sections are evaluated and the maximum ductility demand is noted for each
motion. Considering all these motions, the maximum ductility demand is found to be
6.23, which is from LA 36 motion ( with a PGA of 1.1g) and for the plastic hinge at the
base of the tower.
In this study, fragility curves corresponding to different damage states are
developed following Shinozuka et al., 2000. For a given damage state, the fragility curves
are expressed in terms of lognormal distribution. PGA is considered as Ground motion
intensity. Two fragility parameters, median ( c) and log- standard deviation ( z ) are
estimated through a maximum likelihood method such that fragility curves at different
damage levels do not intersect each other. Therefore, a common z is needed to satisfy
this criterion. Although this method can be used for any number of damage states, for the
ease of demonstration of analytical procedure it is assumed here that there are three states
of bridge damage. Therefore, a family of three fragility curves exists in this case for
damage states of ‘ Level- I’, ‘ Level- II’, and ‘ Level- III’ identified by k = 1, 2, and 3.
Under this lognormal assumption, the analytical form of the fragility function F(•) for the
state of damage k is,
75
( )
( )
 

 

= F
z
z
i k
i k
a c
F a c
ln /
, , ( 4.1)
where ck is median of the fragility function associated with damage state k, z is the
common log- standard deviation, ai is the PGA value to which the bridge is subjected and
F [•] is the standardized normal distribution function. The fragility parameters are
computed by maximizing the likelihood function, L which is given by Eq. ( 4.2),
where xik is 1 or 0, depending on whether or not the bridge sustains damage state k under
ai, and n is the total number of ground motions under which the analysis is performed.
Pik is the probability that the example bridge will suffer from a damage state k when
subjected to ai and is expressed as
1 ( , , z ) 0 1 P F a c i i
= −
( , , z ) ( , , z ) 1 1 2 P F a c F a c i i i
= −
( , , z ) ( , , z ) 2 2 3 P F a c F a c i i i
= −
( , , z ) 3 3 P F a c i i
=
( ) [ ] Õ Õ
= =
=
3
1 1
1 2 3 , , ,
k
n
i
x
ik
ik L c c c z P ( 4.2)
( 4.3)
( 4.4)
( 4.5)
( 4.6)
76
Fragility parameters are obtained by solving the Eq. ( 4.7), by implementing a
straightforward optimization algorithm.
( ) ( )
0
ln , , , ln , , , 1 2 3 1 2 3
=
¶
¶
=
¶
¶
z
z L c c c z
c
L c c c
k
for k = 1,2,3
For the fragility curves, this study proposes performance levels in terms of ductility
demands of critical tower sections, since the damage states related to expected
performance level of suspension bridge is not clearly defined in the literature. Three
different damage states are considered in this study in terms of the maximum ductility
demands of all the critical tower sections. They are ( 1) Level- I ( plastic hinge formation,
ductility > 1) ( 2) Level- II ( ductility ³ 2) and ( 3) Level- III ( ductility ³ 4). Figure 4.5
shows the fragility curves considering these damage states and for before and after
retrofitting of the bridge. One can observe from this figure that for a PGA of 0.9g, the
probability of exceedance corresponding to damage Level- I ( i. e., plastic hinge formation
at one or more locations) is 50%. Similarly, for the same probability of exceedance, a
ground motion with PGA of 1.05g or greater will cause a damage of Level- II. PGA of
1.82g was recorded at the Tarzana Station during the main shock of the 1994 Northridge
earthquake. For that PGA the probability of exceedance to damage Level- II is 90%. The
bridge was retrofitted with total 48 dampers and from the fragility curves it is clear that
the effect of dampers are minimal for low to moderate earthquake and high for strong
earthquake.
( 4.7)
77
Table 4.2 Details of the motions considered in this study for fragility development
SAC Earthquake Distance Scale dt Duration PGA PGV PGD
Name Magnitude ( km) Factor ( sec) ( sec) ( g) ( cm/ sec) ( cm)
LA21 1995 Kobe 6.9 3.4 1.15 0.02 59.98 1.28 142.70 37.81
LA22 1995 Kobe 6.9 3.4 1.15 0.02 59.98 0.92 123.16 34.22
LA23 1989 Loma Prieta 7 3.5 0.82 0.01 24.99 0.42 73.75 23.07
LA24 1989 Loma Prieta 7 3.5 0.82 0.01 24.99 0.47 136.88 58.85
LA25 1994 Northridge 6.7 7.5 1.29 0.005 14.945 0.87 160.42 29.31
LA26 1994 Northridge 6.7 7.5 1.29 0.005 14.945 0.94 163.72 42.93
LA27 1994 Northridge 6.7 6.4 1.61 0.02 59.98 0.93 130.46 28.27
LA28 1994 Northridge 6.7 6.4 1.61 0.02 59.98 1.33 193.52 43.72
LA29 1974 Tabas 7.4 1.2 1.08 0.02 49.98 0.81 71.20 34.58
LA30 1974 Tabas 7.4 1.2 1.08 0.02 49.98 0.99 138.68 93.43
LA31 Elysian Park ( simulated) 7.1 17.5 1.43 0.01 29.99 1.30 119.97 36.17
LA32 Elysian Park ( simulated) 7.1 17.5 1.43 0.01 29.99 1.19 141.12 45.80
LA33 Elysian Park ( simulated) 7.1 10.7 0.97 0.01 29.99 0.78 111.03 50.61
LA34 Elysian Park ( simulated) 7.1 10.7 0.97 0.01 29.99 0.68 108.44 50.12
LA35 Elysian Park ( simulated) 7.1 11.2 1.1 0.01 29.99 0.99 222.78 89.88
LA36 Elysian Park ( simulated) 7.1 11.2 1.1 0.01 29.99 1.10 245.41 82.94
LA37 Palos Verdes ( simulated) 7.1 1.5 0.9 0.02 59.98 0.71 177.47 77.38
LA38 Palos Verdes ( simulated) 7.1 1.5 0.9 0.02 59.98 0.78 194.07 92.56
LA39 Palos Verdes ( simulated) 7.1 1.5 0.88 0.02 59.98 0.50 85.50 22.64
LA40 Palos Verdes ( simulated) 7.1 1.5 0.88 0.02 59.98 0.63 169.30 67.84
Record
Figure 4.7 Before and after retrofit Fragility curves for different damage levels
Probability of Exceeding a Damage State
PGA ( g)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Level I_ Before
Level II_ Before
Level III_ Before
Level I_ After
Level II_ After
Level III_ After
78
4.6 Simulation of Ground Motion Considering Spatial Variability
4.6.1 Generation of Evolutionary PSDF from Given Ground Motion using STFT
This section briefly reviews the work done by Liang et al. ( 2007). The STFT F ( t, w ) of a
function f ( t ) is expressed by the convolution integral in the following form:
( )  ( ) ( ) ¥
− ¥
−
w = t − t t
w t F t f h t e d i , ( 4.8)
where h ( t ) is an appropriate time window. The evolutionary PSDF S ( t w ) f f ,
0 0
can be
written as
( )   ( ) ( ) ( ) ( ) ¥
− ¥
− −
¥
− ¥
= − −
1 2 1 2 1 2
2
, w t t t t 1 2 t t
w t w t F t f f h t h t e e d d i i
( 4.9)
The total energy of f ( t ) can be estimated as
( )
( ) ( ) ( ) ( )
( )
  ( ) ( )
   
 
¥
− ¥
¥
− ¥
¥
− ¥
¥
− ¥
¥
− ¥
− −
¥
− ¥
¥
− ¥
¥
− ¥
= −
= − −
f h t d dt
f f h t h t e d d dtd
F t dtd
i
t t t
t t t t t t w
w w
w t t
2 2
1 2 1 2 1 2
2
1 2
,
( 4.10)
For the derivation of Eq. ( 4.10), the following equation is used:
( )
 ( ) ¥
− ¥
− −
= −
1 2
1 2
w d t t
w t t e d i ( 4.11)
If h ( t ) = d ( t )
2 , the total energy in Eq. ( xx) is
  ( )  ( ) ¥
− ¥
¥
− ¥
¥
− ¥
F t dtd = f t dt 2 2
, w w ( 4.12)
79
This implies that the time window should be chosen such that it satisfies the following
condition
 ( ) ¥
− ¥
= 1 2 h t dt ( 4.13)
The total energy can be kept identical ( Perseval’s identity) in estimating evolutionary
PSDF. Here a Gaussian time window squared with standard deviation s = 0.25 s, is used.
It satisfies the condition in Eq. ( 4.13). The time window function has the following form,
( ) ( 0.25 )
2
1 2 2 2 2
= =
−
s
s p
t s h t e ( 4.14)
Figure 4.8 shows the evolutionary PSDF of LA21 scenario earthquake record estimated
using STFT ( Gaussian window).
Figure 4.8 Evolutionary PSDF of LA21 earthquake record using STFT method
80
4.6.2 Generation of Evolutionary PSDF from Given Ground Motion using Wavelet
Transform
This section briefly reviews the work done by Liang et al. ( 2007). The wavelet transform
( WT) of a function f ( R ) 2 Î L ( finite energy function  f ( t ) dt < + ¥
2
) at time u and
scale s , and the corresponding inverse relationship are given by Daubechies ( 1992)
( ) ( ) dt u s R
s
t u
f t
s
W f u s Î  	


 
 −
=  ¥
− ¥
* , ,
1
, y
y ( 4.15)
and
( )   ( ) ( ) ¥
− ¥
¥
− ¥
Î  	


 
 −
= duds u s R
s
t u
W f u s f t
C s
f t , , ,
1
2
1
2
y
p
y
y
( 4.16)
where
( )
=  < ¥ ¥
− ¥
w
w
y w
y C d
2 ˆ
( 4.17)
In Equations ( 4.15) – ( 4.17), the wavelet function ( R ) 2 y Î L known as ‘ mother” wavelet
with average value equal to zero,
 ( ) ¥
− ¥
y t dt = 0 ( 4.18)
and is centered in the neighborhood of t = 0, and as normalized y = 1. y ˆ ( w ) denotes
the Fourier transform of y ( t ) and is given by
( )  ( ) ¥
− ¥
−
= t e dt i w t
y
p
y w
2
1
ˆ ( 4.19)
It may be noted that the WT decomposes signal f ( t ) over dilated and translated
wavelets. As W f ( u, s ) y is convolution of f ( t ) with ( 1 s ) ( t s ) , W f ( u, s ) *
y
y −
81
represents the contribution of the function f ( t ) in the neighborhood of t = u and in the
frequency band corresponding to scale s . It can be shown that ( Daubechies, 1992)
( ) W f ( u s ) duds
C s
f t dt
2
2
2
,
1
2
1
y
y
p
   ¥
− ¥
¥
− ¥
¥
− ¥
= ( 4.20)
Now, if any wavelet function satisfies the condition
 ( ) ¥
− ¥
ˆ = 1
2
,
y w d w u s ( 4.21)
Then Equation ( 4.20) can be written as
 ( )    ( )  ( ) ¥
− ¥
¥
− ¥
¥
− ¥
¥
− ¥
¥
− ¥
×
  

  

= y w w
p
y
y
W f u s duds d
C s
f t dt u s
2
,
2
2
2 , ˆ
1
2
1
( 4.22)
In Equations ( 4.21) and ( 4.22), y ( w )
u, s ˆ represents the Fourier transform of  	


 
 −
s
t u
y and
can be expressed as ( ) ( )
i u
u s s s e w
y ˆ w y ˆ w ,
= . Then, using Perseval’s identity, one can write
( ) W f ( u s ) ( ) duds
C s
F u s
2
,
2
2
2 , ˆ
1
2
1
y w
p
w
y
y
  ¥
− ¥
¥
− ¥
= ( 4.23)
where F ( w ) = Fourier transform of f ( t ) . As the wavelet coefficient W f ( u, s ) y provides
the localized information of signal f ( t ) at t = u , from Equation ( 4.23) the Evolutionary
PSDF ( , w )
0 0
S t f f can be expressed as
( )  ( ) ( ) ¥
− ¥
= W f t s ds
C s
F t t s
2
,
2
2
2 , ˆ
1
2
1
, y w
p
w
y
y
( 4.24)
It may be noted that the expression of evolutionary PSDF given in Equation ( 4.24) obeys
total energy equilibrium. Therefore, any wavelet basis can be used which satisfies
Equation ( 4.21), for generation of evolutionary PSDF [ e. g., modified Littlewood- Paley
basis proposed by Basu and Gupta ( 1998)] that maintains total energy. Figure 4.9 shows
82
the evolutionary PSDF of LA21 scenario earthquake record estimated using STFT
( Gaussian window).
Figure 4.9 Evolutionary PSDF of LA21 earthquake record using wavelet transform
4.6.3 Simulation of One- Dimensional Multi- Variate ( 1D- mV), Nonstationary
Gaussian Stochastic Process
To generate sample functions of stochastic processes, the spectral representation method
developed by Shinozuka and Jan ( 1972) appears to be most versatile and widely used
today. Spectral representation based algorithm to simulate one- dimensional multi- variate
nonstationary Gaussian stochastic process developed by Deodatis ( 1996b) is used in this
study and described as follows.
83
Consider a one- dimensional, n- variate ( 1D- nV) non- stationary stochastic vector
process with components ( ) , ( ) ,........., ( ) , 0 0
2
0
1 f t f t f t n