natural frequencies of bridge structures from ambient vibration

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FIELD INVESTIGATION OF FUNDAMENTAL FREQUENCY OF
BRIDGES USING AMBIENT VIBRATION MEASUREMENTS
by
Arden Reisham Pradeep Heerah
Department of Civil Engineering and Applied Mechanics
McGill University
Montréal, Canada
August 2009
A thesis submitted to McGill University
in partial fulfillment of the requirements of the degree of
MASTER OF ENGINEERING.
© Arden Reisham Pradeep Heerah, 2009. All Rights Reserved.
ABSTRACT
The transient nature of forces induced in structures during
earthquakes requires the use of dynamic analyses to fully characterise
their behaviour. A modal analysis describes the dynamic response of the
structural system through modal descriptors: natural frequencies, mode
shapes and damping ratios. Efficiently estimating these modal parameters
for bridges allows for better structural integrity assessments and structural
health
monitoring
of
these
structures.
Using
ambient
vibration
measurements to estimate modal parameters is time-saving and efficient.
This research reviews the literature on the application of ambient vibration
testing to the modal characterisation of bridges. Natural frequencies from
ambient vibration measurements are obtained for a typical bridge in the
City of Montréal in Canada.
The MATLAB computing platform is used to execute spectral
analyses of the field measurements. Linear, analytical models of the
bridges are constructed with the SAP2000 structural analysis programme
and Eigenvalue analyses are performed. The experimental and analytical
results are compared and discussed, followed by recommendations for the
application of this procedure to other bridges in the Montréal region.
-i-
RÉSUMÉ
Du fait de la nature transitoire des forces se créant au sein des
structures lors des tremblements de terre, la caractérisation complète de
leur comportement nécessite l’utilisation d’analyses dynamiques. Une
analyse modale décrit la réponse dynamique de la structure à travers ses
modes de vibration : les fréquences naturelles, la forme des modes et le
facteur d’amortissement. Une estimation efficace des paramètres modaux
des ponts permet alors une meilleure évaluation, et donc un meilleur suivi,
de leur intégrité structurale. L’utilisation des mesures de vibrations
ambiantes pour estimer les paramètres modaux est donc une méthode
efficace et économique. Cette étude passe en revue la littérature sur
l’application de l’analyse des mesures de vibrations ambiantes pour la
caractérisation modale des ponts.
Les fréquences naturelles ont été obtenues sur un pont typique de
la ville de Montréal. La plate-forme informatique MATLAB a été utilisée
pour effectuer des analyses spectrales de ces mesures. Des modèles
analytiques linéaires ont ensuite été construits avec le programme
d’analyse structurel SAP2000 pour effectuer des analyses d’Eigenvalue.
Les résultats expérimentaux et analytiques sont ensuite comparés et
discutés, et des recommandations énoncées, pour appliquer ce procédé à
d’autres ponts de la région de Montréal.
- ii -
ACKNOWLEDGEMENTS
I extend sincere appreciation to the following, among others:
My supervisor, Professor Luc E. Chouinard, for his invariable and
multifaceted support and guidance throughout our research activities at
McGill University and my stay in the City of Montreal.
Salman Saeed, Dr. Myriam Belvaux, Dr. Alejandro de la Puente Altez, Dr.
Philippe Rosset, Damien Gilles, Kuei-hua Rebecca Huang and Wendy
Itagawa; their assistance throughout my research has been consequential.
The Ministère des Transports du Québec, for assisting our research
efforts.
Lennartz electronic and LEAS électronique, manufacturers of the
equipment used in this research, for their guidance and clarification.
Fédon Honoré, for translating the abstract into French.
Professors Saeed Mirza, Ghyslaine McClure and Colin Rogers, for their
invaluable contributions toward my development throughout graduate
studies.
C.E.P. Ltd, to whom I am timelessly grateful.
Peter and Michelle Santlal, for their friendship and limitless generosity.
Staff of the Department of Civil Engineering and Applied Mechanics:
Franca Della-Rovere, Anna Tzagournis, Sandy Shewchuk-Boyd, Dr.
William Cook, Jorge Sayat and Ronald Sheppard, among others; and staff
of the Schulich Library of Science and Engineering.
My colleagues at McGill University, for their stimulating, earnest and
sometimes inebriated discussions: Adriana Parada, Ali Ghafari, Hilary
Ingram, Joe Mattar, John-Edward Franquet, Lai Wai Tan, Li Li, Ling
Zhang, Miguel Nunes, Nabil Elias Saliba, Nicolas Desramaut, Nisreen
Balh, Reza Erfani, Tatiana Tobar Valencia, Stephane Villemain, among
others.
I also convey gratitude to my friends and their families for their thoughts
and involvement; especially to Andre Bagoo, Anuradha Gobin, Beena
John, Binta Trotter, Charline Augustine, Dwayne Dubarry, Fadil Sahajad,
Fédon Honoré, Navin Seebaran, Shalu Bujun, Suzette Parillon, Suzanne
Seepersad, Vijay Mohan, Vince Ramlochan, Xue Zeng.
- iii -
ACKNOWLEDGEMENTS
My mum, Gemma, and my brother, Arundel, to whom I am grateful for
their unwavering support, patience and encouragement during my
research efforts and for fostering the resolve within me to strive for more
than only what seems possible.
”Be the change you want to see in the world.”
- from Mahatma Gandhi
“The art of directing the great sources of power in Nature for the use and
convenience of man.”
- from Institution of Civil Engineers’ Royal Charter of 1828.
“Plagiarize, Let no one else’s work evade your eyes.”
- from the song Lobachevsky by Tom Lehrer, Harvard Mathematics
lecturer.
“If we do not maintain our infrastructure, do not upgrade it, we’ll continue
to have spectacular collapses.”
- from Professor Saeed Mirza, McGill University.
“One shall have to undergo suffering to reach truth. That is why it is said
that truth is eternally victorious.”
- from Rig Veda.
- iv -
TABLE OF CONTENTS
Section
Page
ABSTRACT
i
RÉSUMÉ
ii
ACKNOWLEDGEMENTS
iii
LIST OF FIGURES
viii
LIST OF TABLES
xii
LIST OF APPENDICES
xiii
CHAPTER 1 - INTRODUCTION
1
1.1
Developments
3
1.2
Objectives
4
1.3
Organisation of the thesis
5
CHAPTER 2 - LITERATURE REVIEW
7
2.1
Dynamical behaviour of structures
7
2.2
Experimental modal analysis
12
2.2.1 Frequency response function (FRF)
14
2.2.2 Impulse response function (IRF)
15
2.2.3 Vibration tests
18
2.2.4 Modal testing of civil engineering structures
21
2.2.5 Ambient vibration testing of bridges
22
Modal parameter estimation
24
2.3.1 Fourier spectral analysis (FSA)
25
2.3.2 Power Spectral Density (PSD) and Cross-power
28
2.3
Spectral Density (CSD) Analysis
2.4
2.3.3 Auto-regressive moving-average (ARMA) method
30
Signal processing issues in AVT
32
2.4.1 Sampling
32
2.4.2 Averaging
33
-v-
TABLE OF CONTENTS
Section
Page
2.4.3 Spectral leakage
33
2.4.4 Resolution
38
CHAPTER 3 - CASE STUDY
40
3.1
Bridge structure
40
3.2
Construction materials
47
3.3
Equipment
49
3.3.1 Testing
52
3.3.2 Test locations
55
3.3.3 Sampling parameters
55
3.3.4 Tests
57
3.3.5 Synchronisation
57
Analysis of experimental results
60
3.4.1 Fourier method
60
3.4.2 Welch’s PSD method
61
3.4.3 Welch’s CSD method
61
3.4.4 Auto-regressive, Modified Covariance method
62
Analysis of analytical models
62
3.5.1 Models
63
3.4
3.5
CHAPTER 4 - DISCUSSION AND RECOMMENDATIONS
69
4.1
Notes on the experimental analysis
69
4.2
Notes on the analytical analysis
79
4.3
Fundamental frequency estimates
80
4.4
Reviewing the analysis
83
4.5
Recommendations for ambient vibration testing of bridges
84
- vi -
TABLE OF CONTENTS
Section
Page
CHAPTER 5 - CONCLUSIONS AND FUTURE
88
RECOMMENDATIONS
5.1
Conclusions
88
5.2
Reviewing the analysis
90
5.3
Recommendations for ambient vibration testing of bridges
90
5.4
Proposal for future research
91
5.5
Fulfillment of objectives
93
REFERENCES
APPENDICES
- vii -
LIST OF FIGURES
Figure
1-1
Content
Page
Approximate distribution of seismic risk across Canada’s
2
urban population. (Adams et al., 2002)
1-2
Map of Canada showing delimitation of the eastern and
2
western seismic regions and the stable central region.
(Adams and Halchuk, 2007)
2-1
Mechanical models of a multi-story building structure.
10
(Tedesco et al., 1999)
2-2
Mechanical model for a SDOF system. (Tedesco et al.,
10
1999)
2-3
Coupling
theoretically
and
experimentally
derived
13
Definition of the unit impulse forcing function. (Maia and
17
dynamic modelling. (Maia and Silva, 1997)
2-4
Silva, 1997)
2-5
Definition of an arbitrary, non-periodic forcing function.
17
(Maia and Silva, 1997)
2-6
Devices used in FVT. (Cunha and Caetano, 2006)
19
2-7
Shaker devices used to excite. (Cunha and Caetano,
19
2006)
2-8
Random signals. (Ewins, 2000)
28
2-9
Description of Welch PSD analysis. (Trauth, 2007)
30
2-10
Interpretations of multi-sample averaging. (Ewins, 2000)
35
2-11
Finite-length sample and spectral leakage. (Ewins, 2000)
35
2-12
Window functions used frequently in spectral analysis.
36
(Stoica and Moses, 1997)
2-13
The effect of the window functions from Figure 2-15 in
37
the frequency domain. (Stoica and Moses, 1997)
2-14
Demonstration of how original time histories are treated
with select windows. (Ewins, 2000)
- viii -
38
LIST OF FIGURES
Figure
Content
Page
3-1
Views of the monitored bridge.
41
3-2
Sketch views of the bridge deck framing system.
43
3-3
Sketch view of the bridge deck framing system: plan
44
view.
3-4
Sketch
views
of
the
moment-resisting,
reinforced
44
Sketch sections through a typical column in the moment-
45
concrete frame. (De La Puente Altez, 2005)
3-5
resisting, reinforced concrete frame. (De La Puente
Altez, 2005)
3-6
Sketch views of the girders’ connection to the top of the
46
moment-resisting, reinforced concrete frame.(De La
Puente Altez, 2005)
3-7
Profile view (transverse direction) sketch of the beam
46
connected to the top of the abutment. (Ministry of
Transport of Quebec)
3-8
Assumed stress-strain plots for the steel and concrete at
48
the time of construction of the bridge. (De La Puente
Altez, 2005)
3-9 (a)
Experimental equipment: seismometer and DAS.
49
3-9 (b)
Experimental equipment: RCS antenna attached to DAS
50
and GPS sensor mounted on railing.
3-10
Extract from a data file indicating record details followed
50
by five rows of data points.
3-11
Typical measurement time history describing motion in
each orthogonal direction (Record 6: File 27 from DAS
28 on November 30th 2007).
- ix -
53
LIST OF FIGURES
Figure
3-12
Content
Page
Testing configurations: (a) A - along the central
54
longitudinal axis; (b) B - along a transverse line; and (c)
C - along one exterior longitudinal axis.
3-13
Synchronisation of a pair of measurements.
58
3-14
AM1: (a) solid view of the model; (b) schematic of the
67
structural model for AM1, Case 1; (c) schematic of the
structural model for AM1, Case 2.
3-15
AM2: (a) solid view of the model; (b) schematic of the
68
structural model for AM2, Case 1; (c) schematic of the
structural model for AM2, Case 2.
4-1
Typical response spectra from the four experimental
73
approaches using the ambient vibration measurements.
4-2
Frequency estimates and repeatability rates using the
75
FFT approach.
4-3
Frequency estimates and repeatability rates using the
76
Welch’s PSD approach.
4-4
Frequency estimates and repeatability rates using the
76
Welch’s CSD approach.
4-5
Frequency estimates and repeatability rates using the
77
MCOV approach.
4-6
Frequency estimates and repeatability rates in the
77
vertical direction (Z).
4-7
Frequency estimates and repeatability rates in the
78
transverse direction (Y).
4-8
Frequency estimates and repeatability rates in the
longitudinal direction (X).
-x-
78
LIST OF FIGURES
Figure
Content
4-9
Fundamental frequency estimates (frequency at first
Page
mode of vibration) for all experimental and analytical
modal analyses.
- xi -
82
LIST OF TABLES
Table
3-1
Content
Page
Assumed material strengths for the analytical modal
48
analysis.
3-2
Specifications of the seismometers and data acquisition
51
stations used in the ambient vibration testing.
3-3
Sampling parameters used in testing.
56
3-4
Ambient conditions during testing.
58
3-5
Measurement catalogue.
59
3-6
Contributors to the reactive mass.
64
4-1
Summary
of
resonant
frequency
estimates
and
74
repeatability rates by direction and processing method.
4-2
Summary of resonant frequency estimates for all four
75
algorithms in EMA and corresponding directions of
occurrence.
4-3
Summary of natural frequency estimates for AM1.
81
4-4
Summary of natural frequency estimates for AM2.
81
- xii -
LIST OF APPENDICES
Appendix
Content
Page
A
Output spectra from FFT estimation.
A -1
B
Output spectra from Welch’s PSD estimation.
B-1
C
Output spectra from CSD estimation.
C-1
D
Output
spectra
from
AR,
estimation.
- xiii -
Modified
Covariance
D-1
Chapter 1
Introduction
Modern society is highly dependant on its infrastructure. Safe and
efficient transportation is required for economic activities as well as for
emergency response services. Urban environments are highly complex
with interconnected and interdependent transportation networks where
bridges play a prominent role.
Bridges are some of the most critical components of transportation
infrastructure systems. For these structures, failure is defined as any
interruption of pedestrian or vehicular traffic across or under it due to
structural distress. Direct consequences of failure can range from injury to
loss of life and property in the case of collapse, and indirect consequences
such as disruptions to economic activities and reduced access to
emergency facilities in the event of collapse or closure.
One approach towards mitigating bridge failure is to ensure that it is
properly designed to satisfy performance needs associated with traffic and
extreme environmental loads, as well as to meet maintenance and
durability objectives. It can be appreciated that evaluating the condition of
existing bridges and assessing their vulnerabilities to extreme loads are
essential steps towards mitigating bridge failures.
The Montreal Urban Community (MUC), with its population of 3.5
million inhabitants is ranked second (behind Vancouver) among Canadian
cities for seismic risk (Figure 1-1). Figure 1-2 shows that Montreal is
located in a region of moderate seismic activity, an exposure which
demonstrates a need to understand the health of the city’s infrastructure
for the safety and convenience of its inhabitants.
-1-
1 - Introduction
Figure 1-1
Approximate distribution of seismic risk across Canada’s urban
population. (Adams et al., 2002)
Figure 1-2
Map of Canada showing delimitation of the eastern and western seismic
regions and the stable central region. (Adams and Halchuk, 2007)
-2-
1 - Introduction
1.1
Developments
Structural vibrations are a major hazard and design limitation for
civil engineering structures, including bridges (Ewins, 2000). Historically,
there have been notable bridge failures that resulted in significant losses.
Engineers have learnt from these experiences and refined their
approaches to design, construction, monitoring and maintenance of
structures (Lawson, 2005).
In 1879, the Tay Bridge collapsed in Scotland from a combination of
poor quality control, bad construction practices and inadequate design
recommendations. A contributing factor appears to have been a dynamic
system associated with wind gusts. Comparisons have been drawn
between this failure and the 1940 failure of the Tacoma-Narrows Bridge;
the latter bridge failed due to nonlinear mode coupling between wind
forces and the flexibility and inertia of the bridge structure (Billah and
Scanlan, 1991). The Hyatt Regency Skywalk in Kansas City, U.S.A.
collapsed in response to vibrations from overcrowding of the skywalk in
1981. More recently, the Millennium Bridge in London, England was
closed temporarily after resonance was induced by pedestrian traffic
(Lawson, 2005).
These few examples of bridge failures underscore the importance
of determining the dynamic response of bridges to time-varying forces
induced by traffic (pedestrian and vehicular) and wind and earthquake
loads.
The present work is part of a research project on seismic hazard
analysis for the MUC. The objectives of the project are to perform seismic
hazard analysis, assist in developing mitigation plans and improve
emergency preparedness (Chouinard et al., 2004). Projects have already
been completed on seismic microzonation (Chouinard and Rosset, 2007;
Rosset et al., 2004, 2003; Madriz, 2004), seismic deficiencies of existing
bridges (De La Puente Altez, 2005), seismic vulnerability of electric
-3-
1 - Introduction
distribution networks (Jaigirdar, 2005), seismic screening procedures for
buildings and bridges (Lui, 2002), and ambient vibration analysis of
buildings (Huang, 2007).
1.2
Objectives
The transient nature of seismic forces induced in structures
necessitates the use of dynamic analysis to characterise their response.
Modal analysis defines the dynamic response of a structure through modal
descriptors: natural frequencies, mode shapes and damping ratios.
Efficiently formulating these modal parameters for bridges is essential for
efficient and reliable structural assessments. Past work has shown
experimental modal testing to be an efficient means of estimating these
parameters. In particular, ambient vibration measurements provide
accurate estimates of modal parameters that are used for performing
structural assessments and structural health monitoring in a timely and
cost-effective manner.
The objective of this project is to use ambient vibration testing to
characterise a typical overpass bridge in Montreal. The MATLAB
computing platform is used to conduct spectral analyses on the field
measurements. Linear, analytical structural models of the bridge are built
with the analysis programme SAP2000 to perform eigenvalue analysis for
the dynamic response. Experimental and analytical results are evaluated,
compared and discussed.
The main goals of this research are:
1. To review the literature on ambient vibration testing of bridges.
2. To conduct experimental and analytical modal analysis of a typical
bridge in Montreal.
3. To compare and discuss the findings of the modal analyses.
-4-
1 - Introduction
4. To provide guidelines for the ambient vibration testing of other bridges
in Montreal.
1.3
Organisation of the Thesis
Chapter 1 of the thesis presents the background and objectives of
this research. Chapter 2 introduces fundamental concepts used in modal
analysis and structural dynamics theory is reviewed to define modal
parameters
for
system
identification.
The
chapter
also
reviews
experimental modal analysis using vibration testing for civil engineering
structures, especially ambient vibration testing of bridges. This is followed
by a brief description of modal analysis techniques in the frequency and
time domains for civil engineering structures and bridges in particular.
Emphasis is placed on the techniques utilised in this research. The
chapter concludes with a review of data processing issues with ambient
vibration testing.
Chapter 3 describes the bridge structure selected for ambient
vibration testing. The structural characteristics of the bridge pertinent to
modal analysis are presented and details of the testing procedure are
discussed. Subsequently, the spectral analysis techniques used on the
ambient vibration records are presented. Finally, analytical models of the
bridge that were created for comparative purposes are described.
Chapter 4 presents a discussion of the findings obtained from the
analyses that were presented in Chapter 3. Firstly, the results for the four
procedures (FFT, Welch’s PSD, Welch’s CSD and AR-MCOV) used to
analyse the ambient vibration measurements are outlined. This is followed
by the presentation of the results for the analytical models of the bridge for
different support conditions and material properties. The results from all of
the analysis techniques are discussed. Finally, recommendations for the
modal testing of bridges are proposed.
-5-
1 - Introduction
Chapter 5 presents the conclusions from the experimental and
analytical modal analysis of the bridge. A review of the analysis is
discussed briefly and a summary of recommendations for modal testing of
bridges is provided. As well, future research directions in experimental
modal analysis of bridges using ambient vibration measurements are
outlined.
-6-
Chapter 2
Literature Review
This chapter introduces fundamental concepts used in modal
analysis and structural dynamics theory is reviewed to define modal
parameters
for
system
identification.
The
chapter
also
reviews
experimental modal analysis using vibration testing for civil engineering
structures, especially ambient vibration testing of bridges. This is followed
by a brief description of modal analysis techniques in the frequency and
time domains for civil engineering structures and bridges in particular.
Emphasis is placed on the techniques utilised in this research. The
chapter concludes with a review of data processing issues with ambient
vibration testing.
2.1
Dynamical behaviour of structures
The dynamic behaviour of a structure is described by the equation
of motion for a system using the fundamental theory of vibrating systems.
The main structural characteristics of a vibrating system are the mass,
stiffness and damping properties of the structure (Tedesco et al., 1999;
Craig and Kurdila, 2006). These properties determine the inertia, stiffness
and damping forces of the system within the equations of motion that
describe the dynamic response {x(t)} of the system (Maia and Silva, 1997).
Inertia forces {Ft} are derived from the contributing mass of the
system as the product of the mass {m} and the acceleration of the mass
{ &x& (t)} (Equation 2.1). Some systems can vibrate indefinitely even after
the external excitation has ceased. However, almost all structures exhibit
energy dissipation where the magnitude of the structural oscillation
-7-
2 – Literature Review
decreases as a function of time. This damping of the system is
characterised by a damping coefficient {c} which relates the velocity of the
mass { x& (t)} of the system and the damping force {FD} (Equation 2.2).
There are several models for damping but the most commonly employed
are viscous (dash-pot), Coulomb (friction) and Hysteresis (material)
damping models (Maia and Silva, 1999). Elasticity of the system is
characterised by the spring constant {k} that relates the elastic spring force
{Fs} to the displacement of the mass { x(t) } (Equation 2.3).
Ft = -m&x&(t)
2.1
FD = cx& (t)
2.2
Fs = kx(t)
2.3
The characteristic equation of motion is obtained from the direct
application of Newton’s second law, in combination with the D’Alembert’s
principle of virtual work or Hamilton’s direct integral approach (Craig and
Kurdila, 2006; Clough and Penzien, 2003). The resulting equations
account for all three internal forces and the excitation force {f(t)} (Equation
2.4).
[m]{&x&(t)} + [c]{x& (t)} + [k]{x(t)} = {f(t)}
2.4
Dynamic analysis is performed by discretising the structure into a number
of representative degrees-of-freedom (DOF). Figure 2-1 illustrates how a
typical structure is idealised by a single-degree-of-freedom (SDOF) and
three-degree-of-freedom (3-DOF) systems. The SDOF system is the
simplest analytical model that defines the physical structure at a single
spatial coordinate and is characterised by single mass, stiffness and
damping properties. The equation of motion for the system can then be
constructed easily from this analytical model (Equation 2.4). Figure 2-2
-8-
2 – Literature Review
illustrates the analytical model for the SDOF system. The circular
frequency of the single mode of vibration is described by {ω}, in terms of
radians per second, and the eigenvalue is the square of this circular
frequency. The cyclic frequency is defined by {f}, in Hertz, and the period
of the vibration is defined by {T}, in seconds, through Equation 2.5. The
dynamic response of the SDOF system is predicted from the solution of
the equation of motion (Equation 2.4).
f=
ω
2π
2.5 a
1
f
2.5 b
T=
A similar approach is adopted in estimating the dynamic behaviour
of multi-degree-of-freedom (MDOF) systems. The equations of motion for
the 3-DOF model shown in Figure 2-1 (d) are as follows,
m1&x&1 (t) + (k1 + k 2 )x1(t) − k 2 x 2 (t) = f1 (t)
2.6 a
m 2 &x& 2 (t) - k 2 x1 (t) + (k 2 + k 3 )x 2 (t) − k 3 x 3 (t) = f 2 (t)
2.6 b
m 3 &x& 3 (t) − k 3 x 2 (t) + k 3 x 3 (t) = f 3 (t)
2.6 c
, where {f1(t)}, {f2(t)} and {f3(t)} are external forces that act on masses {m1},
{m2} and {m3}, respectively. For MDOF systems, the lowest frequency of
vibration is the natural or fundamental frequency of vibration of the
system.
Most real structures are systems with distributed mass and stiffness
properties, with an infinite number of spatial coordinates. By discretizing
the continuous structure into a countable quantity of coordinates, a
solution of the constituent equations of motion for the idealised
representation can be attempted. Chapra (2005) and Hoffman (2001)
-9-
2 – Literature Review
demonstrated that standard eigenvalue and characteristic-value problems
define vibrating systems.
Figure 2-1
Mechanical models of a multi-story building structure.
(a) Physical representation; (b) continuous (uniform distribution) model; (c)
SDOF discrete model; (d) 3-DOF discrete model. (Tedesco et al., 1999)
Figure 2-2
Mechanical model for a SDOF system. (Tedesco et al., 1999)
Ewins (2000) noted that when the excitation function {f(t)} is
considered in the complex form,
+∞
f(t) = ∫ F(ω )e iωt dω
−∞
- 10 -
2.7
2 – Literature Review
, the system’s response function {x(t)} assumes the form,
+∞
x(t) = ∫ X(ω )e iωt dω
2.8
−∞
, and the resulting eigenvalue problem can be defined for an undamped
MDOF vibrating system by,
{[K] − [ω 2 ][M]}{[x(t)]} = {[f(t)]}
2.9
The solution of these equations can be obtained with any of several
methods outlined by Hoffman (2001) to generate estimates of eigenvalues
and their matching eigenvectors. Eigenvectors {φ} yield the relative
positioning of the coordinates (or mode shapes) and eigenvalues {ω}
characterise the circular frequencies of the system (in radians per
second). Estimates of modal damping may be derived by applying the
Half-Power Bandwidth Method to response signals (Craig and Kurdila,
2006; Feng et al., 1998) and curve-fitting methods applied to crosscorrelation functions of response measurements, among other approaches
(Ewins, 2000; Maia and Silva, 1997).
Craig and Kurdila (2006) demonstrated how the dynamic response
of vibrating systems can be described by a finite number of vibration
modes. Natural frequency, mode shape and damping value are the three
parameters that define each vibration mode; collectively they are called
modal parameters. The fundamental mode of vibration corresponds to the
lowest estimated frequency for MDOF systems. The second lowest
frequency and its modal property estimates describe the second mode of
vibration, and so on. For MDOF systems, the total number of vibration
modes corresponds to the number of mass degrees-of-freedom
comprising the system.
- 11 -
2 – Literature Review
Estimates of these parameters can be extracted from vibration
surveys and the entire process is known as modal testing, vibration testing
or system identification or dynamic characterisation.
2.2
Experimental modal analysis (EMA)
EMA is used to estimate modal parameters of vibrating systems
from data that has been measured on actual structures or full-scale
models. This type of analysis has been performed in the industry since the
early 1930’s and multiple surveys and analysis techniques have been
developed to date. Several publications document modal testing and
applications to civil engineering structures (Huang, 2007; Michel and
Guéguen, 2007; Cunha and Caetano, 2006; Zivanovic et al., 2006; He et
al., 2005; Ren et al., 2004; Dye, 2002; Ewins, 2000; Ivanovic et al., 2000;
Huang et al., 1999; Muhammad, 1999; Feng et al., 1998; Maia and Silva,
1997; Xu et al., 1997; Brincker et al., 1996; Felber et al., 1995; Asmussen,
1994; Farrar et al., 1994; Rainer and Van Selst, 1976).
In certain instances, the frequency of an oscillatory excitation will
coincide with the natural frequency of the structure. This produces the
phenomenon of resonance where there is significant amplification of
vibration of the structure that can lead to damage (Tedesco, 1999). The
fundamental idea behind modal testing is that natural frequencies of the
structure are identified by resonant frequencies (Inman, 2006). Vibration
measurements are recorded in the time domain and the analysis is
performed in either the frequency or the time domain. Classical methods
for system identification from EMA are based on frequency response
functions (FRFs) in the frequency domain and impulse response functions
(IRFs) in the time domain (He et al., 2005). Some of the objectives of
modal testing are:
1. To determine mode frequencies of the structure;
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2 – Literature Review
2. To estimate mode shapes and damping information for the structure;
3. To correlate a finite-element (FE) or other theoretical model of the
structure with measurements from the actual structure.
4. To formulate a dynamic model of the structure that can be used to
investigate potential modifications to the structure;
5. To prepare a dynamic model that is suitable for updating a FE model of
the structure and improve numerical results.
6. To construct a model to extrapolate the structural behaviour to extreme
loads.
7. To add to the body of knowledge and the performance of the structure
- this forms the basis for structural health monitoring.
Maia and Silva (1997) present a succinct flow diagram that
describes the components of numerical and experimental modal analyses
and their relationship (Figure 2-3).
Figure 2-3
Coupling theoretically and experimentally derived dynamic modelling.
(Maia and Silva, 1997)
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2 – Literature Review
2.2.1 Frequency Response Function (FRF)
Dynamic analysis in the frequency domain produces FRFs of the
vibrating system. These model the amount of energy dissipation of the
structure during vibration (He and Fu, 2001) and are commonly used to
identify modal parameters of monitored structures. Ewins (2000) has
detailed the various forms which FRF plots can take, and the theoretical
basis for modal testing has been widely outlined in the literature.
Fourier analysis is the basis for estimating system response in the
frequency domain (Inman, 2006). It uses the Fourier Series to describe
dynamic signals by representing periodic functions as a summation of
harmonic functions. System vibrations are dynamic signals that are
classified as either deterministic (periodic or transient) or random
(stationary or non-stationary). While non-periodic (transient) functions
cannot be properly addressed by the Fourier Series, they have been
viewed with an infinite period (T = ∞) to help the analysis along (Maia and
Silva, 1997). In other words, the Fourier Series approach is extended to a
Fourier Transform as the case of a signal with an infinitely long period.
This assumption allows signals to satisfy the Dirichlet condition (Equation
2.10) (Maia and Silva, 1997),
+∞
∫ x(t) dt < ∞
2.10
−∞
, where the Fourier Transform {F(ω)} of the forcing function {f(t)} can be
computed by,
+∞
1
f(t)e −iωt dt
F(ω) =
2π −∫∞
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2.11
2 – Literature Review
The Fourier Transform of the response of the system {X(ω)} can then be
determined from,
X(ω ) =
+∞
1
x(t)e −iωt dt = H(ω )F(ω )
2π −∫∞
2.12
Subsequently, the dynamic response function of the system { x(t) } is
derived from the Inverse Fourier Transform of {X(ω)}, and this computation
includes the transfer function of the system {H(ω)},
+∞
x(t) = ∫ X(ω )e iωt dω
2.13 a
−∞
+∞
+∞
−∞
−∞
x(t) = ∫ [H(ω )F(ω )]e iωt dω = ∫ X(f)e i(2πf)t df
2.13 b
, which is the same as Equation 2.8.
The Laplace Transform is an alternative method of solving for the
frequency response of vibrating systems subjected to any type of forcing
function. Inman (2001) and Maia and Silva (1997) describe its formulation
and Farrar et al. (1994) employed it in modal testing.
2.2.2 Impulse Response Function (IRF)
The convolution or Duhamel’s Method is the time domain
alternative to Fourier analysis and is based on the formulation of the
dynamic response of vibrating systems from simple (unit) impulses (Ewins,
2000). It has been shown in the literature that system response to an
arbitrary,
non-periodic
forcing
function
can
be
described
as
a
superposition (assuming linearity of the system) of responses to a series
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2 – Literature Review
of impulses which collectively represent the forcing function. Consider the
unit impulse or Dirac δ-function,
+∞
∫ δ (t − τ )dt = 1
2.14
−∞
The unit impulse function, when applied at time (t = τ), occurs for an
infinitesimal period of time and has an infinite magnitude with the integral
of {f(t)δ(t)} equal to unity (Figure 2-4). The corresponding IRF is denoted
by,
h(t − τ )
2.15
When an arbitrary forcing function is decomposed into a series of
impulses (Figure 2-5), the response of the system, at time {t}, to any one
of the incremental impulses, at time {τ}, is described by,
δx(t) = h(t − τ )f(τ )dτ
2.16
, and the total system response is described by integrating all incremental
responses. This is the convolution or Duhamel’s integral,
+∞
x(t) = ∫ h(t - τ ))f(τ )d(τ )
where h(t − τ ) = 0; t ≤ τ
2.17
−∞
, which identifies the system’s response {x(t)} as the convolution {*} of the
IRF {h(t)} and the forcing function {f(t)} in the time domain,
x(t) = h(t) ∗ f(t)
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2.18
2 – Literature Review
, and which is the time domain formulation of the Fourier Transform of the
response of the system {X(ω)} described in Equation 2.12 This
demonstrates the relationship between the IRF and FRF of the system:
FRF is the frequency domain representation of the IRF in the time domain.
Figure 2-4
Definition of the unit impulse forcing function. (Maia and Silva, 1997)
Figure 2-5
Definition of an arbitrary, non-periodic forcing function. (Maia and Silva,
1997)
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2 – Literature Review
2.2.3 Vibration tests
Two classes of tests exist for conducting vibration surveys: forced
vibration testing (FVT) and ambient vibration testing (AVT). The difference
between them is in the nature of the excitation.
In the case of FVT, an external force is imposed upon the structure.
The main advantage of FVT is that precise input excitation may be
controlled and measured (Huang, 2007). Common excitation sources of
forced vibrations include impact hammers, mass vibrators, dynamic
shakers and other impulse excitation devices (Figures 2-6, 2-7). A
frequently used exciter is the eccentric mass vibrator which enables the
application of sinusoidal forces with variable frequency and amplitude.
Brownjohn et al. (2003) and Abdel Wahab and De Roeck (1998) describe
in more detail the various types of excitation devices that are used. In
some instances monitored and unmonitored sources in the form of driven
test vehicles, explosives and reciprocating equipment have been used as
inputs.
This approach to modal testing requires force generation apparatus
which is costly and difficult to transport, install and implement. In addition,
He et al. (2005) note that a major difficulty with this approach is to provide
controlled excitations that generate sufficient response levels. Notably, the
operation is intrusive and disturbs normal activities in the building.
Some testing is based on measurements of the free vibration
response of the structure. To achieve free vibration response, the
structure is first subjected to a predetermined displacement and then
released into free response. Undoubtedly, applying this method in fullscale dynamic testing is challenging, costly and inconvenient. However,
tests of this nature have been executed on civil engineering structures
such as buildings, footbridges, highway bridges and offshore platforms
(Lawson, 2005; Ewins, 2000; Farrar et al., 1994, Srinivasan et al., 1984;
Rainer and Van Selst, 1976).
- 18 -
- 19 Shaker devices used to excite: (a) bridges, vertically; (b) electrohydraulic shaker from Arsenal Research; (c) dams, laterally
(EMPA). (Cunha and Caetano, 2006)
Devices used in FVT: (a) impulse hammer; (b) eccentric
mass vibrator; (c) electrodynamic shaker; (d) impulse
excitation device for bridges (K.U. Leuven). (Cunha and
Caetano, 2006)
Figure 2-7
Figure 2-6
2 – Literature Review
2 – Literature Review
Ambient vibrations result from excitation due to wind, wave action,
human activity (pedestrian and vehicular traffic, construction, etc) and
micro-tremors. The key characteristic of AVT is that the excitation is
unknown and unmeasured, and certain assumptions are required in order
to perform modal analysis.
The forcing function is assumed to be white noise which is a
function containing all frequencies within the frequency spectrum; in other
words the frequency domain representation of white noise is a function
with constant power (Bensen et al., 2007; Bendat and Piersol, 1993). This
assumption is true for random processes that are defined as ergodic,
stationary random force signals with Gaussian distributions (He and Fu,
2001); that is, the average of values across a collection of time histories at
a given instant in time is identical to the average of each record over time
(Maia and Silva, 1997).
Unfortunately, random processes inherently do not satisfy the
Dirichlet condition (Equation 2.10). As a consequence, neither the
excitation nor the response can be subjected to valid Fourier Transform
calculations as in Equations 2.10 to 2.17. Furthermore, Bensen et al.
(2007) advise that ambient noise is not truly flat in the frequency domain.
In spite of this, it has been common practice to carry along the assumption
of the forcing function being white noise and of white noise exciting all
frequencies equally (Maia and Silva, 1997). Additionally, correlation
functions and spectral densities have been introduced to describe ambient
vibrations. They will be elaborated later.
Since the amplitudes of vibration during AVT are small, they
describe the linear behaviour of structures (He and Fu, 2001; Ivanovic et
al., 2000). In fact, random excitations linearize the amount of energy
dissipation of the structure during vibration, thereby modelling many
structures that behave non-linearly through linearized frequency response
functions (He and Fu, 2001).
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2 – Literature Review
Since strict rules must be followed during the analysis, important
assumptions are required for analysing ambient vibration measurements
(Huang, 2007):
1. The forcing function is white noise excitation;
2. The response of the structure is a linear combination of inputs, due
to system linearity.
3. The vibration measurements are ergodic and stationary random
processes.
Much research has been carried out on the use of forced vibration
tests and ambient vibration tests, indicating that estimated modal
parameters from either test have always proven to be consistent with each
other and with corresponding analytical modal analyses (Crawford and
Ward, 1964; Hudson, 1970; Trifunac, 1970; Felber et al., 1995a; Farrar
and James, 1997; Huang et al., 1999; He et al., 2005).
2.2.4 Modal testing of civil engineering structures
Modal testing evolved from mechanical engineering resonance
tests. Cunha and Caetano (2006), Ivanovic et al. (2000), Ewins (2000) and
Farrar and James (1994) presented extensive literature reviews of
developments in modal testing of civil engineering structures and Carder
(1936) identified early work by the U.S. Coast and Geodetic Survey dated
in the early 1930’s.
The research of Crawford and Ward (1964) advanced the use of
spectral techniques for identifying the natural periods of buildings and was
extended to the dynamic testing of other structures, including bridges
(McLamore et al., 1971). Hudson (1970) subsequently provided detailed
reviews on the dynamic testing of full-scale structures. Jennings et al.
(1972) obtained estimates of frequencies, mode shapes and damping
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2 – Literature Review
values for a twenty-two storey steel structure and Ellis and Jeary (1979)
compared forced vibration and ambient vibration test results for a tower.
Later studies by Srinivasan et al. (1981, 1984) examined several dynamic
test results from nuclear power plants, buildings, dams, bridges, towers
and tall chimneys as well as offshore platforms.
An increasing number of structures have been instrumented in
recent years, largely due to a increased awareness of dynamic issues with
structures, better accessibility to equipment and greater computing
capabilities (Ewins, 2000). Recent studies have been performed by Carne
et al. (1988) on a turbine; by Brownjohn et al. (1992), Ventura et al. (1994)
and Ivanovic and Trifunac (1995) on full-scale buildings; by Feng et al.
(1998) on a tower; by Daniel and Taylor (1999) on dams; and by Wahab
and DeRoeck (1998), Brownjohn et al. (1999), He et al. (2005) and
Nielson et al. (2007) on bridges.
2.2.5 Ambient vibration testing of bridges
Varney and Galambos (1966) summarised a series of dynamic
tests performed on highway bridges in the USA between 1948 and 1965.
Farrar and James (1997) reported that one of the earliest applications of
modal testing to bridges using ambient vibrations was performed by
McLamore et al. (1971) using an extension of spectral techniques
developed by Crawford and Ward (1964). Iwasaki et al. (1972) presented
tests performed in Japan while Ganga Rao (1977) conducted field and lab
tests on bridge systems. Rainer and Van Selst (1976) characterized the
dynamic properties of the Lions’ Gate Suspension Bridge in Canada using
ambient vibration measurements. Douglas et al. (1981) use the approach
to monitor a reinforced concrete bridge and a composite girder bridge
while Pardoen et al. (1981) tested a steel truss bridge in New Zealand.
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2 – Literature Review
Tanaka
and
Davenport
(1983)
investigated
ambient
vibration
measurements from wind excitation for a suspension bridge in the USA.
Cantieni (1984) reported on the testing of slab and girder bridges in
Switzerland between 1958 and 1981. Brownjohn et al. (1987, 1989)
characterized the dynamic properties of suspension bridges from ambient
vibration measurements due to traffic and wind excitation. Ventura et al.
(1994) estimated dynamic properties of a medium span, composite girder
bridge in Canada. In 1995, Felber et al. (1995a, 1995b) presented findings
of ambient vibration studies carried out on a medium span, composite
steel box girder bridge in Switzerland and on a long span, steel truss
bridge in Canada. Brinker et al. (1996) presented techniques for extracting
modal parameters for bridges from ambient vibrations.
Huang et al. (1999) studied typical multi-span, highway bridges in
Taiwan and Arup tested the Millenium Bridge in UK in 2000 (Dallard et al.,
2001). Ivanonvic et al. (2000) listed a number of publications addressing
ambient vibration testing of bridges since the 1970’s. Cunha et al. (2001)
presented findings of modal testing of a cable-stayed bridge in Portugal.
Ren et al. (2004) tested a continuous steel girder bridge while He et al.
(2005) presented the findings of an ambient vibration survey of a
suspension bridge in the USA. Zivanovic et al. (2006) monitored a
Montenegrin footbridge using ambient vibrations and Nielson et al. (2007)
presented findings for typical multi-span, concrete girder highway bridges
in the USA.
Much of the modal testing research employed ambient vibration
measurements because they were easily obtainable and did not disrupt
user access to the structure. In fact, AVT was routinely used to validate
the results of FVT and numerical models. In most cases, the results of the
experimental modal tests were consistent and differences with numerical
models were attributed to the incompleteness of structural models
(Zivanovic et al., 2007; Farrar and James, 1997; Rainer and Van Selst,
1976).
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2 – Literature Review
Farrar and James (1994) reviewed modal testing of a number of
bridges that were investigated using ambient vibration measurements. The
more significant details of each test (such as the excitation source,
analytical techniques employed to extract modal properties and the modal
properties that were estimated from the test) are summarised.
For 3-span bridges (with 2 to 4 lanes), spans between 10 and 95
metres, internal supports of 5 to 30 metres above foundation level, and
with a deck skew of less that 30 degrees and comprising a composite
deck of steel reinforced concrete fastened to steel girders, the
experimentally estimated fundamental frequency can range from 0.5 and
2.5 Hertz. Similar bridges with shorter internal supports and less slender
decks have a larger fundamental frequency of vibration (Maia and Silva,
1997). The bridge tested in this project is not classified as slender and has
a skew of 10 degrees.
2.3
Modal parameter estimation
The dynamical properties of a vibrating structure are its natural
frequencies, mode shapes and damping values. Every structure vibrates
according to its modes of vibration. Each mode is defined by unique modal
properties which collectively describe the structure’s dynamic response.
Since the response of vibrating systems can be explained in the
frequency domain as well as the time domain, it is not surprising to find
that techniques have been developed in both of these areas. In the
frequency domain, methods are based on an extension of Fourier analysis
while time domain methods are based on the theory of correlation and
auto-regressive, moving average and transfer-functions for parametric
models. Some of the commonly used methods include the Fourier spectral
analysis (He et al, 2005), power spectral density analysis, cross-power
spectral density analysis (Trauth, 2007) and the curve-fitting method (Maia
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2 – Literature Review
and Silva, 1997). Other extraction methods include the auto-regressive
moving-average approach (He and Fu, 2001), the Ibrahim Time Domain
approach (Maia and Silva, 1997), the Random Decrement technique
(Feng et al., 1998), the Eigensystem Realisation Algorithm (ERA) (He et
al., 2005) and the Complex Exponential methods (Ewins, 2000).
These are by no means the full extent of techniques used for modal
parameter estimation. Ewins (2000) and Maia and Silva (1997) conducted
reviews of techniques for modal parameter estimation from vibration
measurements and further subdivided established techniques according to
the direct (based on spatial models) and indirect (based on modal models)
nature of the processes and also on the number of modes (number of
DOFs) treatable by each process. More recent work has led to the
development of subspace methods of system identification such as the
Stochastic Subspace Identification (SSI) technique (Watkins, 2007).
Four techniques were employed to process the ambient vibration
measurements in this research. They were chosen because they have
been used extensively in modal testing and are user-friendly. They are
discussed here in detail.
2.3.1 Fourier Spectral Analysis (FSA)
FSA is one of several SDOF approaches in which a MDOF system
is analysed at resonant frequencies, one by one. The basis of the method
is peak-picking from FRFs in terms of Fourier amplitude; the natural
frequencies are taken at peak amplitudes (Maia and Silva, 1997), the
damping ratios are calculated from several methods including the
sharpness of the peaks using the half-power method (Craig and Kurdila,
2006) and the mode shapes are estimated from the ratios of peak
amplitudes on FRFs at various positions across the structure to
corresponding peaks on the FRF of a reference position (He et al., 2005).
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2 – Literature Review
The details of FSA are covered extensively in the literature,
however, its important features are summarised here (Bendat and Piersol,
2000). It has been demonstrated that any periodic function {x(t)}, with a
period {T}, can always be decomposed into an infinite series of sinusoids.
The Fourier Series assumes the sum of two infinite cosine and sine series,
∞
1
x(t) = a 0 + ∑ (a n cosωn t + b n sinωn t )
2
n =1
where ω n =
2πn
T
2.19
, with the Fourier coefficients {an} and {bn} related to the function {x(t)}, and
where {n} is the order of cycles,
T
an =
2
x(t)cosωn tdt
T ∫0
bn =
2
x(t)sinωn tdt
T ∫0
2.20 a
T
2.20 b
T
2
a 0 = ∫ x(t)dt
T0
2.20 c
In the instance where a non-periodic function {x(t)} satisfies the condition
of Equation 2.10, the response function can be written as,
+∞
x(t) =
∫ [A(ω )cosωt + B(ω )sinωt ]dω
2.21
−∞
, where the coefficients are defined by,
A(ω) =
1
π
+∞
∫ x(t)cosω tdt
n
−∞
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2.22 a
2 – Literature Review
B(ω) =
1
π
+∞
∫ x(t)sinω tdt
n
2.22 b
−∞
, and alternative forms of the function are expressed as Equations 2.12
and 2.13.
Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)
In practice, the response function {x(t)} is recorded for a finite time
duration at {N} discrete points that are evenly spaced by the sampling
scheme and digitised by the analogue-to-digital conversion process.
Assuming the record is periodic about the length of the sample, the
Fourier Transform can be estimated as a finite series with discrete points
at (t = tk) and (k = 1, N) (Ewins, 2000),
1
x(t k ) ≡ (x k ) = a0 +
2
N (N −1)
or
2
2
∑
N =1
2πnk ⎞
2πnk
⎛
+ b n sin
⎟
⎜ a n cos
N ⎠
N
⎝
2.23
, and the coefficients are defined by,
an =
1 N
2πnk
x k cos
∑
N k =1
N
2.24 a
bn =
1 N
2πnk
x k sin
∑
N k =1
N
2.24 b
2 N
∑xk
N k =1
2.24 c
a0 =
Cooley and Tukey (Pollock, 1999) proposed an optimisation of the
DFT known as the Fast Fourier Transform (FFT) where the method
requires N to be an integral power of two (2) thereby reducing the
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2 – Literature Review
execution time to compute the DFT of the response time history. The
modal parameters of the system are then estimated from Fourier spectra
generated from these relationships.
2.3.2 Power Spectral Density (PSD) and Cross-power Spectral
Density (CSD) Analysis
When the dynamic response function {x(t)} is now taken as a
random vibration, the statistical expectation or average value of the
product {x(t).x(t+τ)} yields the autocorrelation function {Rxx(τ)},
R xx = E[x(t).x(t + τ )]
2.25
, that describes how the function {x(t)} at an arbitrary time {(t+τ)} is
dependent upon the function at a previous time {t}. Here, the original time
history is transformed into a new function of time. Since this function
obeys the Dirichlet condition (Equation 2.10) a frequency description of the
original signal {f(t)} is estimated from the Fourier Transformation of {Rxx(τ)}
as the power spectral density (PSD) or auto-spectral density (ASD)
{Sxx(ω)},
S xx (ω ) =
+∞
1
R xx (τ )e −iωt dτ
∫
2π −∞
2.26
Figure 2-8 illustrates the time history, auto-correlation function and
PSD of a random signal. The same logic, when applied to a pair of
simultaneous functions y(t) and x(t) will generate cross-correlation {Rxy(τ)}
and cross-spectral density (CSD) {Sxy(ω)} functions. The cross-correlation
function is defined as,
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2 – Literature Review
R xy = E[x(t).y(t + τ )]
2.27
, and the CSD is defined by its Fourier Transform as,
S xy (ω ) =
+∞
1
R xy (τ )e −iωt dτ
∫
2π −∞
2.28
Developed in the 1960’s, the Welch’s approach to PSD analysis
divides the time history into overlapping segments, computes the power
spectrum for each segment and generates an average power spectrum
(Trauth, 2007). Figure 2-9 demonstrates the principle of the Welch PSD
analysis which has proven useful by increasing the signal-to-noise ratio of
power spectra despite reducing frequency resolution.
Rxx(τ)
Sxx(ω)
Figure 2-8
Random signals: (a) time history x(t); (b) auto-correlation function Rxx(τ);
and (c) PSD Sxx(ω). (Ewins, 2000)
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2 – Literature Review
Figure 2-9
Description of Welch PSD analysis. (Trauth, 2007)
2.3.3 Auto-regressive moving-average (ARMA) method
An ARMA model is a combination of an auto-regressive (AR) model on
one part and a moving-average (MA) model on another. The basis for the
operation of each of these is the assumption that a dynamic response
function {x(t)} is the random single output of a linear system which is
driven by a random single input. This parametric approach is commonly
applied to systems where only the output is measured and the unknown
input is taken as white noise (Pollock, 1999; Maia and Silva, 1997).
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2 – Literature Review
The AR component expresses the response function as a linear
function of its past values where the order of the AR model {n} indicates
the number of lagged points that are included. It is defined through the
function,
n
x t - ∑ φ r x t −r = e t
2.29
r =1
, where {xt} is the value of the function at time {t}, {φr} is the AR coefficient,
{xt-r} is the value of the function at {r} past points and {et} is the error
residual from the prediction process.
The MA component expresses the function as an unevenly
weighted, moving average of the {et} series, demonstrated by,
m
x t = e t - ∑θ s e t − s
2.30
s =1
, where {xt} is the value of the function at time {t}, {θs} is the MA coefficient,
{et-s} is the value of the function at {s} past points and {et} is the error
residual from the prediction process. Similarly, the order of the model {m}
is indicative of the number of lagged points that are included in the
estimation.
By combining the features of AR and MA modelling, the ARMA
method executes a system identification algorithm through the function,
n
m
r =1
s =1
x t - ∑ φ r x t −r = e t - ∑θ s e t − s
where n > m
2.31
The idea of the method is to identify the system and predict both
present and future responses from past inputs and outputs (He and Fu,
2001; Pollock, 1999). Furthermore, by increasing the order of either
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2 – Literature Review
component of the model (‘n’ or ‘m’), the computational demand of the
model will grow and the accuracy of the estimation will improve.
2.4
Signal processing issues in AVT
Signal processing forms an integral part of the analysis of ambient
vibration measurements. Therefore, it is important to appreciate and
address some pertinent concerns. The key signal processing issues relate
to the data acquisition process as well as to the analysis methods. Some
of these issues regard sampling, averaging, spectral leakage, and
resolution.
2.4.1 Sampling
An alias is an error resulting from the processing of time signals.
One such error is commonly introduced into analyses by improper
sampling of the data. This aliasing can be reduced substantially by
adhering to Shannon’s sampling theorem where the sampling rate (SR) of
the data must be at least twice the maximum frequency of interest (fmax) in
the signal (Equation 2.32). This ensures that the sampling interval (Δt)
(Equation 2.33) is small enough to obtain at least two sample points in
each cycle of the largest frequency of interest. Inman (2006) noted this
alias results from misinterpretation of the analogue signal by digital
recorders during conversion. Otnes and Enochson (1972) further indicated
that at least 2.5 samples per cycle was a better choice. In addition, antialiasing filtering is employed to remove components of the analogue signal
with a higher frequency than can be properly resolved by recording
devices. Low pass filters cut off frequencies that are larger than roughly
half of the maximum frequency of interest; this is the Nyquist frequency
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2 – Literature Review
(fNYQ) (Equation 2.34) (Inman, 2006). The Nyquist frequency and the
maximum frequency of interest influence the selection of the sampling
rate.
SR ≥ 2f max
2.32
SR =
1
Δt
2.33
f NYQ ≈
f max
2
2.34
2.4.2 Averaging
Since Fourier Transforms do not strictly exist for random processes,
the frequency content of time signals is estimated using power spectral
densities and correlation functions. The frequency content is estimated
from the Fourier Transforms of the signals introducing spurious results.
Ewins (2000) described the sequential and overlap averaging techniques
for sampling data from within the time signal to minimise such artefacts in
the results (Figure 2-10). Overlap averaging was employed in the Welch’s
approach to spectral analysis.
2.4.3 Spectral leakage
Spectral plots exhibit the phenomenon of energy leakage at
frequencies immediately below and above true resonant frequencies. It is
the result of the assumption of periodicity of the finite time history
measurement which allows the relative amplitude of spectral side lobes to
be erroneously large. Figure 2-11 presents two time histories where the
only difference between them is the finite length of the measurements. To
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2 – Literature Review
process the time domain measurements into the frequency domain, both
signals were assumed to be cyclic with a period (T) equal to their lengths.
In the first scenario (a), the start and end of the measurement allow
uninterrupted continuity between consecutive cycles; consequently, the
spectral plot correctly indicates a single line at the frequency of the signal.
However, a marked discontinuity between consecutive cycles in the
second instance (b) results in significant energy leakage in the spectrum;
power is leaked around the true frequency of the signal.
Short of attempting the almost impossible task of adjusting the
recorded signal length to reflect the period(s) within the random signal, the
impact of leakage is more ordinarily diminished with a windowing
approach. Here, one of the many constructs of windows is imposed upon
the time signal prior to its transformation to the frequency domain. The
finite time signal is strategically scaled between factors zero and one (0 –
1.0) to promote continuity between consecutive segments of the signal
stemming from the assumption of signal periodicity.
Windowing is important to the generation of an average of
segmented signals, especially when the segments overlap. The
Rectangular window is the simplest type. It is a function with a constant
magnitude of one (1.0); therefore the subjected signal or segment is
wholly repeated. When the segments of the signal overlap, the repetition
of portions of the signal can be depreciated by using windows that
attenuate the extremities and maintain the interior values of the segments.
The Hanning, Hamming and Blackman windows produce such an effect
on signals and are used frequently in signal processing. The Rectangular
window is inappropriate for this task. Figures 2-12, 2-13 and 2-14 illustrate
some of the types of windows used commonly to alleviate the influence of
spectral leakage and the result of using each of them.
Averaging the segments of a signal reduces the level of noise that
can contaminate the resulting spectra, allowing the interesting features of
the signal to be identified (Smith, 1997).
- 34 -
2 – Literature Review
Figure 2-10
Interpretations of multi-sample averaging: (a) sequential and (b) overlap.
(Ewins, 2000)
Figure 2-11
Finite-length sample and spectral leakage. (Ewins, 2000)
- 35 -
2 – Literature Review
Figure 2-12
Window functions used frequently in spectral analysis. (Stoica and Moses,
1997)
- 36 -
2 – Literature Review
Figure 2-13
The effect of the window functions from Figure 2-12 in the frequency
domain. The windows allow differing levels of spectral leakage from
processed signals. (Stoica and Moses, 1997)
- 37 -
2 – Literature Review
Figure 2-14
Demonstration of how original time histories (left and right) are treated with
select windows (centre). Window types are (a) Boxcar (Rectangular); (b)
Hanning; (c) Cosine-taper, and (d) Exponential. (Ewins, 2000)
2.4.4 Resolution
Processing signals through the DFT has been found to yield
spectra with inadequate frequency resolution or spectral resolution. The
problem originates from a combination of the limited number of discrete
points, size of the frequency range of interest and/or length of the time
sample required to generate acceptable results. Ewins (2000) indicated
two typical means of addressing this problem: controlling the size of the
transform and padding the signal with zeros.
The size of the transform (number of data points in the transform)
can be increased to abate the resolution problem, especially with modern
computational advancements and the efficient FFT by Cooley and Tukey
(Trauth, 2007, Pollock, 1999). However longer transforms provide
increased resolution properties while little is done to remove noise levels
- 38 -
2 – Literature Review
in the spectra. A solution is to impose a window on many segments of the
original sample with a smaller size of transform. Alternatively, a larger
transform may be used with a low pass filter, which cuts off higher
frequencies from the signal (Ewins, 2000). Smith (1997) indicated these
two solutions provide similar spectral resolutions though the latter is more
computationally intensive. Advancements in computing capabilities
diminish the significance of this constraint.
Ewins (2000) suggested increasing the length of the measurement
and increasing the sampling rate to improve spectral resolution. This may
not be pragmatic; however, a simple way to increase the spectral
resolution is by padding the measurement with zeros. Here, the number of
data points describing the record is increased to the size of the transform
by appending zero-valued coordinates. The increased number of points in
the signal facilitates spectra with a finer resolution.
- 39 -
Chapter 3
Case Study
This chapter describes the bridge structure selected for ambient
vibration testing. The structural characteristics of the bridge pertinent to
modal analysis are presented and details of the testing procedure are
discussed. Subsequently, the spectral analysis techniques used on the
ambient vibration records are presented. Finally, analytical models of the
bridge that were created for comparative purposes are described.
3.1
Bridge structure
The bridge is an above-grade crossing of St. Jean Boulevard over
Donegani Avenue and the railway lines from the Canadian National (CN)
and Canadian Pacific (CP) railway companies in the city of Pointe-Claire
(Quebec). Figure 3-1 shows the location and a picture of the bridge.
The 2006 edition of the Canadian Highway Bridge Design Code
(CHBDC 2006) (CAN/CSA-S6, 2006) classifies this structure as an
emergency-route bridge which carries and crosses over routes that need
to be open, at minimum, to emergency and security/defence services
immediately following the design earthquake. The CHBDC 2006
recommends the use of a design earthquake with a 10% probability of
exceedance in 50 years (equivalent to 15% probability of exceedance in
75 years, with a return period of 475 years).
The bridge has a total length of 68.89 m (226 ft) with three spans
(21.34 - 26.21 - 21.34 m). The deck is 25.3 m wide and accommodates
traffic flow in two directions (northbound and southbound) which are
separated by a central median. The pavement width is 11 metres in each
- 40 -
3 – Case Study
ni
ega
Don
e
u
n
Ave
N
i
gan
one
eD
u
n
Ave
CP
and
CN y lines
a
w
l
rai
Monitored bridge
le
Bou
enir)
Souv
u
d
(
20
route
Auto
vard
nir)
ouve
(du S
0
2
route
Auto
ea
St-J
A-N
n
Assumed field North
(a)
(b)
Figure 3-1
Views of the monitored bridge: (a) plan view sketch map indicating the
location of the bridge (Not drawn to scale); (b) western elevation of the
bridge. (De La Puente Altez, 2005)
- 41 -
3 – Case Study
direction consisting of three 3.65 m lanes and no emergency shoulder.
Figure 3-1(b) shows a full view of the three spans of the bridge with plan
and elevation details indicated in Figures 3-2 and 3-3.
The composite bridge deck system is a reinforced concrete deck
(190 mm thick) underpinned by twelve structural steel girders (WF846x210
mounted with 20mm-φ-shear stud connectors) aligned along the
longitudinal direction. The deck is supported at four locations – two
internal moment-resisting, reinforced concrete frames and an abutment at
either end. Each of the identical moment-resisting frames (Figure 3-4)
comprises twelve 0.4572 m-square columns (Figure 3-5) that stand 4.88 m
tall on
a 3.66 m high reinforced concrete wall having cross-sectional
dimensions of 0.762 m and 25.654 m. The moment-resisting frames are
completed by a 0.4572 m-square reinforced concrete beam which
connects all twelve columns at the top.
The steel girders were spliced within the internal span - to ensure
structural continuity along the entire length of the bridge - and framed by a
steel joist (WF457x75) above each support location (Figure 3-3). Within
each bay are three lines of transversely aligned channel sections
(C381x51) that help to maintain the girders’ positions. Along with shear
stud connectors atop each girder, these transverse brace elements
increase girder resistance to lateral buckling through reduced effective
span. They also contribute toward the transverse, in-plane stiffness of the
deck system.
- 42 -
3 – Case Study
i
0.190 m
ii
0.8458 m
viii
ix
iii
ix
0.4572 m
vii
iv
4.8768 m
v
3.6576 m
vi
3.6576 m
21.340 m
26.210 m
21.340 m
(a)
i
0.190 m
ii
0.8458 m
ix
viii
x
iii
vii
iv
4.8768 m
v
3.6576 m
vi
3.6576 m
21.340 m
26.210 m
21.340 m
(b)
ID
Structural Details
0.4572 m
ID
Structural Details
i
R.C. Slab (190 mm thick)
vi
R.C. Foundation (mass strip)
ii
Steel Girder (WF 845 x 210)
vii
Moment splice (for continuity)
iii
R.C. Cap beam (457 x 457 mm)
viii
Hinge support (atop column)
iv
R.C. Column (457 x 457 mm)
ix
Roller support (atop abutment)
v
R.C. Wall (762 x 25,654 mm)
x
Hinge support (atop abutment)
Figure 3-2
Sketch views of the bridge deck framing system: (a) profile view: Case 1;
(b) profile view: Case 2. (Not drawn to scale)
- 43 -
3 – Case Study
25.300 m
21.340 m
26.210 m
21.340 m
WF845x210 structural steel beam
C381x51 structural steel channel
WF457x75 structural steel beam
(a)
Figure 3-3
Sketch view of the bridge deck framing system: plan view. (Not drawn to scale)
Centre line
(a)
(b)
Figure 3-4
Sketch views of the moment-resisting, reinforced concrete frame: (a) sectional
view; (b) frontal view. (Not drawn to scale) (De La Puente Altez, 2005)
- 44 -
3 – Case Study
457.2
#8 Ø 25.4 mm
#10 Ø 32.3 mm
457.2
#10 Ø 32.3 mm
#8 Ø 25.4 mm
#4 Ø 12.7 mm
#8 Ø 25.4 mm
(a)
10 Ø 32.3 mm
#10 Ø 32.3 mm
54
114.3
120.7
114.3
54
#8 Ø 25.4 mm
304.8
Stirrups
#4 Ø 12.7 mm
304.8
(b)
Figure 3-5
Sketch sections through a typical column in the moment-resisting, reinforced
concrete frame: (a) plan cut-section; (b) profile cut-section. (Not drawn to scale)
(De La Puente Altez, 2005)
The connection of the girders to the top of each moment resisting
frame is detailed to preclude separation of and moment transfer between
the two elements. A 25 mm-wide expansion joint exists at each end of the
bridge between the bridge deck system and the abutments. They
accommodate the thermal-induced, longitudinal, volumetric changes of the
structure. The connection of the girders to the abutments is also detailed
to preclude separation of and moment transfer between the two. Details of
- 45 -
3 – Case Study
the connection between the bridge deck system to the moment-resisting
frame is shown in Figure 3-6 and to the abutment in Figure 3-7.
(a)
(b)
Figure 3-6
Sketch views of the girders’ connection to the top of the moment-resisting,
reinforced concrete frame: (a) profile view (transverse direction); (b) frontal
view (longitudinal direction). (Not drawn to scale) (De La Puente Altez, 2005)
road level
2 layers of
asphalt paper or
the equivalent
beam
support device
abutment
Figure 3-7
Profile view (transverse direction) sketch of the beam connected to the top of
the abutment. (Ministry of Transport of Quebec)
- 46 -
3 – Case Study
3.2
Construction materials
The 2006 edition of the CHBDC 2006 (CAN/CSA-S6, 2006)
suggests several means by which the properties of the construction
materials comprising an existing bridge can be determined. Two of these
methods recommend estimating the material properties from reviewing asbuilt construction plans and approximating material usage from the date of
construction.
The as-built construction drawings that were provided by the
Ministry of Transport of Quebec (MTQ) were dated 1961 and 1962. From
1955 to 1965, the minimum yield strength (fy) of structural steel ranged
from 250 to 350 MPa with an estimated Modulus of Elasticity (Es) of 2.0 x
105 MPa (Canadian Institute of Steel Construction, 2004). The 28-day
compressive strength of concrete cylinders (fc’) varied from 20 to 25 MPa
(De La Puente Altez, 2005) during the same period. Figure 3-8 illustrates
the corresponding design stress-strain plots of the main materials at the
date of construction (De La Puente Altez, 2005).
The compressive strength of concrete generally increases with long-term
hydration of cement followed by some retrogression. Brinkerhoff (1993)
noted that much concrete construction from the 1960’s, attain compressive
strengths exceeding 35 MPa (5000 psi). Neville (1996) found American
Portland cements made at the start of the 20th century led to an increase
in the strength of concrete stored outdoors where the 50-year strength
was generally 2.4 times the 28-day strength. Alternately, cements made
since the 1930’s attained peak strengths between 10 and 25 years and
then exhibited some retrogression (Neville, 1996). Furthermore, German
Portland cements made in 1941 led to 30-year strengths of 2.3 times the
28-day strengths. Design concrete compressive strengths of 25 MPa
(3500 psi), 35 MPa (5000 psi) and 45 MPa (6500 psi) were used in
consideration of these variations in the unknown, present compressive
strength.
- 47 -
3 – Case Study
Steel Reinforcement Material Information
500
Concrete Material Information
30
Stress [MPa]
Stress [MPa]
450
25
400
350
20
300
Yield Stress: 300 MPa
Initial Elastic Modulus: 200000 MPa
Strain at strain hardening: 7.0 mm/m
Strain at maximum stress: 100.0 mm/m
Ultimate Stress: 450 MPa
250
200
Cylinder Strength: 25.0 MPa
Tensile Strength (auto): 1.63 MPa
Cracking Strain: 0.069 mm/m
Strain at peak stress (auto): 1.90 mm/m
Initial tangent Stiffness: 23499.9 MPa
Tension stiffening factor: 0.00
15
10
150
100
5
50
Strain [mm/m]
Strain [mm/m]
0
0
0
20
40
60
80
100
120
0
0.5
(a) Steel
1
1.5
2
2.5
3
(b) Concrete
Figure 3-8
Assumed stress-strain plots for the steel and concrete at the time of
construction of the bridge. (De La Puente Altez, 2005)
Table 3-1 indicates the material strengths that were assumed for
the analytical modal analysis.
Assumed material strengths
Material
Structural Steel
Concrete
Property
Value (MPa)
minimum yield stress (fy)
300
minimum ultimate stress (fu)
450
modulus of elasticity (Es)
2.0 x 105
minimum 28-day, compressive cylinder 25, 35, 45
strength (fc’)
modulus of elasticity (Ec)
25.0 x 103,
29.6 x 103,
33.5 x 103
Table 3-1
Assumed material strengths for the analytical modal analysis.
- 48 -
3 – Case Study
3.3
Equipment
Field investigations were conducted using two velocimeter
seismometers manufactured by Lennartz Electronic (Figure 3-9). Each
seismometer simultaneously detects motion of the structure in three
orthogonal directions - two perpendicular horizontal axes and one vertical
axis. Coaxial cables link each sensor to a Cityshark II data acquisition
station (DAS) with conditioning and analog-to-digital convertors. The
measurements are stored in ASCII format on 32 MB flashcards. Figure 310 shows an extract from one of the measurement files.
The operational range of parameters and other specifications of the
geophones and recorders are provided in Table 3-2. Global positioning
system (GPS) capability is used to synchronise the internal clocks of each
battery-powered DAS. Several analysis methods required the use of
simultaneous records. A remote control system (RCS) and wireless
communications (walkie-talkie) devices were used to obtain simultaneous
recordings. Details of the sampling parameters of each record are
presented in the following section addressing the testing procedure.
(a)
Figure 3-9 (a)
Experimental equipment: seismometer (left) and DAS (right).
- 49 -
3 – Case Study
(b)
Figure 3-9 (b)
Experimental equipment: (b) RCS antenna attached to DAS and GPS
sensor mounted on railing.
O riginal file nam e: 09110310.023
O riginaled
fileinto:
nam070911_0310.023
e: 09110310.023
T ransform
T ransform
ed into:
R eadC
ity version:
3.2070911_0310.023
R eadCserial
ity version:
S tation
num ber:3.2
027
Station
serial
ber: 027
S tation
softw
arenum
version:
0623
Stationnum
softw
are3 version: 0623
C hannel
ber:
C hannel
num
ber: 3
S tarting
date:
11.09.2007
Starting
11.09.2007
S tarting
timdate:
e: 03:10:03.223
Starting
tim11.09.2007
e: 03:10:03.223
E nding
date:
E nding
11.09.2007
E nding
timdate:
e: 03:10:03.223
E nding
tim100
e: 03:10:03.223
S am
ple rate:
Hz
Sam
rate:
Hz
S am
pleple
num
ber:100
90000
Sam ple num
ber: 90000
R ecording
duration:
15 m n
R ecordingfactor:
duration:
15 m n
C onversion
52428.6
C onversion
factor: 52428.6
G ain:
16
G ain:ic16range: 5 V
D ynam
D ynamsam
ic range:
5V
C lipped
ples: 0.00%
C lipped: sam
ples: 0.00%
L atitude
45 26.632
N
L atitude : 734548.904
26.632WN
L ongitude:
L ongitude:
A ltitude
: 66 m73 48.904 W
: 663 m
N o.A ltitude
satellites:
N o. satellites:
3
M axim
um am plitude:
0/1
M axim um am plitude:
-689
-171 0 / 1
51
-689
-171
51
-661
-177
-114
-661
-177
-580
-148
23-114
-580
-510
62 -148
24 23
-510
-548
48 62
-5424
48
-54
… -548
… … … … … … … ...
… … … … … … … … ...
Figure 3-10
Extract of a data file indicating record details followed by five rows of data
points.
- 50 -
3 – Case Study
Specifications of data extraction system
Sensor
Type
LENNARTZ
ELECTRONIC
Triaxial
velocity
seismometer (LE-3D/5s)
Eigenfrequency
0.2 Hz (5 s eigenperiod)
Frequency Range
0.5 – 50 Hz
Power supply
12 V (DC)
Number
2
Data acquisition station
Type
CITYSHARK II Microtremor Acquisition Station
Analog input
Single 3D channel
Gain Amplifier
14 selectable values (1-8192)
Dynamic Range
108 dB at 100 Hz; 90 dB at 250 Hz
Sampling Rate
18 selectable values (1-1000 Hz)
Temperature Range
-10 0C / +50 0C
Compatibility
Lennartz
Number
2
Other
Global positioning system (GPS) sensors
Coaxial cables
Remote control system (RCS) device
Adaptors
Wireless communications (walkie-talkies)
Table 3-2
Specifications of the seismometers and data acquisition stations used in
the ambient vibration testing.
- 51 -
3 – Case Study
3.3.1 Testing
After identifying the bridge for monitoring, the next step was to
establish an appropriate measurement protocol. The purpose of the
ambient modal testing is to determine the natural frequencies of the
structure from strategically located ambient vibration measurements. In
this study, the sources of ambient vibrations are vehicular and pedestrian
traffic, wind and micro-tremors, and excitation from the trains travelling
beneath the structure.
Tests were conducted during two field trips: September 10th 2007
and November 30th 2007. Since seismometers are sensitive equipment, it
was important to avoid unnecessary oversaturation of the data records
from noisy input. To minimise oversaturation, sampling was planned at
night when vehicular and train activities are minimal. A large proportion of
ambient vibrations are from vehicular traffic and the gain parameter was
adjusted to an appropriate low level to allow for it. In spite of this, each
record had to be reviewed to eliminate portions that were oversaturated
(Smith, 1997; Felber et al., 1995). Figure 3-11 shows a typical time history
measurement.
Post-processing steps were used to synchronise pairs of records
for further analysis. All of the records were de-trended to correct errors
that are introduced by the analog-to-digital conversion in sampling and
storage (Trauth, 2007; Farrar et al., 1994). Subsequently, experimental
modal analyses were performed on the adjusted ambient vibration
measurements.
- 52 -
3 – Case Study
Figure 3-11
Typical measurement time history in three orthogonal directions (Record
6: File 27 from DAS 28 on November 30th 2007).
- 53 -
3 – Case Study
A
B
C
D
E
(a)
F
G
Position of reference sensor
Position of roving sensor
A
B
C
(b)
Position of reference sensor
Position of roving sensor
B
A
(c)
Position of sensors
Figure 3-12
Testing configurations: (a) A - along the central longitudinal axis; (b) B along a transverse line; and (c) C - along one exterior longitudinal axis.
- 54 -
3 – Case Study
3.3.2 Test locations
Test locations were selected at various positions across the bridge
(Figure 3-12). Since only two sensors were available for testing, a
maximum of two simultaneous recordings was possible for each sampling
session; a reference sensor was stationed at one position while a roving
sensor
was
deployed
to
various
positions.
In
configuration
A,
measurements were obtained along the central longitudinal axis of the
bridge. The reference sensor was positioned at mid-span of the outer span
of the bridge. The roving sensor was deployed to the end of the bridge,
over inner supports and the mid-span of the bridge’s middle span.
Additional measurements were obtained from configuration B along the
transverse direction, above an inner support. The reference sensor was
positioned at the outer edge of the bridge deck and the roving sensor was
deployed to the central and the other outer edge points along that
transverse line. In configuration C, measurements were sampled at two
points along an outer longitudinal line of the bridge. Nineteen (19) records
were obtained from all tests (Table 3-5). Of these, five (5) record pairs
were measured.
3.3.3 Sampling parameters
The sampling parameters are the sampling rate, duration of
recording, and sampling sensitivity (gain). The operational range of the
sensors and acquisition units were described in the previous section.
Sampling parameters were identified for each test and are summarised in
Table 3-3.
A sampling rate of 100 Hertz (100 data points per second) was
employed during the tests on September 10th 2007. Synchronised records
were required for some of the experimental modal analysis methods. They
- 55 -
3 – Case Study
were first obtained using the remote control system (RCS). However, the
RCS device was malfunctioning and a higher sampling rate of 1000 Hertz
(1000 data points per second) was selected and combined with a manual
synchronisation of pairs of records using the internal clock of each device.
These tests were performed on November 30th 2007. The synchronisation
procedure is detailed in the following subsection.
Sampling parameters
Sampling rate (Hz)
100, 1000
Duration (minutes)
10, 15
Gain
16, 32, 64
Table 3-3
Sampling parameters used in testing.
The choice on appropriate record duration is a debatable issue.
Past research used record durations ranging from several seconds up to a
number of minutes (He et al., 2005; Felber et al., 1995; Rainer and Van
Selst., 1976). The selection of record duration is influenced by the
sampling rate and the desired level of output resolution (Smith, 1997). For
this research, measurements were sampled for 15 minutes at 100 Hz and
for 10 minutes at 1000 Hz. These are similar to record durations used by
Rosset et al. (2004), De La Puente Altez (2005) and Madriz (2004) in
ground ambient noise monitoring and by Huang (2007) in ambient
vibration testing of buildings, in Montreal.
Measurement sensitivity is controlled by the gain in each acquisition
unit. Increasing the gain amplifies the observed signal whereas decreasing
the gain attenuates it. Appropriate sensitivity level depends on the
sampling location and anticipated amplitude of vibrations during the
recording session. In general, less sensitivity is needed when the sensors
- 56 -
3 – Case Study
are deployed at mid-span locations whereas greater sensitivity is required
above supports.
3.3.4 Tests
The tests were conducted on two dates and relevant ambient
conditions such as weather and traffic level were noted (Table 3-4).
Traditionally, sensors are connected to the structure or positioned
on concrete blocks (Ren et al., 2004; Farrar and James III, 1997). In this
case, the sensors were deployed onto the sidewalk and central divider due
to safe access issues and time constraints (Figure 3-12). Each sensor was
aligned vertically and horizontally. The north-south axis of the sensor was
aligned with the transverse direction of the bridge while the east-west axis
of the sensor was aligned with the longitudinal direction of the bridge
(Figure 3-1).
Table 3-5 shows the measurement catalogue with notes including
dates, sampling rates, gain values, sampling locations and disturbances
during each measurement sampling.
3.3.5 Synchronisation
Synchronisation of the two data acquisition systems became
necessary after failure of the RCS unit during testing. Before testing, the
internal clocks of both data acquisition units were synchronised using the
GPS. This allowed the delay (Δ) between each pair of measurements to
be identified using the start and end times that were stamped into the data
files. A section of the record corresponding to the delay was removed from
the start of an early record and from the end of the late record. Figure 3-12
illustrates the records pre- and post-synchronisation.
- 57 -
3 – Case Study
Ambient conditions during testing
Date
10-Sep-2007
30-Nov-2007
Times
23:00 - 24:00
00:00 - 04:00
Traffic Conditions
Light to moderate
Light
Weather
Cloudy
Mainly clear
Temperature (0C)
17.1
-2.4
Wind speed (km/hr)
0
33
Wind direction ( )
n/a
260
Humidity (%)
71
66
Record duration (min)
15
10
Configurations *
C
A, B
0
Table 3-4
Ambient conditions during testing.
*Refer to Figure 3-12
structural response
Record 1
structural response
Record 1
time / s
Δ
structural response
time / s
Δ
Δ
structural response
Record 2
Δ
Record 2
time / s
time / s
(a)
(b)
Figure 3-13
Synchronisation of a pair of measurements: (a) original time history pair
with a time difference (Δ) attributed to manual initiation; (b) modified
records after post-testing synchronisation of the time histories in (a).
- 58 -
- 59 -
Measurement catalogue.
Table 3-5
3 – Case Study
3 – Case Study
3.4
Analysis of experimental results
Experimental modal analysis (EMA) was performed with the
ambient vibration measurements. Nineteen (19) measurements were
obtained from the tests (Table 3-5). Of these, five (5) were measurement
pairs where two records were obtained simultaneously. The MATLAB
platform (The MathWorks, 2007) was used to process the data files and
execute spectral analyses. In conducting experimental modal testing, the
following assumptions were used:
1. Excitation of the structure is provided by ambient sources inclusive of
pedestrian, and vehicular traffic on the bridge, oncoming wind and
micro-tremors;
2. The excitation is described by a forcing function that was assumed to
be white noise (Gaussian) where the frequency domain representation
of the excitation is observed as a flat line;
3. The response of the structure is a linear combination of inputs, due to
system linearity.
4. The response measurements are ergodic and stationary random
processes and representative of a linear vibrating system.
5. P-Delta effects and material nonlinearities are negligible.
Four analysis methods were used and compared: (1) Fourier spectral
approach; (2) Welch’s PSD spectral approach; (3) Welch’s CSD spectral
approach; (4) Auto-regressive, Modified Covariance spectral approach.
3.4.1 Fourier method
The FFT algorithm was executed with the number of data points in
each record used as the corresponding length of the FFT (approximately
132,000 and 1,000,000 points). None of the records were segmented and
a high resolution Fourier spectrum was constructed from each record in
- 60 -
3 – Case Study
terms of its Fourier amplitude. Resonant frequencies were read directly
from each spectrum. Eighteen records were treated with this approach
(except record No. 13 on Table 3-5).
3.4.2 Welch’s PSD method
First, the time history was cut into 8 segments of equal length with
50% overlap between adjacent segments. A Hamming window (Figure 215) was then applied to each segment to attenuate the overlapping
portions. Windowing reduces spectral leakage from the ensuing analysis
where the segments are assumed periodic. A FFT, as long as the
segments (ranging between 11,250 and 75,000 points), was then applied
to each segment and corresponding periodograms were created. Finally,
the average of the periodograms from all segments was computed. This
periodogram describes the power spectral density spectrum in terms of
power per frequency. All of the nineteen records were treated by this
approach (Table 3-5) and resonant frequencies were read directly from
each averaged spectrum.
3.4.3 Welch’s CSD method
The cross spectral density estimate of simultaneous readings was
computed with Welch’s modifications. Initially, each record was cut into
100 equal segments with 50% overlap between adjacent segments
(Figure 2-13). A Hanning window was then applied to each segment
(Figures 2-15). After applying a FFT to each segment (approximately
8,200 points) and computing the corresponding periodograms, an average
periodogram was constructed to obtain the estimated cross spectral
density spectrum in terms of power per frequency. Resonant frequencies
- 61 -
3 – Case Study
were read directly from the spectra. Five measurement pairs were treated
with this algorithm (records No. 1, 2, 3, 4 and 6 listed in Table 3-5).
3.4.4 Auto-regressive, Modified Covariance method
This parametric method uses an auto-regressive fitting of each
sequence of measurements to a prediction model that is dependent upon
the assumed order of the model. Following preliminary runs and
computational limitations, the order of the prediction model was set to 150.
Lower order models generate spectral plots in which resonance is not
easily discernable. Higher orders increase the resolution of the spectra
with minimal improvements to identifying the resonant frequency peaks.
The Modified Covariance form of the auto-regressive algorithm minimises
forward and backward errors, in the least squares sense, to produce
estimated power spectral density spectra in terms of power per frequency
(The Mathworks, 2007). Resonant frequencies were read directly from
each spectrum. All nineteen measurements were treated with this
algorithm (Table 3-5).
3.5
Analysis of analytical models
Two finite-element models were constructed on the structural
analysis platform SAP2000 (Computers and Structures, Inc, 2007). The
input properties were extracted from construction plans. A modal analysis
was performed on each model, using eigenvalue analysis to obtain modal
parameter estimates for the bridge.
- 62 -
3 – Case Study
3.5.1 Models
The two models were built using the lumped-mass idealisation
concept to represent the as-built configuration of the bridge (Tedesco et
al., 1999; Wilson, 1998). The structural components of the bridge were
idealised by an assemblage of nodes connected by single, mass-less
framing members. Each member was defined between adjacent nodes
and described by sectional and material properties of structural
components between the nodes. Only gross concrete sections were
assumed for reinforced concrete sections since the stiffness of the steel is
typically neglected when conducting an elastic dynamic analysis (Wilson,
1998). The mass of the structure was determined and proportionately
assigned to each node as representative reactive masses. Each reactive
mass defined a mass DOF for the ensuing modal analysis. Only elements
with relevant stiffness properties were modelled and the significant nonstructural components were included in computing the masses (Wilson,
1998). Member fixity and support conditions were guided by the
construction details.
Notable assumptions used for the two analyses were:
1. The skew of the superstructure and the camber of the bridge deck
were considered minimal and neglected;
2. Dynamic soil-structure interactions were not considered;
3. Material and structural nonlinearities were neglected;
4. P-Delta effects were not considered;
5. Deterioration of structural members was not considered.
The two linear models idealise the bridge for analysis in three
orthogonal directions (X-Y-Z). The Z direction was aligned with the vertical
axis of the bridge, the Y direction with the transverse axis of the bridge
(i.e. across the width of the bridge) and the X direction was aligned with
the longitudinal axis of the bridge (i.e. along the length of the bridge from
- 63 -
3 – Case Study
abutment to abutment). The second model describes a more detailed
three-dimensional representation of the bridge.
Reactive masses were determined from all structural elements and
permanently secured fixtures. Table 3-6 itemises the main components
which contribute toward the reactive mass calculations.
Component
Number
Descriptive dimensions
Unit Density
(mm)
(kg/m3)
Reinforced
Concrete
. Deck slab
---
190 mm (thick)
2447
. Columns
24
457 x 457 x 2,440 mm
2447
. Walls
2
762 x 25,900 x 1,830 mm
2447
. Cap beams
2
457 x 457 x 25,750 mm
2447
. Girders
36
W840x295x209.8
7849
. Joists
99
W381x51
7849
. Connectors
4
W457x75
7849
. Connections
---
5% of total steel mass
---
114 mm (average thickness)
Structural Steel
Asphalt
--2400
Table 3-6
Contributors to the reactive mass.
1. Analytical Model 1 (AM1):
This model was formulated purely in one plane, using line
elements along the longitudinal axis of the bridge. Two-node frame
members were used to represent combined sections with translational
and rotational DOF at each node. A total of 17 internal nodes with
reactive masses and 4 support nodes were used. Three types of massless, equivalent sections idealised the framing components, equivalent
- 64 -
3 – Case Study
composite deck of reinforced concrete slab on steel girders, columns
and walls. Of the 20 framing members in the model, 12 are horizontal
and 8 are vertical.
The hinge-rocker supports at the abutments, which allow for 25
mm
of
thermal
induced
movements,
were
modelled
in
two
configurations (Figure 3-2). In Case 1, the abutment supports are
modelled as hinges at both ends in the transverse direction, and
modelled by a hinge at one end with a roller at the other end in the
longitudinal direction. Case 2 assumed degradation of the bearings at
the abutments and these supports were modelled by hinges at both
ends of the bridge in the longitudinal and the transverse directions. The
foundations of the two lines of columns were represented by fixed
bases because of their assumed rigidity.
Moment transfer between the bridge deck system and the
supporting moment-resisting reinforced concrete frame was not
allowed in accordance with the connection details. This model
comprises 21 reactive mass DOFs. Figure 3-14 illustrates model AM1.
2. Analytical Model 2 (AM2):
This larger model is an extension of AM1. Twelve identical
frames were perpendicularly spaced at 2.24 m between the columns
comprising the moment-resisting frames underpinning the bridge deck.
The modelled bridge deck system is a composite floor arrangement of
steel girders rigidly connected to overlaying reinforced concrete slab
sections. Shell sections, representative of the reinforced concrete slab,
were meshed along the top of the girders at nine points. This modelling
approach was used to better reflect the structural characteristics and
the connectivity of the components comprising the bridge deck. Six
typical mass-less sections (deck shell; wall shell; channel, steel beam;
reinforced concrete beam; reinforced concrete column) were used to
describe the bridge. Two-node frame members were used for beams
- 65 -
3 – Case Study
and columns while four-node sections were used for deck and wall
sections. Geometrical and member properties were modified to reflect
equivalent sectional details. Of the 313 frame sections, 265 are
horizontal and 48 are vertical. The model consists of 132 horizontal
deck sections and 44 vertical wall sections. Abutment supports and
internal foundations were described similar to those in AM1; hinges at
the abutments and fixed foundations beneath the walls were used in
Case 1, and hinges at one abutment with a roller at the other abutment
were used in Case 2 (Figure 3-2). Moment transfer between the bridge
deck system and the supporting moment-resisting, reinforced concrete
frame was not permitted in accordance with the construction plans. In
total, this model is built with 204 structural nodes and 48 support
nodes. Figure 3-15 presents AM2.
- 66 -
3 – Case Study
(a)
(b)
(c)
Figure 3-14
AM1: (a) solid view of the model; (b) schematic of the structural model for
AM1, Case 1; (c) schematic of the structural model for AM1, Case 2.
- 67 -
3 – Case Study
(a)
(b)
(c)
Figure 3-15
AM2: (a) solid view of the model; (b) schematic of the structural model for
AM2, Case 1; (c) schematic of the structural model for AM2, Case 2.
- 68 -
Chapter 4
Discussion and Recommendations
This chapter presents a discussion of the findings obtained from the
analyses that were presented in Chapter 3. Firstly, the results for the four
procedures (FFT, Welch’s PSD, Welch’s CSD and AR-MCOV) used to
analyse the ambient vibration measurements are outlined. This is followed
by the presentation of the results for the two analytical models of the
bridge (AM1 and AM2) for different support conditions (two support cases)
and material properties (three values of compressive strength of concrete).
The results from all of the analysis techniques are discussed. Also,
recommendations for the modal testing of similar bridges are proposed.
For similarly constructed bridges with narrower decks and taller
internal supports, the fundamental frequency ranges between 0.5 and 2.5
Hertz. The fundamental frequency for the monitored bridge is expected to
be more than 2.5 Hertz because of its shorter internal supports and
broader deck; both of these characteristics will improve the stiffness of the
bridge. In keeping with this expectation, all discernable resonant
frequencies were scrutinised within this range and up to 10 Hertz since the
fundamental frequency for most civil engineering structures occurs within
these limits.
4.1
Notes on the experimental analysis
The full set of response spectra was scrutinised within the range of
0.5 Hertz up to 10 Hertz to identify the fundamental frequency of the
bridge and any other important frequencies at the lower extremity of the
- 69 -
4 – Discussion and Recommendations
spectra. Resonant frequencies are identifiable in all orthogonal directions.
The directional components for each measurement were aligned with the
principal axes of the bridge (Figure 3-1); the Z component was aligned
with the vertical axis of the bridge, the Y component with the transverse
axis of the bridge (across the width of the bridge) and the X component
was aligned with the longitudinal axis of the bridge (along the length of the
bridge from abutment to abutment).
Figure 4-1 displays typical response spectra with the four
experimental methods while the full set of spectra for each approach can
be found in Appendices A through D. The probability of occurrence of a
resonant frequency within each spectrum and along each axis of motion is
expressed as the repeatability of the resonant frequency. The repeatability
of each resonant frequency was computed in terms of percentage and is
illustrated in Figures 4-2, 4-3, 4-4 and 4-5 for each procedure used.
Figures 4-6, 4-7 and 4-8 present the resonant frequencies estimated by all
of the procedures in each orthogonal direction.
The Fourier (abbreviated by FFT) method produced three resonant
frequencies with at least 80% repeatability in the full set of spectra (Figure
4-2). Interestingly, these values occur solely in one of the orthogonal
directions (3.20 Hz in the vertical direction, 6.12 Hz in the transverse
direction and 3.20 in the longitudinal direction).
The Welch’s PSD (abbreviated by PSD) method produced three
resonant frequencies with more than 80% repeatability in all of the output
spectra (Figure 4-3). Similar to the results of the Fourier method, these
values occur solely in one of the orthogonal directions (3.22 Hz in the
vertical direction, 3.54 Hz in the transverse direction and 3.22 in the
longitudinal direction).
The Welch’s CSD (abbreviated by CSD) method produced twelve
resonant frequencies with at least 80% repeatability (Figure 4-4). In the
vertical direction, resonance is indicated at 3.22, 4.94, 5.95, 6.21 and 7.01
Hertz. In the transverse direction, there are signs of resonance at 3.53,
- 70 -
4 – Discussion and Recommendations
5.17 and 6.13 Hertz; while in the longitudinal direction, resonance is
observed at 3.22, 4.58, 5.00 and 6.18 Hertz.
The Auto-Regressive, Modified Covariance (abbreviated by MCOV)
method indicated resonance at seven frequencies with approximately 80%
repeatability or greater (Figure 4-4). Resonance is indicated at 3.21, 6.08
and 7.02 Hertz in the vertical direction and at 3.53 and 6.11 Hertz in the
transverse direction. In the longitudinal direction, 3.22, 6.08 and 7.00 are
suggested resonant frequencies.
Other resonant frequencies are identified on the spectra but are
evident with much lower repeatability levels (Table 4-1). Despite their
lower incidence rate, these frequencies can be important and must be
analysed further. The amplitude of the peaks at some resonant
frequencies overshadows the amplitudes of other peaks within the same
spectrum. This could explain the disparity in the levels of repeatability of
resonant frequencies. Noise also exhibits an adverse effect on the clarity
of resonant frequencies with difficulty discerning distinct peaks within the
spectral plots. Interference from components of the measuring system,
switching power lines or electromagnetic fields can be ruled out because
their influence occurs in the frequency ranges of 10 to 100, 25 to 40 and
approximately 60 Hertz, respectively (Smith, 1997). However, it still
remains that resonance is suggested at the indicated frequencies although
their lack of repeatability is not fully understood (Table 4-1).
Other works noted that peaks in the spectra often result from nonstationary inputs which are then interpreted as resonant responses of the
structure (Farrar et al., 1999). This observation could also explain the
frequencies occurring with low levels of repeatability in all the spectra
since all sources of the unknown excitation were assumed to be stationary
inputs.
As well, some of the resonant frequency estimates from the four
experimental methods are clustered around select values (Figures 4-6, 4-7
and 4-8; indicated with boxes). The frequencies marked by this
- 71 -
4 – Discussion and Recommendations
observation in the vertical direction of motion (Z) include 3.20-3.22 Hertz,
4.55-4.62 Hertz, 4.93-4.94 Hertz, 5.95 Hertz, 6.08-6.21 Hertz, and 7.017.02 Hertz. In the transverse direction of motion (Y), similar observations
occur at 3.20-3.21 Hertz, 3.53-3.54 Hertz, 5.15-5.19 Hertz, and 6.11-6.13
Hertz. In the longitudinal direction of motion (X), similar observations exist
at 3.20-3.22 Hertz, 4.58-4.60 Hertz, 4.95-5.00 Hertz, 5.97-6.18 Hertz and
6.97-7.00 Hertz. While this occurrence suggests corroboration among the
experimental methods, some uncertainty is introduced by the variability in
the repeatability of those resonant frequency estimates; as much as 45%
variation is noted across all three plots. Furthermore, some of these
clusters include estimates generated by only two of the four experimental
methods. This too requires further investigation.
Table 4-2 summarises the prominent frequencies for all four of the
methods used in the analysis of ambient vibration measurements from the
bridge. The table presents the frequencies that were observed in the full
set of spectra at least 80% of the time, and notes the directional
measurement from which the estimates were extracted.
- 72 -
- 73 (d)
(b)
Date:2007-11-30, DAS:, File:28, Vertical (Z) component.
CSD spectrum of simultaneous measurements; Date: 2007-11-30, DAS:28 and 27, File:24, Vertical (Z) component; (d) AR-MCOV spectrum;
11-30, DAS:28, File:24, Vertical (Z) component; (b) Welch’s PSD spectrum; Date:2007-11-30, DAS:28, File:, (24) component; (c) Welch’s
Typical response spectra from the four experimental approaches using the ambient vibration measurements: (a) FFT spectrum; Date: 2007-
Figure 4-1
(c)
(a)
4 – Discussion and Recommendations
4 – Discussion and Recommendations
Resonant frequencies*
CSD
MCOV
PSD
CSD
MCOV
3.22
3.21
3.21
1.95
3.53
3.53
3.20
1.96
3.22
3.22
100**
44
100
100
50
41
80
63
100
56
80
95
6.20
3.22
3.99
4.55
3.54
3.20
5.17
5.17
---
3.22
4.58
4.60
67
100
40
42
67
71
100
58
100
80
74
7.01
4.57
4.62
4.94
5.15
3.54
6.13
6.11
3.94
5.00
6.08
61
56
60
47
50
88
100
79
56
80
79
---
4.93
4.94
6.08
6.12
5.19
6.85
---
---
4.59
6.18
7.00
39
80
84
89
65
40
61
100
68
---
5.95
5.95
7.02
7.96
5.99
---
---
---
4.95
6.97
---
72
80
84
44
41
56
40
---
6.20
6.21
---
---
6.13
---
---
56
100
---
7.01
7.01
---
---
---
---
---
72
100
---
---
---
---
---
FFT
PSD
1.95
FFT
3.20*
FFT
MCOV
Longitudinal Direction (X)
CSD
Transverse Direction (Y)
PSD
Vertical Direction (Z)
---
---
---
---
5.97
---
---
---
6.17
71
44
56
---
---
---
---
---
---
6.97
56
*
**
Frequencies in Hertz.
Repeatability in percent: the percentage of observable resonance at that frequency in the full set of
spectra from this experimentation.
Table 4-1
Summary of resonant frequency estimates and repeatability rates by
direction and processing method.
- 74 -
4 – Discussion and Recommendations
Resonant frequencies* (up to 10 Hz)
Algorithm
3.20*, 3.20, 6.12
FFT
(z**)
(x)
(y)
3.22, 3.22, 3.54
PSD
(z)
(x)
(y)
3.22, 3.53, 4.58, 4.94, 5.00, 5.17, 5.95, 6.13, 6.18, 6.21, 7.01
CSD
(z)
(y)
(x)
(z)
(x)
(y)
(z)
(y)
(x)
(z)
(z)
3.22, 3.22, 6.08, 6.08, 6.11, 7.02
MCOV
(z)
(x)
(x)
(z)
(y)
(z)
*
Frequencies in Hertz.
**
Direction in which the resonant peak was detected with 80% repeatability in the full set of
spectra.
Table 4-2
Summary of resonant frequency estimates for all four algorithms in EMA
and corresponding directions of occurrence.
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
Frequency (Hz)
FFT - Z
FFT - Y
FFT - X
Figure 4-2
Frequency estimates and repeatability rates using the FFT approach.
- 75 -
10
4 – Discussion and Recommendations
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
PSD - Z
PSD - Y
PSD - X
Figure 4-3
Frequency estimates and repeatability rates using the Welch’s PSD
approach.
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
CSD - Z
CSD - Y
CSD - X
Figure 4-4
Frequency estimates and repeatability rates using the Welch’s CSD
approach.
- 76 -
4 – Discussion and Recommendations
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
MCOV - Z
MCOV - Y
MCOV - X
Figure 4-5
Frequency estimates and repeatability rates using the MCOV approach.
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
FFT - Z
PSD - Z
CSD - Z
MCOV - Z
Figure 4-6
Frequency estimates and repeatability rates in the vertical direction (Z).
- 77 -
4 – Discussion and Recommendations
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
FFT - Y
PSD - Y
CSD - Y
MCOV - Y
Figure 4-7
Frequency estimates and repeatability rates in the transverse direction (Y).
Probability of resonant peak (%)
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
FFT - X
PSD - X
CSD - X
MCOV - X
Figure 4-8
Frequency estimates and repeatability rates in the longitudinal direction
(X).
- 78 -
4 – Discussion and Recommendations
4.2
Notes on the analytical analysis
Finite element models were used in the analytical approach to
modal analysis of the bridge.
Regarding both AM1 and AM2 models, Case 1 considered a hinge
support at one abutment and a roller at the other, and Case 2 considered
both supports as hinges. The results for AM1 show little difference
between Case 1 and Case 2 and estimates of the natural frequencies are
mostly identical (Table 4-3). For AM2, the results for Case 2 are
consistently 1.6% greater than for Case 1 (Table 4-4). Furthermore, the
estimates between AM1 and AM2 are considerably different with AM1
suggesting a stiffer structure with a fundamental frequency as much as
13% larger than for AM2.
As mentioned previously, when describing the analytical AM1 was
constructed with 21 reactive masses while AM2 was developed with 252
reactive masses. Since frequencies are associated with each mass DOF
the frequency estimation for AM2 is more accurate than with AM1. Also,
AM2 is less rigid than AM1 because of a greater use of three-dimensional
modelling.
Current data for the compressive strength of concrete in the bridge
is unavailable. Based on the practice at the time of construction, the
assumed compressive strength of concrete is 25 MPa at 28 days (De la
Puente Altez, 2005). Now, 50 years later, the strength is expected to reach
45 MPa (Brinkerhoff, 1983; Neville, 1996).
The two analytical models (AM1 and AM2) with support conditions
Case 1 and fc’ of 25 MPa simulate the bridge at the time of construction.
The two analytical models, with support conditions Case 2 and fc’ of 45
MPa, better represent the current state of the bridge. Figure 4-9 reveals
that the fundamental frequency increases by as much as 16% from 2.44
Hertz to 2.83 Hertz between immediately-after-construction and at-present
conditions using model AM1; for AM2, the fundamental frequency
- 79 -
4 – Discussion and Recommendations
increases by 4.5% from 2.42 Hertz to 2.53 Hertz for the same two
conditions.
Figure 4-9 also illustrates the relationship between the compressive
strength of the concrete, fc’, and the fundamental frequency of vibration of
the bridge: the frequency of vibration increases when the concrete
strength is increased. This observation is evident for both analytical
models; AM1 and AM2. While Figure 4-9 presents results for the first
mode of vibration, this proportional relationship is also evident for higher
modes of vibration.
4.3
Fundamental frequency estimates
Figure 4-9 indicates estimates of the frequency for the first mode of
vibration from all modal analysis approaches. EMA produced an estimate
of 3.21 Hertz. At the same time, the estimates between AM1 and AM2 are
considerably different with AM1 suggesting a stiffer structure with a
fundamental frequency of vibration as much as 13% larger than estimates
for AM2. AM1 is a conservative model of the bridge that is based on
simplified assumptions. AM2 is more complex, incorporating the threedimensional nature of the bridge to a greater extent. The current
compressive strength of the concrete (fc’) in the bridge is considered to be
45 MPa. Using AM1, the fundamental frequency was estimates at 2.83
Hertz for Case 1 and Case 2. Using AM2, the fundamental frequency was
estimated at 2.49 Hertz for Case 1 and at 2.53 Hertz for Case 2
All four EMA approaches yielded higher estimates of the bridge’s
fundamental frequency than any of the analytical approaches; EMA results
are at least 13% larger than AM1 results and 27% larger than AM2
findings. EMA estimated the fundamental frequency of 3.21 Hertz by FFT
method, 3.22 Hertz by PSD method, 3.22 Hertz by CSD method and 3.21
Hertz by MCOV method.
- 80 -
4 – Discussion and Recommendations
Resonant frequencies*
Mode of
vibration
*
Case 1
Case 2
fc’ (MPa)
fc’ (MPa)
25
35
45
25
35
45
1
2.44*
2.65
2.83
2.44
2.66
2.83
2
3.63
3.95
4.21
3.63
3.95
4.21
3
4.40
4.79
5.10
4.40
4.79
5.10
4
8.80
9.58
10.20
8.80
9.58
10.20
5
10.68
11.61
12.37
12.14
13.22
14.07
Frequencies in Hertz.
Table 4-3
Summary of natural frequency estimates for AM1.
Resonant frequencies*
Case 1
Mode of
vibration
*
Case 2
fc’ (MPa)
fc’ (MPa)
25
35
45
25
35
45
1
2.42*
2.46
2.49
2.46
2.50
2.53
2
2.61
2.66
2.70
2.65
2.70
2.74
3
3.33
3.40
3.45
3.35
3.42
3.48
4
3.43
3.48
3.51
3.50
3.55
3.59
5
3.67
3.73
3.78
3.76
3.82
3.86
6
3.94
3.99
4.03
3.98
4.03
4.07
7
4.21
4.28
4.32
4.35
4.31
4.36
8
4.32
4.41
4.47
4.38
4.47
4.53
9
4.79
4.88
4.94
4.83
4.91
4.97
10
5.04
5.18
5.29
5.05
5.19
5.30
Frequencies in Hertz.
Table 4-4
Summary of natural frequency estimates for AM2.
- 81 -
- 82 analyses.
FFT
3.20
PSD
3.22
CSD
3.22
MCOV
3.21
Modal Analysis Approach
Fundamental frequency estimates (frequency at first mode of vibration) for all experimental and analytical modal
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
Figure 4-9
Frequency of 1st Mode of Vibration (Hz)
AM1, Case1, fc'=25MPa
2.44
AM1, Case1, fc'=35MPa
2.65
AM1, Case1, fc'=45MPa
2.83
AM1, Case2, fc'=25MPa
2.44
AM1, Case2, fc'=35MPa
2.66
AM1, Case2, fc'=45MPa
2.83
AM2, Case1, fc'=25MPa 2.42
AM2, Case1, fc'=35MPa
2.46
AM2, Case1, fc'=45MPa
2.49
AM2, Case2, fc'=25MPa
2.46
AM2, Case2, fc'=35MPa
2.50
AM2, Case2, fc'=45MPa
2.53
4 – Discussion and Recommendations
4 – Discussion and Recommendations
4.4
Reviewing the analysis
The experimental modal results are consistent with each other but
not with the analytical results. Differences between the experimental and
analytical modal analyses can be explained by dissimilarities between the
actual structure and the idealised analytical models. The analytical models
may not accurately and completely reflect the structural properties of the
bridge.
Knowledge on the geometrical specifications of the structure,
material properties and force-transferring mechanisms is required to
develop realistic models of the structure since finite element modelling of
full-scale structures is limited by this information. Better instrumentation
and more accurate construction details could provide better agreement
between the experimental and analytical results. Some influential
parameters include material properties, extent and location of material
deterioration and conditions of connection and joint assemblies.
The results of EMA are traditionally used to calibrate and tune
analytical models of bridges so that structural issues can be identified
(Ewins, 2000; Maia and Silva, 1997). Here, the disagreement between the
EMA and analytical results may be due to possible problems with the
assumptions for the connection of the bridge deck to the column supports,
the accuracy of as-built drawings or other issues associated with field
measurements.
The models AM1 and AM2 were modified to consider a full rigid
connection of the support columns to the continuous deck above it. Under
this assumption AM1 has an increased fundamental frequency of 3.01
Hertz while AM2 increases to 2.73 Hertz, representing increases of 6%
and 8% respectively for results from Case 2 investigations with both
models. This means a 7% discrepancy from the EMA estimate for the
fundamental frequency of 3.21 Hertz.
- 83 -
4 – Discussion and Recommendations
Further calibration of the experimental and analytical models would
permit even better agreement of the results and should include the
analysis for higher modes. Vibration mode shapes and damping ratios can
be estimated from the analysis of the experimental results. These types
are beyond the scope of the current thesis.
4.5
Recommendations for ambient vibration testing of bridges
For EMA using ambient vibration measurements, it is essential to
understand the factors that influence the experimentation. Here are some
notes and recommendations to assist with performing efficient modal
testing of bridges.
1. Problem Definition: Define the results of the modal testing clearly from
the outset of the investigation. Typically, the parameters of interest
include natural frequencies, mode shapes, damping ratio estimates, a
combination of these and other parameters.
2. Structural Details: Review current details of the structure. Important
features include,
i. sectional and material properties,
ii. connections between structural components,
iii. support types,
iv. distribution and estimation of masses.
3. Experimental Modal Analysis Techniques: Be aware of current modal
analysis techniques. Algorithms are available in time and frequency
domains. Do not rely solely on one technique.
4. Research: Review reports of bridges subjected to vibration tests for
guidance on the use of various measurement details and analysis
techniques. Noteworthy measurement details include equipment type,
configurations and sampling parameter settings. Extra focus on bridges
- 84 -
4 – Discussion and Recommendations
of similar construction and scale will suggest an anticipated range of
results. It is also useful to be well-informed of signal processing issues.
5. Instrumentation: Be aware of available equipment. Some equipment
can be too sensitive for the investigation while others can be
insufficiently so. The operational features (gain, sampling rate, sample
duration, data storage capacity, range of weather-resistance, among
others), practicality of measuring with the unit(s) and any other relevant
limitations
are
issues
which
must
be
considered.
European
Commission (2002) evaluated the suitability of 17 sensors and 13
digitisers
for
ambient
noise
measurements.
A
list
of
tested
accelerometers was found insensitive for capturing lower frequency
data; only a few velocimeter transducers were endorsed by the
organisation. An additional recommendation is the mandatory 10minute ‘warm-up’ of sensors and digitisers before sampling to allow for
stabilisation of the equipment.
6. Testing Procedure: Some recommendations for modal testing of
bridges similar to the monitored structure include:
i. Instrumentation:
a. Use a measuring device with a fundamental frequency
lower than the minimum frequency of interest from the
bridge.
Otherwise,
correcting
the
contaminated
measurements could be difficult.
b. Use a minimum of two instruments when testing
bridges. The required number of units must be guided
by the goals of the testing. More instruments may be
installed based on the computational needs, instrument
availability and practicality of the experimentation.
ii. Sampling
a. The sampling rate must satisfy the anticipated range of
resonant frequencies of the bridge and equipment
capabilities (as presented in Table 3-2) and it must
- 85 -
4 – Discussion and Recommendations
meet Shannon’s sampling rate requirement (Equations
2.32 to 2.34). For bridges similar to the structure tested
in this research, a sampling rate of 100 Hertz is
adequate. This is ten times greater than Shannon’s
recommendation. More information can be sampled by
using a sampling rate of 1000 Hertz or more; the
greater precision is especially helpful where post-testing
synchronisation is required.
b. The duration of each measurement must be long
enough
to
capture
the
structure’s
response.
In
combination with an appropriate sampling rate, use a
minimum duration for bridges between 5 minutes and
15 minutes.
c. Saturation of measurements is detrimental since much
of the data can be lost during sampling. Vibration
amplitudes are sensitive to the location of the sensors.
Larger vertical amplitudes occur at mid-span and
greater longitudinal or transverse amplitudes occur at
deck
support
positions.
Select
the
gain
to
accommodate sampling in each individual direction or in
all three directions simultaneously without encouraging
saturation of the recording.
iii. Testing: For typical 3-span bridges with moment-resisting,
reinforced concrete support frames and composite decks of
reinforced concrete slab on steel girders, and with spans limited
to 90 metres and deck width up to 15 metres, the fundamental
frequency ranges between 0.5 Hertz and 2.5 Hertz. This
research found that for a similar 3-span bridge with spans
limited to 30 metres, deck width of 22 metres and total length of
70 metres, the fundamental frequency was 3.21 Hertz.
- 86 -
4 – Discussion and Recommendations
iv. Testing Configurations: Sample measurements along the central
longitudinal axis of the bridge. Also, measure at positions along
one or both edges of the bridge deck and along transverse lines.
These measurements can be analysed to estimate mode
shapes and damping ratios as well as increase the full set of
records.
The
objective
of
the
investigation,
type
of
instrumentation and the expected response (mode shapes) of
the structure are important inputs for designing the sampling
scheme. The instruments should be deployed on the deck
above supports as well as at mid-span. Intermediate positions
should
be
considered
to
increase
the
density
of
the
measurements.
v. Number of measurements: The quality of estimates can be
improved by computing an average of several sets of results.
Record a minimum of three sets of measurements. Greater
numbers of measurements will increase the quality of estimates
by minimising the presence of erroneous data in the
measurement records. Characteristics of the configurations
used will further guide the number of measurements that are
required.
7. Estimating Natural Frequencies: All four algorithms (FFT, PSD, CSD
and MCOV) generated similar estimates for the fundamental frequency
of the bridge. Results for higher vibration modes vary considerably
between the methods. This suggests that several approaches should
be used to ensure that pertinent estimates are not overlooked and are
fully identified by experimental analysis of vibration measurements.
The CSD method produced at least twice as many estimates of natural
frequencies as the other methods.
- 87 -
Chapter 5
Conclusions and Future Recommendations
This chapter presents the conclusions from the experimental and
analytical modal analyses of the bridge. A review of the analysis is
discussed briefly and a summary of recommendations for modal testing of
bridges is provided. As well, future research directions in experimental
modal analysis of bridges using ambient vibration measurements are
outlined.
5.1
Conclusions
1. Field investigation by ambient vibration testing (AVT) estimated the
fundamental frequency of the bridge at 3.21 Hertz.
1.1. FFT method estimated 3.20 Hertz in vertical measures, 3.20
Hertz in longitudinal measures and 6.12 Hertz in transverse
measures.
1.2. PSD method estimated 3.22 Hertz in vertical measures, 3.22
Hertz in longitudinal measures and 3.54 Hertz in transverse
measures.
1.3. CSD method estimated 3.22 Hertz in vertical measures, 3.22
Hertz in longitudinal measures and 3.53 Hertz in transverse
measures.
1.4. MCOV method estimated 3.21 Hertz in vertical measures, 3.22
Hertz in longitudinal measures and 6.11 Hertz in transverse
measures.
- 88 -
5 - Conclusions and Research Prospects
1.5. All methods are relatively equivalent in obtaining the
fundamental mode. Differences are apparent mainly for higher
modes.
1.6. Various configurations of devices were investigated for
measuring ambient noise and provided consistent and
repeatable results for the fundamental frequency.
2. Analytical modal analysis (AMA) using two models of the bridge
yielded fundamental frequency estimates between 2.42 and 3.01
Hertz.
2.1. Various assumptions for restraints at joints and supports and
material properties were used to determine the theoretical
fundamental frequency of the bridge.
2.2. A parametric study of the analytical models indicates natural
frequency estimates for the bridge is proportional to the
compressive strength of concrete.
2.3. Sensitivity of fundamental frequency to the level of detail of a
model, the restraint conditions, material properties and mass
(total and spatial distribution) was investigated. For this bridge,
results were relatively insensitive to support and joint
conditions (deck-dominated or longitudinal response). Material
properties
and
mass
had
the
greatest
influence
on
fundamental frequency.
3. Estimates of fundamental frequency by field investigation exceeded
that from analytical analysis by 7 to 13 percent.
4. Ambient noise has been demonstrated as an efficient response
property for estimating fundamental frequency of bridges.
4.1. Despite using best estimates of material properties, restraints
and
mass,
a
7
to
13
percent
discrepancy
between
experimental and theoretical results remains. This is good
agreement compared with results from other researchers.
- 89 -
5 - Conclusions and Research Prospects
4.2. The most plausible explanation for this discrepancy may be
underestimation of the total mass of the deck, construction
details of the deck and connections or undocumented retrofits
that provide more stiffness than considered by the analytical
models.
5.2
Reviewing the analysis
Extensive knowledge of the present characteristics of the bridge is
required to develop a realistic model of the structure. This would permit
better agreement between experimental and analytical results. Of
particular importance are material properties, extent and location of
material deterioration and conditions of connection and joint assemblies.
5.3
Recommendations for ambient vibration testing of bridges
The following suggestions are recommended for future modal testing of
similar bridges.
1. Use velocimeter sensors to measure translational displacements.
2. Use a measuring device with a fundamental frequency that lies outside
the range of frequencies being investigated to lessen contamination of
ambient vibration measurements.
3. Use at least two setups for modal testing of bridges.
4. Implement a ‘warming’ of sensors and digitisers before each sampling
session to permit stabilisation of the equipment.
5. Use a sampling rate of 100 Hertz or greater for data acquisition,
satisfying Shannon’s sampling rate requirement.
6. Use a measurement duration between 5 and 15 minutes.
- 90 -
5 - Conclusions and Research Prospects
7. Monitor
and
adjust
gain
parameter
to
abate
saturation
of
measurements.
8. Obtain measurements at positions along longitudinal centreline and
edge lines of the bridge.
9. Obtain measurements at positions along transverse lines over supports
and at mid-span and intermediate positions.
10. Obtain several measurements for each recording to compute averaged
estimates.
11. Use several methods to ensure resonant frequencies are not
overlooked and will be fully identified by experimental analysis of the
vibration measurements, especially for bridges with closely spaced
vibration modes.
5.4
Proposal for future research
AVT is beneficial to the dynamic study of bridges. Unlike forced
vibration testing, AVT is non-invasive and does not disturb occupancy; the
equipment is relatively portable and inexpensive; testing procedures are
easily implemented and less time is required for testing. Applications of
AVT include the following: structural health monitoring; hazard analysis;
soil-structure interaction; serviceability monitoring; improved numerical
modelling; design validation.
1. Structural health monitoring: In Canada alone, major crossings such as
the Lions Gate Bridge, Port Mann Bridge and Queensborough Bridge
in British Columbia have already been subjected to vibration testing.
Structural health monitoring of bridges is now widely practiced. With
instrumentation in place, it is possible to repeatedly conduct dynamic
assessments of a monitored bridge whether it is in construction or an
existing structure. The impact of different types of time varying forces
- 91 -
5 - Conclusions and Research Prospects
(resulting
from
human
and
environmental
activities)
and
the
deterioration of material properties on the behaviour of the structure
can be investigated at anytime and be more fully understood.
Furthermore, problems with support mechanisms can be identified.
2. Hazard analysis: Structural vulnerabilities of bridges can be identified
by using ambient vibration testing when performing hazard analysis.
The City of Montreal and researchers at McGill University are currently
implementing a network of permanently instrumented infrastructure for
this purpose. The Jacques Bizard Bridge which connects Montreal
Island to L'Île Bizard is earmarked for such instrumentation.
3. Soil-structure interaction: In general there exists on-going research
aimed at defining soil-structure interactions and its influence on the
dynamic response of foundations and structures. Already, ambient
vibration measurements have proven to be useful to the microzonation
mapping of regions, thus furthering efficiency of the study of seismic
hazard analysis.
4. Serviceability monitoring: Ambient noise measurements can be used to
evaluate engineering structures from a serviceability perspective. While
resonance can lead to structural failure, vibrations can also adversely
affect occupancy and equipment.
5. Improved numerical modelling: AVT helps to characterise the dynamic
behaviour of bridges through estimates of natural frequencies, mode
shapes and modal damping and can be been used to improve
analytical models. When combined with soil-structure interaction
simulation and careful modelling of bridge supports and member
connections, these tuned models produce more reliable estimates of
structural responses.
6. Design validation: The results of EMA can be used to validate and
update design assumptions, identify structural deficiencies and
increase structural efficiency and project economy. This practice of
updating design models is a component of structural health monitoring.
- 92 -
5 - Conclusions and Research Prospects
5.5
Fulfilment of objectives
The goals that were established at the onset of the dissertation have
been achieved as follows:
9 1. Ambient vibration testing of bridges was reviewed.
9 2. Experimental and analytical modal analyses were conducted on a
typical bridge in the City of Montreal in Canada.
9 3. The results for the experimental and analytical modal analyses were
discussed and compared.
9 5. Recommendations for future testing of similar bridges were
provided.
9 6. Applications for ambient vibration testing of bridges were identified.
- 93 -
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APPENDIX A
- OUTPUT SPECTRA FROM FFT ESTIMATION -
Appendix A - Output spectra from FFT estimation -
A-2
Appendix A - Output spectra from FFT estimation -
A-3
Appendix A - Output spectra from FFT estimation -
A-4
Appendix A - Output spectra from FFT estimation -
A-5
Appendix A - Output spectra from FFT estimation -
A-6
Appendix A - Output spectra from FFT estimation -
A-7
Appendix A - Output spectra from FFT estimation -
A-8
Appendix A - Output spectra from FFT estimation -
A-9
Appendix A - Output spectra from FFT estimation -
A - 10
APPENDIX B
- OUTPUT SPECTRA FROM WELCH’S PSD ESTIMATION -
Appendix B - Output spectra from Welch’s PSD estimation -
(b)
B-2
Appendix B - Output spectra from Welch’s PSD estimation -
B-3
Appendix B - Output spectra from Welch’s PSD estimation -
B-4
Appendix B - Output spectra from Welch’s PSD estimation -
B-5
Appendix B - Output spectra from Welch’s PSD estimation -
B-6
Appendix B - Output spectra from Welch’s PSD estimation -
B-7
Appendix B - Output spectra from Welch’s PSD estimation -
B-8
Appendix B - Output spectra from Welch’s PSD estimation -
B-9
Appendix B - Output spectra from Welch’s PSD estimation -
B - 10
APPENDIX C
- Output spectra from Welch’s CSD estimation -
Appendix C - Output spectra from Welch’s CSD estimation -
C-2
Appendix C - Output spectra from Welch’s CSD estimation -
C-3
Appendix C - Output spectra from Welch’s CSD estimation -
C-4
APPENDIX D
- Output spectra from AR-Modified Covariance estimation -
Appendix D - Output spectra from AR-MCOV estimation -
D-2
Appendix D - Output spectra from AR-MCOV estimation -
D-3
Appendix D - Output spectra from AR-MCOV estimation -
D-4
Appendix D - Output spectra from AR-MCOV estimation -
D-5
Appendix D - Output spectra from AR-MCOV estimation -
D-6
Appendix D - Output spectra from AR-MCOV estimation -
D-7
Appendix D - Output spectra from AR-MCOV estimation -
D-8
Appendix D - Output spectra from AR-MCOV estimation -
D-9
Appendix D - Output spectra from AR-MCOV estimation -
D - 10
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