Dynamic response of a steel arch bridge due to traffic load

84 Pages·2015·3.31 MB·English

DEGREE PROJECT, IN CIVIL ENGINEERING , SECOND LEVEL
STOCKHOLM, SWEDEN 2015
Dynamic response of a steel arch
bridge due to traffic load
- A CASE STUDY OF VÄSTERBRON
FREDRIK HILL AND FREDRIK JOHANSSON
KTH ROYAL INSTITUTE OF TECHNOLOGY
KTH ARCHITECTURE AND THE BUILT ENVIRONMENT
Abstract
The purpose of this master thesis was to study the dynamic response of the bridge
Västerbron. The bridge is situated in Stockholm and is considered being of critical
significance for the infrastructure. The thesis consists of both field measurements
and analyses of a finite element model. A stochastic load model was created that
is intended to simulate different realistic traffic situations based on parameters as
velocity, vehicle type and amount of traffic. The traffic load model was implemented
in a finite element model to study if the response was similar to the measurements.
With comparisons of the dynamic properties the validity of the model can be as-
sessed. Parameters as stiffness, mass and boundary conditions also often needs to be
updated to describe the real behaviour of the bridge. With these updates a model
can be created that could better predict problematic behaviour as fatigue.
The field measurements were made with accelerometers and analysed in Matlab.
The stochastic load model is also scripted in this environment. The FE-model was
created using Python scripts that were implemented in BRIGADE/Plus.
No conclusive results regarding the mode shapes of Västerbron could be found,
however possible eigenfrequencies were identified and presented. The load model
was implemented in the FE-model and the influence of different parameters were
discussed. The results were consistent with structural dynamics theory and in the
same order of magnitude as the measurements. This implies that the traffic load
model can be used for further studies regarding dynamic analyses.
Keywords: Dynamics, finiteelementanalysis, measurements, fieldtests, frequency,
accelerations, traffic simulation, moving force
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Sammanfattning
Syftet med detta examensarbete var att undersöka de dynamiska egenskaperna hos
Västerbron. Denna bro är belägen i Stockholm och bedöms vara av stor betydelse
för infrastrukturen. Undersökningen skedde både genom fältmätningar av acceler-
ationer och med hjälp av en finita elementmodell. En stokastisk trafiklastmodell
skapades som var avsedd att simulera olika realistiska trafiksituationer med parame-
trar som hastighet, fordonstyp och trafikmängd. Trafikmodellen implementerades i
en finita elementmodell för att se om resultatet stämmde överens med mätningarna.
Vidjämförelseravdedynamiskaegenskapernakangiltighetenavmodellenbedömas.
Parametrarsomstyvhet, massaocholikarandvillkorbehöveroftauppdaterasföratt
beskriva det verkliga beteendet hos bron. Med dessa uppdateringar kan en modell
skapas som bättre skulle kunna förutsäga problemområden som t.ex. utmattning.
Fältmätningarna utfördes med accelerometrar och analyserades i Matlab. Den
stokastiska trafiklastmodellen skapades också i detta program. FE-modellen byg-
gdes med skript utförda i Python som sedan implementerades i BRIGADE/Plus.
Inga definitiva resultat med avseende på Västerbrons modformer kunde bestäm-
mas, men eventuella egenfrekvenser identifieras och presenteras. Lastmodellen an-
vänds i FE-modellen och påverkan av olika parametrar diskuterades. Resultaten
överensstämmer med teori för strukturdynamik och är i samma storleksordning som
mätningarna. Detta visar på att trafiklastmodellen kan användas för vidare studier
av dynamiska analyser.
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Preface
The topic of this master thesis was initiated by the consultant company Tyréns AB
together with the department of Civil and Architectural Engineering at the Royal
Institute of Technology (KTH).
We would like to thank Tyréns and our supervisor Ph.D. Mahir Ülker-Kaustell for
the guidance and support during the writing of this thesis. We would also like to
thank Joakim Kylén at Tyréns for the good support and inspiring discussions.
We further would like to express our gratitude to the Div. of Structural Engineering
and Bridges at KTH together with Ph.D. Ignacio Gonzalez who helped us with the
field measurements.
A special thank to Prof. Raid Karoumi for his inspirational lectures on dynamics
and bridge design which lead us to the topic of this master thesis.
Last but not least we would like to thank our family and friends for supporting us
throughout the years at KTH.
Thank you!
Stockholm, June 2015
Fredrik Hill and Fredrik Johansson
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Notations
Notation Description
a Mass coefficient
0
a Stiffness coefficient
1
c Viscous damping matrix
C Diagonal viscous damping matrix
d Displacement
E Elastic modulus
f Natural frequency
n
F Nyquist frequency
n
F Sampling frequency
s
F Axial force
1
φ Mode shape
n
Φ Modal matrix
n
g Surface roughness
ligk
g Gravitational acceleration
I Moment of inertia
k Spring constant
k Stiffness matrix
K Diagonal stiffness matrix
L Length
m Mass matrix
M Diagonal mass matrix
µ Mean value
ω Natural circular frequency
n
p(t) Force
P DFT of signal
j
p Inverse of DFT of signal
n
q Modal coordinate
n
q Modal displacement
r
σ Standard deviation
T Natural period of vibration
n
t Time
t Start time
0
u(t) Displacement
w(t) Displacement
ζ Damping ratio
n
v
Abbreviations
Abbreviation Description
2D Two-dimensional
3D Three-dimensional
AMS Automatic Multi-level Substructuring
DAF Dynamic amplification factor
DFT Discrete Fourier Transform
DOF Degree of freedom
FE Finite element
FEM Finite element method
FEA Finite element analysis
FFT Fast Fourier Transform
FLM Fatigue Load Model
GUI Graphical User Interface
GVW Gross Vehicle Weight
MDF Multi degree of freedom system
RPY Python script file
RMS Root Mean Square
ODB Output database file
OMA Operational Modal Analysis
PSD Power spectral density
WIM Weight In Motion
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Contents
Abstract ii
Sammanfattning iii
Preface iv
Notations v
Abbreviations vi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Challenges of structural health . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Aim of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Description of the bridge: Västerbron . . . . . . . . . . . . . . . . . . 3
2 Literature review 4
2.1 Vehicle bridge interaction . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Vehicle induced vibrations . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Probabilistic traffic load simulations . . . . . . . . . . . . . . . . . . . 9
2.4 Summary of literature review . . . . . . . . . . . . . . . . . . . . . . 10
3 Theoretical background 11
3.1 Structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Undamped system . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Systems with damping . . . . . . . . . . . . . . . . . . . . . . 13
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3.1.3 Modal dynamic analysis . . . . . . . . . . . . . . . . . . . . . 14
3.2 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Description of element types . . . . . . . . . . . . . . . . . . . 16
3.3 Signal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Time and frequency domain . . . . . . . . . . . . . . . . . . . 18
3.3.2 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 19
3.3.3 Errors in signal processing . . . . . . . . . . . . . . . . . . . . 20
3.4 Arch Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Methodology 22
4.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 2D Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1 Geometry and properties . . . . . . . . . . . . . . . . . . . . . 25
4.2.2 Connections and boundary conditions . . . . . . . . . . . . . . 27
4.3 Python scripting with BRIGADE/Plus . . . . . . . . . . . . . . . . . 28
4.4 Traffic load model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4.1 Moving force . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4.2 Traffic load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Verification of moving load model . . . . . . . . . . . . . . . . . . . . 32
4.6 Finite element dynamic analysis . . . . . . . . . . . . . . . . . . . . . 34
4.6.1 Linear Eigenvalue analysis . . . . . . . . . . . . . . . . . . . . 34
4.6.2 Implicit and modal dynamic analysis . . . . . . . . . . . . . . 34
4.6.3 Rayleigh damping . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.6.4 Convergence study of 2D-model . . . . . . . . . . . . . . . . . 36
5 Results 38
5.1 Measured dynamic response . . . . . . . . . . . . . . . . . . . . . . . 38
5.1.1 Acceleration response . . . . . . . . . . . . . . . . . . . . . . . 38
5.1.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Frequency content of the load model . . . . . . . . . . . . . . . . . . 43
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5.2.1 Influence of velocity . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.2 Influence of vehicle type . . . . . . . . . . . . . . . . . . . . . 45
5.2.3 Influence of vehicle distance . . . . . . . . . . . . . . . . . . . 46
5.3 Eigenfrequency analysis, FE-model . . . . . . . . . . . . . . . . . . . 47
5.3.1 Initial eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . 47
5.3.2 Sensitivity study of FE-model . . . . . . . . . . . . . . . . . . 48
5.4 Dynamic response, FE-model . . . . . . . . . . . . . . . . . . . . . . 51
5.4.1 Acceleration response . . . . . . . . . . . . . . . . . . . . . . . 51
5.4.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5.2 Comparison of accelerations . . . . . . . . . . . . . . . . . . . 55
5.5.3 Comparison of frequency response . . . . . . . . . . . . . . . . 56
5.5.4 Implementation of traffic load model . . . . . . . . . . . . . . 58
6 Conclusions and further research 60
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Suggested further studies . . . . . . . . . . . . . . . . . . . . . . . . . 61
Bibliography 62
Appendix A - Matlab scripts 65
A.1 Traffic load model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.1.1 Main script . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.1.2 Class, amplitude method . . . . . . . . . . . . . . . . . . . . . 68
Appendix B - Python scripts 70
B.1 Python ODB-reader . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Appendix C - Mode shapes 72
C.1 Mode shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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Chapter 1
Introduction
1.1 Background
Stockholm City is currently running several projects intended to estimate the struc-
tural capacity of bridges of critical significance for the infrastructure. The bridge
Västerbron is one of these bridges, especially as it is the only bridge between Söder-
malm and Kungsholmen.
This master thesis was initiated by Tyréns AB as they together with BROSYS has
carriedoutaninvestigationofthestructuralcapacityofVästerbron, howeverfurther
analyses are of interest to validate the results.
There is a need for validation of produced finite element models by measurements,
especially with regards to dynamic properties. This is common in many countries
butisnotoftencarriedoutintheSwedishbridgeindustry. Measurementsconducted
on a bridge can together with the use of a finite element model give indications of
parameters that need to be updated. Parameters as material properties, boundary
conditions and the stiffness in connections are typically chosen with safety consid-
eration and do not always describe the real behaviour.
1.2 Challenges of structural health
The Swedish transport administration hastogether with the Royal Institute of Tech-
nology (KTH) published a summarizing report, see (Andersson et al., 2007), on the
challenges and needs of monitoring our bridges. As many bridges in Sweden are of
considerable age there is a need for methods that can provide optional maintenance
and repairs intervals.
Amodelthatprovidesrealisticdeformationsandvibrationspresentsagoodbasisfor
decisions regarding a structures health. With estimated load situations it is possible
to predict which areas that are especially subjected to fatigue from a combination
of static and dynamic responses.
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