MASTERS THESIS2009:152 CIV
Universitetstryckeriet, Lule
Jens Hggstrm Frida Martinsson
Numerical Modelling of the Vindel River Railway Bridge
Upgrade of a finite element model from dynamic measurements
MASTER OF SCIENCE PROGRAMME Civil and Mining Engineering
Lule University of Technology Department of Civil, Mining and Environmental Engineering
Division of Structural engineering
2009:152 CIV ISSN: 1402 - 1617 ISRN: LTU - EX - - 09/152 - - SE
Division of Structural Engineering
Department of Civil, Mining and Environmental Engineering
Lule University of Technology
SE - 971 87 LULE
www.ltu.se/web/shb
cee.project.ltu.se/~cam
MASTER'S THESIS
Numerical Modelling of the Vindel River
Railway Bridge
Upgrade of a Finite Element Model from Dynamic Measurements
Jens Hggstrm and Frida Martinsson
Lule 2009
Numerical Modelling of the Vindel River Railway Bridge
Numerical Modelling of the Vindel River Railway Bridge Upgrade of Finite Element Model from Dynamic Measurements
Hggstrm Jens, Martinsson Frida
The Vindel river railway bridge, Martinsson (2009)
Master's Thesis 2009:152
ISSN 1402-1617,
1st Edition
AUTHOR(S) NAME(S), October 2009
Division of Structural Engineering
Department of Civil, Mining and Environmental Engineering
Lule University of Technology
SE-971 87 LULE, SWEDEN
Telephone: + 46 (0)920 491 363
Universitetstryckeriet, Lule 2009
Cover: Photo Frida Martinsson, 30-09-2009
I
PREFACE
Writing a master thesis is usually the final step towards getting a Master of
Science degree in Civil Engineering. However, that is not the case for us. In
order to complete our studies earlier we decided to write our thesis earlier than
it was stated in the course plan, therefore we spent the whole summer of 2009
at LTU. The summer of 2009 was also the summer when Lule finished at
second place on the sun-hour-list after Visby, Gotland. After the thesis is completed we will continue and take our final courses in order to graduate by
Christmas 2009.
The research presented in this thesis was initiated by Banverket and has been
carried out at the Division of Structural Engineering, Department of Civil,
Mining and Environmental Engineering at Lule University of Technology.
Along the way we have faced a variety of problems, and had our ups and
downs. When the road is narrow and the wind is blowing it feels good to have
some people to rely on, who supports both with knowledge, time, and opinions.
There are several people we would like to thank and some are listed below.
Special thanks go to Jens Malmborg and Johan Klfors at Scanscot
Technology for letting us borrow a license for Brigade, and also for all the time
they have spent supporting and teaching us Brigade.
Our supervisor Lic. Ola Enochsson has supported us with great words of
encouragement during the whole project.
Prof. Lennart Elfgren for his never ending knowledge and rewarding
discussions.
Numerical Modelling of the Vindel River Railway Bridge
II
Anders Carolin at Banverket for feedback and discussions.
Lic. Anders Bennitz for the discussions and insights.
Senior Lecturer Ulf Ohlsson for his interest and encouragement in our work.
Ph.D Hendrik Gabrielsson at Reinertsen AB.
Ph.D Palle Anderssen at Structural Vibrations, for letting us borrow licenses
for ARTeMIS
Finally, we would like to thank everyone else that has supported us in this
project.
Lule, September 2009
Jens Hggstrm, Frida Martinsson
III
ABSTRACT
The Vindel river railway bridge is located outside of Vindeln. The bridge is a
concrete arch bridge with a total length of 226 m. It was built in 1952 and was
originally constructed to carry an axle load of 25 tons but has been classified to
carry 22.5 tons until a couple of years ago when the allowed axle load was
increased to 25.0 tons.
In 1997 the bridge was investigated and cracks were found in the sections
closest to the crown, movements were also noticed when trains passed by,
which led to further investigations.
The bridge has been the subject to two earlier reports, Bennitz (2006) and He et
al (2005). Complab at LTU made measurements on the bridge and Bennitz
evaluated these in order to find the eigenfrequencies and their correlating
eigenmodes. Hes work resulted in FE-models with which he analysed the bridge.
The purpose with this report is to continue where Bennitz and He left of and to
continue the analysis in order to find out more about the dynamic behavior of
the bridge. Creating a new FE-model and evaluating new measurements was a
part of this but unfortunately the measuring was delayed and the analysis of
those results will be presented in a report by Ola Enochsson.
From the FEM-analysis that has been performed ten eigenmodes with
frequencies up to 7 Hz was identified, the nine that Bennitz found and one
more to add. The frequencies found are relatively close to the earlier
measurements.
Numerical Modelling of the Vindel River Railway Bridge
IV
The static deflections of the bridge have been simulated with Brigade using
loads representing the heaviest test train, BV-3 in three different positions on
the bridge. The initial deflection based on the self weight of the bridge was 8.3
mm and the largest deflections from the different train loads were 2.0 mm, 2.0
mm and 8.0 mm.
The dynamic live load effect has also been simulated. For the dynamic analysis
have the load from three different trains have been evaluated, the trains are
BV-3, D-2 and the steel train. These trains have been simulated for velocities
between 50-120 km/h with the interval 10. The largest deflections from the
dynamic live load analysis are when the train is positioned covering half the
bridge, and they are 8.3 mm for BV-3 in the speed 80 km/h, 6.6 mm for D-2 travelling in the speed of 90 km/h and for the steel train 6.5 mm at the speed of
90 km/h.
The values from the dynamic live load compared to the values from the static
live load test are almost the same as for the largest deflection. This indicates
that the bridge does not swing with its eigenmodes for the velocities tested.
V
SAMMANFATTNING
Numerisk Modellering av Vindellvsbron Uppgradering av Finit Element-modell frn Dynamiska Mtningar
Vindellvsbron r belgen utanfr Vindeln. Bron r en bgbro i betong med en
total lngd p 226 m. Den byggdes 1952 och var ursprungligen konstruerad fr
att klara en axellast p 25 ton, men har varit klassifierad att bra 22.5 ton tills
fr ngra r sedan d berkningar gjordes och strsta tilltna axellast hjdes till
25 ton, vilket r det som gller idag.
r 1997 undersktes bron och d upptcktes sprickor i farbanebalkarna i
facken nrmast hjssan. Man noterade ocks att bron rrde sig mycket vid
tgpassager, vilket har lett till vidare underskningar.
Bron har drefter varit mne fr flera utvrderingar, bl.a. Bennitz (2006) och
He et al (2005). Complab genomfrde mtningar och Bennitz utvrderade
dessa fr att hitta de egensvngningar som existerade p bron. He har
modellerat och analyserat bron med hjlp av FEM.
Syftet med den hr rapporten r ta vid dr Bennitz avslutade och driva analysen
vidare. Skapandet av en ny FE-modell och utvrdering av ny mtdata var en
del av detta. Tyvrr har de nya mtningarna blivigt uppskjutna men kommer att
presenteras i en senare rapport av Ola Enochsson.
Utifrn den FEM-analys som gjorts kunde tio egenmoder med egenfrekvenser
upp till 7 Hz identifieras. Av dessa har Bennitz identifierat 9, och ytterligare en
har nu kommit till. Egenfrekvenserna stmmer bra verens med tidigare
mtningar och FEM-analyser.
Numerical Modelling of the Vindel River Railway Bridge
VI
Statiska nedbjningar p bron har simulerats i Brigade med last frn det tyngsta
av testtgen, BV-3, vilket sedan har placerats p tre olika stt p bron vid
berkning. Grundnedbjningen med bara brons egenvikt var p 8.4 mm, och de
strsta nedbjningarna med tglaster var 2.0 mm, 2.0 mm respektive 8.0 mm.
I Brigade har ven de dynamiska effekterna av tg simulerats. Fr den
dynamiska analysen har tre olika tg anvnts, BV-3, D-2 och stlpendeln.
Dessa har simulerats med hastigheter mellan 50-120 km/h med intervall 10. De
strsta nedbjningarna vid den dynamiska analysen r 8.3 mm fr BV-3 i
hastigheten 80 km/h, 6.6 mm fr D-2 i hastigheten 90 km/h och fr stlpendeln
6.5 mm vid hastigheten 90 km/h.
Vrdena frn den dynamiska analysen jmfrt med vrden erhllna frn den
statiska analysen r vldigt lika, vilket tyder p att bron inte kommer i
egensvngning vid dessa hastigheter.
VII
NOTATIONS AND ABBREVIATIONS
Explanations in the text of notations or abbreviations in direct conjunction to
their appearance have preference to what is described here.
Roman upper case letters
A Area [m2]
D Displacement dynamic amplification factor [-]
E Modulus of elasticity [N/m2]
F Force [N]
I Moment of Inertia [mm4]
N Normal Force [N]
L Length [m]
T Period time [s]
Fs Spring force [N]
Fd Damping force [N]
K Stiffness matrix [EI/L3]
Roman lower case letters
a Acceleration [m/s2]
m Distributed designing bending moment [kNm]
k Spring constant [N/m]
Numerical Modelling of the Vindel River Railway Bridge
VIII
Displacement [m]
Acceleration vector [m/s2]
c Damping coefficient [Ns/m]
v Velocity [m/s]
fn Frequency [Hz]
Greek lower case letters
Poissons ratio [-]
Damping ratio [-]
d Damped angular frequency [rad/s]
Natural frequency [Hz]
l Natural frequency of the load [Hz]
Abbreviations
DOF Degrees of freedom
CFDD Curve-fit Frequency Domain Decomposition
CVA Canonical Variate Analysis
EFDD Enhanced Frequency Domain Decomposition
FEM Finite Element Method
FDD Frequency Domain Decomposition
GUI Graphical User Interface
LTU Lule University of Technology
PC Principal Component
SLS Serviceability Limit State
ULS Ultimate Limit State
UPC Unweighted Principal Component
IX
TABLE OF CONTENTS
PREFACE ............................................................................................................ I
ABSTRACT ...................................................................................................... III
SAMMANFATTNING ...................................................................................... V
NOTATIONS AND ABBREVIATIONS ....................................................... VII
TABLE OF CONTENTS .................................................................................. IX
1 INTRODUCTION ..................................................................................... 1
1.1 Background ...................................................................................... 1 1.2 Purpose ............................................................................................. 2
1.3 Objectives ......................................................................................... 2 1.4 Limitations ....................................................................................... 3 1.5 Previous work ................................................................................... 3
1.6 Work Process.................................................................................... 4 1.7 Outline .............................................................................................. 5
2 CONDITION ASSESSMENT OF CONCRETE ARCH BRIDGES ........ 7
2.1 Arch Bridges .................................................................................... 7 2.1.1 History of Arch Bridges ....................................................... 7
2.1.2 Properties of Arch Bridges ................................................... 7 2.2 Concrete ........................................................................................... 8
2.2.1 Elasticity ............................................................................... 9
2.2.2 Cracking ............................................................................... 9 2.3 Condition Assessment .................................................................... 10
3 STRUCTURAL DYNAMICS ................................................................. 11 3.1 Natural frequencies ........................................................................ 11
3.2 Free vibration ................................................................................. 12
Numerical Modelling of the Vindel River Railway Bridge
X
3.3 Forced vibration ............................................................................. 13 3.4 Eigenfrequencies ............................................................................ 14
3.5 Damping ......................................................................................... 15 3.6 Modal Analysis .............................................................................. 16 3.7 Measurement .................................................................................. 16
4 GEOMETRY, MATERIALS AND LOADS .......................................... 17 4.1 General ........................................................................................... 17 4.2 Geometry ........................................................................................ 17
4.3 Material Properties ......................................................................... 18 4.3.1 Concrete ............................................................................. 18
4.4 Boundary conditions ...................................................................... 19 4.5 Loads .............................................................................................. 19
4.5.1 Dead load ........................................................................... 19
4.5.2 Dynamic live load .............................................................. 19
4.6 Drawings ........................................................................................ 20
5 FINITE ELEMENT METHOD ............................................................... 21 5.1 General ........................................................................................... 21
5.2 Preprocessing in Brigade ............................................................... 22 5.2.1 Element types ..................................................................... 24
5.2.2 Analysis types .................................................................... 25 5.3 Processing ...................................................................................... 26
5.4 Postprocessing ................................................................................ 31 5.5 FEM-Software ................................................................................ 31
5.5.1 ABAQUS/CAE .................................................................. 31
5.5.2 Brigade ............................................................................... 32 5.5.3 Other FEM Softwares ........................................................ 32
5.6 The FE-Model of the Vindel River Railway Bridge ...................... 32 5.6.1 Convergence test ................................................................ 36
5.6.2 Effective mass and participation factors ............................ 38
6 MODAL IDENTIFICATION ................................................................. 41 6.1 General ........................................................................................... 41 6.2 Modal identification by using output-only information ................. 41
6.3 Creating a model ............................................................................ 42 6.4 Methods for evaluation .................................................................. 42
6.4.1 FDD (Frequency Domain Decomposition) ........................ 42 6.4.2 Stochastic Subspace Identification (SSI) ........................... 43
6.5 Measurement methods ................................................................... 46 6.5.1 Accelerometers ................................................................... 46
XI
7 RESULTS ................................................................................................ 47 7.1 Brigade ........................................................................................... 47
7.2 Eigenmodes and eigenfrequencies from Brigade ........................... 47 7.3 Deflections ..................................................................................... 52
7.3.1 Dead load ............................................................................ 52 7.3.2 Static Live Load ................................................................. 53 7.3.3 Dynamic Live Load ............................................................ 55
8 DISCUSSION AND CONCLUSIONS ................................................... 59 8.1 Discussion ...................................................................................... 59
8.1.1 The Model in Brigade ......................................................... 59 8.1.2 The Results ......................................................................... 60
8.2 Conclusions .................................................................................... 60 8.3 Suggestions for further research ..................................................... 61
REFERENCES .................................................................................................. 63
APPENDIX A CONVERGENCE TEST ....................................................... 65
APPENDIX B CONSTRAINTS TEST ......................................................... 71
APPENDIX C DEFLECTIONS FOR D-2 .................................................... 73
APPENDIX D ARTEMIS INPUT FILE ....................................................... 75
Numerical Modelling of the Vindel River Railway Bridge
XII
Introduction
1
1 INTRODUCTION
1.1 Background
The railway bridge is located outside of Vindeln, a small community, about 60
km northwest of Ume. The bridge was built in 1952, replacing an old bridge
with less capacity. It is a concrete arch bridge with a main span of 112 m and
the sides are divided in to four spans with a total length of 57 m, which gives a
total length of 226 m. The maximum free height of the main span varies
between 28 and 37 m depending on the level of the water surface.
The bridge is operated by trains that transport people and freight from north to
south in Sweden and is located on the line Vnns to Hllns. Since norra stambanan is the only electrified railway covering that area, it turns out that all the heavy transports passes the Vindel river railway bridge. The heaviest
trains are those carrying steel and timber. Originally the bridge was designed
for a maximum axle load of 25 tons for the locomotives and a distributed load
of 85 kN/m for the following carriages according to the design code valid at the
time of the design. An early assessment of the bridge done by strm (1997)
resulted in a load-carrying capacity corresponding to BV-2, Bv Brighet (1997)
with a maximum axle load of 25 tons and a distributed load 72 kN/m. During
strms work cracks were discovered in the beams in the longitudinal
direction, especially in the section closest to the crown. The movements in the
bridge have been experienced as big when standing on the bridge when trains
pass by. The cracks and the movement in the bridge gave rise to doubts
whether the carrying capacity was sufficient. The owner of the bridge wanted
to investigate these doubts more thoroughly and decided to monitor the
structural behaviour of the bridge. In 2005 Complab at Lule University of
Technology performed dynamic and static measurements which were evaluated
in the master thesis Bennitz (2006). At the same time a FE-model was created
Numerical Modelling of the Vindel River Railway Bridge
2
for evaluation of the bridges structural behaviour He et al (2006). The result
from the measurement and the modelling was used to investigate the response
for an axle load of 22.5 and 25 tons.
Figure 1-1 Location of the Vindel river railway bridge
1.2 Purpose
The Division of Structural Engineering at LTU has been working on the Vindel
river bridge for some years. Recently the bridge was upgraded to carry a 25 ton
axle load instead of 22.5 ton.
The purpose with this dissertation is to
Evaluate the dynamic and static behaviour of the bridge due to the increased axle load.
1.3 Objectives
Our objective is to upgrade and calibrate the FE-model to a three dimensional
model in Brigade using beam and shell elements so that the model can be used
for dynamic and static calculations. The calibration will be done by comparing
the results from the model to the measurements Bennitz did in 2005.
Introduction
3
The model is then used to find the deflections for vehicles with axle loads of
22.5 kN and 25 kN both for dynamic and static loading.
Create a model of the bridge in ARTeMIS that can be used to evaluate the
measured values that will be sampled in September 2009.
1.4 Limitations
Since the project is relatively short (20 weeks), there is a limitation in time for
the upgrading of the FEM-model. Therefore the amount of details in the model
will be adapted to the time available for modelling.
The measurement data that will be used in the model is the data collected in
September and December 2005. New measurements are to be done, but not
until September 2009, which will be after this master thesis project is finished
which means that there is no possibility for us to take them into account. These
results will be presented in a report by Ola Enochsson.
The measurements that were done in 2005 did not use a reference point which
might cause trouble when evaluating the data in ARTeMIS. Therefore we will
create a model in ARTeMIS that can be used for the measurements that are to
be done.
1.5 Previous work
There are several reports about the Vindel river railway bridge, considering the
upgrade of maximum axle load, Bennitz (2006), He et al (2006, 2009). Two
different FEM-models have been developed, one beam model and one shell
model that shows the main structure of the bridge, He (2009).
Along with these models measurements were done on the bridge. The
measurement that was done was performed with Vibration sensors (Harbin 891
and 941B) and laser equipment to determine the deflections of the bridge. At
the time the field work was done trains with 25 tons axle load had not yet
started to traffic the bridge. Therefore new measuring will be done during
September 2009 for the heavier trains to complement the earlier measurements.
Numerical Modelling of the Vindel River Railway Bridge
4
1.6 Work Process
Figure 1-2 Description of the work process
Project start
Working on report
Deliver report
Present the results
Training on Abaqus and
Brigade
Create 3D-model of
the bridge in Brigade
Analyse the reulsts
Litterature study
FEMDynamic
loadsArch bridges
Introduction
5
1.7 Outline
Chapter 1 - Introduction. Describes the problem, background, purpose,
objectives, limitations and the previous work.
Chapter 2 - Condition Assessment of Concrete Arch Bridges. This chapter
describes the need and background for condition assessment of concrete
bridges. It also deals with concrete as a material.
Chapter3 - Structural dynamics. Briefly explains about dynamic loads,
eigenfrequencies and eigenmodes.
Chapter 4 - Geometry, materials and loads. Contains the indata for the Bridge
such as geometry, boundary conditions, loads and drawings.
Chapter 5 - Finite Element Method. Description of FEM as a technique for
analysis and also how the FEM-model for the Vindel River Railway Bridge is
designed in this project.
Chapter 6 Modal Identification. Description of ARTeMIS and how to analyse the collected data.
Chapter 7 - Results. The results from Brigade are presented, such as
eigenmodes, eigenfrequencies and deflections for static and dynamic load.
Chapter 8 - Discussion and Conclusions. Discussion about the results and the
FE-model in Brigade. And also suggestions for further research are given here.
Numerical Modelling of the Vindel River Railway Bridge
6
Condition Assessment of Concrete Arch Bridges
7
2 CONDITION ASSESSMENT OF CONCRETE ARCH BRIDGES
2.1 Arch Bridges
2.1.1 History of Arch Bridges
Arches are the second oldest type of bridge structure, the oldest are beams.
Even if the technique of using arches was known by both the ancient Greeks
and the Etruscans the idea of using arches in bridges was first realised by the
Romans, many of them are intact and still in use. A problem when building
solid arch bridges is that the type of construction requires a large amount of
building material.
During the last centuries there have been some major improvements in building
arch bridges. In comparison to the early masonry arches where stone was used,
reinforced concrete is now used instead. By reinforcing concrete which is
strong in compression with steel that is strong in tension it is possible to create
a material that is relatively strong in tension as well as in compression, and
therefore it is possible to design more slender structures.
2.1.2 Properties of Arch Bridges
Arch bridges works by transforming vertical loads and self weight into
horizontal forces which are restrained at either side. Compared to girders, arch
bridges are well suited for the use of stone materials. This is due to the fact that
most of the parts are in compression, Xanthakos (1994).
There are several different types of arch bridges, four of them are described
here; hinge-less, two-hinged, three hinged and tied arch.
Numerical Modelling of the Vindel River Railway Bridge
8
(a) Hinge-less arch bridge (b) Two-hinged arch bridge
(c) Three-hinged arch bridge (d) Tied arch bridge
Figure 2-1 Different types of arch bridges
The hinge-less bridge (a) is very stiff with little deflection compared to the
other types. It is only suitable to build this type of structure on stable ground
since it gives rise to big forces in the foundation.
The two-hinged arch bridge (b) is hinged to the foundation which only gives
horizontal and vertical forces compared to the hinge-less which also have
bending. This is probably the most common type of arch bridge since it is
generally an economical design.
The three-hinged (c) arch bridge is generally the same as the two-hinged except
that it is hinged on the crown as well. The result is a construction less sensitive
to movements such as earthquakes etc. The negative aspect of this type of
bridge is that it suffers from larger deflections and that the hinges can be hard
to maintain. These types of bridges are rarely used today.
The tied arch bridge (d) relies on the girder instead of the ground to take
horizontal forces. Therefore this type of bridge can be used when the ground is
not solid enough to build on for example a two-hinged arch bridge,
http://www.matsuo-bridge.co.jp/english/bridges/basics/arch.shtm(accessed
2009-06-10).
2.2 Concrete
Concrete is a construction material that was invented by the Romans. In the
18th
century it was rediscovered and the mixture was modified to make a
Condition Assessment of Concrete Arch Bridges
9
material stronger and more easy to use. Concrete contains of cement, water,
sand and chemical admixtures depending on the properties that are required. To
get a more stable and reliable construction reinforcement is used. There is a lot
of research on concrete and the technique and material develops continuously.
Concrete is a unique material in many ways. Its compressive strength is
relatively much higher than its tensile strength. The tensile strength is
approximately 10-15 % of the compressive. To handle this problem concrete is
often used in combination with steel as reinforcement, since steel has a high
tensile strength. The use of prestressed concrete is a technique that is
commonly used and means that the concrete is compressed from the start. This
method helps to overcome concretes natural weakness in tension which leads to a stronger material.
2.2.1 Elasticity
The modulus of elasticity of concrete is a function of the modulus of elasticity
of the aggregates and the cement matrix and their relative proportions. The
modulus of elasticity of concrete is relatively constant at low stress levels but
starts to decrease at higher stress levels as matrix cracking develops. The
elastic modulus of the hardened paste may be in the order of 10-30 GPa and
aggregates about 45-85 GPa. The concrete composite is then in the range of
30-50 GPa, Bellander (1982).
2.2.2 Cracking
The fact that concrete is a brittle material with low tensile strength means that
cracks can easily occur. This is something that is needed to take in to account
when designing concrete structures. Reinforcement is usually used in order to
reduce and control the cracks. By using joints it is possible to get the
movement and cracks in the structure to where it is wanted. In many large
structures joints or concealed saw-cuts are placed in the concrete as it sets to
make the inevitable cracks occur where they can be managed and out of sight.
Structures exposed to water pressure and highways are examples of structures
requiring crack control.
There are several ways that cracks can occur in a concrete structure. They can
be divided into categories depending on the main cause that give rise to the
actual crack. The different types of cracks and when they occur are illustrated
in Figure 2-2.
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10
Figure 2-2 Approximate moment for appearance of the different types of cracks
2.3 Condition Assessment
Because of the special behaviour of concrete and the safety aspects working
with bridges, condition assessment becomes necessary and is performed either
with certain time intervals or if there has been some damage on the bridge or if
the purpose for the bridge changes.
Concrete has been used as construction material for bridges for a long time.
This results in that there are several bridges all over the world that needs
surveillance and condition assessment in the future to secure the safety using
these bridges. Developing standards for condition assessment of bridges will
save both time and money.
Sustainable Bridges is a European research project initiated by LTU which
assesses the readiness of railway bridges to meet the demands of 2020 scenario
and provides the mean for upgrades, if they fall short. The 2020 scenario
comprehend heavier loads, longer and faster trains, and mixed traffic. Due to
this scenario it is important to upgrade existing bridges so that they will meet
the present and future demands and behave properly under these conditions,
www.sustainablebridges.net (accessed 2009-07-03).
Structural dynamics
11
3 STRUCTURAL DYNAMICS
3.1 Natural frequencies
The normal mode of an oscillating system is when all the parts move
simultaneously in the same direction with the same frequency, so called natural
frequency or resonant frequencies. Structures have a set of natural modes
which depend on the composition of the structure. If for an example an
earthquake or a dynamic load is to excite a structure near one of its natural
frequencies the displacements may exceed more than the system can tolerate in
order to not collapse. Modelling the structure with modern FE-programs in
order to find its normal frequencies is one way to find and avoid normal
frequencies near to the frequencies it may be exposed to.
The essential physical properties of a linearly elastic structural system
subjected to external loading are its mass, stiffness properties and energy
absorption capability or damping. The principle may be illustrated through a
single-storey structure as shown in Figure 3-1, where f(t) is the time-varying
force, k is the spring constant that relates the stiffness of the structure and the
dash pot relates the damping force due to the velocity by a damping coefficient
c.
= (3.1)
= = (3.2)
= = (3.3)
The equations above combined form the equation of motion, a second order
differential equation for displacement as a function of time.
Numerical Modelling of the Vindel River Railway Bridge
12
+ + = (3.4)
3.2 Free vibration
Free vibration is when a system is set in motion with lack of damping.
Therefore the system will continue to swing at one or more of its natural
frequencies after set in motion. An example of this is when pulling a child back
on a swing and then letting go.
In order to simplify equation (3.4) the parameters and are introduced, where
=c
2 km (3.5)
= k
m (3.6)
The first parameter is called damping ratio and is a dimensionless unit. The second is called the natural frequency of the system and is expressed in radians/s. By using this parameters equation (3.4) can be rewritten as
+ 2x + 2x = 0 (3.7)
The solution to equation (3.7) depends on whether the vibrations are damped or
undamped. If the system is undamped (c = 0) the solution is
x
f(t)
c
m
k
Figure 3-1 Damped 1 DOF structure
Structural dynamics
13
= t + Bcos t (3.8)
The system will oscillate indefinitely with an amplitude of 2 + 2 and the
natural frequency of =
2.
If the system is damped the system will oscillate around its natural position and
decay with time, as illustrated in Figure 3-2. The damped angular frequency is
defined as
= (1 2) (3.9)
Damping with 2 = 1 is called critical damping, this is the case when minimum damping is needed to prevent oscillation, Ryall (2000).
Figure 3-2 Undamped 1-DOF system
3.3 Forced vibration
When a structure is subjected to sinusoidal motion such as ground acceleration,
it will oscillate and after some time reach a steady state. The system will
vibrate at constant amplitude and frequency, this is called steady state
response.
The amplitude of the vibration is equal to the product of the static deformation
multiplied with the dimensionless displacement amplification factor D
-1,5
-1
-0,5
0
0,5
1
1,5
x(t)
t
Numerical Modelling of the Vindel River Railway Bridge
14
=1
1
2
+ 2
2 (3.10)
Where is the natural frequency and is the frequency of the load. The dynamic displacement amplification factor as a function of the damping ratio
( ) and the frequency ratio (/) is shown in Figure 3-3. As seen in the figure a system which is exposed to a dynamic force with a frequency close to one of
its natural frequencies, the displacement increases significantly. Should the
system lack damping ( = 0) its likely to collapse. This phenomenon is called resonance. The most famous example of resonance is probably the collapse of
Tacoma Narrows Bridge which collapsed in 1940 due to resonance from wind
loads.
Figure 3-3 Variation of displacement amplification factor with damping and
frequency
3.4 Eigenfrequencies
For a beam with constant stiffness and mass distribution over the length, its
eigenfrequencies may be expressed as
=
2
4 (3.11)
0
1
2
3
4
5
6
0 1 2 3
D
/n
= 0
= 0,2
= 0,5
= 0,7
= 1,0
Structural dynamics
15
Where is a constant that depends on the boundary condition.
Table 3-1 Eigenvalues.
Boundary condition Eigenvalue kn
n=1 n=2 n=3
Pinned-Pinned 2 (2)2 (3)2
Fixed-Fixed 22,4 61,7 120,9
Fixed-Pinned 15,4 50,0 104,3
Cantilever 3,5 22.0 61,7
The shape of the first four eigenmodes for a pinned-pinned beam is visualised
in Figure 3-4.
Figure 3-4 Pinned-Pinned beam
3.5 Damping
Damping is found to increase with the increasing of amplitude of vibration. It
arises from the dissipation of energy during vibration. The most common
mechanisms that contribute to damping is; material damping, friction at
interfaces between components and energy dissipation due to foundation
interacting with soil, among others.
The amount of damping in a structure can never be predicted precisely, so
design values are generally based on dynamic measurements of structures of a
1
2
3
4
Numerical Modelling of the Vindel River Railway Bridge
16
similar type. Damping can be measured based on the rate of decay of a free
vibration following an impact; by spectral methods based on analysis of
response to windloading; or excitation by a mechanical vibrator at varying
frequencies in order to establish a steady state resonance curve. These methods
may not be easily carried out if there are several modes of vibration close to
each other.
3.6 Modal Analysis
Modal analysis is the process of determining the inherent dynamic
characteristics of a system in forms of natural frequencies, damping factors and
mode shapes in order to create a mathematical formula to describe the dynamic
behaviour of the system. The created model is known as the modal model and
model data is the known information of the system. Modal analysis are based
on the fact that the systems response to vibrations can be described in a
combination of sets of harmonic motions. This is called the natural modes of
vibrations and is illustrated in Figure 3-4.
Modal analysis uses both theoretical and experimental techniques. The
theoretical analysis is based on a physical model that comprises weight,
stiffness and damping ratio. The solution to the equation provides natural
frequencies and mode shapes. Modern finite element analysis makes it possible
to perform analysis on almost any linear dynamic structure and has therefore
enhanced capacity of the theoretical analysis.
The improvement in data acquisition and processing capabilities has improved
the experimental realm of modal analysis (which also is known as modal
testing) significantly.
3.7 Measurement
New measurements have not been done on the Vindel river railway bridge
during this thesis, but are to be done during September 2009. More information
about the earlier measurements can be found in Bennitz (2006).
Geometry, materials and loads
17
4 GEOMETRY, MATERIALS AND LOADS
4.1 General
The dimensions of the bridge are from drawings and previous reports, He et al
(2006) and Bennitz (2006) about the Vindel river railway bridge.
4.2 Geometry
Figure 4-1 The Vindel river railway bridge
The deck is 6,9 m wide and 1,9 m high and made of concrete. On top of the
deck there is macadam which makes the foundation for the railroad. The cross
section of the deck is described in Figure 4-2. There are 14 pairs of concrete
columns with different heights that support the deck in addition to the arch.
Numerical Modelling of the Vindel River Railway Bridge
18
The arch is constructed as a box with two cells as shown in Figure 4-3. The
dimensions of the arch vary along the arch.
Figure 4-2 Cross section of the deck
Figure 4-3 Cross section of the arch (the dimensions varies along the arch)
4.3 Material Properties
4.3.1 Concrete
The Bridge over the Vindel River was designed with concrete of quality K400
which represent C28/35 in the quality class used today. Testing indicates that
Geometry, materials and loads
19
the concrete has hardened with time so that the strength of the material has
increased and now can be classified as C60/75, Enochsson (2009).
Table 4-1 Material properties for the structural parts of the bridge.
Structural Part
Concrete Class
Youngs modulus, E
[GPa]
Mass density, [tons/m3]
Deck C75 40 2,4
Columns C75 40 2,4
Arch C75 40 2,4
Ballast - - 2
4.4 Boundary conditions
The boundary conditions for the construction of the bridge are as follows. The
arch is fixed to the abutments and the columns are attached to the deck and the
arch with a joint that is both fixed and/or moveable. The columns that are on
the side spans are fixed both to the deck and the ground. The entrances are free
to move in the length direction of the bridge and free to rotate in the vertical
direction. There is a more detailed description of the boundary conditions and
the assumptions being done for connections in chapter 5.6.
4.5 Loads
4.5.1 Dead load
The dead load contains the self weight of the construction as well as the ballast
and is for the main span approximately 5000 tons, 50 MN. This includes the
weight of the deck, columns and the arch, He et al (2006).
4.5.2 Dynamic live load
The load that appears when a train passes the bridge is called live load, or
dynamic live load. In this case the dynamic live load is represented by three
different types of trains. Two of them are defined in BVS 583.11
Brighetsberkning av jrnvgsbroar, and one is the train transporting steel.
These trains are presented in Table 4-2 and the load distribution is described in
Figure 4-4 and 4-5. The steel train and D-2 are very similar. The length of the
trains is defined as at least the total length of the bridge, Scanscot (2006).
Numerical Modelling of the Vindel River Railway Bridge
20
Table 4-2 Train loads.
Type of train Axle load [KN]
Distributed load [KN/m]
Length of the train [m]
D2 225 64 236
BV-3 250 80 233
Steel train 225 64 240
4.6 Drawings
Copies of the original drawings are provided by Banverket.
1,5
0 1,50
12,5
1,8
0 1,80
5,90
Q/4 Q/4
Q/4
Q/4
Figure 4-4 Load distribution for load type D-2
Figure 4-5 Load distribution for load type BV-3
1,5
0 1,50
14,05
1,8
0 1,80
7,45
Q/4 Q/4
Q/4
Q/4
1,620
1,6
0
14,2
1,80
1,8
0
7,40
Q/4 Q/4
Q/4
Q/4
Figure 4-6 Load distribution for load type Steel train
Finite Element Method
21
5 FINITE ELEMENT METHOD
5.1 General
To be able to analyse the movements of the bridge the finite element method,
FEM will be used. FEM is a numerical technique for finding approximate
solutions to partial differential equations. The technique used in FEM-
modelling is that the construction is divided into smaller parts, a mesh or finite
elements. A mesh is when a part is divided in to a pattern of squares, triangles
or hexagons over the surface of the construction parts, the calculations are then
performed on these smaller parts. The smaller the pieces are the more points
will be calculated which results in more accurate results.
The principle of finite element method was first given in a paper by a
mathematician Courant in 1943. There was no impact but a couple of years
later the principles were developed independently by aeronautical engineers
R.W Clough in the USA and J.H Argyris in England, Samuelsson (1998).
FEM can be used in many different engineering fields, it all started because of
the need to solve complex structural problems in the aeronautical engineering
field. Now the technique is used for calculations in thermal, electromagnetic,
fluid, and structural working environments. FEM is an effective tool for
visualizing stiffness, strength, and provides great possibilities to minimize the
amount of material and by doing so, save money.
The procedure when working with FEM can be described in six steps. Step 1-2
is preprocessing step 3-4 is processing and step 5-6 is postprocessing. These
steps will be described further on. The procedure is described in Figure 5-1.
Numerical Modelling of the Vindel River Railway Bridge
22
Figure 5-1 Procedure when using FEM-analysis
5.2 Preprocessing in Brigade
When creating a model there are some decisions that needs to be done. It is
important to note that the model is just a model, which means that it can be
more or less accurate, depending on what values are put into it. It is the users
task to define boundary conditions, what type of model, which structural parts
that needs to be included and so on. The access to drawings is important to be
able to make a proper model with correct constraints and boundary conditions.
The first step in preprocessing is generating a model. The model can either be
designed graphically, GUI (Graphical user interface) or by code. It is easier to
create a model of this size using GUI. The first step is to create the parts that
the model contains of in the part module. Three different types of elements are
possible at this time, beam elements, shell elements and solid elements, or a
combination of those. The different element types will be described further on
in the report. Different materials can be defined in the property module and
later on applied on the elements that are present. When defining the material
different properties are applied, for example, if it is a non-linear model values
for elastic and plastic deformations have to be defined, also thermal and
acoustic behaviour can be applied. To create the full model the parts are being
assembled together in the assembly module and boundary conditions are
Step 1: Idealization + Mathematical model
Step 2:Discretization
Numerical model
Step 3: Element analysis
Step 4: Coupling Structure analysis
Step 5: Post-processing
Step 6: Manual control and interpretation of
results
Finite Element Method
23
defined. The analysis procedure is performed in steps which are defined in the
step module, for example one step could be static analysis and another step
could be dynamic analysis. To connect the parts with each other constraints are
set in the interaction module by defining degrees of freedom in the connection.
When shape and boundaries are set the loads are defined and applied in the
load module. The loads and boundary conditions can be defined to vary over
time as well as in different steps.
Step 2 is to create the mesh in the mesh module, there are several techniques
and the mesh can be designed so that the mesh is finer on those areas that are
critical and of more interest by using partitions. This is also useful for
complicated geometries. Creating the mesh is the main step, and is mainly what
FEM-modelling is all about. Depending on the size and type of the mesh that is
created, a certain number of nodes will appear, one in each corner of the
geometrical shape. For each node in the model; strain, stress and deflection will
be calculated. To get reliable results it is important to choose the mesh so that
no important information will be lost. For example if the mesh is designed with
large distance between the nodes the values of interest for shear or moment that
appears in a specific area can be missed. An example of that can be seen in
Figure 5-2 where the shear force is plotted for a plate with a point at x=1, the
mesh is quadratic with element sizes 0.05x0.05 m and 0.5x0.5 m. If the mesh
size is 0.5 m the maximum and minimum values between the nodes are lost.
Due to this it is important to be careful when evaluating the results from a FE-
model, Davidsson (2003).
Numerical Modelling of the Vindel River Railway Bridge
24
Figure 5-2 Shear force vy along the y-axis for x=30m
Depending on what type of calculation that is performed different calculation-
steps are to be created. The type of calculation has to be defined and limitations
in the amount of increments are to be set. When the model is completed it is
time to move to the processing step which is performed by the computer, see
5.3 Processing.
5.2.1 Element types
The element types available are as mentioned earlier beam elements, shell
elements and solid elements and the modelling space is divided into 3D space,
2D planar space and axisymmetric space. The model can consist of a mixture
of these types depending on what outputs are of interest, the limitations of
computer power and the time available for modelling.
Beam elements
A beam element is an element that has a length, direction and a defined cross-
section. It is made with assumptions so that it can be considered a one
dimensional problem. The primary solution variable is then a function of the
length direction, which to be valid must be large compared to its cross-section.
There are two main types of beam element formulations, the Euler-Bernoulli
theory and the Timshenko theory.
-800
-600
-400
-200
0
200
400
600
800
0 1 2 3 4 5 6v y[k
N/m
]
y [m]
Mesh 0.05
Mesh 0.5
Finite Element Method
25
The Euler-Bernoulli theory is based on the assumption that plane cross faces
remain plane and undistorted. All beam elements in Brigade that use linear or
quadratic interpolation are based on this theory. When modelling with thick
beams the Timshenko theory is more useful since it allows the beam to have
transverse shear strain, which means that the cross section does not have to stay
normal to the beam axis, Andersson, Malm (2004).
A beam element generates two nodes, which means four unknown parameters
per element, which are displacement and rotation in each node. The beam is
also assigned a cross section which is defined in the program.
Shell elements
In Brigade there are three different kinds of shell elements, thin, thick and
general-purpose elements. Thin elements are based on the Kirchoff shell theory
and the thick elements are based on the Reissner-Mindlin shell theory. The
Reissner-Mindlin theory assumes that transverse shear deformation occurs
which makes it more suitable for thick shells. General-purpose shells can
provide solutions for both thin and thick shells, Andersson, Malm (2004).
Depending on what type of mesh is generated each element has three or four
nodes, the number of unknown parameters are two in each node, which means
that the number of equations increases rapidly compared to the beam model.
Solid elements
Solid elements can be generated in two or three dimensions. Two-dimensional
solid elements are suitable when modelling plane or axisymmetric problems.
When defining the mesh for solid parts isoparametric hexahedra is the most
common shape for the elements, when it is not possible to create a satisfactory
mesh because of complex geometry, tetrahedron elements can be used. Brigade
provides first-order linear and second-order quadratic interpolation of the solid
elements, Andersson, Malm (2004).
When modelling with solid elements the amount of nodes increases. The total
number of nodes for an element is eight respectively four for hexahedrons and
tetrahedrons, one in each corner of the geometry.
5.2.2 Analysis types
In a model sequences of analysis steps can be created in the steps module.
Using steps makes it possible to change the loads and boundary conditions
Numerical Modelling of the Vindel River Railway Bridge
26
throughout the analysis. The types of analysis that will be used in the model of
the Vindel river railway bridge will be described. In all models created in
Brigade there is an initial step where the boundary conditions, interactions and
other parameters that are to be applicable in the very beginning of the analysis
are defined.
General static linear perturbation
This step is used to calculate the behaviour of the bridge with static conditions
and generates values for deflections, stresses and so on due to static load. A
criterion to be able to run this analysis is that the construction is stable. This
step uses time increments which makes it possible to evaluate the results over
time. The final result is the result that is carried on to the next step. If a
nonlinear result is expected such as a friction, contact, large displacements or
material nonlinearities the NLGEOM command should be used. If the structure
is unstable and a collapse or buckling is expected the modified Riks method
can be used.
Linear Eigenvalue analysis
To perform an eigenvalue extraction and calculate the natural frequencies and
the corresponding mode shapes the linear eigenvalue analysis is used. Three
different eigensolver algorithms can be used, Lanczos, subspace or AMS.
Lanczos is the fastest eigensolver when a large number of eigenmodes are to be
calculated, for smaller systems the subspace might work better. When using
Lanczos it is possible to limit the range of eigenvalues that are of interest,
therefore this is the algoritm that will be used in this analysis.
5.3 Processing
Processing is the step where the calculations are being done. The calculation
steps will be described for a beam element.
Theory
FEM-calculations have become more advanced and accurate as computers
have been improved. In this chapter the calculation steps will be described
which are performed for a beam element using the Euler-Bernoulli theory.
The beam is illustrated in Figure 5-3. In each of the two nodes there are two
variables, rotation around the y-axis and displacement in the z-direction. These
are called w1, 1 , w2 , 2.
Finite Element Method
27
Figure 5-3 Beam element
The idea is to describe the beam deformation in form of the four variables. In
order to do this the deflection of the beam is expressed as a third dimension
polynomial with four variables a1, a2, a3 and a4.
= 1 + 2 + 32 + 4
3 = 1 23
1234
= (5.1)
The idea is to express the a-matrix in a form of the variables w1, 1 , w2 and 2 and use the hypotheses of the minimum potential energy in order to find a
relation to the unknown node displacement. Therefore the vector u is
introduced.
=
1122
(5.2)
The a-matrix can be expressed in forms of the unknown variables.
1 = 0 = 1 (5.3)
1 = 0 = 2 (5.4)
2 = = 1 + 2 + 32 + 4
3 (5.5)
2 = = 2 + 23 + 34
2 (5.6)
Which can be expressed in form of a matrix
W1
1
1 2
z W1
2
EI,L
Numerical Modelling of the Vindel River Railway Bridge
28
=
1 0 0 00 1 0 01 2 3
0 1 2 32
1
2
3
4
= (5.7)
Since the matrix is not singular the coefficients in a may be expressed as
= 1 (5.8)
Equation 5.8 combined with equation 5.1 gives
= 1 = 1 2 3 4 = (5.9)
Where the functions
1 = 1 32
2+ 2
3
3 (5.10)
2 = 22
+
3
3 (5.11)
3 = 32
2 2
3
3 (5.12)
4 = 2
+
3
2 (5.13)
are called form-functions or base-functions, graphical representations of these
are illustrated in Figure 5-4.
C
Finite Element Method
29
Figure 5-4 Form-functions
A load q(x) is applied on the beam as described in Figure 5-5.
Figure 5-5 Load on the beam
In order to determinate the unknown node-displacements the hypothesis of
minimum potential energy is used. Defined as
= (5.14)
Where W is the internal energy and defined as
2 2
0 (5.15)
And A is the energy of the external force which is defined as
-0,5-0,4-0,3-0,2-0,1
00,10,20,30,40,50,60,70,80,9
1
0 0,2 0,4 0,6 0,8 1
x / L
N1
dN2/dx
N3
dN4/dx
1 2
M2
P2
M1
EI,L
P1 q(x)
x
Numerical Modelling of the Vindel River Railway Bridge
30
= + 1 0 + 2 + 1 0 + 2
()
0 (5.16)
The second derivative for the deflection is received from equation 5.9 as
2
2 =
2
2 = () (5.17)
Where
() = 6
2+
12
3
4
+
6
2
6
2
12
3
2
+
6
2 (5.18)
The internal energy of the beam is
=1
2 =
1
2
0
0 (5.19)
Where the K-matrix is a 4x4-matrix representing the stiffness of the structure.
=
3
12 6 12 66 42 6 22
12 6 12 66 22 6 42
(5.20)
The energy from the external load is
=
0 + 1 +
1
=0+ 2 +
2
=0=
0+
1000
1
0100
1
0010
2
0001
2
(5.21)
Where fv is called the consistent nodal load vector and fb called the external
nodal load vector. The potential energy can finally be written on the compact
form of
K
fv fb
Finite Element Method
31
=1
2
(5.22)
The potential energy now needs to be minimized to find the unknown node
displacements which can be found in the u-vector. The minimum value for u is
when the derivative is equal to zero, which gives:
= + (5.23)
This is the linear equation system of which the solution gives the unknown
node-displacements. This will give a linear solution, in order to get more
accurate results it is necessary to divide the beam in to smaller sections, and is
referred to as meshing, Faleskog (2003).
5.4 Postprocessing
In the postprocessing step the results are evaluated. The program generates a
lot of information and depending on what is of interest for the user, different
outputs can be chosen.
One way to show the results are by contour plots. A contour plot shows the
values by colour codes and one colour represents a certain value. It is possible
to present the plots with deformations using scale factors. The outputs available
in contour plots are deformation, stress, strain etc. These plots can be
supplemented by tables with numerical values, but since numerical values are
in one single node it is important to present these together with the plots.
It is also possible to present the results in diagrams where one for example can
pick nodes along the bridge to see the variation in stress or strain along the
bridge.
To ensure that the results are reliable and that the model does not contain
incorrect boundary conditions that cause misleading outcomes a step in the
postprocessing is convergence tests. Convergence tests means that the model is
tested by changing parameters and makes sure that the results correspond to the
changes in a predictable way.
5.5 FEM-Software
5.5.1 ABAQUS/CAE
ABAQUS is a software by Simulia, (www.simulia.com) used for FEM-
modelling and calculations. Abaqus/CAE has an interactive environment where
Numerical Modelling of the Vindel River Railway Bridge
32
the finite element model is created, graphical user interface, GUI. The same
software is used for analysing, monitoring and diagnosing jobs.
5.5.2 Brigade
Brigade is a software by Scanscot Technology, (www.scanscot.com). It is
based on ABAQUS but adapted to simulation of bridges and dynamic loads. It
contains special features for analysing for example arch bridges and long span
bridges. Brigade also includes a special operation for applying dynamic loads
and to evaluate the static response.
5.5.3 Other FEM Softwares
There are several FEM-softwares on the market, for example Strusofts FEM-design, LUSAS, CATIA, ANSYS, DIANA and many more. The softwares are
usually adapted for certain fields and depending on what the purpose with the
FE-modelling different programs are used, Maekawa (2008)
5.6 The FE-Model of the Vindel River Railway Bridge
The Vindel river railway bridge is modelled as a combination of beam and
shell elements. The first attempt was to create a model of solid elements but
due to the size of output files and the type of output data and also the time
available for analysis, the decision was taken to focus on keeping the model
small and still get the results requested. When modelling with solid elements
the amount of nodes are several times larger than for shell and beam models.
The extra information generated is sometimes not necessary.
The bridge is modelled in two parts that are assembled in the assembly module.
The reason for this is to save time and use the facts that the bridge is symmetric
around its middle span. On the side spans the height of the columns differs,
other than that the whole bridge is symmetric. SI-units are used in the entire
model. The coordinate system for the model is located with the x-axis running
along the bridge, the y-axis is in vertical direction and z-axis is in transversal
direction. Origo is located where the arch is attached to the abutment on the
south side.
Finite Element Method
33
Figure 5-6 Type sketch of the bridge showing the symmetry over the mid span and
the differences on the side spans
Parts modelled as shell elements are the arch, the deck, stiffeners and parts of
the column above the abutments. The beams that are a part of the deck and the
columns are modelled as beam elements.
Figure 5-7 Part 1of the model from Brigade
Since the geometric properties for the deck and the arch are not symmetrical,
moment of inertia is calculated and used to give the parts representative values
for the thickness of the shell. For the arch, where the size varies in two
directions moment of inertia is calculated for different cross sections along the
arch. The total cross section area of the arch turns out to be larger in the model
than in reality, due to this fact the density in the arch is reduced to represent the
actual weight of the arch. The parts are defined with nodes representing the
bottom line of the construction part. The deck is divided into three construction
Numerical Modelling of the Vindel River Railway Bridge
34
parts in the model, one that represents the top of the deck, and two that
represents the beams that are located under the deck. The beams are modelled
as beams with a rectangular cross section and the deck is a shell element.
Figure 5-8 Cross section of the deck Figure 5-9 Cross section of the arch
The columns above the abutments can be seen in Figure 5-10. The wall that is
between the columns is modelled as a shell with the thickness 0.3 m. The
columns are modelled with beam elements with a circular cross-section with
the radius of 0.535 m.
Figure 5-10 Cross section of column above the abutments [mm]
Figure 5-11 shows the bridge with numbered columns to easier understand how
the boundary conditions and constraints are set. The columns that support the
bridge are modelled as beam elements with a radius 0.5 m. On the side spans
the constraints for the columns are set to zero degrees of freedom where they
are connected to the deck beams and the same constraint is set where they are
connected to the ground. For columns no 4 and 9 in the middle span the
constraints are set to zero degrees of freedom both where they are connected to
the arch and the deck beams. Column no 5-8 are free to rotate in all directions
where they are connected to the arch, and free to rotate around the z-axis where
Finite Element Method
35
they are connected to the deck beams, these constraints are defined by using
the function coupling. The boundary conditions for the arch are defined with
zero degrees of freedom to the abutments.
Figure 5-11 The bridge with numbered columns
The crown on the bridge is very stiff and the construction parts that are
modeled are the stiffeners and the sides are modeled as shells. This can be seen
in Figure 5-12.
Figure 5-12 The crown of the bridge
Stiffeners located between the deck beams are modelled as shell elements with
a thickness of 0.6 m and a height of 1.040 m. They are connected to the deck
and the deck beams using the constraint tie which gives them no degrees of
freedom. The constraint is created by using the function partition on the deck.
The partitions are the same size as the cross section of the stiffener.
The deck is divided into three sections, mid span and two side spans. On the
land piers the deck is free to rotate and move in the x-direction. Over the
Numerical Modelling of the Vindel River Railway Bridge
36
abutment columns the mid span deck is connected to the column with a joint so
that it is free to rotate around the z-axis and move back and forth along the x-
axis. The deck from the side span is connected with a joint so that it is free to
rotate around the z-axis, but fixed in all other directions. These connections are
set by using the constraint coupling. Due to outer factors such as climate and
friction it is not sure that the connections behave as they are described in the
drawings, it might be that a connection that is supposed to be free behaves as a
combination of fixed and free, which makes it difficult to model. Therefore the
extreme cases will be tested to get a span of values where the true results most
likely are.
The bridge is tested for static response due to live loads. Brigade has a module
for this, but due to lack of computer capacity it was not possible to run the
calculations. Therefore the live load was tested by placing a train at different
positions on the bridge. Three positions were tested with two types of trains,
BV-3 and D-2, to see how the results correspond with the results from the
dynamic live load test.
To be able to test the bridge for dynamic live loads a rail is modelled on the
deck. The distance between the rails is 1435 mm. The trains that are of interest
are created in the dynamic live load module. When defining the trains the
distances between the axles are put in to the program. After that the speeds and
the speed intervals are decided, and also what part of the bridge the results are
to be taken from. Information about these trains is found in Table 4-2 in
chapter 4.
5.6.1 Convergence test
A convergence test is performed to ensure that the results correspond correctly
to changes in the model. The parameters changed are for example material
parameters, boundary conditions and connections. The results from these tests
are presented in Appendix A. In Figure 5-13 the variation for each test
compared to Bennitz (2006) is visualized. In Table 5-1the changes for each
setup are described. What was proven doing these tests was that the model
corresponds in the way we want, the equation for eigenfrequency is
=
2
4 (5.24)
For example by increasing the elastic modulus we expected to get higher
values for the eigenfrequencies. The moment of inertia was changed by
increasing or decreasing the thickness of parts, which also affected the weight.
Finite Element Method
37
To just change the weight the density for the materials was increased or
decreased.
Figure 5-13 Convergence test, values from Brigade-model comparing to measured
values. Test numbers according to table 5-1.
0,00%
20,00%
40,00%
60,00%
80,00%
100,00%
120,00%
140,00%
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Var
iati
on
co
mp
are
d t
o m
eas
ure
me
nts
Test no.
1
2
3
4
5
6
7
8
9
Numerical Modelling of the Vindel River Railway Bridge
38
Table 5-1 Changes for each test setup
Test nr Variation
1 Default
2 Thickness on deck + 25%
3 Thickness on deck - 25%
4 Modular of elasticity + 50%
5 Modular of elasticity + 50% except columns
6 Column on the sidespans pinned instead of fixed
7 Fixed entrance instead of free in bridge-direction
8 Bridge entrance free for rotation
9 All columns on the main span is free for rotation around x-axis (direction of the bridge)
10 Changed value of density on concrete in the arch bow
11 Modul of elasticity + 1/3 and density of concrete as in test 10
12 As in test 11 but used section integration before instead of during
5.6.2 Effective mass and participation factors
If considering a spring in motion with a weight at the end. The weight of the
spring has an influence on the motion but since not all of the spring moves with
the same velocity as the weight the masses cant simply just be added. Therefore effective mass is introduced; in a single Dof-system only a third of
the springs mass (effective mass) would be added to the weight.
In FEM-analysis the effective mass can be used for judging the significance of
natural frequencies. The effective mass is compared to the total mass to
evaluate the significance. Modes with high effective mass can be readily
excited by base excitation while modes with low effective mass cant since these are more theoretical.
The effective mass also helps when trying to determine how many modes that
should be included in the analysis. The effective mass of all considered modes
should together add up to at least 90% of the total weight in x, y and the z-
direction. Some of the total mass should be removed since it cannot move due
Finite Element Method
39
to constraints. In Figure 5-14 is the added effective mass for the modes up to
30 Hz illustrated, Irvinge (2009).
Figure 5-14 Cumulative effective mass
Definition
Consider a discrete dynamic system which is controlled by equation 5.24.
+ = (5.25)
M is the mass matrix
K is the stiffness matrix
is the acceleration vector is the displacement vector is the forcing function or base excitation function
A solution to equation 5.25 can be found in forms of eigenvalues and
eigenvectors where the eigenvectors represent vibration modes.
The systems generalized mass matrix can then be written as
= (5.26)
Where is the eigenvector matrix. is introduced as the the vector which represent the displacement of the masses resulting from static application of a
unit ground displacement, Irvinge (2009).
A coefficient vector is defined as
00,10,20,30,40,50,60,70,80,9
1
0 20 40 60 80 100 120 140 160
% o
f to
tal m
ass
Mode no.
Y-COMPONENT
X-COMPONENT
Z-COMPONENT
Numerical Modelling of the Vindel River Railway Bridge
40
= (5.27)
The modal participation factor can then be defined as
i =L i
m ii (5.28)
And the effective modal mass is defined as
, =
2
(5.29)
Where , is the effective mass in motion for an eigen mode which is compared to the total mass of the bridge for validation of the model.
Modal identification
41
6 MODAL IDENTIFICATION
6.1 General
ARTeMIS stands for Ambient Response Testing and Modal Identification
Software, and is a tool used for modal identification of civil engineering
structures such as buildings, bridges, dams and offshore structures. The
software estimates natural frequencies of vibration and associated mode shapes
and modal damping of a structure from measured responses only, ARTeMIS
(2009)
6.2 Modal identification by using output-only information
Modal identification means to determine modal parameters from experimental
data. The parameters that have been taken in to account are mode shapes,
natural frequencies and damping ratios.
The software that is to be used for evaluation of the measurements is
ARTeMIS Extractor Pro 2010, Release 5.0.
Traditionally structures are measured through input output modal identification which means that the structure is excited artificially, excitation
and response is measured at the same time. In large structures such as
buildings, offshore structures and bridges which usually are exposed to natural
loads that excite the structure, there is however no need for this. Since the
structure already is in motion its unnecessary to deal with the natural excitation as an unwanted noise source. Instead its possible to use output only for modal identification, Brinker (2000).
Numerical Modelling of the Vindel River Railway Bridge
42
6.3 Creating a model
Since ARTeMIS isnt made for modeling a in data script is created in order to define the shape, properties and where measurements are applied. The script is
modified in Notepad and the node-points are calculated by help from MS
Excel. When the node-points are defined, the shape is created by connecting
nodes with lines or surfaces.
Figure 6-1 Bridge-model created in ARTeMIS
The raw measurement files from the field test are stored in a format called .mea
which is used by the program Catman. Catman was therefore used to open
these files and export the measurements of interest to .asc; a file format that is
supported by ARTeMIS. The measurement files are assigned to nodes where
the direction of the acceleration is defined.
In ARTeMIS it is possible to apply constrains and describe how node-points
move in consideration to each other. This is used in order to get fixed nodes at
the supports and in nodes on half the bridge in order to get homogeneous
motion since measurements are only preformed on one side.
6.4 Methods for evaluation
In ARTeMIS there are several ways to evaluate and find eigenfrequencies.
There are two general types of evaluations; Frequency Domain Decomposition
(FDD) and Stochastic Subspace Identification (SSI).
6.4.1 FDD (Frequency Domain Decomposition)
The FDD method is a technique for modal identification of output-only
systems, i.e. in the case where the modal parameters must be estimated without
knowing the input exciting the system. By decomposing the spectral density
Modal identification
43
function matrix, the response spectra can be separated into a set of single
degree of freedom systems, each corresponding to an individual mode. By
using this decomposition technique close modes can be identified with high
accuracy which may prove difficult when using the classic approach when
exciting the structure artificially. In ARTeMIS there are three different types of
FDD analysis described below.
The Frequency Domain Decomposition (FDD) technique where each mode is
estimated as a decomposition of the systems response spectral densities into several single-degrees-of-freedom (SDOF) systems.
The Enhanced Frequency Domain Decomposition (EFDD) emerges as an
improvement of the first technique with the difference that the damping ratio
estimation is available as an extra feature as well as enhanced eigenfrequencies
and averaged mode shapes
The Curve-fit Frequency Domain Decomposition (CFDD) is similar to EFDD.
The extension to the FDD relies on a frequency domain least squares
estimation technique, Brinker (2000).
6.4.2 Stochastic Subspace Identification (SSI)
In the Stochastic Subspace Identification (SSI) a parametric model is created
directly from the raw time series of data. A parametric model is a mathematical
model with parameters which can be adjusted in order for the model to fit the
data. Most commonly a set of parameters which makes the models predicted values correlate to the measured ones is wanted. This process is often called
model calibration. See Figure 6-2.
Numerical Modelling of the Vindel River Railway Bridge
44
Figure 6-2 Model calibration
All known time domain modal identification techniques can be formulated in a
generalized form as an innovated state space formulation
+1 = + (6.1)
= + (6.2)
Where the
A-matrix contains the physical information
C-matrix extracts the information that can be observed in the system response
K-matrix contains the statistical information.
Choosing the right state space dimension is essential in the Stochastic
Subspace Identification techniques. If the dimension is too small, then the
dynamics cannot be modeled correctly. On the other hand, if the dimension is
too high, then the estimated state space model becomes over-specified, and as a
result, the statistical uncertainty on the estimated parameters increases
unnecessarily. This dilemma is illustrated in Figure 6-3.
Modal identification
45
Figure 6-3
The art of parametric model estimation is to determine a model with a
reasonable number of parameters. This means it is crucial to choose the right
model order also known as the state space dimension, which is the dimension
of the A-matrix.
The Stochastic Subspace Identification techniques all use the same estimation
engine for estimation of state space realizations (models). In ARTeMIS there
are three different implementations of the Stochastic Subspace Identification
technique and the difference between these is how the matrix is weighted,
ARTeMIS (2009).
Unweighted Principal Component (UPC)
The Unweighted Principal Component algorithm is the most simple because no
weighting is performed at all. The input to the estimation engine is the
Common SSI Input matrix itself. This algorithm works best with data having
modes with comparable energy level. In such cases it will produce good results
using reasonably small state space dimensions.
Principal Component (PC)
The PCA analysis was invented by Karl Pearson in 1901. The Principal
Component Analysis involves a mathematical procedure that transforms a
number of possible correlated variables into a smaller number of uncorrelated
variables called principal components.
Numerical Modelling of the Vindel River Railway Bridge
46
Canonical Variate Analysis (CVA)
This algorithm typically forces the use of a larger state space dimension than
the two other available algorithms. The reason is its ability to estimate modes
with a large difference in energy levels. In order to see low excited modes
among well-excited modes, it is necessary to force a large state space
dimension. For data with only well-excited modes it is better to use the
Unweighted Principal Component algorithm instead.
These three methods are described mathematically in both the help files in
ARTeMIS and in several papers but this thesis will not cover that,
Brinker (2006).
6.5 Measurement methods
The earlier measurements used in this study was sampled in September and
December 2005, Bennitz (2006). New measurements will be done in
September 2009 but since this master thesis project is finished by then there is
no possibility to apply them on this analysis. The setup for the new
measurements will be done in another way compared to the previous ones. The
new measurements will be measured with two fixed points in order to get
reference-values in all measurements.
6.5.1 Accelerometers
An accelerometer is a device that is used to measure accelerations.
Acceleration is the rate of change in velocity with respect to time. In the
measurements done on the Vindel bridge an electronical device is used which
gives results in form of voltage, proportional to the acceleration. When
calibrating the accelerometer the scale factor for transforming the
electronically output to acceleration is determined. The used accelerometers
can only measure acceleration in one dimension and therefore two sensors are
used in every measuring point.
Results
47
7 RESULTS
7.1 Brigade
In Table 7-1 information about the final model in Brigade is given.
Table 7-1 Data from Brigade.
Data about the model from Brigade
Number of elements 9993
Number of nodes 13495
Number of nodes defined by the user 11047
Number of internal nodes defined by the program 2448
Total number of variables in the program 66354
Total mass of the model 8178076
7.2 Eigenmodes and eigenfrequencies from Brigade
The settings used for the model are based on the convergence test and then
modified with different constraints. According to the drawings some joints are
pinned, but due to friction, cold climate and with relatively small movements
the joints are found to behave more as fixed joints. This conclusion is drawn
based on the measurements done by Bennitz in Dec 2005.
Instead of using semi-fixed constraints the both extreme cases (fixed and free)
have been calculated and compared. Three different constraints have been
modified. These constraints are, for the column above the abutments where the
column is connected to the deck, the constraints for column number 5,6,7 and
8, see Figure 5-11 chapter 5 at the mid span in rotation and also the boundary
Numerical Modelling of the Vindel River Railway Bridge
48
condition for the bridge entrances. See Table 7-2 in order to see each test setup
of boundary conditions. Figure 7-1 shows the frequencies estimated from each
FE-analysis compared to Bennitz (2006) measurements. All frequencies are
illustrated as test results over measured results, 1,0 means the same frequency
and 1,1 10 % higher compared to the measured result.
Figure 7-1 The eight different setups compared to Bennitz. Setup numbers
according to table 7-1.
Setup 6 was found to fit best to the previous results and its frequencies together
with the results from Bennitz are illustrated in Figure 7-2. Mode number 8 is
not presented in Bennitz measurements but is in the range of frequencies where
the modes have been found which makes it interesting for the results. There are
eigenmodes generated by Brigade that are not of interest for us, these have
been excluded from the results.
0,6
0,7
0,8
0,9
1
1,1
1,2
0 1 2 3 4 5 6 7 8 9
Var
iati
on
co
mp
are
d t
o
me
asu
rem
en
ts
Test no.
1
2
3
4
5
6
7
8
9
Results
49
Figure 7-2 Frequencies from Bennitz and setup 6.
Table 7-2 Test setup of boundary conditions.
Test no Column side
span (rotation
around z-axis)
Column main span
(rotation around z-
axis and x-axis)
Bridge entrance
(movable in x-
direction)
1 Free Free Free
2 Free Free Fixed
3 Free Fixed Fixed
4 Free Fixed Free
5 Fixed Fixed Free
6 Fixed Fixed Fixed
7 Fixed Free Free
8 Fixed Free Fixed
The eigenmodes generated by Brig