LTU-EX-09152-SE

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 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.


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.


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.


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]


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


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


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


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.


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.


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.


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.


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.


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


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


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.


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).


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.


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


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).


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


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


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.


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


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


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


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


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


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.


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


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


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


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.


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


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


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


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).


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


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.


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.


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.


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


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

Documents

LTU-EX-09152-SE