steel railway bridges with one central arch sven snauwaertgaining more information about steel...

Sven Snauwaert

Steel railway bridges with one central arch

Academic year 2016-2017Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Peter TrochDepartment of Civil Engineering

Master of Science in Civil EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: Dr. ir. Dries StaelSupervisors: Prof. dr. ir. Hans De Backer, Prof. ir. Bart De Pauw

Acknowledgement

During this master dissertation I was able to enrich myself in a technical way as well as in a personal

way. I was lucky enough to work with bridges as a subject which is a dream topic for a lot of Civil

Engineers, and the admiration of these civil artworks stayed with me throughout the whole research.

However, sometimes it was hard to find a real defined end goal of this thesis, which made me from

time to time at a loss to know how to proceed. Although the process of executing the numerous

amount of simulations and data gathering was time consuming. The realisation of this report does

make me happy and proud. Nevertheless, I would not have been able to complete it without help.

Therefore I would like to thank some people who made the realisation of this dissertation possible.

Firstly I would like to thank Prof. dr. ir. Hans De Backer and Prof. ir. Bart De Pauw to give me the

opportunity to do this master dissertation for which I craved since the subjects were made available.

As well as the TUC RAIL company which provided the subject to the University of Ghent.

Thank you.

Secondly I want to thank both professors as well as Prof. dr. ir. Philippe Van Bogaert for the remarks

and guidelines during the interim presentations throughout the academic year. Although professor Van

Bogaert was actually not related to my dissertation, he did show his interests and shared his opinions

and ideas. These critical views were appreciated and allowed me to create a different point of

perspective on several aspects. Furthermore, they made it more clear to me on which elements I could

focus and go more into detail. In short, this feedback was very helpful and made it possible to make

this dissertation a better work.

Thank you.

Also, I want to thank professor De Pauw additionally to keep in contact with me despite of his busy

schedule. I realize that answering my é-mails and questions was an extra task added to your

professional and teaching activities. Therefore I want to show my gratitude.

Thank you.

Furthermore I would like to thank Dr. ir. Dries Stael to help me anytime I had questions or problems.

He was able to help in solving the struggles I had and comforted me when I had certain doubts about

the work I did. When I was stuck at a certain point, he provided me with new possible ideas. Also the

creation of the models and the simulations with the software program SCIA Engineer were not

possible without him. Although Dries did not need to be at the campus in Zwijnaarde for personal

business, he came anyway when I asked to. I know you have a lot of other activities going on. Hence,

I want to show my appreciation for the time you made free and the efforts you did to help and advise

me.

Thank you.

Lastly I want to thank my parents to give me the possibility to start my Civil Engineering studies and

provide me the support anytime I needed. Even throughout the difficult and stressful times over the

years they were there for me, irrespectively of their own problems. I also want to show my gratitude to

my brother who is closest to me and makes me laugh and comforts me anytime I need it. This helped

me to maintain the positive spirit, certainly while doing my master dissertation.

Thank you

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masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder de bepalingen van het

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het aanhalen van resultaten uit deze masterproef.

The author gives permission to make this master dissertation available for consultation and to copy

parts of this master dissertation for personal use. In the case of any other use, the copyright terms

have to be respected, in particular with regard to the obligation to state expressly the source when

quoting results from this master dissertation.

June 2nd, 2017

Abstract


Author: Sven Snauwaert

Supervisors: Prof. dr. ir. Hans De Backer, Prof. ir. Bart De Pauw

Counsellor: Dr. ir. Dries Stael

Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil

Engineering.

Department of Civil Engineering

Chair: Prof. dr. ir. Peter Troch

Faculty of Engineering and Architecture

Academic year 2016-2017

Summary: No research has yet been done for steel (railway) bridges with one central arch. Such kind

of bridges would allow to create more aesthetically pleasing constructions as well as possibly more

economically ones. Therefore a parameter study will be done to create an idea of the possibilities of

this bridge type. To be able to do the intended study, a short literature research is done as well as a

determination of the different loads and load combinations to be considered according to the

Eurocode. The start-up of the parameter study consists of a three arched bridge designed by TUC

RAIL which functions as intermediate step between the classic tied arch bridges and the one arched

bridge. Hence, the first step is modelling the three arched bridge in the software program SCIA

Engineer. It can be noted that the design has a skewed deck. However, also a straight deck model is

made as the skewness already induces another behaviour of the bridge. Secondly a study is done for

the main parameters of this bridge once the model is ready. In this way an idea of the influence of

each parameter is obtained and the found results are partly compared with what is found in literature

for general steel tied arch bridges. These simulations are done to be able to do a comparison in the

later stage of the research with the results found for the bridge with one central arch. In a third step it

is wanted to know how the parameters influence each other. This is done as a design mind set for the

three arched bridge with each time maintaining the most optimal situation of the parameters. While for

the one arched bridge a combination of two parameters is each time done relative to the base model.

At the end of each of these chapters the general found conclusions are presented. The dissertation is

finalized with a comparison between the three arched bridge and the bridge with one central arch.

Followed by a short discussion of how the research can be consulted.

Keywords: Central arch, stiffness, arch rise, buckling, stress


Sven Snauwaert

Supervisor(s): Dr. ir. Dries Stael, Prof. ir. Bart De Pauw, Prof. dr. ir. Hans De backer

Abstract: A parameter study is done with the purpose of

gaining more information about steel railway bridges with one

central arch. This is done by starting with a three arched bridge

design by TUC RAIL, modelled in the software program SCIA

Engineer, which functions as an intermediate situation between a

classic tied arch bridge and a bridge with one central arch.

Furthermore, also a model for the one arched bridge is created.

Subsequently the most determining parameters are varied for

both types of bridges. This study is done for the parameters

individually as well as a simultaneously variation of some. With

the latter to check whether the influence due to a certain

parameter on the bridge behaviour is altered by another one.

The results are discussed and explanations for different

behaviours are looked for.

Keywords: Central arch, stiffness, arch rise, buckling, stress

I. INTRODUCTION

The classic tied arch bridge with two outer arches which are

laterally connected, is already widely used. However, the use

of a steel (railway) bridge with one central arch can introduce

an aesthetically more pleasing view and possibly reduce the

cost of the bridge. Though, no research has yet been done for

this kind of bridges. Therefore a parameter study is conducted

to create a general idea of the behaviour of this type of

bridges.

A short literature study for tied arch bridges in general

results in the collection of the most determining parameters of

the bridge model.

The study is started, based on a steel bridge design made by

TUC RAIL. This bridge has a steel orthotropic deck which

has a 45° skewness angle. The length is 48,3 m and a

perpendicular width of 18 m is considered to provide space

for three railway tracks. There are furthermore two outer

arches with a rise of 5 m and one larger central arch of which

a height of 8,74 m is present. This is used as start-up to have

an idea of the behaviour of this intermediate type between

lateral braced tied arch bridges and bridges with one central

arch. No bracings are present anymore, but the outer arches

induce a stabilizing function on the total bridge behaviour.

First the implementation of the three arched bridge in the

software calculation program SCIA Engineer is realized.

Secondly a study is done for the individual parameters of

this bridge. Which is executed for a model with a straight

deck and also for one which keeps the skewed deck from the

TUC RAIL model. The consideration of both is done due to

the fact that the skewness of the bridge deck will have an

influence. Furthermore, it should be mentioned that there will

be looked to the global behaviour of the bridge during the

study. And the variations of the parameters will be considered

relative to the base models.

Next a combined variation of several parameters is done to

look for possible different induced behaviours of the bridge.

Lastly the outer arches are removed from the model and

combined parameter simulations are done for the steel bridge

with one central arch. The results of the simulations are

gathered and a deeper look is given into them to try to explain

the different bridge behaviours. Each time several conclusions

are obtained at the end of the different parameter study parts

and these are given here.

II. LITERATURE STUDY - INFLUENCING PARAMETERS

A short literature study is done at first, as mentioned before.

And in this way a list of parameters to be simulated is found.

Deck skewness angle

Central arch shape

Arch rise to bridge span ratio (central and outer)

Arch moments of inertia Iz (central and outer)

Central girder moment of inertia Iy

Outer girder moment of inertia Iy

Bridge length

These parameters are modified throughout the different

simulations for the three arched bridge. Those which are still

possible to vary for the bridge with one central arch are

simulated as well in those models.

It should be noted that the choice of which moment of

inertia is varied is based on the most susceptible situations

found in literature. An idea of the tubular cross-sections used

for both the arches and the girders is given in Figure 1.

Figure 1: Tubular cross-section for the arches and the girders

III. MODEL - LOADS

A. SCIA Engineer model

The base model which will be used as start for each

parameter variation is obtained in different steps.

First a skewed deck is created as a single 2D ribbed slab

element in SCIA.

Next the arches are modelled. These were initially built up

as piecewise linear elements between the hanger summits for

ease of calculations in SCIA. However, these linear elements

caused stress peaks at the transition points between the

different parts. Which induced the need for smooth elements

anyway.

Thirdly the hangers connecting the arches and the

orthotropic deck/longitudinal girders are modelled. Their

height is determined in a way that the arch going through their

summits forms a parabolic shape. A short check for the

stresses in the hanger elements shows almost only tension.

Hence, the traditional tied arch behaviour is approximated

quite well, which approves the model.

The longitudinal girders are added subsequently, and these

form the tension ties between the arch springs for the different

arches. A visualisation of the three arched bridge base model

is given in Figure 2. The following elements can be found

when looked in the positive y-direction. There is the first outer

arch, followed by one railway track and then the central arch.

Next there are two railway tracks and eventually the second

outer arch can be seen. Figure 3 shows the bridge with one

central arch situation.

Lastly the discussion of the boundary conditions can be

done. It is chosen to set the rotations around al axes free at

both abutments. The translations at the left supports are all

fixed. While the one in the longitudinal direction is modelled

freely at the other abutment. It should be noted that a line

support along the whole edge is used instead of only at each

longitudinal girder. However, simulations showed only small

differences in the results for the line and point support

situations.

Figure 2: SCIA Engineer base model for the three arched bridge

Figure 3: SCIA Engineer base model for the bridge with one central

arch

B. The loads

The appropriate loads to which the bridge is subjected have

to be determined before the simulations with the created

models can be done properly. These are found by applying the

Eurocode documents. [1],[2]

The permanent loads given below are all directed vertical.

Self weight of the elements

Self weight of the train tracks and supporting ballast

The variable loads are however directed along all three main

axes. These are varied in location and combined in a way that

the most determining load combinations can be created. The

latter is done based on the rules in the Eurocode. An overview

of the considered loads is given:

Vertical train traffic load - Load Model 71

Traffic induced traction and brake loads

Temperature induced loads (ΔTD = 35 K)

Wind load in the y-direction

It should be mentioned that the variable loads related to

train traffic are multiplied with a classification factor of 1,2

for the considered bridge. This is because it is part of the train

traffic connection between Bruges and the harbour of

Zeebruges. Hence, the trains which will cross this bridge are

more heavy than the traditional cases considered in the

Eurocode.

The following chapters contain the conclusions obtained

throughout the different parameter studies.

IV. PARAMETER STUDY - THREE ARCHED BRIDGE

A. General

The different parameters mentioned earlier are now varied

for the straight and skewed three arched bridge base model.

To have a good idea of the global behaviour of the bridge,

several variables are checked for each simulation. An

overview of the latter is given below.

Maximum central arch compression stress [MPa]

Buckling check of the central arch [-]

Maximum outer arches compression stresses [MPa]

Buckling check of the outer arches [-]

Maximum outer girders tension stresses [MPa]

Maximum central girder tension stress [MPa]

The stresses obtained from SCIA will be the principal ones

and the buckling checks will be presented by a buckling

coefficient given by the software.

For the ease of discussion the abbreviation "StD" is used for

the straight deck situation, while "SkD" indicates the skewed

deck model.

B. Deck skewness angle

The deck skewness angle has a decreasing influence on the

compression stress in the central arch from 45° onwards to

larger angles, more skewed deck. The same can be said for the

buckling coefficient of the arch. Also the central girder feels

this beneficial influence. However, the variables checked in

the other elements are only slightly influenced

C. Shape of the central arch

It is clear that a parabolic shape of the arch has a beneficial

behaviour relative to a circular shaped arch.

D. Central arch rise to span ratio (RTS)

The central arch rise to span ratio variation shows an

interval in which the stability of the central arch has an

optimum around a ratio of 0,18-0,25, see cross-section two in

Figure 4. The smaller the buckling coefficient, the more stable

the arch is. Furthermore, the straight deck is less favourable

for the central arch buckling coefficient than the skewed deck.

Figure 4: Buckling check of the central arch in function of the central

arch rise to span ratio (StD)

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Bu

ck

lin

g c

oeff

icie

nt

[-]

Central arch rise to span ratio [-]

Buckling check of the central arch

Cross-section 1 Cross-section 2 Cross-section 3

Cross-section 4 Cross-section 5

The same progress can be seen for the compression stress in

the central arch, be it that the optimum interval is shifted

somewhat to the larger rise to span ratio values of 0,35-0,45.

The outer arches as well as the outer girders do not feel the

influence of the variations in the central arch rise.

While the tension in the central girder shows a slight

decrease at first which is followed by an increase as the rise to

span ratio increases. The optimum is found around a ratio of

0,452 which approximately corresponds with the optimal

ratios for the central arch compression stress.

E. Central arch moment of inertia Iz

An increasing central arch moment of inertia induces a

decrease in buckling susceptibility of the central arch towards

an asymptote, see Figure 5. The horizontal axis on the figure

shows the ratio of the moment of inertia of the considered

cross-section relative to the initial one in the base model.


arch moment of inertia Iz ratio

Also the compression stress shows this kind of decrease, but

in a less pronoun way.

The outer arches do not feel an influence of the increase in

central arch cross-section. Their compression stress as well as

their buckling coefficient remains quasi constant. Also the

tension stress in the longitudinal girders does not vary.

F. Simultaneous variation in central arch rise to span ratio and central arch moment of inertia Iz

The conclusion for the variation of the rise to span ratio

influence on the stability of the central arch is valid

throughout the simultaneous variation of the central arch

cross-section. However, a horizontal curve is more and more

seen for both decks as the cross-section increases. Which

indicates that the central arch rise variation has less and less

influence on the central arch stability.

The whimsical behaviour of cross-section four in Figure 4,

is due the increase in cross-sectional self weight, while the

resisting moment of inertia Iy against this load is kept quasi

the same. Therefore cross-section five is introduced. Its cross-

sectional width to height ratio is the same as for the initial

cross-section. Hence, the same relative moment of inertia as

cross-section four is used, but with a larger stiffness around

the y-axis. It should be noted that the influence of this

different buckling behaviour is less pronoun present for the

skewed deck.

The same decreasing-optimum-increasing behaviour is

present for the compression stress in the central arch, for each

cross-section as the rise to span ratio increases. The increase

in cross-sections induces a decrease in compression stress and

a smaller increase in stress once the optimum rise is exceeded.

No large differences could be noted for the other variables

when a simultaneous variation of the central arch parameters

is done.

G. Outer arch moment of inertia Iz

A decrease in the compression stress in the outer arches is

created when their Iz is increased. The buckling coefficients

show some random jumps at first. Which is due to the fact that

the self weight of the cross-sections becomes larger, while the

stiffness around the y-axis is kept quasi constant. This is

solved by varying the height of the arch cross-sections

accordingly with their width. A decreasing buckling

susceptibility of the outer arches can then be found as their

cross-section increases. Both variables seem, just like for the

central arch where its cross-section was changed, to go to an

asymptote for large cross-sections.

The central arch does not show variations in its checked

variables due to a change in outer girder cross-section. This is

shown in Figure 6, where the STS cases indicate a variation in

outer arch cross-section. Also no changes in longitudinal

girder tension stress are found.

Figure 6: Buckling check of the central arch in function of the outer

arch rise to central arch rise ratio for different outer arch cross-

sections (StD)

H. Outer arch rise to span ratio

The second outer arch shows a decrease in its compression

stress as the outer arch rise increases. While much less

influence can be found for the compression stress in the first

outer arch.

Both arches show furthermore a decrease in buckling

coefficient, but the advantageous influence is not that large.

A decrease towards an asymptote can be seen for the

buckling check and the compression stress verification of the

central arch, see Figure 6. The asymptote is reached around

the 50% situation. This shows that a higher outer arch rise to

central arch rise ratio than 50% does not come with any

benefits anymore in possible reduction of the central arch

cross-section

The other elements do not show an influence. However, the

influence of totally removing the outer arches shows a large

disadvantageous peak in all of the checked variables.

A simultaneous variation of the outer arch rise and outer

arch cross-sectional moment of inertia Iz does not lead to a

change in behaviour of the different elements relative to the

individual parameter variations.

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0 100 200 300 400 500 600

Bu

ck

lin

g c

oeff

icie

nt

[-]

Central arch moment of inertia Iz ratio [%]


Straight deck Skewed deck

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 20 40 60 80 100 120

Bu

ckli

ng

co

effi

cien

t [-

]

Rise to rise ratio [%]


STS 25 STS 50 STS 75 STS 100

I. Central girder moment of inertia Iy

An increase in stiffness around the y-axis of the central

girder induces a stabilizing effect on the central arch as well

as a reduction of the compression stress in it. This is however

only valid if a very large increase in moment of inertia is

realized. To give an idea, an increase of about 1200% in Iy

relative to the initial situation caused a decrease in buckling

coefficient of about 10% and 15% for the compression stress.

The other elements do not show variations. Except for the

central girder itself which feels a decrease in tension stress as

its cross-section increases.

J. Outer longitudinal girder moment of inertia Iy

Some small drops can be seen in the buckling check and

compression stress verification of the arches. But other than

that no influence is seen, except in the tension stress in the

outer longitudinal girders themselves. This stress decreases

towards an asymptote as their cross-section increases.

A simultaneous variation in both the outer and central

longitudinal girders did not induce any changes from what is

told above. The conclusion found in literature about the axial

longitudinal girder stiffness not having a large influence on

the general bridge behaviour is therefore confirmed.

K. Bridge length

Large differences in the compression stresses as well as the

buckling coefficients can be seen. A decrease in bridge length

shows a more beneficial situation for the three arches. Also

the tension stresses in the outer girders follow this

advantageous progress.

V. PARAMETER STUDY - BRIDGE WITH ONE CENTRAL ARCH

The outer arches are now removed and a parameter study is

done on the bridge with one central arch. The same geometry

and cross-sections are kept furthermore, so the values of the

variables increase as there are less load bearing elements

present. Four parameters are actually left to be varied, see

paragraphs A to D which follow, and each time two of these

were simulated together. Below, the influence of each

parameter on the checked variables is discussed. If the general

induced behaviour is altered when another parameter is

simulated simultaneously, then this will be mentioned. The

mentioned general behaviour concerns the situation for the

base model for which one parameter is varied.

A. Central arch rise to span ratio


An optimum can be seen for the rise to span ratios of about

0,18-0,25 for the buckling coefficient of the central arch. This

is valid for both the StD and the SkD and was also found for

the three arched bridge.

Both decks show a decrease of the influence of the rise to

span ratio as the central arch cross-section increases. For large

cross-sections almost no variation in buckling coefficient can

be seen for a variation in rise to span. Also, when the outer

girder cross-section is increased, the influence of varying the

central arch rise to span ratio decreases. So the optimum is

less emphatically present. Lastly, the increase in central girder

moment of inertia induces smaller influences of the rise to

span ratio variations for the SkD.

Compression stress in the central arch

For the StD an optimum in compression stress is present for

the rise to span ratios of about 0,35-0,45. Just like for the three

arched bridge. The decrease in stress between the most severe

situation and the optimum is about 60%.

A different behaviour is noticed for the skewed deck. Over

there a decrease for the smaller rise to span ratios can be seen

and from a ratio of about 0,352-0,452 onwards, an asymptote

can be found, see Figure 7. Over here the downward jump

from the largest compression stress to the value for the

asymptote is about 40%.

Figure 7: Compression stress in the central arch in function of the

central arch rise to span ratio for different outer girder cross-sections

(SkD)

Tension stress in the outer longitudinal girders

The variation in central arch rise to span ratio does not

induce a variation in the tension stress in the outer girders in

both bridge models.

Tension stress in the central longitudinal girder

The tension stress in the central longitudinal girder for the

StD decreases when the rise to span ratio increases with an

asymptote from a rise to span ratio of about 0,362 onwards.

For a small outer girder cross-section, no asymptotic zone is

present, but a crescent decrease in tension stress can be found.

For the skewed deck an optimum is present for the tension

stress in the central longitudinal girder, see Figure 8. This

occurs around a rise to span ratio of 0,20 which corresponds

with the optimum for the buckling check of the central arch.

Figure 8: Tension stress in the central longitudinal girder in function

of the central arch rise to span ratio for different outer girder cross-

sections (SkD)

0

50

100

150

200

250

300

350

400

450

500

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Co

mp

ress

ion

str

ess

[M

Pa

] Central arch rise to span ratio [-]


STS Iy 44 STS Iy 100 STS Iy 166 STS Iy 221

0

50

100

150

200

250

300

350

400

450

500

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Ten

sio

n s

tress

[M

Pa

]


Tension stress in the central girder


B. Central arch moment of inertia Iz


An increasing central arch cross-section induces a decrease

in buckling susceptibility of the central arch for the StD

situation. From a certain moment of inertia Iz onwards, a

further increase has no influence anymore on the central arch

stability as an asymptote is reached. This value arises around

the STS Iz 200% case for the used model. The same is found

for the SkD, however the asymptote is already found from the

Iz 100% case (initial cross-section) onwards.

The general behaviour throughout the simultaneous

variation in outer girder moment of inertia is kept for the

skewed deck. While the reduction in buckling coefficient to

the asymptotic situation increases though with decreasing

outer girder cross-section.

A small outer girder cross-section alters the general

behaviour induced by the central arch cross-section variation

for the StD. The asymptotic behaviour is faded out and a

continuous decrease of the buckling coefficient is seen as Iz is

increased.


A decreasing progress in compression stress can be seen as

the central arch stiffness increases. A ten times larger moment

of inertia Iz induces a reduction between 30 and 40%. Which

is valid for both deck types. It should be mentioned that this

stress decrease does not happen linearly in function of the

moment of inertia increase.


A variation in central arch cross-section does not have an

influence on the tension stress in the outer girders. Neither for

the straight deck, nor for the skewed deck.


The increase in central arch cross-section has no influence

on the tension stress in the central longitudinal girder for the

straight deck.

While for the SkD an increase in central arch stiffness

induces a decrease in the tension stress in the central girder.

For an increase in Iz with a factor of about ten, a decrease in

stress of 15-20% is present. The non-linear behaviour between

stress decrease and moment of inertia increase should again be

mentioned. Furthermore, the larger the central girder, the

larger this decreasing influence as the central arch cross-

section increases. The differences in decrements are small

though.

C. Outer girders moment of inertia Iy


A larger stiffness of the outer girders induces a decrease in

the buckling susceptibility of the central arch. Hence, a

stiffening effect from the outer girders on the central arch is

present in the StD situation. The smaller and larger central

arch rise to span ratios induce a greater stiffening behaviour

than the intermediate 0,181-0,362 ones, see Figure 9.

Furthermore, the larger the central arch cross-section, the

smaller the gain in buckling coefficient reduction. For very

large cross-sections there is almost no influence of the outer

girder cross-section anymore, see Figure 10.


arch rise to span ratio for different outer girder cross-sections (StD)

Figure 10: Buckling check of the central arch in function of the outer

girder moment of inertia Iy for different central arch cross-sections

(StD)

The buckling coefficient remains quasi constant for the

skewed deck as the outer girder stiffness increases. However,

this constant value is altered in a way that for larger central

arch rise to span ratios also a stiffening effect on the central

arch is present. Moreover, the larger the central arch rise to

span ratio, the more this behaviour is present. Furthermore,

only for small central arch cross-sections an influence can be

seen. The progress found in these situations corresponds with

the behaviour for the straight deck. Also the increase in outer

girder cross-section induces a decrease in buckling coefficient

in those cases.


The compression stress in the central arch for the StD shows

a slight decreasing behaviour throughout the variation in outer

girder cross-section. Furthermore, the same can be seen for

the SkD, but the decrease is more pronoun, almost twice as for

the straight deck.


The tension stress in the outer longitudinal girders decreases

as their cross-section increases for both deck types. This

seems to go to an asymptote for the large outer girder cross-

sections.

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6

Bu

ck

lin

g c

oeff

icie

nt

[-]




0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

0 50 100 150 200 250

Bu

ckli

ng

co

effi

cien

t [-

]

Outer girder moment of inertia Iy ratio [%]


STS Iz 100 STS Iz 200 STS Iz 500


For the increase of the smaller outer girder cross-sections in

the StD situation a decrease in tension stress in the central

girder can be seen. However, this reaches quite quickly an

asymptote as the outer girder stiffness increases. While for the

skewed deck a decrease is present in the tension stress

throughout all of the outer girder cross-sections. So no

asymptote is found.

D. Central girder moment of inertia Iy


There is a quasi constant value as the central girder moment

of inertia increases in the StD situation. While a small

decreasing progress is found for the Skd.

The RTS 0,181 and 0,500 cases induce a slight decrease in

the value of the coefficient as Iy increases for the StD. This

decreasing effect for the larger rise to span ratios is more

pronoun in the SkD model.

Furthermore, the smallest considered central arch cross-

section (STS Iz 50) induces a decreasing behaviour of the

buckling coefficient for the SkD as the stiffness of the central

girder increases.


A slight decreasing progress can be found when the central

girder's stiffness is increased for the SkD. While there are only

small changes for the straight deck.

The smaller rise to span ratios show a constant compression

stress throughout the central girder cross-section variations for

the straight deck. However, the ratios larger than about 0,271

tend to induce an increase in compression stress. This is about

20% between the smallest and largest central girder

configuration considered over here. Furthermore no

differences can be seen for the skewed deck.


A constant tension stress in the outer girders can be found

through the variations for the central girder for both deck

types.


The increase in central girder moment of inertia Iy causes a

decrease of about 15-20% in tension stress between the

smallest, ± 50% Iy ratio relative to the base model, and largest

cross-sections, ± 225% Iy ratio relative to the base model,

considered in the study.

A different situation is present for the SkD. A larger central

longitudinal girder cross-section induces larger tension

stresses in it. This is because the moments around the y-axis

increase and induce this larger tensions tress.

VI. COMPARISON BETWEEN BOTH BRIDGE TYPES

The following table indicates whether differences occur

between the three arched bridge and the type where only one

central arch is present. The top row indicates the varying

parameters, while the first column gives the checked

variables. An "X" placed in the box means that differences

occur between both bridge types. These differences are solely

considered based on the progress of the variables. So not

based on their values. Moreover, the progress objected is the

one induced by the varying of the individual parameters. Thus

not on simultaneous variations of several of them.

There is furthermore referred to the corresponding

paragraphs discussed before in the chapters about the three

arched bridge and the bridge with one central arch to see what

then the actual differences are.

To keep the table clear, the following abbreviations are

used. The parameters:

Central arch rise to span ratio (RTS)

Central arch moment of inertia Iz (CAIz)

Outer girder moment of inertia Iy (OGIy)

Central girder moment of inertia Iy (CGIy)

The variables:

Central arch buckling coefficient (BC)

Central arch compression stress (σc)

Outer girder tension stress (σt,OG)

Central girder tension stress (σt,CG)

Table 1: Comparison between the three arched bridge and the bridge

with one central arch type

RTS CAIz OGIy CGIy

BC StD - - X -

SkD - - - -

σc StD - - - -

SkD X - X -

σt,OG StD - - - -

SkD - - - -

σt,CG StD X - X -

SkD - X - X

REFERENCES

[1] EN 1990. (2002). Eurocode 0: Base of the structural design. CEN, Brussels.

[2] EN 1991-2. (2004). Eurocode 1: Loads on constructions. CEN, Brussels.

Table of contents

LIST OF FIGURES I

LIST OF TABLES X

LIST OF ABBREVIATIONS AND SYMBOLS XIV

INTRODUCTION 1

PART I - SET UP FOR THE PARAMETER STUDY

CHAPTER 1: LITERATURE STUDY 6

1.1 GENERAL 6

1.2 INFLUENCING PARAMETERS 6

1.3 LIMIT STATE CHECKS 9

1.3.1 Ultimate limit state 9

1.3.2 Serviceability limit state 10

CHAPTER 2: SOFTWARE MODEL 11

CHAPTER 3: LOADS AND LOAD COMBINATIONS 15

3.1 PERMANENT LOADS 15

3.2 VARIABLE LOADS 16

3.2.1 Vertical loads 16

3.2.2 Horizontal loads 17

3.2.3 Overview of the different loads 20

3.3 LOAD COMBINATIONS 20

3.3.1 Ultimate limit state 20

3.3.2 Serviceability state 29

PART II - PARAMETER STUDY

CHAPTER 4: INDIVIDUAL PARAMETER STUDY 35

4.1 DECK SKEWNESS ANGLE 35

4.1.1 Maximum compression stress in the central arch 44

4.1.2 Buckling check of the central arch 45

4.1.3 Maximum compression stress in the outer arches 46

4.1.4 Buckling check of the outer arches 46

4.1.5 Maximum tension stress in the outer longitudinal girders 47

4.1.6 Maximum tension stress in the central longitudinal girder 47

4.2 CENTRAL ARCH RISE TO SPAN RATIO 49


4.2.2 Compression stress in the central arch 57

4.2.3 Maximum compression stress in the outer arches 59

4.2.4 Buckling check of the outer arches 60

Table of contents

4.2.5 Maximum tension stress in the outer longitudinal girders 61

4.2.6 Maximum tensions tress in the central longitudinal girder 61

4.3 SHAPE OF THE CENTRAL ARCH 62

4.4 CENTRAL ARCH MOMENT OF INERTIA IZ 64

4.5 CENTRAL ARCH MOMENT OF INERTIA IY 67

4.6 CENTRAL LONGITUDINAL GIRDER CONFIGURATION 69

4.6.1 Case 1 (base model) 69

4.6.2 Case 2 70

4.6.3 Case 3 71

4.6.4 Discussion of the results 72

4.7 OUTER ARCH MOMENT OF INERTIA IZ 73

4.8 OUTER ARCH RISE TO SPAN RATIO 76

4.9 OUTER LONGITUDINAL GIRDER MOMENT OF INERTIA IY 78

4.10 BRIDGE LENGTH 79

4.11 GENERAL CONCLUSIONS 80

CHAPTER 5: COMBINED PARAMETER STUDY - THREE ARCHED BRIDGE 82

5.1 GENERAL 82

5.2 CENTRAL ARCH RISE TO SPAN RATIO AND CENTRAL ARCH MOMENT OF INERTIA IZ VARIATION 82



5.2.3 Other variables 88

5.3 OUTER ARCH RISE TO CENTRAL ARCH RISE AND OUTER ARCH MOMENT OF INERTIA IZ VARIATION 89

5.3.1 Straight deck 90

5.3.2 Skewed deck 91

5.4 LONGITUDINAL GIRDERS MOMENT OF INERTIA IY VARIATION 92


CHAPTER 6: COMBINED PARAMETER STUDY - BRIDGE WITH ONE CENTRAL ARCH 94

6.1 GENERAL 94

6.2 CENTRAL ARCH RISE TO SPAN RATIO AND CENTRAL ARCH MOMENT OF INERTIA IZ VARIATION 96



6.3 CENTRAL ARCH RISE TO SPAN RATIO AND OUTER GIRDER MOMENT OF INERTIA IY VARIATION 101



6.4 CENTRAL ARCH RISE TO SPAN RATIO AND CENTRAL GIRDER MOMENT OF INERTIA IY VARIATION 105



6.5 CENTRAL ARCH MOMENT OF INERTIA IZ AND OUTER GIRDER MOMENT OF INERTIA IY VARIATION 108



6.6 CENTRAL ARCH MOMENT OF INERTIA IZ AND CENTRAL GIRDER MOMENT OF INERTIA IY VARIATION 110



6.7 CENTRAL GIRDER MOMENT OF INERTIA IY AND OUTER GIRDER MOMENT OF INERTIA IY VARIATION 112




Table of contents

CHAPTER 7: A COMPARISON BETWEEN THE THREE ARCHED BRIDGE AND THE BRIDGE

WITH ONE CENTRAL ARCH 121

7.1 THE CENTRAL ARCH RISE TO SPAN RATIO VARIATION 121



7.1.3 Tension stress in the outer longitudinal girders 122

7.1.4 Tension stress in the central longitudinal girder 122

7.2 THE CENTRAL ARCH CROSS-SECTION MOMENT OF INERTIA IZ VARIATION 123





7.3 THE OUTER GIRDER CROSS-SECTION MOMENT OF INERTIA IY VARIATION 124





7.4 THE CENTRAL GIRDER CROSS-SECTION MOMENT OF INERTIA IY VARIATION 125





7.5 OVERVIEW 126

CHAPTER 8: FINAL REMARKS 127

BIBLIOGRAPHY 128

PART III - APPENDIX

A. INDIVIDUAL PARAMETER STUDY A.1

A.1 CENTRAL ARCH MOMENT OF INERTIA IZ A.1

A.2 CENTRAL ARCH MOMENT OF INERTIA IY A.5

A.3 OUTER ARCH MOMENT OF INERTIA IZ A.9

A.4 OUTER ARCH RISE TO SPAN RATIO A.14

A.5 OUTER LONGITUDINAL GIRDER MOMENT OF INERTIA IY A.17

A.6 BRIDGE LENGTH A.20

B. COMBINED PARAMETER STUDY - THREE ARCHED BRIDGE B.1

B.1 CENTRAL ARCH RISE TO SPAN RATIO AND CENTRAL ARCH MOMENT OF INERTIA IZ VARIATION B.1

B.1.1 Straight deck B.1

B.1.2 Skewed deck B.5

B.2 OUTER ARCH RISE TO CENTRAL ARCH RISE AND OUTER ARCH MOMENT OF INERTIA IZ VARIATION B.10



B.3 LONGITUDINAL GIRDERS MOMENT OF INERTIA IY VARIATION B.20



Table of contents

C. COMBINED PARAMETER STUDY - BRIDGE WITH ONE CENTRAL ARCH C.1

C.1 CENTRAL ARCH RISE TO SPAN RATIO AND CENTRAL ARCH MOMENT OF INERTIA IZ VARIATION C.1

C.1.1 Straight deck C.1

C.1.2 Skewed deck C.3

C.2 CENTRAL ARCH RISE TO SPAN RATIO AND OUTER GIRDER MOMENT OF INERTIA IY VARIATION C.5



C.3 CENTRAL ARCH RISE TO SPAN RATIO AND CENTRAL GIRDER MOMENT OF INERTIA IY VARIATION C.10



C.4 CENTRAL ARCH MOMENT OF INERTIA IZ AND OUTER GIRDER MOMENT OF INERTIA IY VARIATION C.16



C.5 CENTRAL ARCH MOMENT OF INERTIA IZ AND CENTRAL GIRDER MOMENT OF INERTIA IY VARIATION C.21



C.6 OUTER GIRDER MOMENT OF INERTIA IY AND CENTRAL GIRDER MOMENT OF INERTIA IY VARIATION C.27



I

List of Figures

Introduction

Figure 1 Initial vierendeel bridge 1

Figure 2 Render design of the new railway bridge 1

Figure 3 Clarification of the deck skewness 3

Figure 4 Side view of the central arch (dimensions in meter) 3

Figure 5 Side view of the outer arches (dimensions in meter) 4

Figure 6 Front view of the bridge (dimensions in meter) 4

Part I - Set up for the parameter study

Figure 7 Visualisation of the rise of the arches 6

Figure 8 The arch rise to span ratio influence on the arch's stability coefficient [2] 7

Figure 9 Simply supported arch 8

Figure 10 The buckling shape which the arch wants to develop 9

Figure 11 The first in-plane buckling mode of the arch due to a counteracting behaviour

of the hangers

9

Figure 12 Top view of the out-of-plane buckling of the arch 10

Figure 13 The model with a straight deck and piecewise linear arch 11

Figure 14 Compression stresses (σ_2) in a hanger 12

Figure 15 The model with a straight deck and parabolic arches 13

Figure 16 The base model with a skewed deck and parabolic arches 13

Figure 17 General cross-section of a classic railway [19] 15

Figure 18 Loads to be applied according to LM71 [17] 16

Figure 19 Eccentric load application [17] 16

Figure 20 Wind loading surface (dimensions in meter) 18

Figure 21 Applied transverse wind load on the bridge (kN/m) 19

Figure 22 List of ultimate limit state combinations with LM71 as the main variable load 22

Figure 23 Overview of the bridge elements 22

Figure 24 LM71 applied symmetrical relative to the transversal centre line of the bridge 23

Figure 25 LM71 applied symmetrical relative to the transversal line at a quarter of the

bridge length

23

Figure 26 General unity check of the elements in the initial model 23

Figure 27 General unity check of the base model 26

Figure 28 List of ultimate limit state combinations with traction as the main variable load 27

Figure 29 List of ultimate limit state combinations with wind load as the main variable

load

28

Figure 30 List of ultimate limit state combinations with thermal load as the main variable

load

28

Figure 31 Most critical un.check of the central arch 29

Figure 32 List of serviceability limit state combinations with LM71 as the main variable

load

30

Figure 33 Vertical deflection of the bridge deck due to the permanent loads (values in

mm)

31

List of Figures

II

Figure 34 Vertical deflection of the central arch due to the permanent loads (values in

mm)

31

Figure 35 List of serviceability limit state combinations with LM71 as the main variable

load (without permanent loads)

31

Figure 36 List of serviceability limit state combinations with wind load as the main

variable load

32

Figure 37 List of serviceability limit state combinations with thermal load as the main

variable load

32

Figure 38 Maximum vertical deflection of the bridge deck (53,4 mm) (values in mm) 33

Figure 39 Maximum vertical deflection of the central arch (14,3 mm) (values in mm) 33

Part II - Parameter study

Figure 40 Simply supported beam as small part of the steel plates 39

Figure 41 Vertical deflection of the uniformly loaded straight plate 40

Figure 42 Vertical deflection of the uniformly loaded skewed plate 40

Figure 43 Internal bending moments of the uniformly loaded straight plate 41

Figure 44 Internal bending moments of the uniformly loaded skewed plate 41

Figure 45 Internal stresses of the uniformly loaded straight plate 42

Figure 46 Internal stresses of the uniformly loaded skewed plate 42

Figure 47 Location of the reference points for the determination of the deck stresses for

the straight deck

43

Figure 48 Location of the reference points for the determination of the deck stresses for

the skewed deck

43

Figure 49 Visualisation of the deck skewness angle α [°] 44

Figure 50 Compression stress in the central arch in function of the deck skewness angle 45

Figure 51 Buckling check of the central arch in function of the deck skewness angle 45

Figure 52 Compression stress in the outer arches in function of the deck skewness

angle

46

Figure 53 Buckling check of the outer arches in function of the deck skewness angle 46

Figure 54 Tension stress in the outer longitudinal girders in function of the deck

skewness angle

47

Figure 55 Tension stress in the central longitudinal girder in function of the deck

skewness angle

48

Figure 56 Buckling check of the central arch in function of the central arch rise to span

ratio

50

Figure 57 Vertical supports of the central arch related to in-plane buckling 51

Figure 58 Horizontal supports of the central arch related to out-of-plane buckling 51

Figure 59 Manual buckling check of the central arch in function of the central arch rise to

span ratio (only normal forces taken into account)

52

Figure 60 Compression stress in the central arch in function of the central arch rise to

span ratio

57

Figure 61 manually calculated compression stress in the central arch in function of the

central arch rise to span ratio (only normal forces taken into account)

58

Figure 62 Central arch cross-section 58

Figure 63 Manually calculated compression stress in the central arch in function of the

central arch rise to span ratio (normal forces and bending moments taken into

account)

59

List of Figures

III

Figure 64 Compression stress in the outer arches in function of the central arch rise to

span ratio

60

Figure 65 Buckling check of the outer arches in function of the central arch rise to span

ratio

60

Figure 66 Tension stress in the outer longitudinal girders in function of the central arch

rise to span ratio

61

Figure 67 Tension stress in the central longitudinal girder in function of the central arch

rise to span ratio

61

Figure 68 Parabolic central arch geometry with a central arch rise to span ratio of 0,500 62

Figure 69 Circular central arch geometry with a central arch rise to span ratio of 0,500 62

Figure 70 Buckling check of the central arch in function of the central arch moment of

inertia Iz

64

Figure 71 The progress of the different terms which are related to the compression

stress in the central arch in function of the central arch moment of inertia Iz

66

Figure 72 Compression stress in the central arch in function of the central arch moment

of inertia Iy

67


inertia Iy

68

Figure 74 Initial geometry of the central longitudinal girder 69

Figure 75 Central longitudinal girder with a constant cross-section 70

Figure 76 Tension stress distribution in the outer part of the central longitudinal girder for

the base model (skewed deck)

70

Figure 77 Tension stress distribution in the outer part of the central longitudinal girder for

case 2 (skewed deck)

71

Figure 78 Compression stress in the outer arches in function of the outer arch moment

of inertia Iz

73

Figure 79 Buckling check of the outer arches in function of the outer arch moment of

inertia Iz

75

Figure 80 Compression stress in the outer arches in function of the outer arch rise to

span ratio

77

Figure 81 Buckling check of the outer arches in function of the outer arch rise to span

ratio

77


inertia Iz for varying central arch rise to span ratios

83


ratio for varying central arch cross-sections

84


inertia Iz for varying central arch rise to span ratios

86



86


span ratio for varying central arch cross-sections

87



88

Figure 88 Most beneficial central arch cross-section according to paragraph 5.2 89

Figure 89 SCIA Engineer model for a steel railway bridge with one central arch 94



96



97

List of Figures

IV



98

Figure 93 k-interaction factors for central arch cross-section case four in function of the

central arch rise to span ratio

99



99

Figure 95 Tension stress in the central longitudinal girder in function of the central arch

rise to span ratio for varying central arch cross-sections

100


ratio for varying outer girder cross-sections

101


ratio for varying outer girder cross-sections

103


span ratio for varying outer girder cross-sections

104

Figure 99 Buckling check of the central arch in function of the outer longitudinal girder

moment of inertia Iy for varying central arch cross-sections

108

Part III - Appendix

Appendix A

Figure A.1 Compression stress in the central arch i.f.o. the central arch moment of inertia

Iz

A.1

Figure A.2 Compression stress in the first outer arch i.f.o. the central arch moment of

inertia Iz

A.1

Figure A.3 Buckling check of the first outer arch i.f.o. the central arch moment of inertia Iz A.2

Figure A.4 Compression stress in the second outer arch i.f.o. the central arch moment of

inertia Iz

A.2

Figure A.5 Buckling check of the second outer arch i.f.o. the central arch moment of

inertia Iz

A.3

Figure A.6 Tension stress in the first outer longitudinal girder i.f.o. the central arch

moment of inertia Iz

A.3

Figure A.7 Tension stress in the second outer longitudinal girder i.f.o. the central arch


A.4

Figure A.8 Tension stress in the central longitudinal girder i.f.o. the central arch moment

of inertia Iz

A.4

Figure A.9 Compression stress in the first outer arch i.f.o. the central arch moment of

inertia Iy

A.5

Figure A.10 Buckling check of the first outer arch i.f.o. the central arch moment of inertia Iy A.5

Figure A.11 Compression stress in the second outer arch i.f.o. the central arch moment of

inertia Iy

A.6

Figure A.12 Buckling check of the second outer arch i.f.o. the central arch moment of

inertia Iy

A.6

Figure A.13 Tension stress in the first outer longitudinal girder i.f.o. the central arch

moment of inertia Iy

A.7

Figure A.14 Tension stress in the second outer longitudinal girder i.f.o. the central arch


A.7

Figure A.15 Tension stress in the central longitudinal girder i.f.o. the central arch moment

of inertia Iy

A.8

List of Figures

V

Figure A.16 Compression stress in the central arch i.f.o. the outer arch moment of inertia Iz A.9

Figure A.17 Buckling check of the central arch i.f.o. the outer arch moment of inertia Iz A.9

Figure A.18 Buckling check of the first outer arch i.f.o. the outer arch moment of inertia Iz A.10

Figure A.19 Buckling check of the second outer arch i.f.o. the outer arch moment of inertia

Iz

A.10

Figure A.20 Tension stress in the first outer longitudinal girder i.f.o. the outer arch moment

of inertia Iz

A.11

Figure A.21 Tension stress in the second outer longitudinal girder i.f.o. the outer arch


A.11

Figure A.22 Tension stress in the central longitudinal girder i.f.o. the outer arch moment of

inertia Iz

A.12

Figure A.23 Compression stress in the central arch i.f.o. the outer arch rise to span ratio A.14

Figure A.24 Buckling check of the central arch i.f.o. the outer arch rise to span ratio A.14

Figure A.25 Tension stress in the first outer longitudinal girder i.f.o. the outer arch rise to

span ratio

A.15

Figure A.26 Tension stress in the second outer longitudinal girder i.f.o. the outer arch rise

to span ratio

A.15

Figure A.27 Tension stress in the central longitudinal girder i.f.o. the outer arch rise to span

ratio

A.16

Figure A.28 Compression stress in the central arch i.f.o. the outer longitudinal girder


A.17

Figure A.29 Buckling check of the central arch i.f.o. the outer longitudinal girder moment of

inertia Iy

A.17

Figure A.30 Compression stress in the outer arches i.f.o. the outer longitudinal girder


A.18

Figure A.31 Buckling check of the outer arches i.f.o. the outer longitudinal girder moment

of inertia Iy

A.18

Figure A.32 Tension stress in the outer longitudinal girders i.f.o. the outer longitudinal

girder moment of inertia Iy

A.19

Figure A.33 Tension stress in the central longitudinal girder i.f.o. the outer longitudinal

girder moment of inertia Iy

A.19

Figure A.34 SCIA Engineer model with a bridge length of 24,15 m A.20

Figure A.35 SCIA Engineer model with a bridge length of 72,45 m A.20

Appendix B

Figure B.1 Compression stress in the first outer arch i.f.o. the central arch rise to span


B.1

Figure B.2 Buckling check of the first outer arch i.f.o. the central arch rise to span ratio for

varying central arch cross-sections

B.1

Figure B.3 Compression stress in the second outer arch i.f.o. the central arch rise to span


B.2

Figure B.4 Buckling check of the second outer arch i.f.o. the central arch rise to span ratio

for varying central arch cross-sections

B.2

Figure B.5 Tension stress in the first outer longitudinal girder i.f.o. the central arch rise to


B.3

Figure B.6 Tension stress in the second outer longitudinal girder i.f.o. the central arch rise

to span ratio for varying central arch cross-sections

B.3

List of Figures

VI

Figure B.7 Tension stress in the central longitudinal girder i.f.o. the central arch rise to


B.4

Figure B.8 Compression stress in the first outer arch i.f.o. the central arch rise to span


B.5

Figure B.9 Buckling check of the first outer arch i.f.o. the central arch rise to span ratio for

varying central arch cross-sections

B.5

Figure B.10 Compression stress in the second outer arch i.f.o. the central arch rise to span


B.6

Figure B.11 Buckling check of the second outer arch i.f.o. the central arch rise to span ratio

for varying central arch cross-sections

B.6

Figure B.12 Tension stress in the first outer longitudinal girder i.f.o. the central arch rise to


B.7

Figure B.13 Tension stress in the second outer longitudinal girder i.f.o. the central arch rise


B.7

Figure B.14 Tension stress in the central longitudinal girder i.f.o. the central arch rise to


B.8

Figure B.15 Compression stress in the central arch i.f.o. the outer arch rise to central arch

rise ratio for varying outer arch cross-sections

B.10

Figure B.16 Buckling check of the central arch i.f.o. the outer arch rise to central arch rise

ratio for varying outer arch cross-sections

B.10

Figure B.17 Compression stress in the first outer arch i.f.o. the outer arch rise to central

arch rise ratio for varying outer arch cross-sections

B.11

Figure B.18 Buckling check of the first outer arch i.f.o. the outer arch rise to central arch


B.11

Figure B.19 Compression stress in the second outer arch i.f.o. the outer arch rise to central


B.12

Figure B.20 Buckling check of the second outer arch i.f.o. the outer arch rise to central


B.12

Figure B.21 Tension stress in the first outer longitudinal girder i.f.o. the outer arch rise to

central arch rise ratio for varying outer arch cross-sections

B.13

Figure B.22 Tension stress in the second outer longitudinal girder i.f.o. the outer arch rise

to central arch rise ratio for varying outer arch cross-sections

B.13

Figure B.23 Tension stress in the central longitudinal girder i.f.o. the outer arch rise to


B.14

Figure B.24 Compression stress in the central arch i.f.o. the outer arch rise to central arch


B.15

Figure B.25 Buckling check of the central arch i.f.o. the outer arch rise to central arch rise

ratio for varying outer arch cross-sections

B.15

Figure B.26 Compression stress in the first outer arch i.f.o. the outer arch rise to central


B.16

Figure B.27 Buckling check of the first outer arch i.f.o. the outer arch rise to central arch


B.16

Figure B.28 Compression stress in the second outer arch i.f.o. the outer arch rise to central


B.17

Figure B.29 Buckling check of the second outer arch i.f.o. the outer arch rise to central


B.17

Figure B.30 Tension stress in the first outer longitudinal girder i.f.o. the outer arch rise to


B.18

Figure B.31 Tension stress in the second outer longitudinal girder i.f.o. the rise to rise ratio

for varying outer arch cross-sections

B.18

List of Figures

VII

Figure B.32 Tension stress in the central longitudinal girder i.f.o. the outer arch rise to


B.19

Figure B.33 Compression stress in the central arch i.f.o. the outer longitudinal girder

moment of inertia Iy for varying central girder cross-sections

B.20

Figure B.34 Buckling check of the central arch i.f.o. the outer longitudinal girder moment of

inertia Iy for varying central girder cross-sections

B.20

Figure B.35 Compression stress in the first outer arch i.f.o. the outer longitudinal girder


B.21

Figure B.36 Buckling check of the first outer arch i.f.o. the outer longitudinal girder moment

of inertia Iy for varying central girder cross-sections

B.21

Figure B.37 Compression stress in the second outer arch i.f.o. the outer longitudinal girder


B.22

Figure B.38 Buckling check of the second outer arch i.f.o. the outer longitudinal girder


B.22

Figure B.39 Tension stress in the first outer longitudinal girder i.f.o. the outer longitudinal

girder moment of inertia Iy for varying central girder cross-sections

B.23

Figure B.40 Tension stress in the second outer longitudinal girder i.f.o. the outer

longitudinal girder moment of inertia Iy for varying central girder cross-sections

B.23

Figure B.41 Tension stress in the central longitudinal girder i.f.o. the outer longitudinal


B.24

Figure B.42 Compression stress in the central arch i.f.o. the outer longitudinal girder


B.25

Figure B.43 Buckling check of the central arch i.f.o. the outer longitudinal girder moment of


B.25

Figure B.44 Compression stress in the first outer arch i.f.o. the outer longitudinal girder


B.26

Figure B.45 Buckling check of the first outer arch i.f.o. the outer longitudinal girder moment

of inertia Iy for varying central girder cross-sections

B.26

Figure B.46 Compression stress in the second outer arch i.f.o. the outer longitudinal girder


B.27

Figure B.47 Buckling check of the second outer arch i.f.o. the outer longitudinal girder


B.27

Figure B.48 Tension stress in the first outer longitudinal girder i.f.o. the outer longitudinal


B.28

Figure B.49 Tension stress in the second outer longitudinal girder i.f.o. the outer


B.28

Figure B.50 Tension stress in the central longitudinal girder i.f.o. the outer longitudinal


B.29

Appendix C

Figure C.1 Tension stress in the first outer longitudinal girder i.f.o. the central arch rise to


C.1

Figure C.2 Tension stress in the second outer longitudinal girder i.f.o. the central arch rise


C.1

Figure C.3 Tension stress in the central longitudinal girder i.f.o. the central arch rise to


C.2



C.3

List of Figures

VIII



C.3

Figure C.6 Compression stress in the central arch i.f.o. the central arch rise to span ratio

for varying outer girder cross-sections

C.5



C.5


to span ratio for varying outer girder cross-sections

C.6



C.6



C.7


to span ratio for varying outer girder cross-sections

C.7



C.8

Figure C.13 Buckling check of the central arch i.f.o. the central arch rise to span ratio for

varying central girder cross-sections

C.10


for varying central girder cross-sections

C.10


span ratio for varying central girder cross-sections

C.11


to span ratio for varying central girder cross-sections

C.11



C.12

Figure C.18 Buckling check of the central arch i.f.o. the central arch rise to span ratio for


C.13


for varying central girder cross-sections

C.13



C.14


to span ratio for varying central girder cross-sections

C.14



C.15

Figure C.23 Compression stress in the central arch i.f.o. the outer longitudinal girder


C.16

Figure C.24 Tension stress in the first outer longitudinal girder i.f.o. the outer longitudinal

girder moment of inertia Iy for varying central arch cross-sections

C.16

Figure C.25 Tension stress in the second outer longitudinal girder i.f.o. the outer

longitudinal girder moment of inertia Iy for varying central arch cross-sections

C.17

Figure C.26 Tension stress in the central longitudinal girder i.f.o. the outer longitudinal


C.17

Figure C.27 Buckling check of the central arch i.f.o. the outer longitudinal girder moment of

inertia Iy for varying central arch cross-sections

C.18



C.18



C.19

List of Figures

IX


longitudinal girder moment of inertia Iy for varying central arch cross-sections

C.19



C.20

Figure C.32 Buckling check of the central arch i.f.o. the central arch moment of inertia Iz for


C.21

Figure C.33 Compression stress in the central arch i.f.o. the central arch moment of inertia

Iz for varying central girder cross-sections

C.21

Figure C.34 Tension stress in the first outer longitudinal girder i.f.o. the central arch

moment of inertia Iz for varying central girder cross-sections

C.22

Figure C.35 Tension stress in the second outer longitudinal girder i.f.o. the central arch


C.22

Figure C.36 Tension stress in the central longitudinal girder i.f.o. the central arch moment

of inertia Iz for varying central girder cross-sections

C.23

Figure C.37 Buckling check of the central arch i.f.o. the central arch moment of inertia Iz for


C.24

Figure C.38 Compression stress in the central arch i.f.o. the central arch moment of inertia

Iz for varying central girder cross-sections

C.24

Figure C.39 Tension stress in the first outer longitudinal girder i.f.o. the central arch


C.25

Figure C.40 Tension stress in the second outer longitudinal girder i.f.o. the central arch


C.25

Figure C.41 Tension stress in the central longitudinal girder i.f.o. the central arch moment

of inertia Iz for varying central girder cross-sections

C.26



C.27



C.27



C.28



C.28



C.29



C.30



C.30



C.31



C.31



C.32

X

List of Tables

Introduction

Table 1 General working of a tied arch [1] 2

Part I - Set up for the parameter study

Table 2 Overview of the applied loads on the railway bridge 20

Table 3 Partial factor values for the different loads 20

Table 4 Combination factors for the different loads [21] 21

Table 5 Partial and combination factors for the ULS combinations with LM71 as the

main variable load

21

Table 6 Overview of the different elements (dimensions in mm) 24

Table 7 Un.check values for the central arch with LM71 as the main variable load 26

Table 8 Partial and combination factors for the ULS combinations with traction as the

main variable load

27

Table 9 Un.check values of the central arch with traction as the main variable load 27

Table 10 Partial and combination factors for the ULS combinations with wind load as

the main variable load

28

Table 11 Un.check values of the central arch with wind load as the main variable load 28

Table 12 Partial and combination factors for the ULS combinations with thermal load as

the main variable load

28

Table 13 Un.check values of the central arch with thermal load as the main variable

load

29

Table 14 Combination factors for the SLS combinations with LM71 as the main variable

load

30

Table 15 Combination factors for the SLS combinations with wind load as the main

variable load

32

Table 16 Combination factors for the SLS combinations with thermal load as the main

variable load

32

Part II - Parameter study

Table 17 Visualisation of the tension stresses within the central arch 36

Table 18 Visualisation of the compression stresses within the central arch 36

Table 19 Visualisation of the vertical deflection of the central arch 37

Table 20 Visualisation of the longitudinal translation of the central arch (deformation in

the x-direction)

37

Table 21 Visualisation of the transversal deflection of the central arch (deformation in

the y-direction)

37

Table 22 Visualisation of the vertical deflection of the bridge deck 37

Table 23 Visualisation of the longitudinal translation of the bridge deck (deformation in

the x-direction)

38

Table 24 Visualisation of the transversal deflection of the bridge deck (deformation in

the y-direction)

38

List of Tables

XI

Table 25 Buckling coefficient of the central arch 38

Table 26 Maximum reaction forces for the most determining ULS combination at the

supports

39

Table 27 Maximum tension stresses at the upper side of the deck 43

Table 28 Tension stresses in the longitudinal girders for the base models 44

Table 29 Different central arch rise to span ratio cases 49

Table 30 Length of the central arch and corresponding critical buckling load 51

Table 31 Normal force in the central arch 52

Table 32 Internal bending moments in the central arch 53

Table 33 Interaction factors for the central arch 54

Table 34 Equivalent uniform moment factors Cmi,0 [28] 55

Table 35 Mz in the central arch (absolute values) 55

Table 36 Deformation along the central arch 56

Table 37 My in the central arch (absolute values) 56

Table 38 Compression stress contribution of the internal forces 59

Table 39 Results for the parabolic central arch geometry with a central arch rise to span

ratio of 0,500

62

Table 40 Results for the circular central arch geometry with a central arch rise to span

ratio of 0,500

63

Table 41 Different central arch cross-sections 64

Table 42 Internal forces in the central arch (skewed deck) 65

Table 43 Geometrical properties of the central arch 65


Table 45 Composing cross-sections of the central longitudinal girder 69

Table 46 Results for the base model 69

Table 47 Results for case 2 71

Table 48 Results for case 3 71

Table 49 Internal forces in the central arch (skewed deck) 72

Table 50 Internal forces in the central girder (skewed deck) 72

Table 51 Different outer arch cross-sections 73

Table 52 Outer arch cross-sectional properties 74

Table 53 Different outer arch rise to span ratio cases 76

Table 54 Internal forces in the outer arches for case 1 and 3 (skewed deck) 76

Table 55 Different outer girder cross-sections 78

Table 56 Different bridge length cases 79


Table 58 Different outer arch rise cases 89

Table 59 Different outer arch cross-sections 90

Table 60 internal forces in the central arch for the STS 50% case (straight deck) 90

Table 61 Internal forces in the first outer arch for the STS 50% case (straight deck) 90

Table 62 Internal forces in the second outer arch for the STS 50% case (straight deck) 90

Table 63 Additional outer arch cross-sections 91

Table 64 Different central girder configurations 92


Table 66 Results for the base model of the bridge with one central arch 95

Table 67 Internal forces in the central arch for both base models (straight deck) 95

Table 68 Internal forces in the central arch for both base models (skewed deck) 95



List of Tables

XII

Table 71 Internal forces in the central arch for several central arch rises and outer girder

cross-sections

102

Table 72 k-interaction factors for the central arch for several central arch rises and outer

girder cross-sections

102

Table 73 Different central girder configurations 105

Table 74 Internal forces in the central arch for the central girder 2 case (skewed deck) 106

Table 75 Internal forces in the central longitudinal girder for the central girder 2 case

(skewed deck)

106

Table 76 Theoretical tension stress in the central longitudinal girder for the central

girder 2 case (skewed deck)

107

Table 77 k-interaction factors for the central arch for the STS Iz 500 case for the

different central girder configurations

110

Table 78 Conclusion of the results for the central arch rise to span ratio as main

parameter

113

Table 79 Conclusion of the results for the central arch cross-section moment of inertia Iz

as main parameter

115

Table 80 Conclusion of the results for the outer girder cross-section moment of i

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