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I
Declaration
I Farhad Huseynov declare that this dissertation study is my own work and that all the sources
that I have used or quoted have been acknowledged by means of complete references.
________________________
Farhad Huseynov 31.08.2012
Signature Date
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II
Abstract
The Bosporus Bridge is one of the two permanent transportation connections between
Europe and Asia in Istanbul. It carries the main arterial transportation link of the city, namely
O-1 motorway. Any broken link due to the bridge failure would totally ruin the whole
transportation system in the city. Due to importance and complexity of the Bosporus Bridge, in
this particular dissertation study special care was given to understand the real behavior of the
structure. The purpose of this research was to develop a sophisticated FE model with less
uncertainty that provides with a clearer understanding and higher confidence in estimating the
real behavior of the Bosporus Bridge.
To develop a FE model commercial software, namely ANSYS V12.1 was used which is a FE
modelling package that numerically solves wide range of mechanical problems. There are two
methods available to use ANSYS. The first method is by means of Graphical User Interface,
which is so called GUI and the second one is by means of script files. For this particular work,
second option was used to produce the FE models.
Dimensions of the structural components play an important role in the modelling process.
To develop an accurate FE model, dimensions have to be adopted as correct as possible. The
Bosporus Suspension Bridge was designed more than 40 years ago as a result softcopy of the
design drawings is not available. Therefore, the bridge major parts (towers, cables hangers and
the suspended box deck section) and overall shape were redrawn in accordance with the
design drawings using the AutoCAD software to get more accurate coordinates for the model.
Details of the structural components, geometric nonlinearities, cable sagging and stress
stiffening and profile of the deck structure are the main factors affecting the vibration
characteristics of the bridge. Sometimes, to define these properties accurately from the first
attempt is impossible during the modelling process. Producing a sophisticated 3-D FE model
of the Bosporus Suspension Bridge requires too much time and a lot of effort. Besides, in terms
of computer processing capacity, analyzing a 3-D FE model takes longer time compared to a 2-D
FE model. To facilitate the modelling process, initially two 2-D FE models were produced to
adopt the correct properties. The first 2-D FE model was restrained in the vertical plane and the
second one in the horizontal plane which provide vertical and lateral modes, respectively. Then
the results obtained from 2-D FE models were confirmed to be accurate by comparing them
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III
with the experimental data available from the past studies. Afterwards, the same properties
were defined to develop a sophisticated 3-D FE model of the Bosporus Bridge.
Since the computer processing capacity is limited, modelling the bridge in three dimensions
with all the structural components is an impossible job. To overcome this issue, degrees offreedom were reduced by introducing the equivalent super elements for the towers and the
suspended deck structure which are explained in detail in the related sections. The 3-D FE
model was divided into 4 major parts being cables, hangers, towers and the suspended deck
structure and was modeled separately. Then all the parts were combined together and
imported into ANSYS to develop a complete model. The model were analyzed both for static
and modal analysis. Finally the results obtained from the 3-D analytical model was compared
with the experiment data available from the past studies and was ensured that a sophisticated
3-D FE model works properly without any major warnings and represents the real behavior of
the Bosporus Bridge.
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IV
Acknowledgement
First and foremost, I would like to thank to my supervisor, Prof. James Brownjohn for the
valuable guidance and advice. This thesis would not have been possible without his help,
support and patience. Besides, I would like to thank to a PhD student, Rahi Rahbari, for his good
advice and friendship who never hesitated to share his knowledge and experience despite his
many other academic and professional commitments.
Last but not least I would like to thank my family. They were always supporting me and
encouraging me with their best wishes.
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V
Contents
Declaration ...................................................................................................................................... I
Abstract .......................................................................................................................................... II
Acknowledgement ........................................................................................................................ IV
Contents ......................................................................................................................................... V
Figure List ..................................................................................................................................... VII
Table List ....................................................................................................................................... IX
1 Introduction ........................................................................................................................... 1
2 Background ............................................................................................................................ 2
2.1 Principal Dimensions and Quantities ............................................................................. 5
3 FINETE ELEMENT MODELLING ............................................................................................... 6
3.1 Dimensions ..................................................................................................................... 7
3.1.1 Suspended Deck Structure ..................................................................................... 7
3.1.1.1 Longitudinal Parts .............................................................................................. 8
3.1.1.2 Main Diaphragm .............................................................................................. 10
3.1.2 Cables ................................................................................................................... 11
3.1.3 Hangers ................................................................................................................ 11
3.1.4 Towers .................................................................................................................. 12
3.1.5 Bridge Profile ........................................................................................................ 133.2 2-D FE Model ................................................................................................................ 14
3.2.1 Deck Modelling .................................................................................................... 14
3.2.2 Cable Modelling ................................................................................................... 14
3.2.3 Hanger Modelling ................................................................................................. 16
3.2.4 Tower modelling .................................................................................................. 17
3.2.5 Complete 2-D FE Model of the Bridge ................................................................. 17
3.2.6 2-D FE Model Analysis .......................................................................................... 17
3.3 3-D FE Model ................................................................................................................ 20
3.3.1 Equivalent Super Element for Suspended Deck Structure ................................... 20
3.3.1.1 Modelling of the Original Box Deck Section..................................................... 20
3.3.1.2 Equivalent Plate Element ................................................................................. 25
3.3.1.3 Equivalent Box Deck Element .......................................................................... 28
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VI
3.3.1.4 Complete 3-D FE Model of the Suspended Deck Structure ............................. 31
3.3.2 Equivalent Super Element for Towers ................................................................. 32
3.3.3 Complete 3-D FE Model of the Bridge ................................................................. 33
3.3.4 3-D FE Model Analysis .......................................................................................... 34
3.3.4.1 Bridge Model with Different Cable Strains ...................................................... 34
3.3.4.2 Bridge Model with Additional Mass ................................................................. 38
3.3.4.3 Bridge Model with Different Boundary Conditions ......................................... 42
4 Model Validation .................................................................................................................. 45
4.1 Comparison of Experimental and Analytical Results for Vertical Modes .................... 45
4.2 Comparison of Experimental and Analytical Results for Lateral Modes...................... 46
4.3 Comparison of Experimental and Analytical Results for Torsional Modes .................. 47
5 Conclusion ............................................................................................................................ 48
6 References ........................................................................................................................... 49
7 Appendix A ........................................................................................................................... 50
8 Appendix B ........................................................................................................................... 51
9 Appendix C ........................................................................................................................... 53
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VII
Figure List
Figure 3-1 Example of original deck section drawing.....................................................................7
Figure 3-2 Example of deck section drawn by AutoCAD................................................................8
Figure 3-3 Deck section divided into 5 main parts......................................................................8
Figure 3-4 Standard upper deck plate drawn by AutoCAD.........................................................8
Figure 3-5 Standard side unit drawn by AutoCAD........................................................................9
Figure 3-6 Standard footway plate drawn by AutoCAD..............................................................9
Figure 3-7 Standard side plate drawn by AutoCAD.....................................................................9
Figure 3-8 Standard bottom flange plate drawn by AutoCAD...................................................10
Figure 3-9 Standard diaphragm drawn by AutoCAD..................................................................10
Figure 3-10 Arrangement of uncompacted cables (Brown & Parsons, 1975).............................11
Figure 3-11 Towers Drawn by AutoCAD.....................................................................................12
Figure 3-12 Side plates labels.......................................................................................................12
Figure 3-13 Suspended deck structure and cable profile.............................................................13
Figure 3-14 Arrangement of cables and hangers.........................................................................13
Figure 3-15 2-D FE model.............................................................................................................17
Figure 3-16 Analysis Results from 2-D FE model restrained on the vertical plane......................18
Figure 3-17 Analysis Results from 2-D FE model restrained on the horizontal plane.................19
Figure 3-18 Original deck section keypoint locations..................................................................20
Figure 3-19 Model of the original deck Section...........................................................................21
Figure 3-20 Diaphragms...............................................................................................................21
Figure 3-21 Complete meshed section (mesh size 500mm)........................................................22
Figure 3-22 Deformed shape of the original deck section for vertical bending...........................22
Figure 3-23 Deformed shape of the original deck section due Lateral bending..........................23
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VIII
Figure 3-24 Deformed shape of the original deck section due to torsion...................................23
Figure 3-25 Equivalent Plate........................................................................................................24
Figure 3-26 Deformed shape of equivalent plate under different loading conditions................26
Figure 3-27 Deformed shape of the equivalent box deck due to vertical bending (1st
Case)....28
Figure 3-28 Deformed shape of equivalent box deck due to lateral bending (1st Case)............28
Figure 3-29 Deformed shape of equivalent box deck due to torsion (1st
Case)..........................29
Figure 3-30 Deformed shape of equivalent box deck due to vertical bending (2nd Case)...........29
Figure 3-31 Deformed shape of equivalent box deck due to lateral bending (2nd
Case).............30
Figure 3-32 Deformed shape of equivalent box deck due to torsion (2nd
Case)..........................30
Figure 3-33 Complete 3-D FE model of the Bridge.......................................................................32
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IX
Table List
Table 3-1 Tower structure plate thicknesses...............................................................................12
Table 3-2 Strain values for main cables........................................................................................15
Table 3-3 Strain values for hanger elements................................................................................16
Table 3-4 Equivalent plate displacements obtained for different arrangements........................26
Table 3-5 Equivalent box deck element displacements obtained for different arrangements....28
Table 3-6 Comparison of vertical mode shapes and frequencies between case 1,2 and 3..........34
Table 3-7 Comparison of lateral mode shapes and frequencies between case 1,2 and 3...........35
Table 3-8 Comparison of Torsional mode shapes and frequencies between case 1,2 and 3.......36
Table 3-9 Comparison of vertical mode shapes and frequencies between case 1 and 2.............38
Table 3-10 Comparison of lateral mode shapes and frequencies between case 1 and 2............39
Table 3-11 Comparison of torsional mode shapes and frequencies between case 1 and 2........40
Table 3-12 Comparison of vertical mode shapes and frequencies between
case 1, 2, 3 and 4.......................................................................................................42
Table 3-13 Comparison of lateral and torsional mode shapes and frequencies
between case 1, 2, 3 and 4......................................................................................43
Table 4-1 Comparison of Experimental and Analytical results for vertical modes.......................45
Table 4-2 Comparison of Experimental and Analytical results for lateral modes........................46
Table 4-3 Comparison of Experimental and Analytical results for torsional modes....................46
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1
1 Introduction
Suspension bridges are the structures with the large dimensions and long service life.
Throughout the history of the suspension bridges, their behavior under different dynamic
loadings such as wind, earthquake and traffic loads was always a matter of concern. Before
1930s, suspension bridges were designed to resist only the static loadings however failure of
the Tacoma Narrows Bridge in 1940 gave a clue to researchers that the suspension bridges are
vulnerable to dynamic loading. To understand the behavior of suspension bridges under
dynamic loadings many remarkable theoretical and experimental studies were carried out by
different authors. Major advances have been achieved through the advances in computer
process capacity and the use of Finite Element (FE) Method. Ambient field measurements were
carried out for the Bosporus suspension bridge in Istanbul and data collected was compared
with already developed analytical model. It was proven that an accurate FE model is a useful
tool to simulate the real behavior of the existing suspension bridges (Brownjohn, et al., 1989).
Furthermore, a precise FE model can be helpful in regular inspections and modifications.
(Merce, et al., 2007)
Long span suspension bridges are very flexible and lightly damped structures. Throughout
their service life, traffic load may significantly change their dynamic behavior and affect the
fatigue life of the bridge. In fact 80-90% of steel structures failures are related to fatigue and
fracture. Therefore, British standards recommend FE method as an accurate method for fatigue
stress analysis in suspension bridges. (Chan, et al., 2003)
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2
2 Background
Istanbul, being the largest and commercial city of Turkey is situated on the NW shore of Sea
of Marmara and is divided between two continents, Europe and Asia, by 22km long and
minimum of 1km wide stretch of water, namely Bosporus strait, which links the Black Sea with
the Sea of Marmara. For centuries, it was a challenging task for the communities to provide a
permanent crossing over the Bosporus strait. It is believed that the very first idea of a bridge
crossing the Bosporus dates back to the ancient times as recorded by the Greek writer
Herodotus in his histories. Once an engineer named Mandrocles designed a boat type bridge
(480 BCE) that stretched across the Bosporus, linking Asia to Europe, so that Darius I, the king
of the Achaemenid Empire (also called Darius the Great) move his army into position in the
Balkans to overcome the Macedon (Pericles, 1987). However, after that, till the half of the 20th
century no permanent link existed over the Bosporus and the transportation between two
parts of the city was provided by ferries but, following the rapid development in the city,
permanent link became mandatory. Although several engineering solutions were proposed
before 1950s, due to several reasons none of them drew a serious attention. Later with the
increase in demand on transportation, government started to give a serious consideration on
this issue and in 1956 feasibility study was performed by De Leuw Cather where it was
concluded that a permanent crossing is feasible and economically viable. The report was
accepted by government and following this several design proposals were presented bydifferent international companies mainly based in US by 1960. Unfortunately, later due to the
political issues in Turkey the project was postponed. Meanwhile the amount of traffic was
continuously increasing and after a while ferries could not cope up with the amount of traffic
and long delays become an issue. Thus in 1967, after a very careful assessment, mainly
considering its economical sides, Turkey government included the construction of the Bosporus
Bridge in its forthcoming 5 year plan of highway construction programme. (Brown & Parsons,
1975)
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Prior to 1960 all the existing major suspension bridges had been built in USA but by 1966,
with the completion of Forth Road and Severn Bridges, UK experience was also available in the
market. Therefore, companies invited for tendering were both from USA and UK. British
structural engineering company, Freeman Fox and Partners, presented their preliminary
proposal in 1967 and formal agreement was made with them in January 1968 (Brown &
Parsons, 1975). Bosporus Bridge was designed as gravity anchored suspension bridge made of
steel with hollow towers, and inclined hangers, carrying the shallow box deck structure which is
located between the villages of Ortakoy and Beylerbeyi.
The Bosporus Bridge consists of one main span and two side spans. Only the main span was
designed as suspended structure which is spanning 1074 m over the Bosporus strait and is
carried by cables and hangers that transfer the load to the massive towers, having 165m height
on each end. Side spans, each 231m and 255m long on Ortakoy and Beylerbeyi sides,
respectively was designed independent of the cable and are carried by piers. The Construction
of the bridge was performed by the Turkish company, namely “Enka Construction & Industry
Co.” along with the contractors which are “Cleveland Bridge & Engineering Co. Ltd.” (UK) and
“Hochtief AG” (Germany). The Construction started in 20th of February 1970 with the big
ceremony and finished in 30th
of October 1973. When the bridge was opened to traffic it was
accounted the first bridge connection between Europe and Asia and had the 4th
longest
suspension bridge span. However at the present it is the 19th longest suspension bridge span in
the world ranking. (General Directorate of Highways, Turkey, 1973)
The bridge total width consists of 8 lanes. Each direction has 3 lanes for daily vehicle traffic
and additional one emergency lane and one for pedestrians. After four years the bridge had
been opened f or use, the pedestrians’ walk over the bridge was prohibited. Previously, they
could walk over the bridge reaching in it through the elevators inside the towers. Recently,
pedestrians are allowed to walk over the bridge only one day in a year (usually in October)
during the “Intercontinental Istanbul Eurasia Marathon”. Visitors to Istanbul in October can
sign up for the Marathon and have a chance to enjoy the view from the bridge.
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The Bosporus Bridge reaches its 40 years of service life in 2013. As per maintenance
schedule, the bridge has to go through a full maintenance programme every 40 years where all
the hangers are needed to be replaced. Therefore, the bridge is planned to be closed for the
vehicular traffic for a year in early 2013 to carry out the maintenance works.
The traffic intensity is continuously increasing in Istanbul. As a result, both Bosporus Bridge
and Fatih Sultan Mehmet Bridge, which is the second bridge spanning the Bosporus, are
exposed to heavy traffic load for which they were not designed. To overcome this issue, the
Turkish government started to consider the construction of the third bridge over the Bosporus
strait in early 2010. Following this, in 29th
of May 2012 it was officially announced that the “IC
Ictas-Astaldi” consortium was awarded a contract for the “Northern Marmara Highway Project”
which includes the third bridge construction over the Bosporus. The site was located between
the Poyraz and Garipce villages and expected completion date is planned by the end of 2015.
The cost of the project was estimated as 4.5 billion Turkish Liras which is equivalent to 2.5
billion USD. (Exchange rate based on the Central Bank of Turkish Republic: 1 USD-1.8 TL. 26th of
August 2012)
The Bosporus Bridge is one of the two continuous transportation connections between
Europe and Asia in Istanbul. It carries the arterial transportation link of the city, namely O-1
motorway. Any broken link due to the bridge failure would totally ruin the whole
transportation system in the city. Due to importance and complexity of the Bosporus Bridge
special care was given by many researchers to understand the real behavior of the structure. In
this particular dissertation study, two 2-D FE models and a sophisticated 3-D FE Model of the
Bosporus Suspension Bridge will be developed using ANSYS V12.1 commercial software. The
purpose of this research is to develop a model with less uncertainty that provide with a clearer
understanding and higher confidence in estimating the real behavior of the Bosporus Bridge.
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2.1 Principal Dimensions and Quantities (General Directorate of Highways, Turkey,
1973)
Total length of a bridge : 1 560 m
Main Span length : 1 074
Approach Spans (Ortakoy) : 231 m
(Beylerbeyi) : 255
Clearance over the sea : 64 m
Height of the towers : 165 m
Design Loads
Live load : 1.33 tons/m
Wind Load : 45 m/s
Ground acceleration : 0.1 g
Main cable sagging : 93 m
Tension in the main cables : 15 400 tons/cable
Some Manufacturing quantities
Excavation : 63 000 m3
Concrete : 71 000 m3
Concrete reinforcement : 4 000 tons
Steel : 17 000 tons
Cables : 6 000 tons
Cost of the bridge : 191 785 265 TL (Turkish Lira)
23 213 666 USD
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3 FINETE ELEMENT MODELLING
Suspension bridges are the complex structures with large dimensions. Details of the
structural components, geometric nonlinearities, cable sagging and stress stiffening and profile
of the deck structure are the main factors affecting the vibration characteristics of the bridge
(Apaydin, 2010). Sometimes, to define these properties accurately from the first attempt is
impossible during the modelling process. Producing a sophisticated 3-D FE model of the
Bosporus Suspension Bridge requires too much time and a lot of effort. Besides, in terms of
computer processing capacity, analyzing a 3-D FE model takes longer time compared to 2-D FE
model. Thus, to facilitate the modelling process, initially, two 2-D FE models were produced to
adopt the correct properties. The first 2-D FE model was restrained in the vertical plane and the
second one in the horizontal plane which provide vertical and lateral modes, respectively. Then
the results obtained from the 2-D FE models were confirmed to be accurate by comparing
them with the experimental data available from the past studies. Afterwards, same properties
were defined to develop a sophisticated 3-D FE model of the Bosporus Suspension Bridge.
In Bosporus Bridge side spans are not connected to cables and are carried by piers. Apart
from the small mass contribution to towers they do not have any significant influence on the
bridge behavior. Thus in all models the side spans were excluded in the model. The procedures
followed to produce both 2-D and 3-D FE models later will be covered in detail in the related
sections.
To develop the FE models commercial software, namely ANSYS V12.1 were used which is a
FE modelling package that numerically solves wide range of mechanical problems. There are
two methods available to use ANSYS. The first is by means of Graphical User Interface, which is
so called GUI and the second is by writing the script files. For this particular work, second
option was used to produce the FE models. Additionally, there are no predefined set of units
specified in ANSYS. It is the responsibility of the user to adopt the consistent set of units. The
units defined within the script files are as follows;
-Length (keypoint coordinates) - in mm -Area-in mm2 -Mass- in tons
-Density- in tons/mm3 -Force- in Newton -Modulus of Elasticity- in MPa
-Second Moment of Area- in mm4
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Dimensions of the structural components play an important role in the modelling process.
To develop an accurate FE model, dimensions have to be adopted as correct as possible. The
Bosporus Bridge was designed more than 40 years ago as a result softcopy of the design
drawings is not available. Therefore, the bridge major parts (towers, cables, hangers and the
suspended box deck section) and overall profile of the bridge were redrawn in accordance with
the design drawings using the AutoCAD software to get more accurate coordinates for the
models.
3.1 Dimensions
Due to the absence of the softcopy of the bridge design drawings, bridge major parts and
overall profile of the structure were redrawn using the AutoCAD software as described in the
below sections
3.1.1 Suspended Deck Structure
Suspended deck structure consists of 60 box girders, each of 17.9 m long, 33.4m wide and
3m deep. Each box section is made up of 22 stiffened plates. 2 types of stiffeners were used
which are V shaped 6mm thick pressed through members for upper deck plates and single
sided bulb flat (S.S.B.F) stiffeners for elsewhere in the deck structure. Diaphragms are placed at
every 4475 mm apart along the length of the deck structure to prevent local buckling.
Figure 3-1 Example of original deck section drawing
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Figure 3-2 Example of deck section drawn by AutoCAD
3.1.1.1 Longitudinal Parts
Box deck section was divided into 5 main parts as shown in the figure below and was drawn
separately using the AutoCAD software.
Figure 3-3 Deck section divided into 5 main parts
Part 1
Part 1 consists of 10 plates each of 2470mm wide and 17900mm long, stiffened with 6mm
pressed V shaped stiffeners and 12mm thick diaphragm plate which connects to the main
diaphragm and forming the upper part of the deck structure.
Figure 3-4 Standard upper deck plate drawn by AutoCAD
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Part 2
Part 2 plays and important role in
deck section. It connects the upper
and lower deck and footway section. It
is made up of 2 plates, each of 9mm
thick and 17900mm long, stiffened
with 150x8x17750 S.S.B.F stiffeners
and 6mm thick diaphragm plate that
connects to the main diaphragm.
Figure 3-5 Standard side unit drawn by AutoCAD
Part 3
Part 3 is a footway section which consists of 3 plates, each of 8mm, 10mm and 12mm thick
and stiffened with 150x8x17750 S.S.B.F stiffeners and 8mm thick diaphragm plate
Figure 3-6 Standard footway plate drawn by AutoCAD
Part 4
Part 4 is bottom side section of
the deck structure which is made
up of 9mm thick and 17900mm
long plate, stiffened with
150x8x17750 S.S.B.F stiffeners and
9 mm thick diaphragm plate that
connects to the main diaphragm
Figure 3-7 Standard side plate drawn by AutoCAD
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Part 5
Part 5 consists of 7 plates, each of 9mm thick and 17900mm long welded together and
stiffened with 150x8x17750 S.S.B.F stiffeners and 9mm thick diaphragm plate which is
connected to the main diaphragm and forming the bottom deck section
Figure 3-8 Standard bottom flange plate drawn by AutoCAD
3.1.1.2 Main Diaphragm
Main diaphragm is made up of 6mm plate and stiffened with 2 types of stiffeners. In the
horizontal direction, 150x8 S.S.B.F stiffeners were provided and in the vertical direction, 75x6
flat stiffeners were used to achieve the required resistance. After all, the main diaphragms are
connected to the main deck diaphragm plates using 16mm black bolts.
Figure 3-9 Standard diaphragm drawn by AUTOCAD
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3.1.2 Cables
The cables are built up from parallel wires, each 5mm in diameter. Initially, the main span
cables were designed to have 10414 wires (82 strands each having 127 wires) and both side
spans 11176 wires (88 strands each having 127 wires). However, during the tendering the
number of strands in the main cables was reduced to 19, each having 448 wires and an
additional 4 strands each of 192 wires, in the backstays. The final arrangement of cables for
main span and backstays became as shown in Figure 3-10 (Brown & Parsons, 1975)
Figure 3-10 Arrangement of uncompacted cables (Brown & Parsons, 1975)
After compaction the diameter of the main cables and the backstays became approximately
511mm and 528mm, respectively.
3.1.3
Hangers
The suspended deck structure is connected to the cables with the inclined hangers. Each
hanger is built up from single spiral galvanized wire strand with the approximate diameter of
52mm. The type of connections both with cables and suspended deck structure is pinned in
longitudinal direction. (Brown & Parsons, 1975)
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3.1.4 Towers
The towers, being 165 m high, are made up from hollow steel
sections. Each tower has two columns which are connected by three
portal beams. Cross section dimensions and plate thicknesses
change over the height. Cross section dimension is 7000x5200 mm
at the bottom and 7000x3000mm at the top of the towers.
Table 3-1 tabulates the plate thicknesses for each plate (shown in
figure 3-12) along the height of the tower. To get an accurate
keypoint coordinates, the towers were drawn in three dimensions
using the AutoCAD software as shown in figure 3-11
Table 3-1 Tower structure plate thicknesses
Figure 3-12 Side plates labels
SectionHeight
(mm)
Plate A
(mm)
Plate B
(mm)
Plate C
(mm)
Plate E
(mm)
1 25000 5570X224677/
4423.5X22800X22
200X
100X15
2 19500 5570X224423.5/
4156X22800X22
200X
100X15
3 19500 5570X224156/
3888.5X22800X22
200X
100X15
4 19500 5570X223888.5/
3621.5X20800X22
200X
100X15
5 19500 5570X223621.5/
3354X20 800X22200X
100X15
6 19500 5570X223354/
3087X20800X22
200X
100X15
7 19500 5570X223087/
2819.5X20800X22
200X
100X15
8 18500 5570X222819.5/
2566X20800X22
200X
100X15
Figure 3-11 Towers
Drawn by AutoCAD
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3.1.5 Bridge Profile
Bridge profile was derived in accordance with the design drawings. Deck shape was drawn
in a way that it forms a part of a circle in the vertical plane with the radius of 17900 m. All the
cable coordinates was calculated assuming it is catenary element under dead-load conditions.
Figure 3-13 Suspended deck structure and cable profile
Figure 3-14 Arrangement of cables and hangers
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3.2 2-D FE Model
As already mentioned, because it is faster and easier to develop a 2-D FE model compared
to a sophisticated 3-D FE model, initially, two 2-D FE models of the bridge were produced using
ANSYS commercial software. First model restricts the vibration in the vertical plane, so thatallowed degrees of freedom are translations in longitudinal (UX) and vertical (UY) directions,
and rotation about Z-axis (ROTZ), whereas the second 2-D FE model allows vibration to take
place only in the horizontal plane, so that allowed degrees of freedom are translations in
longitudinal (UX) and lateral (UZ) directions, and rotation about Y-axis (ROTY). Otherwise,
modelling of the elements, which will be explained in detail in the sections below, is completely
similar for both 2-D FE models.
3.2.1
Deck Modelling
2-D suspended deck structure model was produced in ANSYS using BEAM 4, 3-D elastic
beam element. Keypoint coordinates were extracted from AutoCAD drawing, showing the
overall shape of the bridge, into EXCEL spreadsheet, to make them more accessible. Real
constants were assigned based on the provided axial area of steel, second moment of area for
vertical bending and second moment of area for lateral bending values which are 0.851 x 106
mm2, 1.238 x 1012 mm4 and 63.61 x 1012 mm4, respectively (The factors were attributed to
Dumanoglu (1985)). Materials were defined as linear isotropic with the Modulus of Elasticity of
205 x 103 MPa and Poisson’s ratio of 0.3. Based on the provided box deck structure’s mass,
10.84 tons/m, the equivalent density was calculated as 1.276 x 10-8 tons/mm3 (The factors were
attributed to Brown & Parsons (1975)). Finally, elements were meshed with the size of 500 mm
and two ends of the deck structure were restrained to move only in the longitudinal direction
(UX).
3.2.2
Cable Modelling
Cables were divided into different segments based on the hanger connection points and
taking the advantage of long geometry, were modeled as straight lines using LINK 10 element.
Key option 3 were activated as zero using KEYOPT command to define the cables as tension
only elements. Each cable element was labeled, using NUMSTR command, from 1001 to 1030
for west side and from 10001 to 10030 for east side, starting from the centerline of the main
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span towards the towers. Backstay cables were labeled as 1031 and 10031 for Ortakoy and
Beylerbeyi side, respectively. Modulus of Elasticity, Poisson’s ratio and density were defined as
193 x 103 MPa, 0.3 and 7.8 x 10-9 tons/mm3, respectively. Area of main span and backstay
cables were defined as 2.05x105 and 2.19x10
5 mm
2, respectively. Cables were meshed in a way
that each cable segment formed one element.
To calculate the initial strains for each cable element, horizontal component of tension
force was calculated using the H= WxL2/(8xd) formula as described below;
W=142.64 Total weight of the suspended structure calculated along the length (KN/m)
L=1074 Length of main span (m)
d=93 The sag in the cable (m)
H=221140 Total horizontal component of tension force for pairs of cables (KN)
Based on the calculated total horizontal component of tension force and angle of inclination
of each segment, initial strains were calculated as follows;
T=H/cosθ Tension in each segment=Horizontal component of tension/ cosine of the angle
σ=T/A Stress in the cross section= Tension force/ Area
ε=σ/E Strain=Stress/Modulus of Elasticity
Table A-1 provided in Appendix A shows the list of strain values calculated for each cable
elements. For simplification, seven different strain values, from 0.002802 to 0.002955, were
used to define the initial strains in main span cable elements as shown in the below table
Line Number Cable
Strain
Values
West Side East Side
From To From To
1001 1011 10001 10011 0.002802
1012 1015 10012 10015 0.002825
1016 1019 10016 10019 0.002847
1020 1023 10020 10023 0.002876
1024 1026 10024 10026 0.002906
1027 1028 10027 10028 0.0029321029 1030 10029 10030 0.002955
Table 3-2 Strain values for main cables
The initial strain values for backstays will be calculated later during the static analysis which
will be explained in later sections.
Boundary conditions were defined as fixed at tower saddles and pinned in anchorages.
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3.2.3 Hanger Modelling
Hangers were modeled as inclined lines using LINK 8 element. The keypoint coordinates
were already defined by cable and deck elements. Each hanger element was labeled, using
NUMSTR command, from 2001 to 2059 for west side and from 20001 to 20059 for east side,
starting from the centerline of the main span towards the towers. The Modulus of Elasticity,
Poisson’s ratio and density was assigned as 162x103 MPa, 0.3 and 7.8x10
-9 t/mm
3, respectively.
Area of each hanger was defined as 2.1x103 mm2. The vertical component of tension force in
each hanger was calculated by assuming that single hanger element carries it is own self-weight
and half weight of the deck structure between two adjacent hangers. Then the resultant
tension force was calculated based on the hanger inclination and initial strain values were
obtained as follows;
T=V/cos(θ) Resultant tension force=Vertical component of tension force/cosine of the angle
σ=T/A Stress in section= Tension force/Area
ε=σ/E Strain=Stress/Modulus of Elasticity
Table B-1 in Appendix B shows the strain values calculated for each hanger element. For
simplification, seven different strain values, from 0.001408 to 0.002810, were used to define
the initial strains in hanger elements as shown in the below table
Line Number Hanger
strain
Values
West Side East side
From To From To
2001 2029 20001 20029 0.001540
2030 2031 20030 20031 0.001473
2032 2035 20032 20035 0.001450
2036 2041 20036 20041 0.001430
2042 2047 20042 20047 0.001417
2048 2058 20048 2058 0.0014082059 - 20059 - 0.002810
Table 3-3 Strain values for hanger elements
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3.2.4 Tower modelling
Towers were modeled with the same principals used in deck modelling. 3-D elastic beam
element (BEAM 4) was used to model the elements. The real constants were assigned based on
the provided axial area of steel, second moment of area for vertical bending and second
moment of area for lateral bending values which are 1.36 x 106 mm2, 9 x 1012 mm4 and 271 x
1012
mm4, respectively (The factors were attributed to Dumanoglu (1985)). The Modulus of
elasticity, Poisson’s ratio and equivalent density were assigned as 205 x 103 MPa, 0.3 and 1.07 x
10-8
t/mm3, respectively. Elements were meshed with 500 mm size and bottom of the towers
were defined as a fixed support.
3.2.5
Complete 2-D FE Model of the Bridge
The complete 2-D FE model of the bridge was generated by combining the script files
written for each main part of the bridge and inserting them into ANSYS.
Figure 3-15 2-D FE model
3.2.6 2-D FE Model Analysis
To complete the model, backstay initial strain values are needed to be defined. They were
obtained by trial and error approach during the static analysis. The backstay initial strain values
were defined in a way that, deflection in the longitudinal direction at the top of the towers
becomes negligible. Several values were tried and the initial strains for the backstays were
adopted as 3.068x10-3
and 3.003x10-3
for Ortakoy and Beylerbeyi sides, respectively. Finally,
each 2-D FE model was analyzed both for static and modal analysis. To verify the accuracy of
the properties defined within the script files, the results obtained from the modal analysis were
compared with the experimental data available from the past studies. The validation of 2-D FE
model later will be covered in detail under “Model Validation” chapter, however, for
illustration purposes; figure 3-16 and 3-17 sequentially compare the vertical and lateral mode
shapes and frequencies of the first five modes with the ambient vibration test results carried
out for the Bosporus Bridge by Brownjohn et al.(1989).
X
Y
Z
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Static Analysis
Maximum Displacement-1394 mm
2-D FE model Restrained in the Vertical Plane
Vertical Modes
V Mode 1: Theoretical frequency: 0.124 Hz
Experimental frequency: 0.129 Hz
V Mode 2: Theoretical frequency: 0.162 Hz
Experimental frequency: 0.160 Hz
V Mode 3: Theoretical frequency: 0.202 Hz
Experimental frequency: 0.182 Hz
V Mode 4: Theoretical frequency: 0.228 Hz
Experimental frequency: 0.217 Hz
V Mode 5: Theoretical frequency: 0.281 Hz
Experimental frequency: 0.277 Hz
Figure 3-16 Analysis Results from 2-D FE model restrained on the vertical plane
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
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2-D FE model Restrained in the Horizontal Plane
Lateral Modes
L Mode 1: Theoretical frequency: 0.069 Hz
Experimental frequency: 0.070 Hz
L Mode 2: Theoretical frequency: 0.197 Hz
Experimental frequency: 0.209 Hz
L Mode 3: Theoretical frequency: 0.316 Hz
Experimental frequency: 0.284 Hz
L Mode 4: Theoretical frequency: 0.319 Hz
Experimental frequency: 0.294 Hz
L Mode 5: Theoretical frequency: 0.407 Hz
Experimental frequency: 0.365 Hz
Figure 3-17 Analysis Results from 2-D FE model restrained on the horizontal plane
Comparison of the results obtained from the 2-D FE models and the experimental data
assures that the input data used to produce the 2-D FE models represents the real behavior of
the bridge. Therefore, similar properties were used to develop a sophisticated 3-D FE model for
the Bosporus Bridge as described in detail in the below sections.
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
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3.3 3-D FE Model
Representation of cables and hangers by finite elements in 3-D FE model is same as the 2-D
FE model that is by LINK 10 and LINK 8 elements, respectively, except that the full six degrees
of freedom are allowed at each node. Therefore, in this chapter steps to model the hangers
and the cables will not be covered. The major difference in 3-D FE modelling is that the
suspended box deck structure and the towers are now modeled by SHELL 63 elements, having
six degrees of freedom at each node instead of the BEAM 4 elements used in 2-D FE models.
Since the computer process capacity is limited, modelling the towers and the suspended deck
structure with all the details is an impossible job. Therefore, necessary actions should be taken
to reduce the degrees of freedom for the towers and the suspended deck structure. To
overcome this issue, equivalent super elements are now introduced and will be discussed in
the below sections.
3.3.1
Equivalent Super Element for Suspended Deck Structure
To develop a sophisticated 3-D FE model with the less degree of freedom the equivalent
deck structure was designed. To achieve more realistic 3-D FE model of the bridge, the
equivalent deck element should be designed in a way that it represents the actual properties of
the original deck structure. To do so, the real behavior of the deck structure is needed.
Therefore, a box deck section of 17.9m long was modeled in ANSYS to understand the real
behavior of the original deck structure.
3.3.1.1 Modelling of the Original Box Deck Section
To develop a model in ANSYS keypoint coordinates were exported from already drawn
AutoCAD drawing into Excel spreadsheets to make them more accessible. The box deck section
was divided into 6 main areas and the keypoint locations were defined as shown in Figure 3-18
(Next page). Each area was modeled separately within the different script files and later
combined together to generate the whole model of the original deck section.
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X
Y
Z
Figure 3-18 Original deck section keypoint locations
Taking the advantage of the symmetric section, keypoints are labeled in a way that only
loop command (*DO) were used to define the areas for the model, which made the modelling
process faster and simpler. The model was developed in accordance with the design drawings
only with the minor differences. S.S.B.F type stiffeners were modeled with the exact
dimensions but as flat stiffeners. In the design drawing gap was provided between V-shaped
stiffeners and a diaphragm plate to avoid stress concentration in the weld connection which
was excluded in the model. Additionally, at the end of the deck section extra diaphragm was
provided to prevent local buckling and provide smooth stress distribution. Except that, the
model includes all the necessary details from the design drawings, including the horizontal and
the vertical stiffeners for the main diaphragms. All the areas were modeled with SHELL 63
element and meshed with the size of 500mm. The materials are defined as linear isotropic with
the Modulus of Elasticity of 205 GPA and Poisson’s ratio of 0.3. Density for all elements is
defined as zero to exclude the self-weight of the structure.
Figure 3-19 Model of the original deck Section
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X
Y
Z
Figure 3-20 Diaphragms
To calculate the properties of the box deck section all the nodes at 17900mm in Z direction
were fixed and force was applied at the other end of a section as shown in the figure 3-21.
Point loads were applied at two keypoints coinciding with hanger connection. Each point load
was assigned as 1000KN and the direction was depending on the type of the required
displacement.
Figure 3-21 Complete meshed section (mesh size 500mm)
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1
MN
MX X
Y
Z
-
-49.266-43.708
-38.15-32.591
-27.033-21.475
-15.917-10.359
-4.8.757818
AUG 24 2012
13:20:31
NODAL SOLUTION
STEP=1SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =49.292
SMN =-49.266
SMX =.757818
1
MN MX
X
Y
Z
-
-49.266-43.708
-38.15-32.591
-27.033-21.475
-15.917-10.359
-4.8.757818
AUG 24 2012
13:21:22
NODAL SOLUTION
STEP=1SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =49.292
SMN =-49.266
SMX =.757818
1
MN
MX X
Y
Z
-
-1.131
-.991962
-.852722
-.713483
-.574243
-.435004
-.295764
-.156525
-.017285
.121954
AUG 24 2012
14:02:17
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.329
SMN =-1.131
SMX =.121954
1
MN
MX
XY
Z
-
-1.131
-.991962
-.852722
-.713483
-.574243
-.435004
-.295764
-.156525
-.017285
.121954
AUG 24 2012
14:04:47
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.329
SMN =-1.131
SMX =.121954
Vertical Displacement
For the vertical displacement, the point load of 2000KN was applied in the negative
Y-direction and the maximum vertical displacement was obtained as 48.4mm from the analysis.
Figure 3-22 Deformed shape of the original deck section due vertical bending
Based on the obtained displacement the second moment of area for vertical bending (Ixx)
was calculated as follows;
Lateral Displacement
For lateral displacement 2000KN load was applied in the negative X-direction and themaximum lateral displacement was obtained as 1.1mm from the analysis.
Figure 3-23 Deformed shape of the original deck section due lateral bending
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1
MN
MX
X
Y
Z
-
-27.474-21.369
-15.263-9.158
-3.0533.053
9.15815.263
21.36927.474
AUG 24 2012
14:23:05
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =27.476
SMN =-27.474
SMX =27.474
1
MN
MXX
Y
Z
-
-27.474-21.369
-15.263-9.158
-3.0533.053
9.15815.263
21.36927.474
AUG 24 2012
14:39:44
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =27.476
SMN =-27.474
SMX =27.474
Based on the obtained displacement the second moment of area for lateral bending (Iyy)
was calculated as follows;
Torsional Deformation
For torsional deformation, coupled load each 1000KN was applied in the positive and
negative Y-direction and the deflection at the point load location was obtained as +/-23.1 mm.
Figure 3-24 Deformed shape of the original deck section due to torsion
Based on the obtained displacement, torsional constant (J) was calculated as follows;
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1
X
Y
Z
1
X
Y
Z
1
X
Y
Z
1
X
Y
Z
1
X
Y
Z
AUG 24 2012
15:35:12
A-E-L-K-N
U
ROT
F
3.3.1.2 Equivalent Plate Element
The equivalent plate element was modeled in ANSYS to fit with the original deck section
properties. The problems were encountered while matching the relatively similar properties in
original deck section for the equivalent plate, which will be covered in detail later in this
section. As a starting point in producing the equivalent plate, the width (b) of the plate was
taken as 28 m, the distance between the hanger points, and the length equal to the original
deck section length as 17.9 m. Areas were defined with 4 keypoints and SHELL 63 element was
assigned with the six degrees of freedom at each node. Plate was meshed with the size of
15 000mm which divided the plate area into four rectangular elements. The same principals,
used in the original deck section analysis, were applied for the equivalent plate. 1000KN point
loads were assigned at one end of a section, at two keypoints which are 28m apart and thick
diaphragm plate was attached to provide the uniform stress distribution. Nodes at the otherend of a section were fixed to behave as a cantilever. Initially, the material properties were
defined as linear isotropic with the Modulus of Elasticity of (Ex) 205GPa and Poisson’s ratio (ν)
of 0.3.
Figure 3-25 Equivalent Plate
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To make the model ready for the analysis, thickness of the plate had to be defined, which
was the challenging part. To get an idea about the approximate thicknesses that would satisfy
each property of the original deck section (Ixx, Iyy, J), the equivalent thicknesses for each
property were estimated by following observations.
For bending about X-axis, the corresponding Ixx value for the equivalent plate is “b*hb3/12”
where the hb is the equivalent thickness for the bending about X-axis and was calculated as 548
mm. The equivalent plate was analyzed with the same thickness for bending about X-axis and
the displacement was obtained as 48.5 mm, which is same with the displacement obtained for
the original deck section. Referring now to second moment of area for the lateral bending (Iyy)
the corresponding quantity for equivalent plate is “hL *b3/12” where the hL is the equivalent
thickness for the bending about Y-axis and was calculated as 9.3mm. The equivalent plate was
analyzed with the thickness of 9mm and the displacement was obtained as 2.7mm, which is
very close to 1.1mm obtained for the original box section. Looking now to the last property
obtained for the original deck section, which is the torsional constant (J), the standard
expression of the torsional constant for solid rectangular sections is equal to
”
(
) “. For a thin rectangular solid section the
expression simplifies to “b*ht3/3”, where the ht is the equivalent thickness for torsion and
calculated as 744mm. The equivalent plate was again analyzed with the thickness calculated
for torsion and the displacement was obtained as 11.7 mm, which was still close to the value
obtained for the original deck section. Above observation shows that the equivalent
thicknesses corresponding to each property vary significantly and it is impossible to satisfy all
three properties together. Later, to extend the observation further, the equivalent plate was
modeled with the linear orthotropic material properties and different cases were tried. A trial
and error approach was used where the Modulus of Elasticity in Y-axis (Ey), In-plane Shear
Modulus (Gxy) and the thickness of the section (h) were changed. To match the properties of
the equivalent plate with the original deck section, displacement was taken as the commonproperty. Several analyses were carried out and the displacement values obtained from the
analysis were compared with the original deck section displacements as shown in Table 3-4.
The second row from the bottom, highlighted, has the closest displacement values, however,
not all the properties are satisfied as accurate as required. Displacement due to torsion varies
by 29 mm (~230%) from the displacement obtained for the original box deck section.
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1
MN
MXX
Y
Z
-48.89-43.457
-38.025-32.593
-27.161-21.729
-16.297-10.864
-5.4320
AUG 24 2012
16:46:19
NODAL SOLUTION
STEP=1SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =48.896
SMN =-48.89
1
MN
MX XY
Z
-1.223
-1.087
-.951256
-.815362
-.679468
-.543575
-.407681
-.271787
-.135894
0
AUG 24 2012
16:47:26
NODAL SOLUTION
STEP=1SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.432
SMN =-1.223
1
MN
MX
X
Y
Z
-52.107-40.528
-28.948-17.369
-5.795.79
17.36928.948
40.52852.107
AUG 24 2012
16:48:43
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =52.115
SMN =-52.107
SMX =52.107
Figure 3-26 shows the deformed shape of equivalent plate under different loading
combinations.
EY
(GPa)
GXY
(GPa)
H
(mm)
Vertical
displacement (mm)
Lateral displacement
(mm)
Torsional
displacement (mm)
required obtained required obtained required obtained
205 1.3 530 48.4 57.49 1.1 1.87 23.1 59.09
205 1.4 530 48.4 57.48 1.1 1.73 23.1 59
205 1 530 48.4 57.524 1.1 2.42 23.1 59.337
205 1 600 48.4 40 1.1 2.21 23.1 45.73
205 2 600 48.4 39.96 1.1 1.14 23.1 45.02
205 2 550 48.4 51.53 1.1 1.25 23.1 54.12
205 2 560 48.4 48.9 1.1 1.22 23.1 52.11
50 2 560 48.4 195.65 1.1 1.29 23.1 127.76
Table 3-4 Equivalent plate displacements obtained for different arrangements
Figure 3-26 Deformed shape of equivalent plate under different loading conditions
Above discussed observations show that it is not possible to design an equivalent plate
element in ANSYS with all the required properties. To achieve more accurate properties an
equivalent box deck element is now introduced which will be covered in detail in the next
section.
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3.3.1.3 Equivalent Box Deck Element
To get more accurate equivalent super element for the suspended deck structure, the
rectangular equivalent box deck section was designed using ANSYS. The same principles that
were used to model the original box deck section were applied for the equivalent box section.
The width of the equivalent box section was defined as 28m and the length equal to the
original box deck section length, 17.9m. To understand the contribution of diaphragms to
section resistances (except from preventing local buckling), two different cases are modeled. In
the first case, diaphragms are placed 4475mm apart similar with the original deck section and
in the second case, spacing of diaphragms was reduced to half as 2237.5mm. The thickness for
the inner diaphragms was assigned as 20mm and the outer diaphragm, which was placed at the
end of a section to provide a smooth stress distribution, was modeled with the thicker
dimension. The areas were defined with 4 keypoints and SHELL 63 element was assigned withthe six degrees of freedom at each node. Same boundary conditions and loading scenarios that
were used for the equivalent plate were defined for the equivalent box deck section. Areas
were meshed with the size of 5567 mm and the material properties were defined as linear
isotropic with the Modulus of elasticity (Ex) of 205GPa and Poisson’s ratio of 0.3. Again, to
match the section properties, displacement was taken as a common property. Trial and error
approach was used to match the properties of the equivalent box section with the original box
deck section, where the thickness of the bottom and top plates (t1), the thickness of the side
plates (t2) and the height of the section (H) were changed. Table 3-4 shows the results obtained
for different arrangements where the last row, highlighted, had the closest displacement
values. Based on the analysis results, the equivalent box deck section was designed 2m deep
with 13 mm and 12 mm thick plates for the upper and bottom plates, and side plates,
respectively.
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1
MN
MXX
Y
Z
-55.506-49.339
-43.172-37.004
-30.837-24.67
-18.502-12.335
-6.1670
AUG 24 2012
17:53:17
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =55.515
SMN =-55.506
1
MN
MXX
Y
Z
-55.506-49.339
-43.172-37.004
-30.837-24.67
-18.502-12.335
-6.1670
AUG 24 2012
17:53:49
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =55.515
SMN =-55.506
1
MN
MXX
Y
Z
-1.054-.937085
-.819949-.702814
-.585678-.468542
-.351407-.234271
-.1171360
AUG 24 2012
17:55:46
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.872
SMN =-1.054
1
MN
MX XY
Z
-1.054-.937085
-.819949-.702814
-.585678-.468542
-.351407-.234271
-.1171360
AUG 24 2012
17:56:08
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.872
SMN =-1.054
top/bottom
plates
thickness
(mm) t1
Side
plates
thickness
(mm) t2
Section
Height
(mm) H
Vertical
Displacement
(mm)
Lateral
Displacement
(mm)
Torsional
Displacement (mm)
requir-
ed
obtain
-ed
requir-
ed
obtain-
ed
requir-
ed
Obtain-
ed
16 6 2250 48.3 48.53 1.1 1.04 23.3 20.06
16 6 2000 48.3 59.08 1.1 1.05 23.3 24.4
15 8 2000 48.3 56.46 1.1 1.08 23.3 23.5
14 10 2000 48.3 55.97 1.1 1.13 23.3 23.48
14 12 2000 48.3 53.77 1.1 1.118 23.3 22.62
16 12 2000 48.3 48.91 1.1 1 23.3 20.4
15 12 2000 48.3 51.2 1.1 1.05 23.3 21.44
Table 3-5 Equivalent box deck element displacements obtained for different arrangements
1st Case: 4475mm Diaphragm Spacing
Vertical Displacement
Vertical displacement was obtained as -51.2mm
Figure 3-27 Deformed shape of the equivalent box deck due to vertical bending ( 1st
Case)
Lateral displacement
Lateral displacement was obtained as -1.05 mm
Figure 3-28 Deformed shape of the equivalent box deck due to lateral bending ( 1st
Case)
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1
MN
MX
X
Y
Z
-21.443-16.678
-11.913-7.148
-2.3832.383
7.14811.913
16.67821.443
AUG 24 2012
17:58:08
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =21.519
SMN =-21.443SMX =21.443
1
MN
MX
X
Y
Z
-21.443-16.678
-11.913-7.148
-2.3832.383
7.14811.913
16.67821.443
AUG 24 2012
17:58:40
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =21.519
SMN =-21.443SMX =21.443
1
MN
MXX
Y
Z
-55.478-49.314
-43.15-36.985
-30.821-24.657
-18.493-12.328
-6.1640
AUG 24 2012
18:19:24
NODAL SOLUTION
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =55.487
SMN =-55.478
1
MN
MXX
Y
Z
-55.478-49.314
-43.15-36.985
-30.821-24.657
-18.493-12.328
-6.1640
AUG 24 2012
18:20:02
NODAL SOLUTION
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =55.487
SMN =-55.478
Torsion
The displacement due to torsion was obtained as +/- 21.4 mm
Figure 3-29 Deformed shape of the equivalent box deck due to torsion (1st
Case)
2nd Case: Diaphragms with 2237.5mm spacing
To understand the contribution of diaphragms to section resistances except from
preventing local buckling, the same equivalent box deck section was analyzed with the reduced
diaphragm spacing. The results showed that the diaphragms do not have any extra contribution
to section resistances. They are only designed to prevent local buckling and provide uniform
stress distribution.
Vertical Displacement
Vertical displacement was obtained as -51.3 mm
Figure 3-30 Deformed shape of the equivalent box deck due to vertical bending (2nd
Case)
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1
MN
MXX
Y
Z
-1.055-.938005
-.820755-.703504
-.586253-.469003
-.351752-.234501
-.1172510
AUG 24 2012
18:24:55
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.871
SMN =-1.055
1
MN
MX XY
Z
-1.055-.938005
-.820755-.703504
-.586253-.469003
-.351752-.234501
-.1172510
AUG 24 2012
18:25:13
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UX (AVG)
RSYS=0
DMX =1.871
SMN =-1.055
1
MN
MX
X
Y
Z
-21.406-16.649
-11.892-7.135
-2.3782.378
7.13511.892
16.64921.406
AUG 24 2012
18:26:32
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =21.482
SMN =-21.406
SMX =21.406
1
MN
MX
X
Y
Z
-21.406-16.649
-11.892-7.135
-2.3782.378
7.13511.892
16.64921.406
AUG 24 2012
18:26:45
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY (AVG)
RSYS=0
DMX =21.482
SMN =-21.406
SMX =21.406
Lateral Displacement
Lateral Displacement was obtained as -1.06 mm
Figure 3-31 Deformed shape of equivalent box deck due to lateral bending (2nd
Case)
Torsional Displacement
Displacement due to torsion was obtained as -/+21.4 mm
Figure 3-32 Deformed shape of the equivalent box deck due to torsion (2nd
Case)
3.3.1.4 Complete 3-D FE Model of the Suspended Deck Structure
The suspended deck structure was modeled using the equivalent box deck sections. The
keypoint coordinates were extracted from 3-D AutoCAD drawing into EXCEL, to make them
more accessible. The areas were defined with four or six keypoints and SHELL 63 element was
assigned with six degrees of freedom at each node and meshed with the size of 5567mm. The
similar material properties and dimensions which were used for equivalent box section were
assigned for the suspended deck structure elements. Diaphragms are located with variable
spacing up to 4475mm apart and modeled as weightless elements. The weight of the
suspended structure was assigned by defining the equivalent density for upper, bottom and
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side plates. The weight per meter of the original box deck section was provided as 10.84
tons/m. Based on this, the equivalent density was calculated as 1.226E-8 tons/mm3.
Boundary Conditions
The suspend deck structure connects to the towers by the rocker bearings which have agreat impact on bridge mode shapes and frequencies. Therefore, two A-frame rocker bearings
were modeled at each end of the suspended deck structure using BEAM 4 element. Geometric
quantities of a single frame element were calculated in accordance with the design drawings
and obtained as following
Area= 16x104 mm
2
Izz= 60.4 x 106 mm4
Iyy=42.35 x 106 mm
4
At the end of each BEAM 4 element COMBIN 7 revolution joint element was assigned to
work as pin connection in the longitudinal direction.
3.3.2
Equivalent Super Element for Towers
Another challenging task encountered during the 3-D FE modelling was producing an
accurate model for the towers. As already been mentioned, modelling the towers with all the
details, in terms of computer processing capacity, would be an impossible job to analyze.
Therefore, the equivalent structure for the towers had to be designed with the less degrees of
freedom. The same approach used in developing the equivalent super element for the
suspended deck structure is a convenient approach, however, it requires the in detail modelling
of at least one tower structure which is obviously too complicated and time consuming.
Therefore, a new approach was introduced at this point and explained step by step as
described below.
The towers were modeled as hollow sections. Keypoint coordinates were extracted from3-D AUTOCAD drawing into EXCEL spreadsheet to make them more accessible. Same cross-
sectional dimensions that were used in the design drawings were adopted for the tower model.
The areas were defined with four or six keypoints and SHELL 63 element was assigned with the
six degrees of freedom at each node. Materials were modeled as linear isotropic with the
Modulus of Elasticity of 205 GPa and Poisson’s ratio of 0.3. The diaphragms were modeled with
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the thickness of 60mm as weightless elements. The weight of the towers was assigned by
defining the equivalent density for main plates, which was calculated as 1x10-8 tons/mm3. In
the original design drawings opening was provided on the diaphragm plates for the elevator
shaft which was excluded in the model. Besides that, to reduce the degrees of freedom, towers
were modeled without any stiffeners. Instead, equivalent thickness was adopted for the main
plates by trial and error approach during the static analysis. Several different thicknesses were
tried in a way that, using the similar cable strain values that were defined during the 2-D FE
modelling, longitudinal deflection at the top of the towers becomes negligible. 30 mm
thickness end up with 0 mm deflection at the top of the towers in the longitudinal direction
hence was taken as the equivalent thickness for the tower model.
3.3.3 Complete 3-D FE Model of the Bridge
The complete 3-D FE model of the bridge was produced by combining the script files for the
different parts and importing them into ANSYS
Figure 3-33 Complete 3-D FE model of the Bridge
X
Y
Z
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3.3.4 3-D FE Model Analysis
To understand the factors affecting the bridge behavior, a sophisticated 3-D FE model was
analyzed under different cases both for static and modal analysis and discussed in detail in the
next sections. The cases that were analyzed are as follows;
Bridge model with;
Case 1: Different cable strains,
Case2: Different mass,
Case 3: Different boundary conditions.
3.3.4.1
Bridge Model with Different Cable StrainsAs already mentioned in the previous chapters, each cable and hanger elements have their own
strain values. However, to simplify the model 7 strain values were defined both for the cable
and the hanger elements within the script file. To verify this simplification, the model was
analyzed both with correct strains and with 7 different strain values. Besides that, to
understand the influence of the initial strains on the bridge behavior, strain values were
modified slightly that would provide less static deflection. To summarize, the bridge model was
analyzed both for static and modal analysis for three different cases which are as follows;
1.
Correct strain values for each cable and hanger elements
2.
Seven different values for both cable and hanger elements
3.
Modified strains for case 2 that would provide less static deflection
Tables from 3-6 to 3-8 shows the natural frequencies and the mode shapes for vertical, lateral
and torsional modes of each case. Looking at case 1 and case 2, results show that the
difference between the relevant mode frequencies is negligible for all modes, thus using 7
different strain values for the cable and the hanger elements is a reliable simplification.
Referring now to case 3, the least static deflection that was possible to achieve is 1129 mm
which is very close to the previously achieved static deflection (1395mm- Case 2). Therefore,
there was no significant change in the natural frequencies which ensures that the initial strain
values already calculated and defined within the script files are close to reality.
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X
Y
Z
X
Y
Z
.
.
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
.
.
X
Y
Z
.
.
X
Y
Z
-
.
.
X
Y
Z
-
.
.
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
1st Case (Correct strains) 2
nd Case (7 different strains) 3
rd Case (modified strains for case 2)
Static Analysis
Max. Static deflection: 1390 mm Max. Static Deflection: 1395 mm Max. Static Deflection: 1129 mmModal Analysis- Vertical Modes
V Mode 1: 0.125 Hz V Mode 1: 0.125 Hz V Mode 1: 0.125 Hz
V Mode 2: 0.163 Hz V Mode 2: 0.163 Hz V Mode 2: 0.163 Hz
V Mode 3: 0.227 Hz V Mode 3: 0.227Hz V Mode 3: 0.227 Hz
V Mode 4: 0.280 Hz V Mode 4:0.281 Hz V Mode 4: 0.281 Hz
V Mode 5: 0.366 Hz V Mode 5: 0.366 Hz V Mode 5: 0.366 Hz
Table 3-6 Comparison of vertical mode shapes and frequencies between case 1,2 and 3
X
Y
Z
-
.
.
X
Y
Z
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XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
1st
Case
(Correct strains for both cables and hangers)
2nd
Case (7 different strains for both cables
and hangers)
3rd
Case
(modified strains for case 2)
Modal Analysis- Lateral Modes
L Mode 1: 0.070 Hz L Mode 1: 0.070 Hz L Mode 1: 0.070 Hz
L Mode 2: 0.203 Hz L Mode 2: 0.203 Hz L Mode 2: 0.203 Hz
L Mode 3: 0.300 Hz L Mode 3: 0.300 Hz L Mode 3: 0.300 Hz
L Mode 4: 0.306 Hz L Mode 4: 0.306 Hz L Mode 4: 0.307 Hz
L Mode 5: 0.398 Hz L Mode 5: 0.398 Hz L Mode 5: 0.399 Hz
Table 3-7 Comparison of lateral mode shapes and frequencies between case 1,2 and 3
XY
Z
XY
Z
XY
Z
XY
Z
XY
Z
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X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
1st
Case
(Correct strains for both cables and hangers)
2nd
Case (7 different strains for both cables
and hangers)
3rd
Case
(modified strains for case 2)
Modal Analysis- Torsional Modes
T Mode 1: 0.326 Hz T Mode 1: 0.327 Hz T Mode 1: 0.327 Hz
T Mode 2: 0.484 Hz T Mode 2: 0.484 Hz T Mode 2: 0.484 Hz
T Mode 3: 0.632 Hz T Mode 3: 0.631 Hz T Mode 3: 0.630 Hz
T Mode 4: 0.842 Hz T Mode 4: 0.842 Hz T Mode 4: 0.842 Hz
T Mode 5: 1.042 Hz T Mode 5:1.043 Hz T Mode 5: 1.043 Hz
Table 3-8 Comparison of Torsional mode shapes and frequencies between case 1,2 and 3
X
Y
Z
X
Y
Z
X
Y
Z
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3.3.4.2 Bridge Model with Additional Mass
Suspension bridges are the important structures with long term service life. To maintain the
integrity of the bridge, careful inspection and proper maintenance is compulsory. Sometimes,
during the maintenance works such as a road restoration, an extra weight might be added to
the dead load of the structure. To understand the importance of the change in bridge mass, in
terms of the bridge dynamic behavior, the model was analyzed in two cases. In the first case,
bridge mass was defined in accordance with the design stage whereas in the second case,
1tons/m extra distributed mass was added to the dead weight of the structure, by defining a
new equivalent density for the deck structure which was calculated as 1.3946x10-8
ton/mm3.
Each case was analyzed both for static and modal analysis and the results are tabulated in
tables 3-9 to 3-11.
The analyses results show that the vertical modes are the least affected mode due to the
extra mass added to the dead weight of the structure. No significant variations were observed
except slight change in the mode frequencies by up to 3% difference. Looking now to lateral
modes, slight variations (up to 5%) were observed here as well. However, the major difference
is, the model with the extra mass experiences the 3rd
lateral mode (frequency: 0.309 Hz) with
both lateral and torsional deformation as shown in table 3-10.
Similar to vertical and lateral modes, the torsional mode also displayed a slight change in
mode frequencies. However, the main difference is that single noded asymmetric mode shape
appears at two slightly different mode frequencies (0.467 Hz and 0.472 Hz) in case 2 (model
with the different mass) whereas only one single noded asymmetric torsional mode shape was
observed in case 1. Overall, comparison of the modal analysis for case 1 and 2 shows that the
bridge behavior is sensitive to changes in bridge mass and any modifications that could change
the dead weight of the structure should be taken into account.
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X
Y
Z
-
1st Case 2nd Case (Different Mass)
Static Analysis
Maximum Displacement: 1395 mm Maximum Displacement: 1821 mm
Modal Analysis – Vertical Mode
V Mode 1: 0.125 Hz V Mode 1: 0.124 Hz
V Mode 2: 0.163 Hz V Mode 2: 0.158 Hz
V Mode 3: 0.227 Hz V Mode 3: 0.221 Hz
V Mode 4: 0.281 Hz V Mode 4: 0.277 Hz
V Mode 5: 0.366 Hz V Mode 5: 0.359 Hz
Table 3-9 Comparison of vertical mode shapes and frequencies between case 1 and 2
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-X
Y
Z
-
X
Y
Z
-X
Y
Z
-
X
Y
Z
-X
Y
Z
-X
Y
Z
-
X
Y
Z
-X
Y
Z
-
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1st Case 2nd Case (Different Mass)
Modal Analysis – Lateral Mode
L Mode 1: 0.070 Hz L Mode 1: 0.068 Hz
L Mode 2: 0.203 Hz L Mode 2: 0.194 Hz
L Mode 3: 0.300 Hz (Fig. above is in horizontal
below is in vertical plane)
L Mode 3: 0.309 Hz (Fig. above is in horizontal
below is in vertical plane)
L Mode 4: 0.306 Hz L Mode 4: 0.318 Hz
L Mode 5: 0.398 Hz L Mode 5: 0.389 Hz
Table 3-10 Comparison of lateral mode shapes and frequencies between case 1 and 2
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
XY
Z
-
X
Y
Z
-
X
Y
Z
-
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X
Y
Z
-
X
Y
Z
-
T Mode 2: 0.467 Hz
1st Case 2nd Case (Different Mass)
Modal Analysis – Torsional Mode
T Mode 1: 0.327 Hz T Mode 1: 0.321 Hz
T Mode 2: 0.484 Hz T Mode 3: 0.472 Hz
T Mode 3: 0.631 Hz T Mode 4: 0.615 Hz
T Mode 4: 0.842 Hz T Mode 5: 0.815 Hz
T Mode 5: 1.043 Hz T Mode 6: 1.015Hz
Table 3-11 Comparison of torsional mode shapes and frequencies between case 1 and 2
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
X
Y
Z
-
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3.3.4.3 Bridge Model with Different Boundary Conditions
A-frame rocker bearings were designed to allow the movement only in the longitudinal
direction but resist the lateral and the vertical translations. However, throughout the bridge
service life there was a significant increase in traffic load in Istanbul, for which the bridge was
not designed. This issue raises a concern that the rocker bearings at the end of the suspended
deck structure might be jammed in due to overloading. To understand the impact of the
different boundary conditions in terms of the bridge behavior, different cases were analyzed
which are as follows;
Case 1: All the rockers free to move in longitudinal direction
Case 2: Both rockers are restricted to move in the longitudinal direction at ORTAKOY side
Case 3: One of the rockers is restricted to move in the longitudinal direction at ORTAKOY side
Case 4: All the rockers are restricted to move in the longitudinal direction
Tables 3-12 and 3-13 tabulate the results obtained both from static and modal analysis for
cases one to four, respectively. From the modal analysis first four mode shapes and the
corresponding frequencies are provided for each vertical, lateral and torsional mode.
Looking at the static analysis, results show that there are no significant changes in static
displacements and all the obtained values are very close. However, the modal analyses provide
some interesting results which worth’s to mention. For vertical mode, the results show that
depending on the boundary conditions, the first mode shape might be symmetric or
antisymmetric. In case one where the rockers were allowed to move freely in the longitudinal
direction, the first vertical mode shape is obtained as symmetric with 0.125 Hz frequency,
however, analyses results for cases 2,3 or 4, where the rockers either on one side or on both
sides restricted to move in the longitudinal direction, provide the first vertical mode
antisymmetric. This shows that throughout the bridge service life, the first vertical mode shape
might change from symmetric to antisymmetric shape depending on the boundary conditions.
Referring now to the lateral mode, analyses results show that the boundary conditions mostly
influence the second mode where around 10% increase was observed in mode frequencies.
Apart from this, there is no remarkable change in lateral modes. The torsional mode is the least
influenced mode due to the change in boundary conditions. Some of the mode frequencies
were changed slightly by up to 1%. .
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X
Y
Z
-
: :
=
=
=
=
X
Y
Z
-
: :
=
=
=
=
X
Y
Z
-
: :
=
=
=
=
X
Y
Z
-
: :
=
=
=
=
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
= =
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
X
Y
Z
-
: :
=
=
=.
=.
1st Case 2
nd Case 3
rd Case 4
th Case
Static Analysis
Max. Displacement: 1395 mm Max. Displacement: 1395 mm Max. Displacement: 1395 mm Max. Displacement: 1366 mm
Modal Analysis- Vertical Mode
V Mode 1: 0.125 Hz V Mode 1: 0.162 Hz V Mode 1: 0.162 Hz V Mode 1: 0.164 Hz
V Mode 2: 0.163 Hz V Mode 2: 0.193 Hz V Mode 2: 0.182 Hz V Mode 2: 0.206 Hz
V Mode 3: 0.227 Hz V Mode 3: 0.228 Hz V Mode 3: 0.228 Hz V Mode 3: 0.228 Hz
V Mode 4: 0.281 Hz V Mode 4: 0.281 Hz V Mode 4: 0.281 Hz V Mode 4: 0.281 Hz
Table 3-12 Comparison of vertical mode shapes and frequencies between case 1, 2, 3 and 4
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1st
case 2nd
Case 3rd
Case 4th
Case
Modal Analysis- Lateral Mode
L Mode 1: 0.070 Hz L Mode 1: 0.070 Hz L Mode 1: 0.071 Hz L Mode 1: 0.086 Hz
L Mode 2: 0.203 Hz L Mode 2: 0.215 Hz L Mode 2: 0.208 Hz L Mode 2: 0.225 Hz
L Mode 3: 0.300 Hz L Mode 3: 0.301 Hz L Mode 3: 0.300 Hz L Mode 3: 0.301 Hz
Modal Analysis- Torsional Mode
T Mode 1: 0.327 Hz T Mode 1: 0.327 Hz T Mode 1: 0.327 Hz T Mode 1: 0.327 Hz
T Mode 2: 0.484 Hz T Mode 2: 0.484 Hz T Mode 2: 0.484 Hz T Mode 2: 0.484 Hz
T Mode 3: 0.631 Hz T Mode 3: 0.632 Hz T Mode 3: 0.632 Hz T Mode 3: 0.632 Hz
Table 3-13 Comparison of lateral and torsional mode shapes and frequencies between case 1, 2, 3 and 4
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4 Model Validation
So far, the procedures followed to develop 2-D and 3-D FE models of the Bosporus
Suspension Bridge and different type of analysis that were performed to understand the bridge
dynamic behavior, were covered in the previous chapters. As a last step, to validate the
accuracy of the foregoing FE models, the analysis results will be compared with experimental
data available from the past studies.
Due to the importance of the Bosporus Bridge, remarkable theoretical works and full scale
dynamic tests were carried out by Brownjohn et al. (1989) and Tezcan et al. (1975) to estimate
the dynamic characteristics of the Bosporus Bridge. The first dynamic test was carried out by
Tezcan et al. (1975) using ambient vibration measurements in 1973, just before the bridge was
opened to traffic. Due to the limitations on the equipment used, only four vertical and one
torsional modes were identified between 0.2-05 Hz. However, later in 1987 Brownjohn et al.
(1989) carried an ambient vibration survey in the Bosporus Bridge where the vertical, lateral
and torsional modes between 0-1.1 Hz were identified. Using these past available results,
comparison were carried out for each vertical and lateral modes available from 2-D and 3-D FE
models and torsional mode available from 3-D FE model and reported in the below sections.
4.1
Comparison of Experimental and Analytical Results for
Vertical Modes
Analysis results obtained both from 2-D and 3-D FE models have shown that the predicted
vertical modes are quite close to the experimental results. Table 4-1 tabulates results obtained
both from current analytical and past experimental studies. The last two columns compare
sequentially 2-D and 3-D FE model analysis results with experimental studies.
Experimental studies carried out by Brownjohn et al. (1989) showed that the bridge
experiences its first vertical asymmetrical mode at two slightly different frequencies, one above
and one below the first symmetric mode. A Possible explanation given by the authors
(Brownjohn, et al., 1989) was that bridge had a dual character between two bearing conditions
and it was presumable changing depending on the traffic intensity. Accordingly, in the previous
sections, the analyses were carried out for similar cases where it was shown that the first
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vertical asymmetric mode might appear before or after the first vertical symmetric mode
depending on the boundary conditions.
Vertical Mode
Theoretical
frequency (Hz)Symmetry Nodes Antinodes
Experimental frequency
(Hz)
Percent
difference (%)
2-D FE
Model
3-D FE
Model
Brownjohn
et al.
Tezcan et
al.2-D 3-D
0.124 0.125 a 1 2 0.129 - 4 3
0.162 0.163 s 2 3 0.16 - 1 2
0.202 0.182 a 1 3 0.18 - 11 1
0.228 0.227 s 2 3 0.217 0.233 5 5
0.281 0.281 a 3 4 0.277 0.282 1 1
0.371 0.366 s 4 5 0.362 0.357 2 10.453 0.444 a 5 6 0.446 0.44 2 1
0.561 0.543 s 6 7 0.544 - 3 0
0.665 0.636 a 7 8 0.637 - 4 0
0.775 0.728 s 8 9 0.739 - 5 1
0.9 0.833 a 9 10 0.83 - 8 0
0.9 0.833 a 9 10 0.852 - 5 2
1.032 0.938 s 10 11 0.959 - 7 2
Table 4-1 Comparison of Experimental and Analytical results for vertical modes
4.2 Comparison of Experimental and Analytical Results for
Lateral Modes
Comparison between experimental and analytical results is not simple for lateral modes.
Among the 9 modes identified during the test, only 4 modes (1st, 2nd, 5th and 7th modes) have
appreciable movement of deck structure. However, in other modes, the towers moved in
lateral direction together with main cables thereby the deck moved comparatively little
(Brownjohn, et al., 1989). Similar behavior was observed during the analysis for both 2-D and
3-D FE models. Table 4-2 (next page) shows theoretical frequencies, obtained from 2-D and 3-D
FE models, and the experimental frequencies. Looking at last two columns, percent differences
show that the theoretical frequencies are still close to the experimental frequencies.
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Lateral Mode
Theoretical
frequency (Hz)Symmetry Nodes Antinodes
Experimental
frequency-
Brownjohn et al.
Percent
difference (%)
2-D FE
Model
3-D FE
Model2-D 3-D
0.069 0.070 s 0 1 0.07 1 0
0.197 0.203 a 1 2 0.209 6 3
0.3163 0.2999 s 0 1 0.284 10 5
0.3193 0.306 a 1 2 0.294 8 4
0.4073 0.398 s 2 3 0.365 10 8
- 0.456 - - - 0.382 - 16
0.524 0.495 s 2 3 0.44 16 11
0.533 0.552 s 2 3 0.525 2 5
0.746 0.737 a 3 4 0.762 2 3
Table 4-2 Comparison of Experimental and Analytical results for lateral modes
4.3 Comparison of Experimental and Analytical Results for
Torsional Modes
Table 4-3 shows the frequency comparison for torsional modes. There is a good agreement
between experimental and theoretical frequencies except that experimental results show two
single noded asymmetric torsional modes however, only one single noded asymmetric
torsional mode was obtained from 3-D FE model. Similar behavior was observed during the
modal analysis where 3-D FE model was analyzed with extra mass. It worth’s to mention that
the bridge FE models were developed based on the design stage and there is no solid
information available whether the mass changed throughout its service life.
Torsional Mode
Theoretical
frequency (Hz)Symmetry Nodes Antinodes
Experimental
frequency (Hz)
Percent
difference (%)
3-D FE Model
Brownjohn
et al.
Tezcan
et al. 3-D
0.327 S 0 1 0.324 0.331 1
0.484 A 1 2 0.474 - 2
0.484 A 1 2 0.492 - 2
0.631 S 2 3 0.649 - 3
0.842 A 3 4 0.877 - 4
Table 4-3 Comparison of Experimental and Analytical results for torsional modes
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5 Conclusion
Comparison of analytical and experimental results assures that the procedures followed
during the modelling process, particularly replacing the box deck section with the equivalent
box element, are the reliable approaches and provide the accurate results. Thus it can be
concluded that the models discussed so far represent the real behavior of the Bosporus Bridge.
However, the calibration of the model still might need some extra work to achieve better
results. As already discussed previously, the bridge model is sensitive to several factors such as
strains in the cables, mass of the structure, boundary conditions and etc. By tuning these
factors, the accuracy of the model could be slightly improved. Therefore, the following
recommendations are proposed for further studies.
First of all, its worth’s to mention that the boundary conditions for the FE models were
defined in accordance with the bridge initial condition which possibly changed throughout the
bridge service life. Thus defining the related properties after detailed inspection to represent
the current condition would enhance the accuracy of the model
Besides, the mass of the structure is another important factor that could be improved by
calculating the bridge mass in detail and defining the model in accordance.
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6 References
Apaydin, N. M., 2010. Earthquake Performance Assessment and Retrofit Investigation of Two
Suspenion Bridges in Istanbul. Soil Dynamics and Earthquake Engineering, Volume 30, pp. 702-
710.
Brownjohn, J., Blakeborough, A., Dumanoglu, A. A. & Severn, R. T., 1989. Ambient Vibration
Survey of the Bosporus Suspension Bridge. Earthquake Engineering and Structural Dynamics,
Volume 18, pp. 263-83.
Brown, W. C. & Parsons, M. F., 1975. Bosporus Bridge, Part I, History and Design. s.l., Institution
of Civil Engineers.
Chan, T. H., Guo, L. & Li, Z. X., 2003. Finite Element Modeling for Fatigue Stress Analysis of
Large Suspension Bridges. Journal of Sound and Vibration, Volume 261, pp. 443-464.
Dumanoglu, A. A., 1985. Asynchronous Seismic Analysis of Modern Suspension Bridges- Part I:Free Vibration, s.l.: University of Bristol. Department of Civil Engineering.
General Directorate of Highways, Turkey, 1973. Record book: Istanbul Bogazici Koprusu
(Bosporus Suspension Bridge), Istanbul: KGM matbaasi.
Merce, R. N. et al., 2007. Finite Element Model Updating of a Suspension Bridge Using ANSYS
Software. Miami, Florida, U.S.A, Inverse Problems, Design and Optimization Symposium.
Pericles, G., 1987. Darius in Scythia: The Formation of Herodotus' Sources and the Nature of
Darius' Campaign. American Journal of Ancient History, 12(ISSN 0362-8914), pp. 97-147.
Tezcan, S., Ipek, M. & Petrovski, J., 1975. Forced Vibration Survey of Istanbul Bogazici
Suspension Bridge, Skopje, Yugoslavia: Institute of Earthquake Engineering and Engineering
Seismology.
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7 Appendix A
3-D FE Model 2-D FE ModelCable
strain
Values
Line Number (from CL towards the towers)
West Side East side West East
1 31 101 131 1001 10001 2.795E-03
2 32 102 132 1002 10002 2.795E-03
3 33 103 133 1003 10003 2.796E-03
4 34 104 134 1004 10004 2.797E-03
5 35 105 135 1005 10005 2.798E-03
6 36 106 136 1006 10006 2.800E-03
7 37 107 137 1007 10007 2.802E-03
8 38 108 138 1008 10008 2.805E-03
9 39 109 139 1009 10009 2.807E-03
10 40 110 140 1010 10010 2.811E-03
11 41 111 141 1011 10011 2.814E-03
12 42 112 142 1012 10012 2.818E-03
13 43 113 143 1013 10013 2.822E-03
14 44 114 144 1014 10014 2.827E-03
15 45 115 145 1015 10015 2.832E-03
16 46 116 146 1016 10016 2.838E-03
17 47 117 147 1017 10017 2.844E-03
18 48 118 148 1018 10018 2.850E-03
19 49 119 149 1019 10019 2.857E-03
20 50 120 150 1020 10020 2.864E-03
21 51 121 151 1021 10021 2.871E-03
22 52 122 152 1022 10022 2.879E-03
23 53 123 153 1023 10023 2.888E-03
24 54 124 154 1024 10024 2.897E-03
25 55 125 155 1025 10025 2.906E-03
26 56 126 156 1026 10026 2.916E-03
27 57 127 157 1027 10027 2.927E-03
28 58 128 158 1028 10028 2.938E-03
29 59 129 159 1029 10029 2.949E-03
30 60 130 160 1030 10030 2.961E-03Table A-1 Main span cable strain values
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8 Appendix B
3-D FE Model 2-D FE ModelCable strain
ValuesLine Number (from CL towards the towers)
West Side East side West East
201 301 401 501 2001 20001 1.543E-03
202 302 402 502 2002 20002 1.543E-03
203 303 403 503 2003 20003 1.543E-03
204 304 404 504 2004 20004 1.543E-03
205 305 405 505 2005 20005 1.543E-03
206 306 406 506 2006 20006 1.543E-03
207 307 407 507 2007 20007 1.543E-03
208 308 408 508 2008 20008 1.543E-03
209 309 409 509 2009 20009 1.543E-03
210 310 410 510 2010 20010 1.543E-03
211 311 411 511 2011 20011 1.543E-03
212 312 412 512 2012 20012 1.543E-03
213 313 413 513 2013 20013 1.543E-03
214 314 414 514 2014 20014 1.543E-03
215 315 415 515 2015 20015 1.543E-03
216 316 416 516 2016 20016 1.543E-03
217 317 417 517 2017 20017 1.543E-03
218 318 418 518 2018 20018 1.543E-03
219 319 419 519 2019 20019 1.543E-03
220 320 420 520 2020 200201.543E-03
221 321 421 521 2021 20021 1.543E-03
222 322 422 522 2022 20022 1.543E-03
223 323 423 523 2023 20023 1.543E-03
224 324 424 524 2024 20024 1.543E-03
225 325 425 525 2025 20025 1.543E-03
226 326 426 526 2026 20026 1.543E-03
227 327 427 527 2027 20027 1.520E-03
228 328 428 528 2028 20028 1.492E-03
229 329 429 529 2029 20029 1.493E-03
230 330 430 530 2030 20030 1.472E-03231 331 431 531 2031 20031 1.473E-03
232 332 432 532 2032 20032 1.457E-03
233 333 433 533 2033 20033 1.456E-03
234 334 434 534 2034 20034 1.445E-03
235 335 435 535 2035 20035 1.446E-03
236 336 436 536 2036 20036 1.436E-03
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237 337 437 537 2037 20037 1.437E-03
238 338 438 538 2038 20038 1.430E-03
239 339 439 539 2039 20039 1.430E-03
240 340 440 540 2040 20040 1.424E-03
241 341 441 541 2041 20041 1.424E-03
242 342 442 542 2042 20042 1.420E-03
243 343 443 543 2043 20043 1.420E-03
244 344 444 544 2044 20044 1.416E-03
245 345 445 545 2045 20045 1.417E-03
246 346 446 546 2046 20046 1.414E-03
247 347 447 547 2047 20047 1.414E-03
248 348 448 548 2048 20048 1.411E-03
249 349 449 549 2049 20049 1.411E-03
250 350 450 550 2050 20050 1.410E-03
251 351 451 551 2051 20051 1.410E-03
252 352 452 552 2052 20052 1.408E-03
253 353 453 553 2053 20053 1.408E-03
254 354 454 554 2054 20054 1.407E-03
255 355 455 555 2055 20055 1.407E-03
256 356 456 556 2056 20056 1.406E-03
257 357 457 557 2057 20057 1.406E-03
258 358 458 558 2058 20058 1.405E-03
259 359 459 559 2059 20059 2.809E-03
Table B-1 Hanger elements strain values
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9 Appendix C
Main
Span
Cables 3.32
Cable wrapping 0.08
Cable bands 0.11
Handropes 0.02
Protective treatment 0.01
Hangers and sockets 0.16
Suspended box steel work 8.02
Roadway surfacing 2.25
Footway surfacing 0.08
Parapets and crash
barriers 0.2
Services 0.22
Protective treatment 0.07
Total design dead load 14.54
Table C-1 Dead load of the main span measured along the length (tons/m) (Brown & Parsons,
1975)