Avdelningen f¨or Konstruktionsteknik Lunds Tekniska H¨ogskola
Box 118 22 100 LUND Sverige
Division of Structural Engineering Faculty of Engineering, Lund University Box 118
S-22 100 LUND Sweden
Optimizing pontoon spacing for floating bridge over Bjørnafjorden.
Josefin Havnevik Giske June 5, 2019
Preface
This master thesis marks the finalization of the five-year-long study for Master of Science in civil engineering. The thesis has been written during the spring semester of 2019, and the extent of the thesis is 30 credits (ECTS). I have written the thesis in cooperation with the Norwegian Public Roads Administration, NPRA, and the Division of Structural Engineering at the Faculty of Engineering, LTH, at Lund University.
First of all, I would like to thank my supervisor at the NPRA, Senior Engineer Xu Xiang, for his guidance and patience throughout the whole project. I have also received valuable inputs from my supervisor at LTH, Senior lecturer Oskar Larsson Ivanov. I am very grateful for his constructive comments and thoughts about my work, and for assisting me to define the scope of the project.
Johs Holt, a consulting engineering company specialized in bridge construction, which will also be my future workplace, gave me the opportunity to write the thesis at their office in Oslo.
Sharing the workday and getting useful input from the competent colleagues at Johs Holt has made the work with this thesis rewarding and enjoyable. Special thanks to Henric Thompsson at Johs Holt, for always taking time to answer my questions and sharing his knowledge with me.
Finally, I would like to thank Mathias Egeland Eidem, project manager at the Fjord Crossing Project at the NPRA, for giving me the opportunity to take part in the challenging engineering project of making the coastal highway route E39 ferry free.
Oslo, May 2019
Josefin Havnevik Giske
Abstract
The Norwegian government aims to develop a ferry-free, and improved, E39 between Kris- tiansand in the south and Trondheim in the north. Today this distance of approximately 1100 km takes 21 hours to travel, including seven ferry-connections. A ferry-free E39 would decrease the travel time to about half.
This thesis is focusing on a topic concerning the longest crossing of the project, Bjørnafjorden, which is 5000 meters wide. Due to the large width and great depth of the fjord, up to 650 me- ters, a conventional bridge is not feasible. Instead, several concepts of floating bridges are being investigated. This thesis analyses one of the investigated concepts, the end-anchored bridge.
The construction material in the proposed end-anchored bridge will mainly consist of steel. For a structure of this magnitude, the amount of steel will be the driving cost of the project. It is therefore of great importance to decrease the amount of steel in the structure.
The goal of this thesis is to find the optimal pontoon spacing with the purpose to reduce the total amount of steel in the structure. In order to optimize the pontoon spacing, an analysis of the static load actions has been performed. The cable-stayed bridge and the high part of the floating bridge are omitted in this study. The focus of this study is upon the low floating bridge, which constitutes the major part of the structure.
The static analysis performed in this thesis consists of two parts. In the first part, the re- sponses in the structure due to static loading are determined with simplified 2D calculations.
A number of span lengths between the pontoons are evaluated, and the required stiffness of the girder and the pontoons is determined for the investigated span lengths. From the 2D calculations the optimal span length, which gives the lowest amount of steel in the structure, can be determined. The minimum allowable amount of steel is governed by recommendations in the Eurocode. The results from the 2D calculations give an optimal span length of 110 meters.
The second part of the static analysis consists of a 3D analysis in the software RM bridge. In this part, the results from the 2D calculations are verified. In order to get a reliable verification, three different span lengths are evaluated, 100, 110 and 120 meters.
The 3D analysis proved to be consistent with the results from the simplified 2D calculations.
The results from this study indicate that the proposed optimal span length of the end-anchored bridge is 110 meters. However, also a dynamic analysis of the proposed span length must be performed in order to verify the results of this study.
An analysis of the dynamic loads acting on the structure is not included in this study. To be able to determine the optimal span by only performing a static analysis, an approximate share of the responses due to static loading has been assumed. Based on the results from a previous study of the end-anchored bridge, the share of responses due to static loading is assumed to be 60% of the yield capacity.
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 Purpose . . . 2
1.3 Scope . . . 2
1.4 Structure of the report . . . 3
2 Description of the bridge 5 2.1 Key figures . . . 5
2.2 Behaviour of the floating part of the structure . . . 6
2.3 Boundary conditions . . . 7
2.4 Pontoons . . . 7
2.5 Girder . . . 7
2.6 Columns . . . 8
3 Method 10 3.1 Static analysis . . . 10
3.2 Simplifications . . . 11
3.3 Model and software . . . 12
3.4 Requirements . . . 12
3.4.1 Motion limitations . . . 13
3.4.2 Buoyancy of pontoon . . . 13
3.4.3 Capacity check of girder . . . 13
3.5 Contribution of static loads . . . 15
3.6 Parameters . . . 18
3.7 Load combinations . . . 19
4 Design loads 20 4.1 Permanent load . . . 20
4.2 Traffic load . . . 21
4.2.1 Traffic - ULS . . . 21
4.2.2 Traffic - SLS . . . 22
4.3 Static wind . . . 23
4.3.1 Static current . . . 24
5 Hydrodynamic stiffnesses 26 6 2D calculations 27 6.1 Permanent loads . . . 27
6.1.1 Moments . . . 27
6.1.2 Reaction force . . . 27
6.2 Traffic loads . . . 28
6.2.1 Moments . . . 28
6.2.1.1 Moment in span . . . 28
6.2.1.2 Moment over support . . . 29
6.2.1.3 Torsional moment . . . 31
6.2.2 Reaction force . . . 32
6.2.3 Rotation . . . 33
6.2.4 Vertical deflection . . . 34
6.3 Static wind load . . . 34
6.3.1 Moment . . . 34
6.3.2 Normal force . . . 36
6.3.3 Rotation . . . 36
6.4 Static current . . . 37
6.5 Iterative calculation process . . . 38
6.6 Summary of responses . . . 38
7 3D analysis 40 7.1 Geometry . . . 40
7.2 Boundary conditions . . . 41
7.3 Properties . . . 42
7.4 Loads . . . 42
7.4.1 Permanent loads . . . 43
7.4.2 Traffic loads . . . 43
7.4.2.1 ULS . . . 43
7.4.2.2 SLS . . . 44
7.4.3 Static wind . . . 45
7.4.4 Static current . . . 45
8 Summary of results 47 8.1 Summary of responses . . . 47
8.1.1 Span length 100 meter . . . 47
8.1.2 Span length 110 meter . . . 48
8.1.3 Span length 120 meter . . . 53
8.2 Summary of Von Mises stresses . . . 54
9 Analysis and discussion 55 9.1 Comparison of results . . . 55
9.1.1 Vertical load actions . . . 55
9.1.2 Horizontal load actions . . . 55
9.1.3 Comparison of Von Mises stresses . . . 56
9.2 Control of requirements . . . 56
9.2.1 Capacity of the girder . . . 57
9.2.2 Buoyancy . . . 57
9.2.3 Motion limitations . . . 58
9.3 Load actions . . . 58
9.3.1 Permanent load . . . 58
9.3.2 Traffic . . . 58
9.3.3 Static wind . . . 59
9.3.4 Static current . . . 59
9.4 Boundary conditions . . . 59
9.4.1 Roller support . . . 60
9.4.2 Pinned connection . . . 62
9.5 Global buckling . . . 62
9.6 Dynamic analysis . . . 64
9.6.1 Assumption of contribution from static loading . . . 64
9.6.2 Results from previous studies . . . 65
9.6.3 Modal response . . . 68
9.7 Fatigue . . . 69
9.8 Other assumptions . . . 69
10 Conclusions 71
11 Further work 72
Nomenclature and coordinate system
Coordinate system
The coordinate system used in this report is defined as the coordinate axis in the software RM bridge. The local x-axis is located in the bridge axis, the y-axis in the vertical direction and the local z-axis in the transverse direction of the bridge. In figure 1 and 2 below, the global coordinate axis is shown.
Figure 1: Side view of the model in RM bridge.
Figure 2: The model of the bridge seen from above.
In figure 3 below, the terminology of the end-anchored bridge is illustrated.
Figure 3: Terminology of the bridge. (Aas Jakobsen, COWI, Global Maritime, Johs Holt, 2016)
The degrees of freedom of the pontoons are illustrated in figure 4 below. These denominations are used in the report.
Figure 4: Illustration of different motions of pontoon (Aas-Jakobsen, COWI, Global Maritime, Johs Holt, 2016)
Abbreviations
ALS Accidental Limit State LM1 Load model 1
LMV Internordic traffic model
NPRA Norwegian Public Road Administration NTP National Transport Plan
SLS Serviceability Limit State UDL Uniformly distributed loads ULS Ultimate Limit State
Symbols
A Area of the steel plates in the girder AD Drag area in sway
Am Enclosed area of the cross-section Awp Waterplane area of the pontoon cD Dragfactor
C44 Roll stiffness
g Gravitational constant e Eccentricity
E Young’s modulus f Rise of arch Fc Current force
Fm Mean 10-min wind force
ga+g Self-weight of asphalt and girder gc+p Self-weight of column and pontoon H Height above water surface
Hcr Critical horizontal force
Iy Second moment of inertia about strong axis Iz Second moment of inertia about weak axis I0 Factor depending on arch type
K Stiffness of the girder
L Span length
l Chord length of arch
m Weight of the pontoon in the base case mcolumn Mass of column
mgirder Mass of girder mpontoon Mass of pontoon
Mx Torsional moment in the girder
My Bending moment in girder about strong axis
My,2 Bending moment in girder about strong axis considering second order effects Mz Bending moment in girder about weak axis
Nx Normal force in the arch
qU DL Uniformly distributed traffic load, LM 1 qLM V Uniformly distributed traffic load, LMV QT L Tandem load
Qy Reaction force R Buoyancy of pontoon
S Stiffness of pontoon in heave
t Minimum thickness of steel plates in point of interest uy Deflection of the girder
v0 Current velocity
vm1 1-year static wind velocity vm100 100-year static wind velocity Vy Displacement of pontoon Wx Section modulus of torsion Wy Section modulus y-axis Wz Section modulus z-axis
y Distance between z-axis and investigated stress point z Distance between y-axis and investigated stress point α Profile factor for wind loading
β Angle at the abutment of the arch θx Rotation of girder about x-axis λ Scale factor for sizing of pontoon µ Stiffness ratio
ρa Density of air ρw Density of water σvm Von Mises stress σx Normal stress τx Shear stress
1 Introduction
1.1 Background
Along the Norwegian west coast, the highway E39 runs from Kristiansand in the south to Trond- heim in the north. Today this distance, of approximately 1100 km takes 21 hours to travel, including seven ferry connections (Statens vegvesen, 2019).
The Norwegian government has a long-term goal to develop a ferry-free and improved E39. This would reduce travel time to about half. To achieve this goal, seven fjords have to be crossed (ibid.). The fjords are from 1650 to 5000 meters wide and up to 1250 meters deep. To be able to cross these fjords, new innovative technology must be combined with existing knowledge and expertise (Statens vegvesen, 2012b).
The plans for a ferry-free connection along the coastal highway started in 2010, when the Ministry of Transport and Communications commissioned the Norwegian Public Road Admin- istration, NPRA, to investigate the feasibility of crossing the fjords along E39 (Statens vegvesen, 2012a). In 2017 the project ”Ferry Free E39” was confirmed by the Norwegian government in the National Transport Plan, NTP, for 2018-2029 (Statens vegvesen, 2019).
This thesis is focusing on the longest crossing of the project, Bjørnafjorden, which is 5000 me- ters wide (Statens vegvesen, 2012b). Bjørnafjorden is situated south of Bergen, and its depth is over 600 meters where the fjord is to be crossed (ibid.). Due to the large width and depth of the fjord, conventional bridge design is not feasible for the crossing. Instead, several concepts of floating bridges are being investigated. For instance, a suspension bridge with floating towers, a submerged floating tube bridge and the end-anchored bridge investigated in this thesis (Statens vegvesen, 2018).
Two floating bridges have already been built in Norway, Nordhordlands bridge and Bergsøysund bridge. The longest of these bridges is the Nordhordlands bridge, with a total length of 1610 meters (Brun, 2003). Design of floating bridges is hence proved to be feasible. It is, however, the combination of the great width and depth of Bjørnafjorden, which make this project particularly challenging. An illustration of the proposed end-anchored bridge over Bjørnafjorden and the two existing floating bridges in Norway is shown in figure 5 below.
Figure 5: Comparison of the bridge over Bjørnafjorden with the existing floating bridges in Norway (Norconsult, 2017a).
The investigation of the remaining concepts for the crossing of Bjørnafjorden has been initi- ated. Two concepts of floating bridges are investigated by independent consultant groups. The remaining alternatives are the curved end-anchored bridge and a straight side-anchored bridge with mooring lines. Both types are combined with a high cable-stayed bridge in the south, where the navigation channel is situated (Statens vegvesen, 2018).
This thesis is focusing on structural optimization of the end-anchored bridge, with the purpose to reduce the amount of steel in the structure.
1.2 Purpose
The purpose of this thesis is to contribute to the investigation of the crossing over Bjørnafjorden by optimizing the structure with regard to the amount of steel. The goal is to determine the span length between the pontoons which will give the lowest amount of steel for the complete structure. The estimated cost of the floating bridge over Bjørnafjorden is directly connected to the amount of steel in the bridge. Therefore it is of great importance to decrease the steel weight of the structure (Norconsult, 2017b).
In a previous study of the end-anchored bridge, an optimization study of the span length be- tween the pontoons was performed. The optimization study, conducted by Norconsult, is a rough calculation where two span lengths have been evaluated (ibid.). The intention of this thesis is to perform a more detailed analysis of the optimal span length between the pontoons.
The elements that are made out of steel are the girder, the pontoons and the columns. In the previous analysis of the end-anchored bridge, conducted by Norconsult, the steel weight of the girders constitute 67% of the total steel weight in the structure (ibid.). Hence, a reduction of steel in the girders will have a large effect on the total cost of the bridge. The spacing between the pontoons is optimized with focus on decreasing the steel weight of the girder. However, since the purpose is to reduce the steel weight in the structure as a whole, a reduction of the steel content in the pontoons is also of interest in this study. The steel amount in the columns is minor, about 5% of the structure proposed by Norconsult (ibid.).
The steel weight of the girder has a lower limit due to recommendations in the Eurocode con- cerning permissible thicknesses of the steel plates. This is governing for the choice of optimal span length.
Previous studies of the crossing of Bjørnafjorden are used as references in this study. These are described in section 3.
The knowledge obtained from this study will be useful for all of the investigated concepts. The relevant reports are provided by the NPRA.
1.3 Scope
In order to be able to perform this study within the limited time period available for a master thesis, several simplifications have been made. The general simplifications are stated below, however, simplifications are also stated in corresponding sections throughout the report. In section 9 the limitations of the simplifications made in this study are discussed.
The end-anchored bridge proposed by Norconsult in the third phase of the project is used as a reference object, and it is referred to as the base case (Norconsult, 2017a).
The cable-stayed bridge and the high part of the floating bridge are omitted in the analysis.
Only the low floating part of the bridge is analysed, illustrated in figure 3. The choice to focus on the low floating part of the bridge implies that the boundary conditions at the connection between the high part and the low part of the bridge must be investigated. This is done in section 9.4.
The draft of the pontoons is assumed fixed to 5 meters. Draft is defined as the distance from the lower side of the pontoon to the water surface. This implies that the size of the pontoon is not determined by the motion limitations of the girder. Instead, the buoyancy requirements are governing for the pontoon size.
Focus is upon the behaviour in the major part of the bridge girder in the low floating part of the bridge. The responses at the connection between the high part and the low floating part, and at the north abutment are omitted in this study. These responses must be considered in further analyses of the structure.
Only the static load actions assumed to induce the largest responses in the low floating part of the bridge are considered in this study. Static load actions neglected in this study are discussed in section 3.2. The included load actions are:
• Permanent load
• Traffic load
• Static wind
• Static current
Dynamic load actions, wind and wave loading, are not included in this study. An approximate share of the responses due to static loading is determined based on the results from a previous analysis of the bridge. This is discussed more in detail in section 3.5.
1.4 Structure of the report
• Section 2 - Description of the bridge
In this section, the general geometry of the end-anchored bridge is described, in addition to the structural elements of the low floating part of the bridge.
• Section 3 - Method
The method used to perform the analysis in this study is described.
• Section 4 - Design loads
The load actions considered in this analysis are described and calculated.
• Section 5 - Hydrodynamic stiffnesses
The required hydrodynamic stiffnesses of the pontoons are calculated.
• Section 6 - 2D calculations
In this section the simplified 2D calculations of the structure are performed.
• Section 7 - 3D analysis
The input and modelling for the 3D analysis of the structure in RM bridge are described.
• Section 8 - Verification
A summary of the results from the 2D and 3D calculations is presented.
• Section 9 - Analysis and discussion
The results summarized in section 8 are discussed together with the assumptions and limitations of this study. The requirements are controlled with the responses from the 3D analysis.
• Section 10 - Conclusions
The conclusions of this study is presented.
• Section 11 - Further work
Suggestions for further work on the topic are presented.
2 Description of the bridge
The proposed end-anchored bridge over Bjørnafjorden consists of three parts, see illustration in figure 3. From the south abutment, the bridge starts with a cable-stayed bridge followed by the high part of the floating bridge connecting the high bridge to the low floating bridge. The low floating bridge runs all the way to the north abutment. The bridge is curved with the apex of the arch turned towards the east. Both abutments are located at dry land, and the bridge is not anchored to the seabed at any location. The girder is resting on columns which are connected to steel pontoons (Norconsult, 2017b). The proposed end-anchored bridge is shown in figure 6 below.
Figure 6: The proposed end-anchored bridge over Bjørnafjorden (Norconsult, 2017b).
2.1 Key figures
Key figures for the end-anchored bridge are presented in table 1 below (Norconsult, 2017a):
Bridge length between abutments 5439 m
Total arch length 5525 m
Arch length cable-stayed bridge 830 m Arch length high part of floating bridge 1000 m Arch length low part of floating bridge 3695 m
Table 1: Key figures
2.2 Behaviour of the floating part of the structure
The girder in the low floating part of the bridge will behave as a continuous beam. To achieve the desired behaviour, sections of the girder are welded together to one continuous girder. The sections are about 25 meters and the welding will take place on an assembly barge at the site.
By using a barge, the girder is subjected to minimum external forces during installation. The longer section is then supported by temporary mooring while a pontoon is installed by welding.
The pontoons can be transported on own keel with the columns pre-installed. The operation is repeated until the complete low floating part is in place (Norconsult, 2017b).
The arch structure is resisting the horizontal loads acting on the bridge and transferring the forces, through the girder to the abutments situated at both ends of the bridge. Since only the low part of the floating bridge is analysed, the boundary conditions at the connection between the low part and the high part of the floating bridge must be investigated. A study of the influence of the boundary conditions at this connection is performed in section 9.4.
For vertical loading, the bridge will behave as a flexible beam on spring supports. The pontoons will act as linear springs with stiffness in heave and roll, see figure 4 for directions (ibid.). The ratio between the stiffness of the girder and the pontoon expresses how the vertical loads are carried (Multiconsult, 2017). Zero indicates stiff supports and 100 soft supports (ibid.).
The stiffness ratio,µis calculated as:
µ= K S Where:
K Stiffness of girder
S Stiffness of pontoon in heave
The pontoon stiffness in heave,S, is calculated as described in section 5.
The stiffness of the girder, K, is calculated as (Norconsult, 2017b):
K = 48EI L3 Where:
E Young’s modulus
I Second moment of inertia for weak axis L Span length
For the case with only permanent loading, i.e. during the construction phase, the floating part will be in equilibrium with the buoyancy of the pontoons. It has been decided to construct the bridge, so that the moments in the floating part, equals the response from a bridge constructed on fixed supports. This is accomplished by ballasting the pontoons until the desired behaviour is achieved. The ballasting of the pontoons will be permanent, and it will be executed so that roll around the bridge axis is prevented during the construction phase (Multiconsult, 2017).
Both horizontal and vertical loads will induce rotation around the bridge girder. As a simpli- fication, the columns and the pontoons are assumed to have high stiffness. Hence, the whole structure is assumed to rotate as a rigid body around the centre of gravity of the bridge girder.
2.3 Boundary conditions
The bridge is monolithically connected to the abutments at both ends (Norconsult, 2017b). The boundary conditions at the connection between the low part and the high part of the floating bridge are investigated. The influence of the choice of boundary conditions at this location is discussed in section 9.4.
2.4 Pontoons
The pontoons carrying the structure are constructed as steel plate ships. In Norway, this is a well-known technique from ship and offshore practice. In the base case, four different types of pontoons are used in the bridge. As a simplification, pontoon type 1 is used in all axes in this study. This is the pontoon used in the major part of the floating bridge in the base case (ibid.).
The relevant properties of pontoon type 1 are listed in table 2 below. (Norconsult, 2017a) Parameter Value unit
Length 58 m
Width 10 m
Height 9 m
Draft 5 m
Total volume 5027 m3 Displacement 2793 m3 Area waterplane 558.6 m2 Drag coeff surge 0.8 - Drag coeff sway 1.0 -
Table 2: Cross-sectional properties of the base case pontoon type 1.
The pontoon is illustrated in figure 7.
Figure 7: Pontoon type 1 seen from above.
2.5 Girder
The single steel box girder in the base case is used in this analysis (Norconsult, 2017b). Two types of cross-sections are used in the floating part of the bridge. In this study the cross-section in the major part of the low floating bridge is used, denominated cross-section type 1. This
cross-section constitutes over 95% of the length of the low floating bridge. The other cross- section, type 2, is used in the high part and close to the north abutment (Norconsult, 2017b).
The outer dimensions of the chosen girder are fixed, and the girder has a constant stiffness. The height of the girder is 3.5 meters and the structural width is 27.5 meters (Norconsult, 2017a).
The geometry of the steel girder is illustrated in figure 8 below (Norconsult, 2017b).
Figure 8: Cross-section steel girder (Norconsult, 2017b).
Recommendations in the Eurocode, concerning the permissible thicknesses for the steel plates, control the lowest possible steel weight of the girder. Table 3 below summarizes the minimum allowable thicknesses of the steel plates (ibid.).
Location of plate Minimum thickness
Deck 14 mm
Vertical web 35 mm
Bottom 12 mm
Table 3: Minimum thicknesses of steel plates in the girder.
Trapezoidal stiffeners are placed with a certain distance in the girder. These are taken into account by calculating an equivalent thickness. The equivalent thickness of the stiffeners is added to the thickness of the steel plates in the calculations of the cross-sectional properties of the girder.
2.6 Columns
The steel girder is connected to single columns placed at the centre of each pontoon. The size of the columns varies along the bridge, in total four different types are used. In the major part of the low floating bridge, column type 1 is used (ibid.). For simplicity, this column is used in all axes in this study.
Properties of column type 1 are summarized in table 4 below (ibid.).
Parameter Type 1 unit
Length 8 m
Width 6 m
Height 7.5 m
Radius 3 m
Weight 65 tonnes
Table 4: Cross-sectional properties of column type 1.
3 Method
In this section the methods used for performing the analysis in this study are described. Gen- eral simplifications are stated, and the requirements of the structure are presented (Statens vegvesen, 2017a).
The proposed bridge will be situated in a fjord where both wave and wind loading are of sig- nificant magnitude. For a floating bridge of this size, the dynamic load actions will induce large responses in the structure (Norconsult, 2017a). As mentioned in section 1.3, this study is based on a static analysis and the responses from the dynamic load actions are not evaluated.
To be able to perform this study based on a static analysis, the share of the stresses due to static loading must be assumed. In the study ”K7 end-anchored floating bridge” conducted by Norconsult, the stresses in the girder due to static and dynamic loading are presented. The results from the study by Norconsult are studied, and an approximate share of stresses due to static loading is assumed based on these results. However, this assumption is only a rough approximation and must be verified in further studies.
In addition to the analysis of the end-anchored bridge by Norconsult, two more studies of floating bridges over Bjørnafjorden are used as a reference in this study. In the report ”Brief description of static load carrying system for bridge over Bjørnafjorden” by Multiconsult, a side-anchored bridge with mooring lines is analysed. This is a straight bridge resisting the horizontal loads through mooring lines to the seabed. Besides this difference, the bridge has many similarities to the end-anchored bridge studied in this thesis. The end- and side-anchored bridge will resist vertical loading in a similar way.
The third study used as a reference is a study of the proposed end-anchored bridge. The analysis is conducted by Aas-Jakobsen et al. and the title of the report is ”Curved bridge - Navigation channel in south” (Aas Jakobsen, COWI, Global Maritime, Johs Holt, 2016). The general geometry of the bridge is consistent with the analysed bridge in this thesis, but there are differences in the choice of structural elements. Instead of a single steel box girder, a Vireendeel beam with two parallel box girders is used. The pontoons in the study by Aas-Jakobsen et al.
are larger and made out of concrete (ibid.). Due to the significant differences in geometry and material of the structural elements, comparison with this study must be done with caution.
The optimization study conducted by Norconsult, mentioned in section 1.2, concluded that a span length of around 100 meters is reasonable (Norconsult, 2017b). Based on the results from the study by Norconsult, the span lengths investigated in this study are chosen to be within 80 to 130 meters.
3.1 Static analysis
The static analysis performed in this study is divided into two parts. The first part consists of simplified 2D calculations which result in a proposed optimal span length. In the second part, the responses from the 2D calculations are verified in a 3D-model using the software RM bridge.
The static load actions included in this study are:
• Permanent loads
• Traffic loads
• Static wind
• Static current
As mentioned in section 1.3, focus is upon the responses in the major part of the low floating bridge. Large responses will be induced at the north abutment and at the connection between the high part and the low part of the floating bridge. These responses are neglected in this study but must be considered in further analyses of the bridge.
The first part of the static analysis is an iterative calculation process, where the requirements of the structure described in section 3.4 are controlled. The requirements are checked for span lengths in the chosen range, 80-130 meters, with an interval of 5 meters.
The stiffness of the pontoons and the girder is varied until the requirements of the structure are fulfilled for the investigated span lengths. The variation of the stiffnesses is described more in detail in section 3.6.
The first part of the analysis results in a proposed optimal span length, with regard to reducing the total steel weight of the structure. Since only the low floating part of the bridge is con- sidered, the total weight of the whole structure is not calculated in this study. However, the proposed span length determined in this study is assumed to give the lowest steel weight for the complete structure. This is considered to be a reasonable assumption since the low floating part of the bridge constitutes about 70% of the total length of the arch.
In the second part of the analysis, the proposed span length from the 2D calculations is verified by a 3D analysis in RM bridge. In addition to verifying the response of the proposed span length, the responses for two additional span lengths are evaluated. This is done to get a more reliable verification of the results.
3.2 Simplifications
In this section the static load cases neglected in this study are discussed. Other simplifications are also stated in this section.
In addition to omitting the cable-stayed bridge and the high part, the following simplifications are made:
• The bridge is modelled with constant height and cross-section along the whole length.
• The pontoon shape is fixed.
• The height of the pontoon is fixed to 9 meters.
• Only one type of pontoon, column, and girder is used in the analysis.
The static load cases neglected in the analysis are:
• Temperature
• Marine growth
• Tide
• Rise of water level
The influence of the omitted load actions is considered as minor for the investigated responses in this study. The reasons for neglecting the load actions are discussed below.
Differences in temperature will force the girder to elongate and shorten. This longitudinal movement is caused by the change in average temperature in the girder. The movement of the bridge will cause large responses at the abutment where the bridge is restricted from moving (Multiconsult, 2017). As mentioned earlier, the large responses at the north abutment are not of interest in this study, and the temperature load is therefore neglected.
The weight of the marine growth is 60 tonnes for pontoon type 1 in the base case (Norconsult, 2017b). This is about 3% of the self-weight of the structure resisted by each pontoon. The ma- rine growth will increase with increasing pontoon size, however, a larger pontoon also signifies that the mass of the girder is increased. The contribution from marine growth is considered as minor, and it is therefore neglected in this study.
In the mentioned study conducted by Multiconsult, the influence of tidal loads on the structure has been analysed. The results from the analysis show that the tide will induce large responses in the high bridge and close to the north abutment (Multiconsult, 2017). However, along most of the low floating part of the bridge, the water level variation will not cause large responses.
The structure will instead follow the water level, and only minor responses will be induced in the girder (ibid.). The tide is therefore neglected in this study since the focus is to evaluate the responses in the low floating part of the bridge.
Due to climate changes, the mean water level will rise. According to the Design basis, the rise is estimated to 0.8 meters, including the effect of land elevation (Statens vegvesen, 2017a).
It is reasonable to believe that the rise of the water level will induce responses similar to the responses caused by tide, described above. This should implicate that the overall responses, due to the rise of water level, in the low floating bridge will be minor. The water level variation is therefore neglected in this study.
3.3 Model and software
The simplified 2D calculations are performed in excel and the program CALFEM, a toolbox in Matlab for finite element calculations.
To analyse and verify the static responses of the structural system, 3D models of the bridge are created in the software RM bridge. RM bridge is a design and analysis program used in bridge design.
3.4 Requirements
The Eurocode does not cover all the required rules and regulations for a floating bridge of this magnitude. Therefore, the NPRA has issued a Design basis with specific design rules for the proposed end-anchored bridge over Bjørnafjorden (ibid.). The bridge is designed according to the Design basis, together with the Norwegian handbook N400 and the Eurocode. The Nor- wegian handbook N400 consists of supplements to the Eurocode for construction of bridges in Norway (Statens vegvesen, 2015). Design for static wind and static current is performed according to the Design basis MetOcean, issued by the NPRA (Statens vegvesen, 2017b).
The following requirements are checked in this study:
• Motions in the bridge girder due to static loading
• Buoyancy of the pontoons
• Capacity of the steel girder 3.4.1 Motion limitations
The motion limitations considered in this analysis are presented in table 5 below, together with the corresponding load case (Statens vegvesen, 2017a).
Motion Load Maximum allowed motion
Vertical deformation 0.7xtraffic uy ≤1.5m from traffic loads
Rotation about bridge axis 0.7xtraffic θx ≤1.0deg eccentric traffic loading
Rotation about bridge axis 1-year static wind θx ≤0.5deg from static wind load
Table 5: Motion limitations for end-anchored bridge
The rotation and deflection of the structure are determined in the serviceability state, SLS. In SLS the Internordic load model, LMV, is used in accordance with the Design basis (ibid.).
3.4.2 Buoyancy of pontoon
The buoyancy of the pontoons is checked for the investigated span lengths. The buoyancy de- pends on the displacement of water, i.e. it will vary with the waterplane area of the pontoon since the draft is fixed.
The buoyancy, R, is calculated with the following equation (Nationalencyklopedin, 2019):
R=ρw·V ·g Where:
g Gravitational constant V Displacement of the pontoon ρw Density of water
3.4.3 Capacity check of girder
The capacity of the steel girder is checked in accordance with clause 6.2.1 (5) in NS-EN 1993- 1-1:2005+NA:2008 (Standard Norge, 2008b). The Von Mises stress is calculated in relevant stress points and compared to the capacity of the steel plates. Based on a previous study, an approximate share of stresses due to static loading is assumed. This is discussed in section 3.5.
The stresses are calculated in eight points in the girder, in both span and support. In figure 9 below, the stress points are illustrated.
Figure 9: Relevant stress points. Figure adapted from (Norconsult, 2017b).
Buckling of stiffeners and shear lag effects are not accounted for in this simplified analysis.
In compliance with the Design basis, issued by the NPRA, the structural steel plates are of capacity S 420 N (Statens vegvesen, 2017a). The material factor of the structural steel is 1.1 according to the Design Basis (ibid.). This gives a yield capacity of 382 MPa. The equations for calculating the Von Mises stress are retrieved from the study by Norconsult (Norconsult, 2017a).
The Von Mises stress, σvm, is calculated with the following equation:
σvm=p
σx2+ 3τx2 Where:
σx Normal stress τx Shear stress
The normal stress, σx, is calculated with the equation:
σx=±Nx
Ax ±My
Wy ± Mz
Wz Where:
Ax Cross-sectional area My Moment about strong axis Mz Moment about weak axis Nx Normal force in arch Wy Section modulus y-axis Wz Section modulus z-axis
The shear stress, τx, is determined as:
τx = Mx Wx Where:
Mx Torsional moment
Wx Section modulus of torsion
The Von Mises stress is calculated with the responses in the local coordinate system.
The section modulus is calculated for the relevant stress points. The calculations are performed in the same excel sheet as the second moment of inertia, described in section 4.1. The excel sheet used for calculating the properties is adapted from an excel sheet provided by Henric Thompsson at Johs Holt.
The section modulus, Wy and Wz, are calculated with the following formulas (Isaksson et al., 2017):
Wy = Iy
z Wz= Iz y Where:
Iy Second moment of inertia about strong axis Iz Second moment of inertia about weak axis y Distance from the z-axis to the point of interest z Distance from the y-axis to the point of interest
For a hollow thin-walled cross-section, the section modulus for torsion,Wx, is determined using Bredts first formula (Bruhns, 2003):
Wx = 2·Am·t Where:
Am Enclosed area of the hollow thin-walled cross-section
t Minimum thickness of steel plates, meeting in the point of interest
3.5 Contribution of static loads
In order to optimize the span length by performing a static analysis of the structure, the share of stresses due to static loading must be determined. The maximum allowable Von Mises stress due to static loading is determined based on the results from the analysis by Norconsult. This is a rough approximation which must be verified by a dynamic analysis.
The share of stresses due to static loading can be determined based on different ratios. Either, as the ratio of the stresses due to static loading and the total loading, or as the share of the yield capacity of the steel. The normal stresses from the analysis by Norconsult are studied.
It is observed that the ratio of stresses, due to static loading and total loading, is differing considerably for the investigated stress points (Norconsult, 2017a). The ratio, in span and over support, is shown in table 6 below.
Stress Ratio Ratio point span [%] support [%]
1 79 46
2 21 38
3 21 24
4 83 80
5 83 79
6 74 55
7 17 17
8 21 37
Table 6: The ratio of stresses due to static loading and total loading.
The ratio of the stresses due to static loading and total loading varies considerably for the investigated stress points, from 17% to 83%. It is difficult to base an assumption on this ratio when it is varying without a clear pattern between the stress points. Instead, the maximum allowable stress, due to static loading, is determined as the share of the yield capacity.
A rough calculation, using the responses from the analysis by Norconsult, shows that the con- tribution from the shear stress to the Von Mises stress is minor, about 5% (Norconsult, 2017a).
For simplicity, the shear stress is neglected when determining the approximate share of stresses due to static loading. This simplification implies that the Von Mises stress equals the normal stress. In the tables below, the stresses are presented as Von Mises stresses, i.e. as the absolute value of the normal stress.
The results from the analysis by Norconsult show that the stress points exposed to the governing stress points are 4, 5 and 6, see figure 10 below for location of the stress points.
Figure 10: Relevant stress points. Figure adapted from (Norconsult, 2017b).
In table 7 below the total stress in the governing stress points is presented, together with the utilization rate of the yield capacity.
Total stress Utilization rate of Stress point σvm [MPa] yield capacity [%]
4 300 78
5 290 76
6 310 81
Table 7: Governing total stress in the girder.
When neglecting the contribution of the shear stress, the governing stresses in the girder utilize about 80% of the capacity of the steel. Since the shear stress is estimated to only contribute with about 5%, the utilization rate could be increased with some percentages to better utilize the capacity of the steel in the girder.
The governing stresses, due to static loading, from the analysis by Norconsult are presented in table 8 below, together with the utilization rate of the yield capacity.
Stresses from static Utilization rate of Stress point loads σvm [MPa] yield capacity [%]
4 240 63
5 230 60
6 210 55
Table 8: Share of governing stresses due to static loading.
As shown in table 8 above, the static loads in the analysis by Norconsult induce stresses of about 55-60% of the yield capacity in the governing stress points.
The share of stresses due to static loading, shown in table 7 and 8, can be used as a reference point for this study. However, it is important to keep in mind that the load cases considered in this study and the study by Norconsult are different. The static loads included in the analysis by Norconsult are listed below (Norconsult, 2017a):
• Self-weight
• Pretension
• Temperature
• Marine growth
• Traffic
• Tide
• Current
In the analysis by Norconsult, temperature loads, marine growth and tide are included. These load actions are not considered in this study. However, the static wind load included in this study is not a part of the static analysis conducted by Norconsult (ibid.). Pretension is not of relevance for the responses in the low floating part of the bridge.
As mentioned in section 1.2, the optimization study conducted by Norconsult is only a rough calculation. The analysis performed in this thesis is more detailed even though fewer load cases are included. The load cases with minor relevance for the investigated responses are neglected in this analysis.
In section 3.2 the relevance of the additional load cases in the static analysis by Norconsult are discussed. The temperature load is not of significance for the responses investigated in this analysis and the contribution from the marine growth is considered as minor. Tidal loads will induce large responses at the north abutment, but only minor responses in the girder in the major part of the low floating bridge.
The share of stresses, due to static loading of the yield capacity, from the results of the analysis by Norconsult is used as a reference value. However, the uncertainties and differences between the two studies have to be considered. As mentioned above, the share of stresses due to static loading, about 55-60%, can be increased with a few percentages to better utilize the capacity of the steel in the girder. It is, however, important to keep in mind that the contribution from the shear force is neglected in the stresses from the analysis by Norconsult. The contribution from the shear force is roughly estimated to 5%, based on the results from the analysis by Norconsult (Norconsult, 2017a). As a rough approximation, it is assumed that the share of stresses due to static loading is 60% of the yield capacity, i.e. 229 MPa.
The limitations of this assumption are discussed in section 9. This assumption must be verified in further studies by a dynamic analysis of the structure.
3.6 Parameters
The required stiffness of the girder and the pontoons have to be established for each investigated span length. The stiffness is varied until all requirements described in section 3.4 are fulfilled.
The stiffness of the girder is varied by increasing or decreasing the thickness of the steel plates and the trapezoidal stiffeners. The equivalent thickness of the stiffeners is assumed to vary at the same rate as the steel plates. This assumption must be checked in further studies.
The rotational stiffness and the buoyancy of the pontoon depend on the waterplane area. The waterplane area is the horizontal area of the pontoon at the water level. Since the draft is assumed to be constant, the freeboard of the pontoon is fixed to 4 meters. The freeboard is defined as the distance from the water level to the upper edge of the pontoon (Statens vegvesen, 2017a). To achieve the required stiffness of the pontoons, the waterplane area is varied. A scale factor,λ, is introduced to scale the size and mass of the pontoon. The width and length of the pontoon are multiplied with the scale factor, λ. An increase of the waterplane area increases the rotational stiffness and buoyancy of the pontoon. The required stiffness of the pontoons is found for the scale factor which fulfils the buoyancy and the rotational requirements for the investigated span length.
The weight of the pontoon is varying with the scale factor, λ. Since the height of the pontoon is fixed the scaled weight is calculated as:
mpontoon=m·λ2 Where:
m Weight of the pontoon in the base case λ Scale factor for sizing of pontoon
3.7 Load combinations
In this section, the load combinations in ULS considered in this study are presented.
According to the Design Basis, the following load combinations shall be considered, see figure 11. The table is taken from the Design basis (Statens vegvesen, 2017a).
Figure 11: Load combinations in ULS (Statens vegvesen, 2017a).
Three load combinations are considered in this study, with permanent load, traffic load and 100-year environmental load as dominant load. These are assumed to give the largest responses in the structure, based on the results from previous studies of the end-anchored bridge (Aas Jakobsen, COWI, Global Maritime, Johs Holt, 2016; Norconsult, 2017a).
Load cases included in the considered load combinations are presented in table 9 below.
G-EQk Q-Trfk Q-Env(100y)
Permanent load Permanent load Permanent load
Traffic Traffic -
1-year wind 1-year wind 100-year wind 1-year current 1-year current 100-year current Table 9: Load cases included in the considered load combinations.
The load combination with 100-year environmental load as dominant load is calculated without traffic loading, as the bridge will be closed when exposed to 100-years environmental loads (Statens vegvesen, 2017a).
4 Design loads
In this section the loads considered in this study are described and calculated.
4.1 Permanent load
The permanent loads considered in this analysis are the weight of the steel girder, pontoons, columns and the weight of the asphalt and railings. As a simplification, the weight of the ballast is not included in this study. This implies that the reaction force from the permanent loads will be underestimated and hence the buoyancy of the pontoons will be exceeded. This will not affect the choice of optimal span length, but it will influence the size of the pontoon.
The cross-sectional properties of the steel girder are calculated in an excel sheet. The plates in the steel girder are divided into smaller segments, separated by the investigated stress points. In total, the box girder is divided into eight plates. The properties of each plate are calculated and then combined to global properties of the complete cross-section. To account for the trapezoidal stiffeners, an equivalent thickness of the stiffeners is calculated and added to the thickness of the steel plates. The location of the stress points and the centre of gravity are illustrated in figure 12 below.
Figure 12: The plates of the girder and the investigated stress points.
The equivalent plate thickness of the trapezoidal stiffeners is calculated for the case with mini- mum plate thickness. The result is presented in table 10 below.
Location of plate Equivalent thickness
Deck 12 mm
Vertical web 11 mm
Bottom 9 mm
Table 10: Equivalent plate thicknesses of the stiffeners.
The second moment of inertia, I, is calculated for each plate around its local axis. To get the second moment of inertia of the complete cross-section, Steiner’s theorem is used (Isaksson et al., 2017):
I =X
(Ii+Aie2i) Where:
Ai Area of the plates
ei Eccentricity to axis of interest
Ii Second moment of inertia of the plates
As a simplification, the eccentricity, e, is taken as the distance between the centre of gravity of the plate element and the axis of interest.
The dead load from railings, asphalt and other equipment is 54kN/m(Multiconsult, 2017). The weight of the column is assumed to be fixed to 65 tonnes, see table 2 in section 2.6.
The weight of the structural steel is 77kN/m3 in accordance with the Design basis (Statens vegvesen, 2017a). The mass of the steel girder is varying with the thickness of the steel plates and the stiffeners. For the girder with minimum permissible thicknesses of the plates, the mass is calculated to 11.0 tonnes per meter.
4.2 Traffic load
In ULS the bridge is to be designed according to the Eurocode load system, using Load model 1, LM 1. For design in SLS, the Internordic traffic model, LMV, is used (ibid.).
4.2.1 Traffic - ULS
Load model 1 consists of uniformly distributed loads, UDL, and tandem loads. The number of notional lanes is determined using figure 13 below. An illustration of the terminology is shown in figure 14 (Norconsult, 2017a).
Figure 13: Lane definitions for traffic loads, LM 1.
Figure 14: Illustration of notional lanes.
The magnitude and placement of the uniformly distributed loads are stated in table 11 (Nor- consult, 2017a).
Placement UDL [kN/m2] Notional lane 1 5.4
Notional lane 2 2.5 Notional lane 3 2.5 Remaining area 2.5
Table 11: Placement and magnitude of uniformly distributed traffic loads.
Loads shall be placed on all driveable areas, i.e. areas restrained by railings. Pedestrian lanes parallel to carriageways shall be loaded with 2.5kN/m2 (Statens vegvesen, 2017a).
The chosen girder, described in section 2.5, is divided into two carriageways, each 10 meters wide. One lane, 3 meters wide, is loaded with 5.4kN/m2, while the remaining driveable surface is loaded with 2.5kN/m2. The uniformly distributed load is determined:
qU DL,1= 3·5.4 + (2·10−3)·2.5 = 58.7kN/m The pedestrian lane is loaded with 2.5kN/m2:
qU DL,2= 3·2.5 = 7.5kN/m The total uniformly distributed load is determined to:
qU DL,tot= 58.7 + 7.5 = 66.2kN/m
The magnitude and placement of the tandem loads in LM 1 are described in table 12 below (ibid.):
Placement Tandem load [kN] Notional lane 1 2 x 300
Notional lane 2 2 x 200 Notional lane 3 2 x 100 Remaining area 0
Table 12: Placement and magnitude of tandem loads.
The tandem loads consist of two axle loads with a distance of 1.2 meters. In the 2D calcula- tions, the tandem loads are simplified to one concentrated load acting in the middle of the girder.
The total tandem load acting on the girder is calculated:
QT L= 2·300 + 2·200 + 2·100 = 1200kN 4.2.2 Traffic - SLS
The Internordic load model, LMV, consists of uniformly distributed loads. All defined traffic lanes are loaded with 9kN/m. Pedestrian lanes shall be loaded with 2kN/m according to the Design basis (ibid.). The chosen girder has four traffic lanes and one lane intended for pedes- trians.
The total uniformly distributed load acting on the girder is calculated:
qLM V = 4·9 + 1·2 = 38kN/m
4.3 Static wind
For the static wind load, two loading cases shall be considered in ULS according to the Design basis (Statens vegvesen, 2017a).
1. Constant
2. Linearly varying from 0.6q at the north end to 1.0q at the south end.
In this analysis only the first load case is considered, constant uniformly distributed load over the whole structure. A sensitivity study conducted by Norconsult shows that it is a reasonable simplification to focus on this load case when analysing a combination of load actions (Norcon- sult, 2017a).
When combining static wind and static current the direction of the loads is either inwards or outwards of the fjord (ibid.). In the study by Norconsult, the dominating wind response when excluding the effects of reduction factors is obtained for wind from 60°to 240°(ibid.). Based on the result from this sensitivity study, the direction of the wind load is chosen to 90°, i.e. acting outwards of the fjord from the east.
As a simplification, the wind load is assumed to act on the girder and the railing, which has a minimum height of 1.2 meters (Statens vegvesen, 2015). The wind load, acting on the columns and the freeboard of the pontoons is neglected. For the load combinations where static wind is to be combined with traffic load, the wind load acting on the vehicles is neglected.
In accordance with NS-EN 1991-1-4:2005+NA:2009, the 10-min wind velocity is used in the calculations (Standard Norge, 2008a). The wind velocity is determined at the height of the girder of the low floating part of the bridge, approximatelyz= 15m (Multiconsult, 2017). For simplicity, the magnitude of the wind load is assumed to be constant over the surface area. In reality, the wind load is increasing with height.
The mean 10-min wind velocity with a return period of 100 years,vm100, is measured to 31.7m/s, 10 meters above the water surface in Bjørnafjorden (Statens vegvesen, 2017b). The 1-year mean 10-min wind is 22.9m/s at the same height (Statens vegvesen, 2017a).
The wind velocity at the desired height, 15 meters above the water surface, is determined by interpolation using the following formula (ibid.).
vm(H2) vm(H1) =
H2 H1
α
Where:
H Height above water surface vm Wind velocity
α Profile factor, taken as 0.127 (Aas Jakobsen, COWI, Johs Holt, 2016)
The equations for calculating the wind forces are retrieved from the study conducted by Mul- ticonsult (Multiconsult, 2017).
The mean 10-min wind force, Fm, is determined with the following equation (Multiconsult, 2017):
Fm= 1
2 ·ρa·v2m·cD·H Where:
cD Drag factor of the pontoon in sway H Height above water surface
vm Wind velocity
ρa Density of air, 1.25kg/m3 (Statens vegvesen, 2015)
The wind forces are calculated and summarized in table 13 below.
Load case Fm [kN/m]
1-year static wind 0.9 100-year static wind 1.7
Table 13: Mean 10-min wind force.
4.3.1 Static current
The current velocities in Bjørnafjorden in the directions 90° and 270° i.e. in- and outwards of the fjord, are given by the Design Basis and presented in table 14 below (Statens vegvesen, 2017a). As described in section 4.3, the current load and the wind load are acting in the same direction when combined. The direction of the evaluated loads is chosen to be 90°, i.e. outwards of the fjord.
v0[m/s]
Depth [m] 1-year 100 year
0-5 1.0 1.4
15 0.6 1.0
Table 14: Current velocities in Bjørnafjorden.
For the current load, acting in- or outwards of the fjord, the following load cases shall be considered according to the Design basis (ibid.):
1. The current is constant V0 along the bridge.
2. The current increases linearly from 0.5xV0 in the south to V0 in the north.
3. The current is constant in or out the fjord in the south half of the bridge and constant acting in the opposite direction in the northern half. The velocity shall be taken as 2/3xV0.
4. The current is V0 in the mid half of the bridge and 0.5xV0 in the rest.
In this study only the first load case is considered, i.e. constant current load along the bridge.
The load is applied to the complete structure in the global z-axis acting outwards of the fjord, i.e. in the same direction as the wind load. In a previous analysis of the bridge conducted by Norconsult, a sensitivity study of the current load is performed. The results from the sensitiv- ity study show that it is reasonable to focus on the uniformly distributed load when combining several environmental load actions (Norconsult, 2017a).
The current force,Fc, acting on each pontoon is calculated with the following equation (Multi- consult, 2017):
Fc= ρw
2 ·cD·AD·v02 Where:
AD Drag area in sway, i.e. draft times width of the pontoon cD Drag factor of the pontoon in sway
v0 Current velocity ρw Density of water
In table 15 below, the current forces acting on the base case pontoon, i.e. λ= 1.0, are presented.
Load case Fc [kN]
1-year static current 20.5 100-year static current 40.2
Table 15: Static 1-year and 100-year current force.
5 Hydrodynamic stiffnesses
In this section the required hydrodynamic stiffnesses of the pontoons are calculated. In figure 4 the terminology of the different motions of the pontoon is illustrated.
The hydrodynamic stiffness of the pontoon depends on the size of the waterplane area. The larger the waterplane area, the higher the stiffness. The stiffness of the pontoon in heave and roll are calculated according to Faltinsen, 1990. The stiffness in pitch is not required for this analysis and the pontoon has no stiffness in yaw, surge or sway (Norconsult, 2017a).
The stiffness in heave, denominatedS, is calculated as (Multiconsult, 2017):
S =Awp·ρw·g Where:
Awp Waterplane area g Gravitational constant ρw Density of water
The roll stiffness of the pontoon, denominated C44, is calculated with the following equation (Faltinsen, 1990):
C44=ρw·g·V(zb−zg) +ρw·g Z Z
Awp
y2ds
Where:
V Displacement of water
zb Distance from bottom of pontoon to centre of buoyancy
zg Distance from lower edge of pontoon to centre of gravity of pontoon ρw Density of water
To get the total roll stiffness of the complete structure, the mass of the pontoon, column and girder must be added to the equation. The total roll stiffness, C44, is calculated with the fol- lowing equation (Norconsult, 2017a):
C44=ρw·g·V ·zb−mpontoon·g·zg+ρw·g Z Z
Awp
y2ds−
mcolumn·g·dcolumn−mgirder·g·L·dgirder Where:
dcolumn Distance from lower edge of pontoon to centre of gravity of the column dgirder Distance from lower edge of pontoon to centre of gravity of the girder g Gravitational constant
L Span length
mcolumn Mass of column mgirder Mass of girder mpontoon Mass of pontoon V Displacement of water
zb Distance from bottom of pontoon to centre of buoyancy
zg Distance from lower edge of pontoon to centre of gravity of pontoon ρw Density of water
6 2D calculations
In this section the responses induced by static load actions are determined by simplified 2D calculations. This is an iterative calculation process, where the required stiffness of the girder and the pontoons is found for the investigated span lengths. Span lengths with an interval of 5 meters are investigated in the chosen range 80-130 meters. The structural elements which fulfil the requirements of the structure are determined for each investigated span length. The requirements of the structure are described more in detail in section 3.4. The iterative calcu- lation process results in a diagram, showing the total steel weight of the low floating part of the bridge as a function of the span. Finally, a proposed optimal span length based on the 2D calculations is presented. The low floating part of the bridge constitutes about 70% of the complete structure, i.e. a reduction of steel weight in the low bridge will have a large effect on the complete structure.
The static loads included in the analysis are:
• Permanent loads
• Traffic loads
• Static wind
• Static current
The results from the 2D calculations are summarized in section 8.
6.1 Permanent loads
The relevant responses from the permanent loading are calculated using the loads presented in section 4.1.
6.1.1 Moments
The response from the permanent loads is calculated for a beam on fixed supports, as described in section 3.1. The moments, in span and over field, are determined using the following formulas for standardized loading cases (Isaksson et al., 2017):
Mspan= qL2 24 Msupport =−qL2
12 Where:
q Includes weight of steel girder, asphalt, and railings L Span length
6.1.2 Reaction force
The permanent load cases which contribute to the reaction force, at the supports in the low floating part of the bridge, are the self-weight of the girder, pontoons, columns and additional dead load from asphalt and railings.
Since the girder is assumed to act as a continuous beam on fixed supports, the reaction force, R, from permanent load is determined using the following equation:
R=ga+g·L+gc+p
Where:
ga+g Self-weight of asphalt and girder gc+p Self-weight of column and pontoon L Span length
6.2 Traffic loads
6.2.1 Moments
The governing moments in the ultimate limit state, induced by traffic loading, are calculated using LM 1, as described in section 4.2.
6.2.1.1 Moment in span
When submitted to traffic loading, the bridge behaves as a flexible beam on spring supports, as described in section 3.1. To determine the moment in span, tables for continuous beams on spring supports are used (Wahlstr¨om, 1972). To calculate the largest moment in span the load case illustrated in figure 15 is used as an approximation.
Figure 15: Load case for largest moment in span.
The moment is calculated with a simple moment equilibrium for the load case presented above.
The required reaction forces are calculated according to the table in figure 16 below (ibid.).
Figure 16: Reaction forces for continuous beams on spring supports (Wahlstr¨om, 1972).
The reaction force, R, for a uniformly distributed load is calculated as:
R=kqL
For tandem loads, the reaction force,R, is calculated with the following equation:
To determine the constants,kAandkB, required for the calculations, the stiffness ratio between the pontoon and girder, described in section 2.2, must be determined. The stiffness ratio,µ, is calculated with the following equation (Multiconsult, 2017):
µ= K S Where:
K Girder stiffness
S Stiffness of pontoon in heave
The stiffness of the bridge girder,K, is calculated as described in section 2.2, while the pontoon stiffness in heave,S, is calculated with the equation presented in section 5.
Linear variation is assumed between the values of the stiffness ratio,µ. The constants,kA and kB, are determined by interpolation.
6.2.1.2 Moment over support
The approach used for the moment in span will not give a good approximation of the bending moment over support, due to the insufficient number of spans in the tables. Influence lines of the responses in the low floating part of the bridge, from an earlier stage of the project, are studied to identify the governing load case for the response. The influence lines have been provided by Henric Thompsson at Johs Holt, who is working in one of the consultant groups investigating the floating bridges over Bjørnafjorden. The influence line of the moment about weak axis, at the position of a pontoon, is shown in figure 17 below. In Appendix B, the influence lines of all responses are presented.
Figure 17: Influence line for bending moment over support.
The influence line above shows that the largest moment over support, induced by the uniformly distributed load, is obtained when a certain length on both sides of the support is loaded and an area close to the support is unloaded. This is illustrated as the length under the yellow area in figure 17 above.
The largest moment over support, caused by the tandem load, occurs when placing the load at the position where the height of the yellow area is largest. This is illustrated by the red horizontal line in figure 17 above.
When determining the largest moment over support, several factors influence the length of the loaded area and the position of the tandem load. In addition to the span length, the following factors are influencing the response:
• The stiffness of the girder
• The stiffness of the pontoon in heave
The largest moment over support, caused by the uniformly distributed load, is found by varying the parameters in the list above for each investigated span length.
To get a rough approximation of the response, a continuous beam on 9 spring supports is mod- elled in CALFEM. The response at the mid-support is evaluated.
The model in CALFEM is created using 22 beam elements. Two elements between supports in the outer spans, and 5 elements in the two middle spans. The extra elements in the midspans are added to get higher accuracy of the loaded length around the mid support. The model was chosen after an iteration study investigating the magnitude of the response for different span lengths. The model is illustrated in figure 18 below.
Figure 18: Illustration of the model created in CALFEM.
After an iteration study of potential loaded lengths 12 load cases were chosen. These load cases are assumed to give the largest moment over support, caused by traffic loading, for the investigated span lengths. The chosen load cases are shown in figure 19 below.
Figure 19: Load cases for moment over support.
All 12 load cases are evaluated for the span lengths in the chosen range and different stiffness of the girder and the pontoons. This implies that the total length of the model is varying with the investigated span lengths. For the shortest span length, 80 meters, the model has a total length of 640 meters. For the longest span, 130 meters, the investigated length of the bridge is 1040 meters. In section 9.3.2 the influence of the varying span length is discussed.
