Side-Anchored floating bridge at Bjørnafjorden : Design analysis for plate anchors in clay and cyclic bearing capacity evaluation

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Side-Anchored Floating Bridge at Bjørnafjorden

Design Analysis for Plate Anchors in Clay and Cyclic Bearing Capacity Evaluation.

Karoline England Reppen

Civil and Environmental Engineering

Supervisor: Gudmund Reidar Eiksund, IBM Co-supervisor: Heidi Kjennbakken, Statens Vegvesen

Muhammad Adeel Mazhar, Statens Vegvesen Submission date: June 2018

Norwegian University of Science and Technology

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Report Title:

Design Analysis for Plate Anchors in Clay and Cyclic Bearing Capacity Evaluation

Dato: 08.06.2018 Number of pages: 104 Master

Thesis x

Name: Karoline England Reppen

Professor in charge/supervisor: Gudmund Reidar Eiksund

Other external professional contacts/supervisors: Adeel Mazhar (NPRA)

Abstract:

This thesis discusses the anchoring system for a planned side-anchored floating bridge crossing the Bjørnafjord. The anchoring system consists of 32 anchors divided into eight groups, four on each side of the bridge alignment. Due to the risk of submarine landslides hitting one or several anchor groups by eroding the sediment basin plate anchors installed in the seabed is considered instead of the now planned suction, mixed and gravity anchors. The sediments on the seabed basin consists of soft clay with strength increasing linearly with depth. Plate anchors has been considered for locations having a sediment thickness greater than 16 m. A design code published by DNVGL is used for the anchor design. The landslide scenarios evaluated are slides eroding 5 m, 10 m and 15 m of the sediments at the anchor locations. It has been determined that smaller plate anchors penetrated deep will have a lower risk of getting influenced by a landslide than bigger anchors at a shallower depth. How small the plate anchors can be depending on how thick the sediment layer is. The anchor areas are ranging from 25 m

²

to 64 m

²

penetrated at respectively deep and shallow depth.

A cyclic bearing capacity analysis considering the cyclic shear strength for the soil due to its impact on the anchor design has been computed. A dynamic time series accounting for wave loads have been scaled up to represent both dynamic wind and wave loads. The Bjørnafjord clay has not got any cyclic laboratory tests completed therefore cyclic laboratory tests for the clay at Gjøa are computed. The cyclic amplitude from the scaled-up times series was too low to have an impact on the design of the plate anchors.

Keywords:

1. Sub Sea 2. Plate Anchors 3. Contour Diagrams 4. Cyclic Bearing Capacity

_________________________________

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

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Preface

This master thesis is a part of the MSc in Civil and Environmental Engineering at NTNU at the Departement of Geotechnics. The thesis counts 30 ECTS and is written during the spring term of 2018.

The main problem for the thesis was carried out by Heidi Kjennbakken in the Norwe- gian Public Road Administration. During the process the problem has been modified in agreement with supervisor Professor Gudmund Reidar Eiksund and co-supervisor Adeel Mazhar (NPRA).

Trondheim, 2018-06-08

Karoline England Reppen

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Acknowledgment

I would like to show my gratitude to the professors at the Geotechnical division at NTNU, especially my main supervisor Professor Gudmund Reidar Eiksund for always having the office door open and, giving me good guidance and challenging tasks for my thesis. I would also like to thank Professor Gustav Grimstad for taking the time to help me with the understanding of the cyclic behavior soil.

My gratitude also goes out to my co-supervisor Adeel Mazhar at the NPRA for good guidance and inputs during this thesis. Heidi Kjennbakken and Stian Moe Johannesen at the NRPA deserve credit for helping me get a wider understanding of the thesis.

Without my study colleagues and our good working environment during this semester the motivation for this thesis would not have been as good.

K.E.R.

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Summary

This thesis discusses the anchoring system for a planned side-anchored floating bridge crossing the Bjørnafjord. The anchoring system consists of 32 anchors divided into eight groups, four on each side of the bridge alignment. Due to the risk of submarine landslides hitting one or several anchor groups by eroding the sediment basin plate anchors installed in the seabed is considered instead of the now planned suction, mixed and gravity anchors.

The sediments on the seabed basin consists of soft clay with strength increasing linearly with depth. Plate anchors has been considered for locations having a sediment thickness greater than 16 m. A design code published by DNVGL is used for the anchor design. The landslide scenarios evaluated are slides eroding 5 m, 10 m and 15 m of the sediments at the anchor locations. It has been determined that smaller plate anchors penetrated deep will have a lower risk of getting influenced by a landslide than bigger anchors at a shallower depth. How small the plate anchors can be depends on how thick the sediment layer is.

The anchor areas are ranging from 25 m²to 64 m²penetrated at respectively deep and shallow depth.

A cyclic bearing capacity analysis considering the cyclic shear strength for the soil due to its impact on the anchor design has been computed. A cyclic time series accounting for wave loads have been scaled up to represent both dynamic wind and wave loads. The Bjørnafjord clay has not got any cyclic laboratory tests completed therefore cyclic laboratory tests for the clay at Gjøa are computed. The cyclic amplitude from the scaled up times series was too low to have an impact on the design of the plate anchors.

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Samandrag

Denne masteroppg˚ava diskuterer forankringssystemet for ei planlagt sideankra flytebru over Bjørnafjorden. Ankersystemet best˚ar av 32 anker fordelt i ˚atte grupper, fire p˚a kvar side av den planlagde brulinja. Grunna ein risiko av undersjøiske skred som han treffe ei eller fleire ankergrupper ved ˚a erodere sedimenta p˚a sjøbotnen er plateanker installert i sjøbotnen vurdert som forankringsmetode i staden for dei planlagde gravitasjons- og sugeankra. Sedimenta best˚ar av blaut leire med styrke som aukar lineært med djubden.

Plateanker har blitt vurdert p˚a stader der sedimenttjukkelsen er større enn 16 m. Ein berekningsmodell publisert av DNVGL er brukt for ankerdimensjonering. Dei forskjel- lige skredscenarioa som er vurdert eroderer henholdsvis 5 m, 10 m og 15 m av sedimenta ved ankerlokasjonane. Etter dimensjonering har det vorte bestemd at mindre anker pene- trert djupt har ein mindre risiko for ˚a verte p˚averka av skred enn større anker dimensjonert grunnare. Kor sm˚a ankera kan bli vert bestemd av sedimenttjukkelsen p˚a den planlagde ankerstaden. Ankerareala varierer fr˚a 25 m2 til 64 m2.

Ein syklisk bæreevneanalyse har blitt gjennomført, den har vurderert den sykliske skjærstyrken til leira og dens p˚averknad p˚a ankerdimensjoneringa. Ein syklisk tidsserie har blitt skalert opp for ˚a gjelde som b˚ade vind- og bølgelast. Ingen sykliske laboratorietestar har blitt gjennomført for Bjørafjordleira derfor er diagram for slike testar gjennomført p˚a Gjøaleire brukt. Den sykliske amplituden for tidsserien var for lav til ˚a ha p˚averknad p˚a leira.

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Nomenclature

Latin Symbols

A0 Input plate area AP late Plate area Fa Average Load Fcy Cyclic amplitude Ip Plasticity index Nc Bearing capacity factor Neqv Equivalent number of cycles Nf Number of cycles to failure Rd(zi) Design anchor resistance

Rs(zi) Static characteristic anchor resistance s^c_u Undrained shear strength

s^DSS_u Undrained shear strength from DSS test

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su,mean(zi) Mean undrained shear strength

Sc Shape factor

Sf Factor of safety

Sf.dyn Dynamic factor of safety Sf.stat Static factor of safety Ucy Cyclic loading factor

z Depth

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Greek Symbols

Soil weight

cu Undrained shear strength material factor

cy Cyclic shear strain at failure

m Partial safety factor for the design code

mean Partial safety factor for the pre-tension

p Permanent shear strain at failure

⌘ Empirical reduction factor

⌧a Average shear stress

⌧a,f Average shear stress at failure

⌧cy Cyclic shear stress

⌧cy,f Cyclic shear stress at failure

⌧f,cy Cyclic shear strength

⌧max Maximum shear stress

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Abbreviations

AHV Anchor Handling Tug Supply Vessel ca ka BP Calibrated kilo annum Before Present CPTU Cone Penetration Test with pore pressure DSS Direct Simple Shear

FFT Fast Fourier Transformation IFFT inverse fast Fourier transformation NGI Norwegian Geotechnical Institute NGU Geological Survey of Norway

NPRA Norwegian Public Road Administration PPA Position Penetrated Anchor

ROV Remotley Operation Vehicle RP Recomended Practice SBP Subbottom profiler ULS Ultimate Limit State

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Chapter 1 Introduction

1.1 Background

E39 is the major highway on the west coast of Norway, it starts in Kristiansand and goes through Stavanger, Bergen and Molde before it ends up in Trondheim. All together six coastal counties are included in E39. Along the road there are several longer fjord crossings. The smaller crossings have for the most part got bridges or tunnels, and the loger crossings have got ferries. From Stavanger to Bergen the longest fjord crossing is from Sandviksv˚ag to Halhjem and takes almost one hour and twenty minutes with the ferry.

This crossing goes over Bjørnafjorden. In 2010 the Transport Minister at the time Magn- hild Meltveit Kleppa started a project to look into the potential for aFerry-Free E39. The newNational Transport Plan(NTP) from 2018-2029 which was confirmed in June 2017 has a long term goal to further develop the E39 (NPRA, 2017).

In 2012 The Norwegian Public Road Administration (NPRA) started working one the possibilities of a ferry-free fjord crossing over Bjørnafjorden. Two bridge solutions have been considered by Multiconsult for this crossing. Due to the large water depth in the fjord of approximately 550 m (Multiconsult et al., 2017a) at the deepest the anchoring of the bridge at seabed is considered challenging. One of the bridge solutions is a side- anchored floating bridge with anchors installed in the seabed basin for transverse stiffness.

The anchors considered are suction, gravity and mixed anchors. This thesis focuses on two topics regarding anchor design.

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Chapter 1. Introduction

1.2 Problem Formulation

Topic One - Plate Anchors

Seabed investigations show a risk of submarine landslides at the already planned anchor locations. The anchor holding capacity may be reduced in case of a submarine landslide hitting one or several anchors. Plate anchors may be installed to a safe depth below seabed to reduce the risk of losing capacity in case of a slide event. When the plate anchors are penetrated deep enough into the sediments the risk of a landslide eroding and cutting them off is lower than if the anchors were present at sea bottom. A submarine landslide occurring could have different eroding depths and make the initial sediment thickness lower. Landslides with different eroding depths are considered for the plate anchor design, these depths are 5 m, 10 m and 15 m. Analysis considering the anchor area needed and its required penetration depth for no landslide and the different eroding scenarios occurring is the basis of this thesis.

Topic Two - Cyclic Bearing Capacity

The bridge will experience environmental loads from wind, waves and currents. These loads vary with time and may be described as dynamic or cyclic loads. Plate anchors penetrated into the sediments will also be exposed for these cyclic loads. The holding capacity may be influenced by the cyclic loads. Analysis considering the cyclic behavior of soils and how this can be used to calculate anchor capacity is the second topic discussed in this thesis.

1.3 Limitations

All the loads taken out for the design of the plate anchors are based on calculations done for the suction, mixed and gravity anchors. This implies that the loads given at anchor position is the position for these anchors. The load at anchor position is assumed to be the load in the line at seabed for the plate anchors. Results from CPTU tests at the seabed basin may not be valid for every anchor location, but these are used for all plate area calculations.

1.4 Structure of the Report

• Chapter 2 - Bridge Descriptiongives an introduction of the static system for the two floating bridges evaluated by Multiconsult. More details about the mooring

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1.4 Structure of the Report system and the anchor system planned for the side-anchored floating bridge is described.

• Chapter 3 - Seabed Descriptionintroduces the geology history for the Bjørnafjord basin and a description of the sediment thickness of each planned anchor group, soil conditions and slope stability in the sediment basin.

• Chapter 4 - Design Loadsexplains how the design loads are considered for each anchor group. The characteristic loads are taken out from several reports published by the NPRA. From (CEN, 2013) og (DNV-GL, 2015) the load factors for the different loads are considered.

• Chapter 5 - Plate Anchor Design Analysisgives an introduction of the calcula- tion method given by the DNV-GL in DNV-GL (2017). All the parameters in the resistance equation and the method used for calculations is described.

• Chapter 6 - Cyclic Analysisdescribes the cyclic behavior of soil according to observations made by Andersen and Lauritzsen (1988). The theory behind the methodology of how to consider the cyclic bearing capacity of a soil is introduced.

• Chapter 7 - Results and Discussion Plate Areapresents results of the plate anchor design. Required plate anchor area and its needed penetration depth for different anchor groups is discussed.

• Chapter 8 - Results and Discussion Cyclic Bearing Capacitypresents the results from the cyclic bearing capacity analysis based on the cyclic loading on line five and cyclic diagrams from the clay at Gjøa.

• Chapter 9- Summary and Conclusions, and Recommendations for Further Work discussion of the whole thesis and conclusions based on each result. A recommen- dation for further work is presented.

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Chapter 1. Introduction

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Chapter 2 Bridge Description

Two different floating bridge systems are considered for the Bjørnafjord crossing. The two planned bridges are both a combined cable stayed and floating bridge. On the south side the bridge starts as a cable stayed bridge with a span of 470 m, where 400 m is classified as navigation clearance for sea vessels. The foundation of the cable stayed bridge is planned to be placed on the bedrock close to the water surface. After 980 m as a cable stayed bride the construction continues as a floating bridge. A longitudinal section-drawing of the planned bridge alignment is presented in figure 2.1.

Figure 2.1:CAD-drawing of the planned bridge alignment (Multiconsult et al., 2017b) p. 6.

2.1 Static System

The cable stayed bridge is not symmetrical as there is a longer span on the north side. An A-shaped pylon, in compression, is anchored to the bedrock on shore. Cables in tension goes from the pylon on to the bridge deck. With this solution the deck is in compression

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Chapter 2. Bridge Description

towards the pylon from both sides. The pontoons against the north will, as a result of the bridge deck being in compression not be exposed to large anchoring forces from the cables such as in a suspension bridge.

The end-anchored bridge is pre-stressed and fixed at each abutment with large anchoring forces, thereby forcing the whole bridge alignment to be in compression. To make this reliable a stiff reinforced concrete cross section is needed. The whole bridge alignment is constructed as a curve from end-to-end as the right image in figure 2.3 shows. The other planned solution is a side-anchored bridge. These bridges have bearings at the abutments which allows the bridge to move. The bridge beam does not need to be as stiff as an end- anchored bridge when it is constructed to only carry weight from pontoon to pontoon, this makes the bridge beam much lighter. The traversal axis is stiffened by anchors placed in the sediment basin. An illustration is shown on the left side in figure 2.3. Mooring and Anchor system for the side anchored bridge is further described in section 2.2 and 2.3.

Figure 2.3: Illustration of the planned side- anchored (left) and end-anchored (right) floating bridge (NPRA, 2018).

2.2 Mooring Line System

The mooring line system consists of eight mooring groups connected to four pontoons.

All together 33 pontoons makes the bridge float. Axis A6, A14, A22 and A30 have four cables connected on the east and west side, every cable is connected to an anchor placed in the sediment basin. An illustration of the bridge alignment and the planned axis for the mooring lines are shown in figure 2.4. Each mooring line should have minimum static horizontal, i.e. x- and y direction, forces onto the pontoon. The mooring lines have a top and bottom chain and a longer wire in the mid part. Due to the low compression and bending stiffness in the chain and wire bigger cross section than necessary is chosen to maintain stiffness in the lines. The mooring lines are designed as a taut to semi-taut

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2.2 Mooring Line System system, in static condition no line should lie on the seabed. A taut mooring system has a relatively straight line from the pontoons and down to the seabed. To get the best utilization of the lines the length should be approximately 1000 m for the deepest part of the fjord.

This has been difficult to achieve due to no feasible anchor locations with this length. The anchors have been planned on places where the risk of a landslide hitting more than two lines in a group is minimized (Multiconsult et al., 2017b).

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Chapter 2. Bridge Description

Figure 2.4: Illustration of the planned side-anchored bridge alignment with the cable stayed part, pontoons and mooring lines (Multiconsult et al., 2017b) p. 24.

2.3 Anchor System

An anchor design performed with suction, gravity and mix anchors just finished phase two of the project. The system consists of eight anchor groups with four single anchors in each group. Four groups are placed on the east side of the bridge alignment and four on the west. On the shallower areas where the sediment thickness is under 5 m, gravity anchors are preferred. These anchors consist of big concrete cubes which get their holding capacity because of their weight. Where there are thicker sediments, over 5 m, suction anchors are for planned for installation. Suction anchors are tall cylinders penetrated down into the sediments either by gravity or hydraulics as the driving force A mixed anchor is a combination of the two just mentioned. The analysis done on these anchors will be used for the design of the plate anchors. Different types of plate anchors are described in Appendix C.

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Chapter 3 Seabed Description

3.1 Geological History

The Quaternary is a time period which covers the geological history from 2.6 million years ago until today. During the Quaternary numerous glaciations and deglaciations have taken place across the northern hemisphere. Under the glaciations the glacier eroded and formed the current landscapes by depositing sediments. During the deglaciation, sediments were deposited in fjord basins by meltwater (Aarseth, 1997). Approximately 150 km³of sediments were deposited during the deglaciation in the Norwegian fjords between 59-63 N (Aarseth, 1997). 90 percent of the fjord sediments were deposited during the late Weich- selian and early Holocene. Only less than 10 percent of the sediments observed are older than the Lower Weichselian (Jansen and Sjøholm, 1991). The glacio marine sediments consists of silt and clay which reflects the observations made in the Bjørnafjord (Solli et al., 2017; Aarseth et al., 1989)

Observations consider the outer Bjørnafjord as ice-free since the Early Allerød from 13.5 cal ka BP Mangerud et al. (2016). During the last deglaciation a cold climate deteri- oration called the Younger Dryas Chronozone made the glacial ice sheet once again reach Fusafjorden, an inner part of the Bjørnafjord, see illustration in figure 3.1 (Aarseth, 1997;

Mangerud et al., 2016). The deglaciation after Younger Dryas took only a few hundred years from about 11.6 cal ka BP (Aarseth et al., 1975). This explains why the sediments in the Bjørnafjord are normally consolidated. After every deglaciation isostatic uplift made the newly deposited sediments unstable. This caused a high frequency of submarine landslides at the fjord basin, which is a work in progress (NPRA, 2018).

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Chapter 3. Seabed Description

Figure 3.1:Reconstruction of the glacial position during the YD in Fusafjord. The land configuration is based on Mangerud et al. (2016) and references therein. The marine configuration is based on bathymetry data from the Norwegian Mapping Authorities (Figure produced by H. Kjennbakken, 2018)

3.2 Seabed Data

All data of the seabed in Bjørnafjorden is taken from chapter 1, 3 and 4 in (Multiconsult et al., 2017a) and chapter 2 in (Multiconsult et al., 2017c) written for the NPRA. The seabed crossing of Bjørnafjod is approximately 5000 m wide. On the southern part of the fjord the inclination is steep and goes directly from the shore and down into the fjord basin.

The north side has also got a steep inclination from sediment basin and up to a plateau at about 70 m water depth. From there and up to the northern shore the inclination gradient is low. In these shallower areas there is more exposed bedrock with some sediments in the troughs. The seabed is deepest in the southern central part of the fjord, here the water depth is roughly 550 m. This area is not as flat as seen in figure 3.2.

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3.3 Sediment Thickness

Figure 3.2: 3D illustration of the bathymetry in Bjørnafjorden. North to the right (Multiconsult et al., 2017a) p. 7.

3.3 Sediment Thickness

The sediment thickness is based on interpretation of geophysical data and geotechnical boreholes. The geophysical mapping is based on a 50 m grid collected with a subbottom profiler (SBP). On the north-south lines the SBP was mounted on a remotely operation vehicle (ROV) flying 25 m above the seabed. The resolution was about 30 cm and the penetration about 30 m. The east-west lines were collected with a hullmounted SBP with lower resolution but higher penetration (Solli et al., 2017).

In the deepest area of the fjord the sediment thickness is approximately 60 m. These areas are marked dark blue on both the west and east side of the planned bridge alignment as shown in figure 3.3. No anchors are planned in this basin with thick sediment layer.

The southern side is very steep, as presented in both figures 3.2 and 3.4, and here there are no sediments. In the lower parts of the basin at the north side, the thickness varies from 5 to 20 m. Further CPTU testing is planned to be performed on each anchor location, but these results are not ready before October 2018. Analyzed data for each anchor group is described below. All the odd- numbered anchor groups are placed on the east side of the bridge and the even numbered groups on the west (Multiconsult et al., 2017a). A detailed description of the water depth, inclination and sediment thickness is shown in table 3.1.

The sediment thickness is also shown as an illustration on the isopach map in figure 3.3.

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Figure 3.3:Isopach map of the interpreted sediment thickness (DOFSubsea, 2015).

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3.3 Sediment Thickness

Figure 3.4:Batemetry map with anchor locations (Multiconsult et al., 2017a) p. 17.

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Table 3.1:Seabed inclination and sediment thickness for all anchor locations.

Anchor

group Anchor Water depth [m]

Seabed inclination at anchor position

[deg]

Sediment thickness

[m]

1 1 - 528,2 3,5 36,0

2 - 527,0 7,0 31,5

3 - 535,7 0,5 30,0

4 - 538,6 4,5 34,5

2 5 - 532,1 2,0 27,0

6 - 534,9 3,5 28,0

7 - 534,9 4,0 38,0

8 - 530,1 5,0 30,5

3 9 - 505,2 7,5 22,5

10 - 515,5 7,0 23,0

11 - 533,1 4,0 25,5

12 - 534,9 5,5 23,5

4 13 - 458,5 6,5 19,0

14 - 444,9 4,5 21,0

15 - 482,7 4,0 21,0

16 - 456,9 3,0 20,5

5 17 - 541,3 1,5 22,0

18 - 538,4 1,0 24,0

19 - 367,6 1,0 16,0

20 - 368,3 1,0 16,0

6 21 - 132,0 2,0 3,0

22 - 124,9 4,0 0,0

23 - 538,8 4,5 27,0

24 - 547,3 7,0 25,0

7 25 - 269,9 3,0 13,0

26 - 267,1 2,0 16,0

27 - 175,7 1,0 9,0

28 - 174,9 1,5 17,0

8 29 - 61,0 3,0 0,5

30 - 64,9 5,0 0,0

31 - 63,6 4,0 0,0

32 - 59,9 2,0 0,0

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3.4 Sediment Thickness Described by Group

The sediment thickness for anchor group one is about 30 m at 525 m water depth on a relatively flat area where boulders may occur in the sediments. Plate anchors are preferred as anchor type since previous landslides have been observed in the area and this area is potentially highly susceptible to landslides in the future.

Anchor group two has a sediment layer of about 30 to 40 m. No previous landslides have been observed, but this area is also highly landslide-susceptible. The area is described as relatively flat with a water depth of 540 m. This thick sediment layer along with a flat seabed make good conditions for suction and plate anchors.

Group 3 is located on a ridge east of the planned bridge alignment. The ridge has an inclination of seven degrees towards north east and north west. Further north of the ridge the seabed inclination is reduced. No landslides have been observed in the area. The sediment thickness is approximately 20 m and threrfore along with the known inclination suction and plate anchors can be installed in the area. The water depth in the south is about 505 m and 535 m in the north.

The conditions for anchor group four consist of a 15 - 20 meters layer of sediments.

Both suction and plate anchors can be used, even though the seabed inclination is about 12 degrees towards south east. The water depth varies from 440 m in the west to 480 m in the east. No previous landslides have been observed in this area.

Two subgroups represent group five, on deep water in the south anchor number 17 and 18 is planned to be installed and in the north at a shallower depth anchor 19 and 20 is planned installed. In between these two subgroups a steep area with little or no sediments is observed. At the south anchor group previous landslides have been observed. Local variations in both soil mechanical conditions and statigraphy need to be considered along with boulders in the sediments. For the northern subgroup the installation area is relatively flat. Sediment thicknesses of 15 to 20 m at approximately 370 m water depth with no previous landslides observed. Both suction and plate anchors can be used for the northern sub-group, but for the two southern anchors plate anchors are preferred.

Group number six is also divided into two subgroups. Anchor number 21 and 22 are planned to be installed as gravity anchors due to low sediment thickness in shallower depths. The south subgroup with anchor number 23 and 24 is placed on a thick sediment layer of about 20 m at a 540 m water depth. Previous landslides have been observed for

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the south group, here plate anchors are the preferred solution.

Mixed anchors are planned for two of the anchors in group seven. The planned area for the mixed anchors 25 and 27 has got limited sediments. Anchor number 26 and 28 have a medium thick sediment layer from 16 to 17 m. The maximum seabed inclination for the whole group is three degrees. Anchor number 25 and 26 are planned to be installed in the south at 270 m water depth. The two anchors in the north are placed at 180 m water depth. There are no previous landslides observed in the area.

Anchor group eight is placed in shallow water with limited sediment thickness. The whole group is planned to be gravity anchors.

3.5 Soil Conditions

Several sub-marine geotechnical investigations have been carried out in the Bjørnafjord.

The first investigation started in 2012 by Multiconsult, the latest was performed by Nor- consult in 2017. Results from field investigations combined with laboratory analysis show homogeneous conditions. The seabed soil consists of soft clay. Between the clay and the bedrock a layer of sand silty clayey material is found. The clay is described as normal to moderately over consolidated with very low to high strength.

From several CPTU tests the undrained shear strengths^c_uis discussed. The analyses show an undrained shear strength increasing linearly with depth, as the results from several CPTUs shows in figure 3.5. Twos^c_uprofiles are represented in the diagram, the black line shows a lows^c_uprofile used for anchor capacity analyses. At seabed the undrained shear strength is2.3kPa, at 20 m depth 44 kPa and 106 kPa at 46 m depth. After further analysis of the shear strength, two linear equations represents the lows^c_u profile. This profile is used for further calculations, and are presented in equations 3.1 and 3.2. In Eurocode 7 (CEN, 2016) a material factor cuis demanded to use on the undrained shear strength, this factor is set to be 1.4 from the National Annex A.3.2. The plasticity indexIpof the clay has some variations ranging from 25%to 45%(NGI, 2016b).

• Depthz0 - 20 m

s^c_u= 2.085z+ 2.3

cu (3.1)

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3.6 Slope Stability

• Depthz20 - 46 m

s^c_u= 2.3846(z 20) + 44

cu (3.2)

Figure 3.5:Undrained shear strength from CPTU tests in the Bjørnafjord (NGI, 2016a).

3.6 Slope Stability

Investigations of the seabed show a history of several submarine landslides. It is assessed that all anchor groups on the seabed can potentially be exposed to future landslides. The required safety factor for global static slope stability and global dynamic slope stability are respectivelySf.stat = 1.4andSf.dyn = 1.1. The safety factor for many slopes over and under the anchor locations is now below required factor of safety as 1.4 and 1.1.

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In places with too little data, the safety factor is set to be under 1. It has been discussed if a safety factorSf = 1.4given in (NPRA, 2014) should be used. This safety factor is based on total failure of the main foundation and the whole bridge if a landslide occurs.

For the side anchored floating bridge up to two mooring lines can fail without the whole main structure collapsing. A new proposed solution for the design basis is states that the global static stabilitySf = 1.4can be set for certain areas. In these areas a landslide will fail two or more anchors in the same anchor group.

Required safety factors as described in (SVV, 2016) are argued to be too high as the bridge structure is planned to be built with a certain redundancy to withstand the global collapse in case of foundation failure. For the planned bridge critical condition should not be achievable even if two anchor lines in one group fail due to a landslide or a collision.

It has been suggested to only use these safety factors for those anchor locations which are assessed to be critical for global stability of the bridge structure and exceeds the acceptable risk given in form of redundancy of mooring system.

3.7 Slide Scenarios

Norwegian Geotechnical Institute (NGI) has evaluated three submarine landslide scenarios on behalf of NPRA (Multiconsult et al., 2017a). These scenarios are based on flow around the planned suction anchors, the same evaluation will be used for the plate anchors. Slide scenario number one and two have a debris flow at the seabed basin where the flow stops at the end of the suction anchor. The height of the flow depends on how much material is released in the failure. Slide scenario number one has all the soil underneath the flow intact whereas slide number two has remoulded soil 10 m into the sediments. The remoulded soil is caused by a ploughing landslide, and its height is considered conservative.

Slide scenario number three has the same flow thickness on both sides of the anchor with a remoulded soil height of 10 m.

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Chapter 4 Design Loads

The characteristic loads and responses considered in the design of the anchors are pre- tension of the mooring lines, static wind and sea current and dynamic wind and wave at anchor position according to the anchors already planned. The environmental loads are divided into static and dynamic responses. All the responses listed in table 4.1, 4.2 and 4.3 are characteristics, only table 4.4 has load factors included.

4.1 Pre-tension

The pre-tension of the mooring lines is adjusted to achive a taut to semi-taut mooring system (Multiconsult et al., 2017b). At anchor position the pre-tension ranges from 1344 kN for anchor number 8 to 2763 kN for anchor number 22. Anchor number 22 is planned as gravity anchor. The highest tension for a planned plate anchor group is group four with a maximum tension at 2596 kN for line 14. This group is placed on approximately -460 m and has a mooring line length at around 1200 m. All the loads at anchor position is shown in table 4.1. Anchor group 2 has the smallest tension loads on average, while group six has the biggest.

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Chapter 4. Design Loads

Table 4.1:Characteristic pre-tension at anchor position (Multiconsult et al., 2017a) p. 41.

Anchor group

Line no.

Pre-tension at anchor [kN]

1 1 1849

2 1860

3 1802

4 1831

2 5 1406

6 1401

7 1352

8 1344

3 9 2537

10 2525

11 2509

12 2506

4 13 2584

14 2596

15 2516

16 2582

5 17 2211

18 2213

19 2343

20 2376

6 21

22

23 2553

24 2548

7 25

26 2178

27

28 1907

4.2 Environmental Loads

All the environmental responses both static and dynamic load components are taken from Multiconsult et al. (2017d) chapter 9 Mooring lines. The environmental responses are

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4.2 Environmental Loads simulated in a global analysis model called Orcaflex. For the 100 year return period wind, wave, sea current and storm conditions are assumed fully correlated (Multiconsult et al., 2017d). The east and west faced mooring lines take loads from different directions since the lines have low compression and bending stiffness. The direction for the environmental loads are divided into six sections. Four of these sections are used for the anchor design.

The extreme 100 year wind speed has a mean value of 29.5 m/s from the west sector, the east sector is reduced by a factor of 0.85 (Multiconsult et al., 2017d).The east faced lines are all the odd numbered anchor groups. These lines only respond to the loads simulated from the east section hence, 75, 90 and 100 degrees north east. The same counts for the west faced group at 280 and 335 degrees north west. The simulations of dynamic wind and waves from 75 and 90 degrees north are the ruling responses for the eastern groups along with static wind and current from 100 degrees north. All the environmental loads from different sections are listed in tables 4.2 and 4.3. The bold values are used for the design load basis.

4.2.1 Dynamic Responses

A global response simulation is done for wind and wave. The wave analysis is based on hydrodynamic inputs simulated in Orcaflex. Aerodynamic theory in Orcaflex is used for the wind analysis. Anchor group number one has the greatest strain from the east section and anchor group six has the biggest strain from the west section shown in table 4.2.s

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Table 4.2: Environmental characteristic dynamic wind and wave responses (Multiconsult et al., 2017e).

Anchor group

Line

no. Dynamic wind and waves 75 deg 90 deg 335 deg

[kN] [kN] [kN]

1 1 1033 1072 718

2 1036 1077 667

3 1272 1161 793

4 1157 1050 766

2 5 347 344 608

6 328 308 596

7 329 317 694

8 369 347 745

3 9 764 723 561

10 781 689 570

11 786 684 589

12 788 735 579

4 13 613 534 730

14 555 497 732

15 599 534 925

16 600 516 736

5 17 646 622 485

18 658 663 530

19 929 867 600

20 829 776 566

6 21

22

23 666 620 908

24 653 546 844

7 25

26 1291 1185 754

27

28 869 665 543

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4.2 Environmental Loads

4.2.2 Static Responses

In addition to dynamic wind static wind is taken into account as a separate load. Static wind and current are considered to be permanent responses appearing from two different sections in the design analysis. Anchor group one has the greatest stresses from the east and group four has the biggest stresses from the west.

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Table 4.3: Environmental characteristic static wind and current responses (Multiconsult et al., 2017e).

Anchor group

Line

no. Static wind Sea current

100 deg 280 deg 100 deg 280 deg

[kN] [kN] [kN] [kN]

1 1 578 -571 154 -144

2 629 -601 161 -149

3 830 -761 190 -180

4 604 -647 166 -157

2 5 -84 187 -37 40

6 -86 195 -37 40

7 -125 319 -51 56

8 -131 345 -54 61

3 9 233 -260 88 -83

10 247 -273 91 -86

11 246 -272 91 -86

12 246 -274 93 -87

4 13 -192 341 -83 88

14 -194 347 -83 88

15 -271 531 -115 124

16 -189 340 -82 87

5 17 107 -114 50 -47

18 117 -124 54 -50

19 233 -233 98 -89

20 134 -146 62 -57

6 21

22

23 -184 300 -88 91

24 -160 257 -79 81

7 25

26 364 -348 144 -129

27

28 178 -203 79 -74

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4.3 Load Factors

The guidelines for the design loads are based on the rules from ISO 19901-7:2013 annex B.2Offshore Norway(CEN, 2013) for the environmental responses and DNVGL-OS-E301 Offshore Standard - Position Mooring(DNV-GL, 2015) for the pre-tension of the mooring lines.

The annex contains rules relating the petroleum activities, but as there are no clear guidelines for offshore structures not concerning petroleum activities of such consequence class these are used. For permanent mooring systems in ULS design a 100 year analysis should be considered based on the environmental conditions. From table B.1 (CEN, 2013) the procedure to set the safety factor is described. The highest consequence class is set for design calculations based on these assumptions; the planned locations for the anchors are in vicinity to each other and are facing away from other installations in standby condition.

In standby condition the bridge is closed due to weather conditions. The analyzes set for the consequence class should be done in intact condition. All these assumptions gives a safety load factor of Sf = 2.20. For the ULS analysis a three-hour storm will be considered.

Table 1 (DNV-GL, 2015) gives partial safety factors for the characteristic pre-tension load in ULS condition. According to (Multiconsult et al., 2017b) consequence class 2 and dynamic analysis which gives a load factor mean= 1.4.

4.4 ULS Loads

The characteristic loads has been multiplied with the given partial safety factor and added together for every mooring line, these loads are listed in table 4.4. The ULS-load for every line and the maximum load within an anchor group is listed. Anchor group one and four is exposed for the biggest loads at anchor position.

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Table 4.4:ULS load for at each anchor position and maximum ULS load for each anchor group.

Anchor group

Line

no. ULS

Maximum for each anchor group

[kN] [kN]

1 1 6557 7565

2 6711

3 7565

4 6803

2 5 3805 4414

6 3790

7 4245

8 4414

3 9 5939 5997

10 5997

11 5983

12 5988

4 13 6167 6998

14 6202

15 6998

16 6173

5 17 4862 6052

18 4933

19 6052

20 5581

6 21

22

23 5411 5411

24 5148

7 25

26 6136 6136

27

28 4384

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4.5 The Cyclic Responses

To perform a cyclic bearing capacity analysis for the soil a time series from Orcaflex is computed for anchor line five. This time series contains only wave responses with a duration of 1850 seconds counting 30 minutes. The peak value for the signal is 3221 kN at 44 sec, after multiplied with a load factor according to CEN (2013). It is assumed that the time series consists of the dynamic responses counting both wind and wave. For this matter the series has been scaled to fit the ULS peak load 3805 kN for line five shown in table 4.4.

As specified by Andersen (2007) the amplitude of the signal should be⌧cy as shown in figure 6.1. The maximum amplitude Fcyfor the signal has been scaled up to be half the value of the dynamic responses for wind and wave 668 kN.

The series was first scaled down to a mean value of zero. To blow up the signal the amplitude got divided with the value from 44 seconds which lead to a factor of 1.4567.

This factor got multiplied with the series which was scaled to zero. To reach the peak ULS for value at 44 seconds 6.4 the amplitude Fcygot subtracted from the ULS load. In figure 6.4 the orange line shows the ULS load, the blue line shows the scaled and blown up signal and the yellow line the average value of the signal representing Faat 3137 kN. Fcy is the maximum cyclic amplitude from the mean to the peak with a value of 668 kN.

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Chapter 5 Plate Anchor Design Analysis

Det Norske Veritas published in 2002 aRecommended Practice(RP)forDesign and Instal- lation of Plate Anchors in Clay. This is the same RP as (DNV-GL, 2017), only with a new logo after DNV merged with Germanischer Lloyd and became DNV-GL. The RP is meant as a guidance for design analysis and installation, and not as a definite rule. The design principles are taken from (DNV-GL, 2017) Chapter 3Design Code for Plate Anchors.

5.1 Design Code

Anchor Capacity

The characteristic anchor capacity is given from the equations in (DNV-GL, 2017) Chapter 3. These formulas are listed under as a characteristic resistance 5.1 and a design resistance 5.2 with a partial safety factor included. All the parameters used in equation 5.1 and 5.2 are shown in table 5.1.

RS(zi) =Nc·Sc·⌘·su,mean(zi)·Aplate (5.1)

R (z) =RS(zi)

(5.2)

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Chapter 5. Plate Anchor Design Analysis

Table 5.1:Parameters for equation 5.1

Nc Bearing capacity factor

Sc = 1 + 0.2^W_L_F^F Shape factor

WF Plate width

LF Plate length

⌘ Empirical reduction factor

su,mean Mean undrained shear strength

Aplate Plate area

zi Penetration depth

m Partial safety factor

Bearing Capacity Factor N

c

The formula for the bearing capacity factorNc is shown in equation 5.3. This formula presents a deep flow around and a shallow failure. To qualify for the deep penetration failure the penetration depth needs to be zmin = 4.5·WF. The failure could here be mobilized as a spherical volume around the plate with a diameter 3·WF. A shallow failure up to the seabed can occur with a penetration depth less thanzmin.

Nc= 2 +⇡

!

1 + 0.987·arctan

✓ zi

WF

◆!

(5.3)

Shape factor S

c

The shape factor is adjusting the plate geometry for the bearing capacity factorNcthrough a ratio between the equivalent plate width and plate length, given by equation 5.4.

Sc= 1 + 0.2WF

LF (5.4)

Empirical Reduction Factor and Partial Safety Factor

The empirical reduction factor⌘ is set to 0.75 based on onshore field tests for normally consolidated clay. The partial safety factor mis set as 1.4 in ULS (Ultimate Limit State) from table 3-1 (DNV-GL, 2017).

Plate Area A

plate

When the anchor capacity is known different plate areas can be calculated for different penetration depth. Equation 5.1 and 5.2 can therefore be written like 5.5 and 5.6 as shown

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5.2 Limitations under. In equation 5.6 the bearing capacity factor is written as a function of the square root of theAplateinstead ofWF. This makes it easier to see that iteration is needed to find different plate areas at different penetration depth.

Aplate= Rd(zi)· ^m

Nc·Sc·⌘·su,mean(zi) (5.5)

Aplate= Rd(zi)· ^m

2 +⇡

!

1 + 0.987·arctan

✓

zi

pA0

◆!

·Sc·⌘·su,mean(zi)

(5.6)

5.2 Limitations

The cyclic loading factorUcywith the cyclic shear strength⌧f,cyis neglected. This causes only the static anchor resistance to be the valid one. The equation for the static anchor resistance 5.1 assumes the ideal angle of 90 degrees from the mooring line onto the fluke, see figure 5.1. This gives the highest capacity for the anchor. The shape factorScof the anchor depends on plate lengthLF and widthWF. To make this variable constant the area of the plate considered to be quadraticLF =WF. This makes it possible to calculate different plate areas with the needed penetration depth when the needed resistance is known. The angle of the mooring lines at seabed for the already planned anchors are not considered for the plate anchor design.

Figure 5.1: Illustration of the ideal angle, showed by the curved arrow, perpendicular from the mooring line onto the plate. Modified figure from (Bruce, 2017)

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Chapter 5. Plate Anchor Design Analysis

5.3 Methodology

In chapter Anchor Design Loadsthe anchor resistant loads are listed in table 4.4 as the ULS-load. An anchor design according to the design code described above has been done for each anchor group where the sediment thickness is greater than 15 m. This counts for all groups except anchor group eight. The highest ULS-load and the smallest sediment thickness in one anchor group is used for the design analysis. To determine the plate area for a known resistance equation 5.1 and 5.2 is used. The slide scenarios that may occur in the installation area are considered for the design as eroding masses over the anchor position. The eroding masses tend to be 5 m, 10 m and 15 m. For each of these scenarios a new initial sediment thickness were taken into account, but with the undrained shear strength from the depth of the originals^c_u-profile. The penetration depth for the anchor is given at anchor center. The deepest an anchor can be penetrated form the center isWFand the shallowest isWF down in the sediments when the anchor is penetrated perpendicular to seabed. A chosen plate area and the needed penetration depth for a solution with no landslide occurring and slide occurring have been determined.

To visualize how the needed penetration depth and various plate areas rely on each other equation 5.6 has been used by iteratind different areas. A chosen plate areaA0for a certain depth was set as an input in equation 5.6 for finding theAplate. To consider the correct plate anchor for a depth the absolute value|A0-Aplate|needed to get minimized.

This was done by using a problem solver function inExcelwhich was seeking the minimum value of a specific cell. The iterations were made for all depths in the different slide scenarios.

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Chapter 6 Cyclic Analyses

A response analysis for waves is computed in Orcaflex for line five taken at the anchor location on the seabed basin with a duration of 1850 seconds counting 30 minutes. This response can be used to determine the cyclic bearing capacity of the soil due to Andersen and Lauritzsen (1988) and Andersen (2007).

6.1 Cyclic Behavior of Soil

It is important to consider the cyclic behavior of the soil when this can have a decreasing impact on the soils bearing capacity. Cyclic loading can cause volumetric reduction of the soil by breaking down the soil structure. In the case of undrained conditions the effective stresses will decrease when the normal stresses carried by the soil get transferred to the pore water due to the low volumetric compressibility of water. Dyvik et al. (1989) have done several tests on gravity platforms under cyclic loading. These tests show higher dis- placements of the soil with cyclic loading compared to static loading. The cyclic failure load is also lower than the static.

Tests done on marine Drammen clay by Andersen (2007) with plasticity indexIp = 27% show different results depending on the testing method. Two tests were run under symmetrical loading with⌧a = 0. In the Direct Simple Shear (DSS) test the shear strain accumulated symmetrical on extension and compression side. The triaxial tests show a non symmetrical shear strain development. The permanent shear strain has the same magnitude as the cyclic shear strains due to a higher compression than extension strength. A third test was run in a triaxial cell with the average stress and the cyclic amplitude as the same, . The dominant shear strain development here was caused by the static loading,

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Chapter 6. Cyclic Analyses

and only small cyclic shear strains were caused by the cyclic loading. Failure in the soil is defined by a permanent por cyclic cyshear strain at 15%. Other tests done by Andersen (2007) on the same Drammen clay shows that cyclic behaviour does not directly affect the maximum shear stress⌧max. This is presented in figure 6.1. All three tests run in a triaxial cell have the same⌧max, but with different average stress and cyclic amplitude. The two tests B and C had a lower⌧cy and much lower shear strains than test A which failed at 10 cycles. Hence, considering the cyclic amplitude when analyzing the cyclic behavior of the soil is important.

Figure 6.1:Results from cyclic triaxial tests with the same maximum shear stress,⌧max(Andersen, 2007) p. 519.

6.2 Contour Diagrams and Equivalent Number of Cycles

Andersen (2007) plotted test results on normally consolidated Drammen clay as a function of average and cyclic shear stress. A contour diagram for DSS tests is shown in figure 6.2, where y-axis represents the factor between the cyclic shear strength at failure⌧cy,f and the static DSS shear strengths^DSS_u and x-axis represents the factor between the average shear stress at failure⌧a,f ands^DSS_u . The numbers along the linesNf describes the number of cycles in the soil at failure for por cy at 15%. If⌧a = 0the clay will fail for one cycle when the factor⌧a,f/s^DSS_u is about 1.5 and for 1000 cycles when the factor is 0.55.

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6.2 Contour Diagrams and Equivalent Number of Cycles

Figure 6.2: Number of cycles to failureNf and failure mode as a function of average and cyclic shear stresses for cyclic DSS tests on Drammen Clay (Andersen, 2007) p. 520.

The number of cycles to failure in the mentioned tests have all had the same cyclic amplitude during the whole test. In practical cases where wind and waves determine the stress the amplitude will vary much over time. These time series will not look like any done in the laboratory. To be able to use the contour diagrams the equivalent number of cycles Neqv that gives the same effect as the real cyclic load history must be determined.

Andersen (1976) and Andersen (2004) describes procedures to determine Neqv by using pore water pressure or shear strain contour diagrams. These diagrams are built on the same basis as figure 6.2. The average shear stress for diagram in figure 6.3 is⌧a = 0, but same diagrams can be made for other stress situations. Astrain accumulationprocedure can determine the Neqvfor undrained conditions. The different cyclic strain measured will during the cyclic load history be observed and used as a memory for the cyclic effect to make a diagram shown in figure 6.3.

Figure 6.3: Cyclic shear strain as a function of number of cycles on Drammen clay (Andersen, 2007) p. 521.

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6.3 The Fourier Transformation

The responses from Orcaflex are original in time-domain where the it shows how the signal changes over time. For analysis the signal needs to get converted into the frequency- domain. A Fourier transformation converts the time-domain signal into the frequency- domain by decomposing the signal function to a number of sinus wave frequency components. The frequency-domain signal consists of a real and imaginary part which has the information about the magnitude and the phase at each frequency. The magnitude can be referred to as the absolute value of the complex number, and describes the strength of the frequency for each component. The phase describes how the frequency components align in time and represents the angle of a component on the unit circle (M¨uller, 2015). The time-domain signal needs to be converted to the frequency-domain in order to sort out the low frequencies to make the signal less noisy.

6.4 The Rainflow - Counting Algorithm

The rainflow-counting algorithm counts the range of all the half cycles in a signal by counting all thevalleysand all thepeaksin two separate rounds. This method is used for fatigue analysis in steel, but can also be used to determined theNeqv needed to consider the cyclic shear strength according to Andersen and Lauritzsen (1988). From the starting point follow the signal to a peak or valley, when reaching the peak/valley ask the question, is the next peak/valley bigger than the current one? If the next peak/valley is bigger than the current continue to this point and ask the question again. As long as the question could be answered withyescontinue, if the next peak/valley is lower than the current stop the counting and write the range off the stress area counted. Then start again in the next point after the first start. This algorithm continues through the whole signal. All the different ranges are half a cycle, when adding all the same ranges together the number of cycles for a certain stress range is know (International and Indexed by mero, 1997). This algorithm is an embedded function in Matlab.

6.5 Limitations

The first five seconds in the time series has been cancelled due to unreasonable high tension. It is assumed that the time series consists of the dynamic responses counting both wind and wave. No cyclic tests has been performed on the Bjørnafjord clay, an approach for this clay have been to consider the cyclic test done on the clay from Gjøa. The location of Gjøa is north east of the Bjørnafjord assumed to have nearly the same sediments as in the Bjørnafjord.

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6.6 Methodology

6.6.1 The Cyclic Responses

The peak value for the time series is 3221 kN at 44 sec, after multiplication with a load factor as stated in CEN (2013). The time series consists of the dynamic responses counting both wind and wave. For this purpose the series has been scaled and the maximum amplitude got bigger to fit the ULS peak load at 3805 kN for line five shown in table 4.4.

Based on Andersen (2007) the amplitude of the signal should be⌧cyas presented in figure 6.1. The maximum amplitude Fcyfor the signal has been scaled up to be half the value of the dynamic responses for wind and wave 668 kN.

The series was first scaled down to a mean value of zero to get positive and negative values for the multiplication. To scale up the signal the cyclic amplitudeFcygot divided with the peak value for the signal at 44 seconds which lead to a factor of 1.4567. This factor got multiplied with the series which was scaled to zero. The signal has now reached the peak ULS at 3805 kN at 44 seconds. The mean value for the signal is the average load Faat 3137 kN shown as the yellow line in figure 6.4. The red line presented in the diagram represents the Peak ULS-load, and the blue signal is the scaled up time series with the biggest cyclic amplitude Fcyat 44 seconds counting 668 kN.

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Figure 6.4:Scaled and blown up cycle responses for the time series from Orcaflex.

6.6.2 Equivalent Number of Cycles N

eqv

The needed shear strength for the chosen plate area at a certain penetration depth is computed by using equation 5.1 and solving forsu,mean without⌘ the empirical reduction factor. The resistance forces needed wereFa andFcy. The plate areaAplatefor line five is set to be 25 m²with a needed penetration depthzat 14 m. The bearing capacity factor Nc at this depth is 11.37 and the undrained shear strengthsu22.5 kPa. All these values has been taken from the anchor design for line five given in appendix B. Andersen (2007) uses the undrained shear strength measured from the DSS test and triaxial cell. The laboratory tests and diagrams used for these procedures are taken from the clay at Gjøa. The undrained shear strength used for the contour diagrams is taken from equation 3.1 and 3.2 and scaled to as^DSS_u with the same ratio as for the Gjøa clays^c_u= 0.85s^DSS_u (Christensen and Eiksund, 2009). In appendix B the calculations for determined⌧cy/s^DSSu and⌧a/s^DSSu

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6.6 Methodology

is presented.

The diagrams used to determined the Neqv for anchor line five are from the Gjøa clay and has⌧a = ⌧cy which is not similar to the conditions in the Bjørnafjord. Figure 6.5 shows the ratio between⌧a/s^DSS_u and⌧cy/s^DSS_u at x and y-axis when the soil goes to failure at = 15%for a different number of cycles. The black line in diagram 6.5 represents the ideal load case for the Gjøa clay⌧a =⌧cy. For the case in the Bjørafjord when only the static load is considered⌧a/s^DSS_u is 0.48, this is the starting point making the Neqv-diagram. A red line with gradient 1:1 has been drawn from 0.48 and through the blue lines representing the number of cycles to failure in figure 6.5. The ratio between the black and red line shown crossing the blue line forN = 1 in figure 6.5 got multiplied with the y-values for the red line crossing for N 1, 10, 100 and 1000 shown as the green lines. These values gave the = 15%line in figure 6.6. The different strain curves has been placed in the diagram 6.6 with the same ratio between them as for the Gjøa clay in (Christensen and Eiksund, 2009) since nostrain accumulationprocedure as suggested by Andersen (2007) has been done for the Bjørnafjord clay.

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Figure 6.5: Cyclic shear stress⌧cy/s^DSSu average shear stress⌧a/s^DSSu and number of cycles to failure represented by the blue lines for Gjøa clay (Christensen and Eiksund, 2009) p. C-3.

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6.6 Methodology

Figure 6.6:Cyclic shear stress⌧cy/s^DSS_u and equivalent number of cycles Neqvfor different strains given by the colored lines.

6.6.3 Cycle Count

To count the cycles in the time series from Orcaflex the signal was put into an already embedded Matlab function FFT. The output was a vector of complex numbers wherein each value is frequencies from the time-domain. Figure 6.7 shows the frequencies plotted against the magnitude of the signal. Here a visualization of which frequencies not wanted can be seen. The filter will cancel all frequencies over 0.2 and under 1.8. These signals are only noisy for the original signal and will disturb the cycle counting. Several embedded filter functions in Matlab were tried out such as a low-pass filter and signal smoothing, but without working properly. The filter made in the Matlab script is ideal when cutting out all the unwanted frequencies.

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Figure 6.7:The absolute values of the FFT in the frequency domain

A new vector with the filtered Fourier transformations is now an input in the IFFT function. This function converts the filtered frequency signal back to the time-domain.

The filtered signal is now a smoother signal than the original and a cycle counting can be performed, this is shown in figure 6.8 where both the original and filtered signal is plotted from a section of 200 seconds. To count the cycles the rainflow-counting algorithm in Matlab is used. From this counting a histogram, see figure 6.9, shows how many cycles the signal has for a certain stress range. The cycles in the histogram represent a series lasting half an hour. The ULS conditions demand a storm duration of three hours, therefore the cycles from the histogram get multiplied by six.

Figure 6.8: Filtered and original signal shown the small cancelled amplitudes before the cycle counting.

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6.6 Methodology

Figure 6.9:The number of how many cycles there is for different stress ranges.

6.6.4 Baste in the Equivalent Number of Cycles

The highest ratio between⌧cy/s^DSS_u is 0.1026, this value has been set into the rainflow count table as the ratio for the peak tension that gives failure at 3 cycles. A distribution of the ratio⌧cy/s^DSS_u as a percentage of the cycles and their tension has been computed to perform a baste in diagram 6.6 this distribution is shown in table 6.1. The basting could not be performed when the maximum ratio⌧cy/s^DSS_u was lower than the beginning of the strain curves in figure 6.6.

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Table 6.1:Table with the tension range based on the histogram in figure 6.9.The table represents the distribution of⌧cy/s^DSSu with the highest value marked bold as 0.2026.

Tension Range [kN]

Cycles 30 min

Cycles

3h storm % ⌧cy/s^c_u.DSS

20 30 180 0,05 0,0054

40 42,5 255 0,11 0,0108

60 56 336 0,16 0,0162

80 48,5 291 0,21 0,0216

100 41 246 0,26 0,0270

120 33,5 201 0,32 0,0324

140 24,5 147 0,37 0,0378

160 10,5 63 0,42 0,0432

180 8 48 0,47 0,0486

200 4 24 0,53 0,0540

220 2,5 15 0,58 0,0594

240 1,5 9 0,63 0,0648

260 2,5 15 0,68 0,0702

280 1 6 0,74 0,0756

340 0,5 3 0,89 0,0918

360 1 6 0,95 0,0972

380 0,5 3 1,00 0,1026

Side-Anchored floating bridge at Bjørnafjorden : Design analysis for plate anchors in clay and cyclic bearing capacity evaluation