Numerical analysis of wave-induced responses of floating bridge pontoons with bilge boxes

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Mas ter Thesis

Numerical analysis of wave-induced responses of fl oating bridge pontoons with bilge boxes

Espen Kleppa June 2017

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Technical University of Denmark

Nils Koppels Allé, Bld. 403 DK-2800 Kgs. Lyngby Denmark

Phone (+45) 4525 1360 Fax (+45) 4588 4325

www.mek.dtu.dk

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Abstract

In the offshore industry it has become more common to install damping ”devices” on large displacement structures like FPSO and SPAR platforms. The damping devices are often circular discs on the bottom of the structure and it is installed to reduce the motion of the structure, mainly the heave motion. In this thesis a bottom disc is installed on a bridge pontoon supposed to be located in a fjord in Norway. The objective is to study the motion behavior of the pontoon, study how it is dependent on the C_d-coefficient and develop a simple numerical model for estimating the KC- number by vertical oscillation velocity of the pontoon.

A surface model is created in the hydrodynamic analysis tool Ansys Aqwa and run for several different disc ratios in a Sea State given by the Norwegian Public Road Administration (NPRA). The main purpose of the flange is to increase the added mass of the structure and shift the Natural Period away from the wave spectrum.

Several simulations with different disc ratios is performed and the natural period is shifted away from the wave spectrum when the disc ratio is increasing. However, a downside by adding the bottom disc is that a cancellation effect of the damping load is occurring at certain periods.

By adopting KC-numbers andCd-coefficients from earlier studies on similar structures, a procedure is developed for estimating the KC-number for the pontoon. By using linear potential theory and Morison drag linearization, motion analysis is performed in Aqwa by adding Morison drag elements on to the disc to obtain the effect of drag loads on the pontoon.

A second and harsher sea state is tested for the same pontoon for comparison and the same procedure is carried out for this sea state. It is found that the pontoon is quite independent of theC_d- coefficient for the initial moderate sea state, where the resulting vertical velocities are very small and are giving very small values of KC-number. For the second sea state the peak period is shifted and interacting more with the resulting motion RAOs of the pontoon, resulting in larger response and hence larger vertical velocities. The resulting KC-numbers for the second sea state is within range of the initially assumed KC-numbers.

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Preface

This Master thesis was prepared at the department of Engineering Design and Applied Mechanics at the Technical University of Denmark in fulfillment of the requirements for acquiring a Nordic Master degree in Maritime Engineering. The thesis is submitted at the Technical University of Denmark and Chalmers University of Technology for obtaining a double degree as part of the Nordic Master program.

Kongens Lyngby, June 23, 2017

Espen Kleppa (s151091)

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Acknowledgements

This Master thesis is written as a final closure of my Nordic Master degree in Maritime Engineering the spring of 2017. It is assumed that the reader has basic knowledge in hydrodynamic potential flow theory, viscous drag forces and motion response analysis.

It is primarily intended to engineers working within the offshore and maritime industry, as well as researchers and students seeking to research the subject matter further.

In the process of writing this thesis, I have gotten support and guidance from several people that I would like to mention.

I would sincerely like to thank my main supervisor Ass. Professor Yan-Lin Shao for his patience, guidance and support while working on this project. We have had many interesting discussions during the last months.

Further I would like to thank my company supervisor Xu Xiang at Statens vegvesen in Norway. He has taught me and guided me through the software used in the thesis and given good input and support on several matters.

I would also like to thank the people working at the fjord crossing department at Statens vegvesen for taking me in while I was working there for 6 weeks during the project period.

A thank also goes to my co-supervisors, Ass.Professor Wengang Mao and Professor Emeritus Preben Terndrup Pedersen for their support during the project period.

Lastly I would like to thank my parents. They always have faith in me and have supported me whenever I have needed it trough out my whole education.

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CHAPTER 1 Introduction

As a part of the Norwegian Public road administrations (NPRA) project to improve the road system between Kristiansand and Trondheim, there are a lot of fjords that has to be crossed either by tunnels or bridges. One of the fjords to be crossed is Bjørnafjorden between Stord and Os in Hordaland county. It is decided that a bridge is to be built for this crossing. At the decided location of this crossing, the bridge has to be more then 5 km wide and the depth of the fjord can reach over 500 meter, which implies that the best solution will be some kind of floating bridge instead of pier foundations. Different bridge concepts has to be investigated to find the best solution for the crossing. One of the bridge designs that are the most promising, is a long floating bridge supported by pontoons with span widths between 100 meter and 200 meter.

((a)) Potential graphical appearance of the bridge over Bjørnafjorden Baezeni) (2017)

((b)) Existing bridge, Nordhordalandsbrua, over Salhusfjorden/Osterfjorden Salhusgjen- gen (2016)

Figure 1.1: Graphical appearance of the bridge concept investigated in this thesis.

In the south end, the girder is supported by a cable-stayed bridge to increase the height from the sea surface to the girder, this to meet the requirement of a navigation channel for large ships going up the fjord. This bridge type can either be side moored to the seabed at different locations along the girder, or end anchored in both ends of the bridge Xiang et al.. In Figure 1.2 the bridge over Bjørnafjorden is shown graphically how it can potentially look compared to the existing Nordhordalandsbrua.

As Nordhordalandsbrua is showing, this type of bridge concept has been built before so the concept is not new. However the dimensions are tripled compared to

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the length of the bridge and pontoon size and the sea environment is also harsher in Bjørnafjorden with wave periods between 3 - 7 seconds and wave heights up to 3 meter.

1.1 Scope of the thesis

In the offshore industry it has been more and more common to install ”damping devices” on large displacement offshore structures. These ”damping devices” can be damping plates on Spar or Tension Leg platforms, it can be bilge keels on ship shaped FPSO or it can be circular bottom flanges on a circular FPSO or a floating wind turbine structure. The application of damping devices is wide and use full.

((a)) Eni Norges Sevan 1000 FPSO, Go- liat, installed with bottom flange. Thomas (2016)

((b)) Typical SPAR platform with damping plates. News (2017)

Figure 1.2: Two different cases where damping devices are used in the offshore industry.News (2017)

The damping devices are installed on these structures to decrease the motion of the structure, most importantly the heave motion. The damping devices will trigger flow separation, hence viscous damping on the pontoon motion similar to other floating vessels with bilge keels attached. The pontoons on Nordhordalandsbrua had no damping device installed when it was built in in the early 1990’s. It was no need for it as the sea environment in the fjord was pretty modest compared to common north-sea offshore environment. For Bjørnafjorden the sea environment is estimated to be a bit harsher then the case for Nordhordalandsbrua, and NPRA is investigating the possibility of using a bottom flange on the pontoons for the bridge crossing Bjørnafjorden.

In this thesis, the effects of the bottom flange on the pontoon will be investigated mainly when it comes to heave motion response. An important parameter when it

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1.2 Earlier studies 3

comes to heave motion analysis is the drag coefficient, i.e. C_d-coefficient. The C_d- coefficient is closely related to theKeulegian−Carpenter-number, or KC-number, which can be defined in Equation 1.1

KC = V T

D (1.1)

where V is the flow velocity oscillation amplitude, T is the period of the oscillation and D is the characteristic length dimension of the structure. The investigation will include motion response analysis and the effect ofC_d-coefficient on the pontoon in addition to developing a procedure for estimating KC-number of the pontoon based on data from earlier similar structures. The investigation will also include the tuning of the natural period of the structure in regards to the width of the flange.

1.2 Earlier studies

There have been several earlier studies where the effect of heave damping plates has been investigated. To get an idea of the scope of the problem a literature study has been done on several papers written by highly rated naval architects.

Figure 1.3: Behaviour of Damp- ing Coefficient Z against diameter ratio D_d/D_c from paper of Tao and Cai(2004)

Tao and Cai(2004)has done experiments on a circular cylinder with a bottom plate attached which is similar to a Spar or TLP platform with bottom plate. In offshore environment with long wave periods, resonant heave oscillation might occur and cause damage to risers and mooring sys- tems.

The goal for their study is to investigate the effects of a systematic disc ratio variation,D_d/D_c, on the vortex shedding pattern, and study the corresponding associated heave damping effect origi- nating from the vortex shedding of the cylinder and damping plate.

In the paper they are taking a close look on the KC- number and the frequency parameterβ, and their effect on the vortex shedding pattern. It was found in the study that the diameter ratio of the disc and cylinder was strongly influencing the results of the vortex shedding and viscous damping, but also that the the diameter ratio was strongly

dependent on the KC-number and that the form drag on the structure is highly dependent on the KC- number.

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In the case study presented in the study they did numerical simulation on a full size Spar platform with a draft of 198.1 meter, 39 meter diameter hull, and a range of five different disc diameters ranging form 42.1 meter to 78 meter. They run 6 different configurations for the five ratios, and one bare hull configuration, for three different amplitudes of oscillation and at a period of 28 sec. The corresponding KC-numbers where 0.15, 0.44 and 0.74. The resulting behaviour of damping coefficient Z with the diameter ratio is shown in Figure 1.3. One can see from the curve that theZagainst Dd/Dcshow a weak non- linear trend at the different KC-numbers, and when the disc diameter is getting larger the curves seem to flatten out and converge to a specific Z-value.

The OMAE2016-55059 paper by Shao et al. (2010) considering motion analysis of cylindrical floating structure with heave damping bilge boxes attached, more specifically the Sevan 1000 FPSO. The application used here is stochastic linearization of the drag term in Morison Equation, and an inhouse code is developed and verified by comparing with WADAM software results. The linearized drag forces is added in to the standard panel method and generates RAO’s which is corresponding well to the model test results of the FPSO shown in Figure 1.4.

((a)) Heave RAO Comparison, Hs=8.3m, Tp=12.6s,γ = 2.0. Cylindrical FPSO with bilge box, Ballast condition.

((b)) Heave RAO Comparison, Hs=17.2m, Tp=16.8s, γ = 3.0. Cylindrical FPSO with bilge box, Ballast condition.

Figure 1.4: RAO comparison from Shao et al. (2010) for The Sevan 1000 FPSO.

Figure 1.4 shows the model test RAO’s compared to the numerical analysis RAO’s.

The red line has the estimated Cd included, and the green has both the Cd and a percentage ofCm added. The application presented in this paper is estimating the RAO’s quite well. Two sea states are considered in the analysis and they are shown in Table 1.1.

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1.2 Earlier studies 5

Sea state Hs [m] Tp[m] Wave spectrum

1 8.3 12.6 JONSWAP,γ = 2.0

2 17.2 16.8 JONSWAP,γ = 3.0

Table 1.1: Discription of sea state The KC-numbers for the

two sea states is calculated by using the significant relative heave motion at the centre of the FPSO, and they are quite low. 0.21 for sea state 1 and 0.42 for sea states 2, which

agrees well with other studies done for large displacement structures. The estimated KC number andCd-coefficients are used to compare to the study of TTao and Cai (2004) reported earlier in this thesis. The damping coefficients for KC = 0.74 showed good agreement with the numerical results in Tao and Cai (2004), but for KC = 0.15 and 0.44 they are underestimated.

In both of the studies of Tao and Cai(2004) and Shao et al.(2010), they conclude that flow separation viscous effect has an impact on the added mass and natural period of the structure and that the KC-number clearly has e big influence on the drag coefficient for large displacement structures. It would be interesting to study these same aspects on the bridge pontoon with the bottom flange.

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CHAPTER 2 Theory

A floating structure has 6 degrees of freedom and for the pontoon investigated in this thesis, the 6 degrees are oriented as shown in Figure 2.1.

((a)) Global body motion modes for the bridge et.al.

((b)) Local body motion modes for the pontoon.

Figure 2.1: Definition of body motion modes for the global bridge and the local pontoon.

As can be seen from Figure 2.1 the orientation for the local coordinate system on the pontoon is different then that for the global coordinate system on the bridge.

The global coordinate system has defined its surge motion in the direction of the longitudinal bridge girder, while the surge motion for the pontoon is defined perpendicular to the bridge girder in the longest length parameter of the pontoon. This make logic sense if the pontoon is considered as a ship in a hypothetical situation. In this thesis only the pontoon will be considered and the local coordinate system and motion definition will be used.

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2.1 Basic assumptions

The following theory outlined in this chapter will have its reference mainly from the Aqwa Theory Manual ANSYS (2013a) and the text book ”Sea loads on ships and offshore structures” by O.M.Faltinsen (1990).

If a fluid is assumed incompressible, invicid and irrotational, potential theory can be used for calculating a hydrodynamic problem. The pontoon in this thesis is assumed to be a large volume structure and thus the potential flow effects are the dominating effects. There will also be viscous effects present, but these are secondary effects and can be included empirical. It is convenient regarding mathematical analysis to use the velocity potential to describe the velocity vector in a irrotational fluid.

In an incompressible fluid the velocity potential has to satisfy the Laplace equation in equation 2.1.

∂²ϕ

∂x² +∂²ϕ

∂y² +∂²ϕ

∂z² = 0 (2.1)

The pressure in the fluid originates fromBernoulli^′sEquationand can be written as in Equation 2.2 for a incompressible, invicid and irrptational flow, where C is a arbitrary function of time andVis the velocity vectorV(x, y, x, t)=(u, v, w).

p+ρgz+ρ∂ϕ

∂t +ρ

2V∗V=C (2.2)

Relevant boundary condition for solving problems in potential theory are the Kinematic Free Surface Condition (KFSC) and the Dynamic Free Surface Condition (DFSC). The KFSC basically states that the fluid particles on the free surface remain on the free surface, while the DFSC states that the water pressure is equal to the constant atmospheric pressure p0 on the free surface. In linear form based on the velocity potential being proportional to the wave elevation,ξ, KFSC and DFSC can be written:

∂ξ

∂t = ∂ϕ

∂z on z= 0 (kinematic condition) (2.3)

gξ= ∂ϕ

∂t on z= 0 (dynamic condition) (2.4) The linear wave theory and its derivation can be seen in many books and papers.

If a structure is considered floating in incident regular waves with amplitude ofξa

and the wave steepens is small. Basically linear theory means that the wave induced motion and load amplitudes are linear proportional to ξa. The derivation of the

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2.2 Response 9

theory will not be done in this paper, but the reader is referred to take a look in O.M.Faltinsen (1990), page 16, Table 2.1 if it is of interest.

Moving on to the introduction of environmental conditions, statistical measured data is important when it comes to designing a vessel for the correct conditions it will operate in. It is assumed that the sea can be considered as a stationary random process, and it can be referred to as a short term description of the sea. Calculating a wave spectrum for a sea state is highly dependent on the significant wave height and period. The method used describing the wave spectrum will be derived later in the report.

2.2 Response

The response of the pontoon is considered operating in steady state waves, this implies that the linear dynamic loads and motions acting on the a floating structure are harmonically oscillating at the same frequency as the exciting wave loads on the structure. The hydrodynamic wave problem can mainly be divided in to two kind of forces;

1. Wave excitation loads, also calledFroud-Krylov forceandDiffraction force.

These loads occur when the structure is restrained from oscillating and there are incident regular waves acting on the structure.

2. Added Mass, Damping and restoring forces are the forces and moments on the structure when the structure is forced to oscillate at the same frequency as the wave excitation frequency. This counts for every single rigid-body motion mode, and there are no incident waves.

2.2.1 Added Mass and Damping

When a floating structure is forced to oscillate, the structure is generating radiation waves that are outgoing from the structure. This forced motion results in an oscillating fluid pressure on the body surface, and by integrating this fluid pressure over the body surface one can obtain the the resulting forces and moments on the body, also called added mass and damping loads. The added mass is the force due to the water that has to be displaced as the structure oscillates, and the damping is the force due to the energy carried away from the structure through radiated waves from the oscillating body.

In this thesis the heave motion of the pontoon is to be studied more thorough.

To find the fluid motion when a structure is free to move in a fluid domain, it is convenient to use the velocity potential. When the velocity potential is determined, the pressure can be found by using the linearized Bernoulli’s equation. By excluding hydrostatic pressure and integrating the remaining linearized pressure over the mean position of the body, a vertical force on the body will be obtained. The linear force obtained can be written as in Equation 2.5.

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F₃=−A₃₃d²η₃

dt² −B₃₃dη₃

dt (2.5)

whereA33is the added mass in heave direction andB33is damping in heave direction.

2.2.2 Froude-Krylov and Diffraction

The unsteady fluid pressure form the incident waves when a structure is restrained from moving can be divided into two effects. One is the unsteady pressure induced by the undisturbed waves, while the other is the force due to the fact that the structure is changing this unsteady pressure field. The forces are called Froude-Krylov and Diffraction force respectively. The diffraction problem can be solved in a way similar to the added mass and damping problems where one has to solve the boundary value problem for the velocity potential. The boundary condition where the normal deriva- tive of the diffraction velocity potential for the submerged body has to be opposite and of identical magnitude as for the normal velocity of the undisturbed wave system.

This ensures that the normal component of the total velocity on the structure is zero.

2.3 Viscous drag forces

Figure 2.2: Illustration of the strip dz on the on the pontoon.

When it comes to calculating loads on circular and cylindrical structures or members where viscous forces are active, using the Morison’s equation is a nice approach. O.M.Faltinsen (1990) is describing the Morison’s equation for a horizontal force acting on a strip of length dz of a circular cylinder to be:

dF =ρπD²

4 dzC_ma₁+1

2ρC_dDdz|u|u (2.6) where ρis the mass density of water, D is the cylindrical diameter,a1and uis the accel-

eration and velocity of the undisturbed horizontal fluid,CmandCd is the mass and drag coefficients determined empirically by different parameters like Reynolds num- ber, KC-number etc. Applying this theory to the pontoon shown in Figure 2.2, an approximation ofdz will be the diameter/beam of the hull projected in the longitudinal x-direction. In a case where the structure is is moving, equation 2.6 can be extended to Equation 2.7.

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2.3 Viscous drag forces 11

dF =ρπD²

4 dz(Cm+ 1)a1−ρπD²

4 dzCmη¨+1

2ρCdDdz|u−η˙|(u−η)˙ (2.7) η is the displacement of the moving body at the strip dz, hence η˙ andη¨ are the velocity and acceleration due to the body motion respectively, or in other words, the relative acceleration and velocity. In this thesis the numerical model will be run in a potential flow theory solving software, hence the vertical viscous effect on the flange has to be modeled in. Therefore there will be added linearized Morison Elements in the numerical model to act as the viscous effects on the flange of the pontoon. This vertical drag force due to the flange can be expressed with a Morison-like formula, using the area of the bottom disc:

dFdrag =1

2ρSCd|u|u (2.8)

where S = ((π*D²_d)/4 +Dd*Ld) is the area of the whole bottom plate, whereDd

is the disc diameter andLdis the length of the mid section of the bottom disc, Xiang et al.(2017). The drag term in Morison’s equation is important in order to get correct viscous effect.

2.3.1 Added mass coefficient C

m

In Section 2.3 the Morison’s Equation was given as:

dF =ρπD²

4 dz(C_m+ 1)a₁−ρπD²

4 dzC_mη¨+1

2ρC_dDdz|u−η˙|(u−η)˙ (2.9) According to the paper of Shao et al.(2010) the first item in Equation 2.9 is connected to the added mass of the structure while the middle item with the term (Cm+ 1) in it is connected to excitation loads. Further, when the damping devices are modeled into the numerical panel model it is important to not double book the potential flow contribution of the added mass coefficient. The added mass coefficient can be seen like this: Cm =C_m^pot+C_m^visc. When the damping devices are modeled into the numerical panel model,C_m^pot= 0should be used. This is indeed the case for the pontoon investigated in this thesis. For larger displacement structures the added mass can have great effect on the structures RAO’s and Natural period.

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2.3.2 Morison Drag Linearization

There are two linearization methods in the literature. One is Stochastic linearization, the other is regular wave linearization which is applied in this thesis. The quadratic relative velocity term in the drag part of the Morison’s Equation can be written

|ur|ur=|(uw−ub)|(uw−ub)whereuwis the velocity of the water particle motion in the fluid, andub is the velocity of the floating body motion. In the ANSYS (2013a), the Morison Drag Linearization described for a submerged tube/cylinder, and it is done by replacing the non-linear term|(uw−ub)|by a factor multiplied by the root mean square of relative velocity in order to create an equivalent linear term. This factor is according to Borgman ,α= (8/π)^1/2. The linearized drag force at a cross section of a circular cylinder can then be expressed as

dF_drag= 1

2ρDC_Dαu_rms(u_w−u_b)dl (2.10)

dFdrag =1

2ρDCDαurmsuwdl−1

2ρDCDαurmsubdl (2.11) whereu_rmsis the root mean square of the transverse directional relative velocity, and practically theb= ¹₂ρDC_Dαu_rmsdl is the damping coefficient for the structure.

The first item on the right hand side of Equation 2.11 is representing the excitation loads, while the last item on the right hand side is representing the damping loads on the structure in question. Then the resulting linearized drag force can be found by integrating the linearized velocity terms over the whole submerged length of the structure, whereL1 is the draft of the tube and (L^′+L1) is the whole submerged tube:

dFdrag =1 2ρDCD

∫ L^′+L₁ L₁

αurmsuwdl−1 2ρDCD

∫ L^′+L₁ L₁

αurmsubdl (2.12)

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2.4 The Equation of Motion 13

2.4 The Equation of Motion

The basic model for the wave-induced motions of a floating structure can be described by the Linear Harmonic Oscillator. If a floating body can only respond in one single degree of freedom, the vertical heave motion, the equation of motion can be written as:

F_ex(t) = (m+a)¨x₃(t) +bx˙₃(t) +cx₃(t) (2.13) whereF_exis the exciting force,mandais the mass and added mass,bis damping, cis the hydrostatic stiffness andx¨₃,x˙₃andx₃is the responding acceleration, velocity and displacement of the body respectively Bingham.

When a floating body is oscillating and it is assumed that both the force of the body and the response of the body are time harmonic at frequency, the force and response can be written as:

Fex(t) =A[X^Rcos(ωt)−X^Isin(ωt)] =ℜ{

AXe^iωt}

(2.14)

x₃(t) =ξ^R₃cos(ωt)−ξ₃^Isin(ωt) =ℜ{ ξ₃e^iωt}

(2.15) By inserting Equation 2.14 and2.15 into Equation 2.13, the equation of motion can be rewritten as:

AX=ξ3[−ω²(m+a) +iωb+c] (2.16) and the resulting transfer function can be written as:

ξ2

A = X

−ω²(m+a) +iωb+c (2.17)

x2(t) from Equation 2.15 is the resulting displacement of the oscillation. When differentiating it the velocity and acceleration of the oscillation can be found:

˙

x3(t) =ℜ{

iωξ3e^iωt}

(2.18)

¨

x3(t) =ℜ{

−ω²ξ3e^iωt}

(2.19)

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2.5 Natural period

Every structure has its own Natural Period and is an important parameter when assessing a structure’s motion amplitudes. If a floating structure is oscillated by waves with a period that lies in the vicinity of the structures natural period, the structure might start to oscillate within the resonance period range, which can give dangerously large motions to the structure. This is something that has to be avoided at all cost.

Due to high damping or low excitation levels it might be hard to evaluate the response at resonant periods. The natural period of a structure is highly dependent of the added mass and the water plane area, and it is easier to get a high natural period for a semi-submersible then for a FPSO. According to O.M.Faltinsen (1990), the natural period can be given for any structure in any motion mode as

T n_i= 2π

√A_ii+m Cii

(2.20) where Aii is the added mass,m mass of the structure andCii is the hydrostatic stiffness. For the pontoon in question, in addition to the heave hydrostatic stiffness of the structure there has to be added a stiffness from the bridge girder resting on the pontoon.

Figure 2.3: Illustration of the bridge girder section for each pontoon.

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2.5 Natural period 15

According to et.al., the girder can be considered as a simply supported beam and calculated according to Equation 2.21. In Figure 2.3 is a simple illustration of how the bridge girder is likely to look while resting on the pontoon.

kwb=48EIy

L³ (2.21)

whereE = 200 GPA is the elastic module for alloy steel,Iy= 18.7m² is the area moment about lateral axis andL= 200metersis the distance between two pontoons.

All of these properties are fetched from the et.al.. For heave motion Equation 2.20 can be rewritten in to Equation 2.22

T n3= 2π

√ A33+m C33+kwb

(2.22) The pontoon has a high added mass when the flange is added, and this added mass can be used to shift the pontoons natural period away from the range of the peak period (Tp) of the sea state. This way the risk of reaching resonance level will minimized.

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CHAPTER 3 Methodology

Calculating forces and motions in a dynamic system can be very challenging and there are many factors that play a role on the final results. To estimate heave motion for a relatively large full scale pontoon with a bottom flange with proper accuracy, obviously a CFD analysis is the best choice. For a master thesis it might be to computationally expensive to do a proper CFD analysis in 3-D at this design stage.

Therefor it is more common to use an approach that involves the Morison Equation in early design stages to model the viscous contributions for a standard panel model.

This method has shown itself to be very efficient with acceptable accuracy in the design stage of large volume offshore structures. With this method the analysis can operate in frequency domain with linearized Morison drag. Some of the uncertainties of the adopted method will also be addressed in chapter 3 and 4.

3.1 Ansys Aqwa software

In order to get data to use in the evaluation of the pontoons characteristics, establish- ment of a proper panel model has to be done. The software used to generate motion analysis results in this thesis is Ansys Aqwa. This is a Hydrodynamic Analysis tool and it is a modularised, fully integrated hydrodynamic analysis suite based around 3-D diffraction/radiation methods and estimate the first order wave loads on a floating structure. Aqwa is assuming an ideal fluid with an existing velocity potential and is using linear hydrodynamic theory described in Chapter 2. Ansys Workbench implementation provides a modern interface to develop and solve Aqwa panel models.

The panel models are modeled in a sub-software called Spaceclaim and it gives out both surface models and solid modeles. For the hydrodynamic diffraction analysis to work, surface panel models has to be used. Aqwa is using basic hydrodynamic potential theory and definitions to calculate the motion results, and it also uses Morison’s Equation and Morison linearization to calculate the hydrodynamic forces, which is explained in Section 2.2.

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3.2 Design phase

Through this thesis period, 6 weeks has been spent in Norway at the offices of NPRA’s, getting input, inspiration and good discussions on many aspects of the thesis. As mentioned earlier, the pontoon design and dimensions is not finally set when this thesis is written. At the modeling stage of the thesis it is decided on a set of design parameters that will be used, with dimensions on both the the pontoon and flange. On studies NPRA and their consulting firms has done, a flange width of between 5-6 meters has been most commonly used. The design parameters used to model the pontoon in Spaceclaim is shown in Table 3.1.

Diameter hull 25 [m]

Total length of hull 62 [m]

Total width of hull 25 [m]

Draft 9.25 [m]

Weight ca 14000 [ton]

Diameter flange 37 [m]

Flange thickness 0.5 [m]

Flange width 6 [m]

Table 3.1: Parameters pontoon

Figure 3.1: Pontoon modeled in Spaceclaim with properties as in Table 3.1

The model in Figure3.1 is the surface model given from Spaceclaim. Spaceclaim is pretty similar to many other modeling software’s, where a sketch is made and the sketch can be pulled/extruded to make a 3-D model.

Figure 3.2: Morison element circle beams modeled in Spaceclaim.

It is possible to model as many surfaces as one requires, and it is possible to group the surfaces together. There where some important functions in the modeling that had to be done in order to make a functional panel model. The x-y-plane in the coordinate system seen in Figure 3.1 is corresponding to the fictive horizontal water plane in the analyzing software Aqwa, so everything modeled in negative z-direction would technical be submerged and a part of the draft of the structure. Another important modeling function is to check that all the surface normal’s is pointing outwards.

The surface normal indicates what is the outside and inside of the model, and the

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3.3 Aqwa Analysis Tool 19

direction the normal arrow points is the outside. If some of the normal’s are pointing inwards, an error will occur when motion analysis is run on the model.

In order to account for the viscous effect in this potential theory based Aqwa software, one has to model some circle beam sketch elements in Spaceclaim. The elements are sketched with 0.5 meter vertical clearance to the flange. The elements are positioned in the middle of the flange on the longitudinal parts. The beams can only be sketched in straight lines, hence this would cause some difficulties in the circular ends of the pontoon. The way this was done was to split the half circle into three equally long long beams and connect them as shown in Figure 3.3. Of course this elements could be split into several more smaller beams to fit the circular flange better, but due to the time consuming changing of parameters in a later stage of the analysis it was found that splitting the beam into three beams was sufficient.

3.3 Aqwa Analysis Tool

Further when the surface model is ready, it can be used as input to the analysis tool Aqwa. There are a lot of parameters that has to be assigned before running the diffraction analysis, which is using the source distribution method ANSYS (2013a) to calculate the first order loads on a structure. By introducing the source distribution over the mean wetted surface, the fluid potential can be expressed:

ϕ(X⃗) = 1 4π

∫

S₀

σ(⃗ξ)G(X, ⃗⃗ ξ, ω)dS where X⃗ ∈Ω∪S₀ (3.1) Here ϕ(X)⃗ s the velocity potential ξ is the position of a source, G, is the Green’s Function andωis the angular frequency. By applying the surface boundary condition where:

∂ϕ

∂n =

{ −iωnj for radiation potential

∂ϕ

∂n for diffraction potential

in Equation 3.1, the source strength over the mean wetted hull surface can be determined. The Hess-Smith constant panel method is deployed into Aqwa to solve for the source strength, and the mean wetted surface is divided into quadrilateral or triangular panels.

It is assumed that the potential and the source strength within each panel are constant and taken as the corresponding average values over that panel surface.

A point mass has to be assigned for the panel model to have any weight at all.

one can assign a value or let Aqwa calculate it by the draft of the structure. Then by solving for the hydrostatic solution, the mass of the structure is calculated. From Spaceclaim the Morison elements has been modeled as simple circle line beams. In Aqwa these circle line beams can be assigned height, width, weight, Inertia, Added Mass coefficient and Drag coefficient. It is important that the added mass and mass

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properties do not get double booked, so mass, inertia properties and Added Mass coefficient is set approximately to zero. The Drag Coefficient is set to whatever value is desired. In Figure 3.3 the Morison Drag elements are visual as the gray rectangle’s along the flange. These plates will work as the viscous drag force on the structure in motion analysis.

Figure 3.3: Rectangular Morison Drag elements along the flange in Aqwa.

It is important to set a correct and us- able frequency for which the analysis will run for. The range of frequency and number of frequency that should be analyzed can be in- putted after preference, and can be program controlled. A frequency in the range of: ω ∈ 0.1<ω < 2.5 [rad/s] and 50 frequencies to estimate for would be sufficient for this thesis.

In hydrodynamic diffraction analysis like this where surface piercing hulls are considered, there is the occurrence of irregular frequencies.

These irregularities might cause large errors in the solution over a substantial frequency band around these frequencies. These errors cause abrupt variations in the calculated hydrody-

namic coefficients, and this is the case in all forms of surface piercing hulls. In Aqwa there is a function that can be selected to remove these irregular frequencies by using the internal lid method for the source distribution approach. In this method it is assumed that the fluid field is existing interior to the mean wet body surface, which is is satisfying the same free surface condition as the flouting structure. This interior mean free surface is represented in Aqwa by a series of internal LID panels, removing the irregular frequencies, ANSYS (2013b).

3.4 Verification of method

As mentioned earlier, hydrodynamic diffraction analysis including the source distribution method is used to analyze the structure presented in this thesis. In order to identify the validity of this method, a comparison of results from earlier studies has to be done. In the Doctoral thesis of Dr. Yan-Lin Shao, there is a section on diffraction of a simple truncated vertical circular cylinder. Resulting amplitudes of the linear wave excitation forces in surge and heave directions is presented in plots, and compared to results from Kinoshita et al. (1997). In order to validate the hydrodynamic diffraction analysis method used in this master thesis, the results from the reference truncated vertical cylinder Shao has to be replicated in that method.

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3.4 Verification of method 21

((a)) Excitation force Surge motion ((b)) Excitation force Heave motion Figure 3.4: Results for Excitation force in surge and heave motion for a truncated vertical cylinder in finite water depth Shao

Figure 3.5: Mesh of truncated cylinder in finite water depth from Ansys Aqwa

The cylinder is analysed with the same parameters as in the reference case: Radius = R, Draft = R, Water depth = 2R. The same methood is used to present the results in the plots, non- dimensionalizing the resulting force on the Y-axis with F/ρgR²A and the wave number multiplied with radius on the X-axis. To get the curves to fit as good as they do, an approximation of wave dispersion relation from the paper of Zai-Jin You was used You 2008. The most commonly use wave dispersion relation can be written as in Equation 3.3 and 3.2, and are derived under that assump- tion that there is no current, the water depth is constant, the waves are linear and small and that the flow is irrotational. A more explicit solution of the wave dispersion number was needed to fit the curve accurate.

k₀h=kh∗tan(h)∗kh (3.2)

ω²=gk∗tan(h)∗kh (3.3)

where ω is the wave angular frequency, k is the wave number, k0 is the wave number in deep water, g is acceleration of gravity and h is the mean water depth.

The following equation for the wave dispersion relation is one of the most accurate solutions presented in the paper of Zai-Jin You, and the method is developed by J.N.

Hunt You 2008.

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kh= (k₀h) vu ut1 +

[ (k₀h)

( 1 +

∑6

n=1

D_n(k₀h)ⁿ )]₋1

(3.4) whereD1=0,6666666666,D2=0,3555555555,D3=0,1608465608,D4=0,0632098765, D5=0,0217540484, and D6=0,0065407983. khis then multiplied with the radius of the cylinder and we obtain the the curves in Figure 3.6

((a)) Resulting excitation force Surge motion

((b)) Resulting excitation force Heave motion

Figure 3.6: Results for Excitation force in surge and heave motion for a truncated vertical cylinder in finite water depth.

Seen from Figure 3.6 one can see that the results from the validation analysis is fitting the results from the paper of Shao(2010) quite good. The shallow water results is fitting almost perfectly. It is now safe to say that the hydrodynamic diffraction analysis proceadure that is going to be used in this thesis will have good agreement with real model test results and that the method used is valid.

3.5 Correlation Analysis

A mesh has to be defined and the computational time is heavily dependent on the size of the mesh. There are pointers given in O.M.Faltinsen (1990) of what are reasonable mesh sizes and number of elements. The number of elements should be between 500-1500, and the mesh size should not be larger then 1/8 of the wave length. The Aqwa Theory Manual ANSYS (2013a) states the size relation ship should be 1/7 of the wave length, so its more conservative. For the sea state considered in this thesis, the wave length,λ, is calculated to be 61.6 meter at Tp = 6 sec. Then the element length can not be longer then 8/61.6 meter = 7.7 meter according to the pointer in

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3.5 Correlation Analysis 23

O.M.Faltinsen (1990). A correlation test was done to see how different mesh sizes was affecting the results.

The idea is to run a simulation in Aqwa for the same pontoon with mesh sizes with element lengths of1,2,3,4,5 [meter]. The simulation will be run for Froud-Krylov and Diffraction force, and Added Mass and the results will be presented in curves to be analyzed. The results from the analysis is shown in Figure 3.7.

((a)) Resulting Added Mass for correlation analysis

((b)) Resulting excitation Diffraction and Froud-Krylov force for correlation analysis Figure 3.7: Results for Excitation force in surge and heave motion for a truncated vertical cylinder in finite water depth.

Figure 3.8: Element size at 2 meter for the mesh of the panel model in Aqwa.

In Figure 3.7 a, there seem to be some dis- turbance on the results at approximately 0.2 Hz. A sudden peak like this in the results might indicate that the internal lid functions for removing irregular frequency was of when this analysis was done. A new simulation was run for the 3 meter element length with the internal lid on. The results from this simulation was nice and even like it was expected to be, and the results for this simulation can be Found in Appendix??Obviously the simulation with the smallest element length will give the most accurate results, but when ten’s simulations is going to be run, the computational time matter. According to the resulting curves, the 3, 4 and 5 meter length elements gives quite unre-

liable results for the Added Mass, while the 2 meter length element is giving quite similar results as the 1 meter length element. For the Diffraction and Froud-Krylov

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force, the results are all over quite close to the 1 meter length element. It is on decided on basis of the results for the Added Mass and computational time that the mesh element length used in further simulations is2 meter.

3.6 Environmental conditions

This thesis is written in cooperation with NPRA, and the bridge is supposed to be designed for a specific fjord which has a set of more or less specific environmental conditions. NPRA is still measuring the conditions in the fjord, but the preliminary data given for this thesis is given in Table 3.2 and corresponds to waves with 100-year return period for Bjørnafjorden.

Peak Period Tp 6 [sec]

Significant wave height Hs 3 [m]

Non-dimensional peak shape parameter 1.8 - 2.3 Table 3.2: Sea state parameters in Bjørnafjorden (”Sea State 1”)

In NPRA’s and their consultants investigation on the bridge, they have given out several OMEA-papers regarding several aspects of the interest. In their analyses the Jonswap Wave Spectrum is used to describe the sea state, a wave spectrum that will be adopted in this thesis from the DNV-RP-C205 rules and regulations DNV-GL (2010). Definition of a wave spectrum is also required in Aqwa when running motion analysis, and it is possible to choose which type to input. The Jonswap is based on the Pierson-Moskowitz Spectrum which is defined as

S_{P M}(ω) = 5 16

H_s²ω_p² ω⁵ exp

(−5 4

(ω ω_p

)₋4)

(3.5) where ω is the angular frequency and ω_p=2π/Tp is the angular spectral peak frequency. The Jonswap Spectrum can then be found according to Equation 3.6:

S_{J W}(ω) =A_γ∗S_{P M}(ω)∗γ^exp (

−0.5(_ω−ωp

σ ωp

))

(3.6) whereAγ is a normalizing factor,γis the non-dimensional peak shape parameter andσis the spectral width parameter. In general the Jonswap Spectrum has higher spectral density peak then the Pierson-Moskowitz Pectrum, and the peak varies in height whenγ is changed. From the wave spectrum it is possible to calculate the response spectrum of a floating body, by multiplying the wave spectrum by the squared Transfer Function of the body.

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3.7 KC-number estimation 25

3.7 KC-number estimation

As mentioned earlier the KC-number can be used to determine the drag coefficient of a floating structure, referring to Section 1.2. For a big structure with large displacement, as the pontoon, the KC- number is expected to be a low value, approximately

KC = 0.1 < KC < 0.5.

Based on the literature studies in Section 1.2 and the theory outlined in Section 2.2, a procedure has been developed to estimate the KC-number for the pontoon. The procedure is visualized in Figure 3.9 and indicates that a KC-number first has to be assumed in order to perform the analysis. If a constant KC-number is to be used in irregular waves, there are no rational way to decide it because of the varying period of the waves and motion of the structure. KC-numbers has to be assumed based on earlier literature and studies on similar structures.

Figure 3.9: Basics of the procedure used to identify the KC-number.

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The KC-number, or theKeulegian-Carpenter number, can be expressed:

KC = V T

D (3.7)

where V is the flow velocity oscillation amplitude, T is the period of the oscillation and D is the characteristic length dimension of the structure. The period and the length dimension are given by the sea state parameters and design parameters. The Characteristic length dimension is taken to be the beam of the bottom disc, as a lot of the flow separation will happen perpendicular to the length of the pontoon.

The oscillating velocity is not known until it is solved for, and is a key component to estimate the correct KC-number. There are several ways to determine a motion velocity, and using linear theory to simulate irregular sea, it is possible to find statistical estimates for the motion amplitude. In this thesis a 3 hour Most Probable Maxima (MPM) and Significant Value (H¹

3) will be used for the motion amplitude.

Figure 3.9 is visualizing the procedure used to identify the KC-number:

A sea state has been enlightened in Section 3.6, and the wave spectrum can be used together with the heave motion RAO’s to estimate a frequency domain response spectrum for heave motion shown in Equation 3.8:

SR(ω) =SJ W(ω)|H3(ω)|² (3.8) Here S(ω) is the Jonswap wave spectrum and|H₃(ω)²|is the heave motion RAO’s estimated in Aqwa. In all Equations in this section,ωrepresents the angular frequency.

The spectral momentsM0andM2can then be calculated from the response spectrum.

M₀=

∫ _∞

0

ω⁰(S_R)dω (3.9)

M₂=

∫ _∞

0

ω²(S_R)dω (3.10)

Spectral moments in 0^th and2^nd order are useful to estimate statistical parameters. The Standard DeviationSD is calculated in Equation 3.11 and represents the variation in the response data set.

SD=√

M₀ (3.11)

The spectral moments can also be used to find the mean response period,T2for the response data set.

T2= 2π

√M0

M2

(3.12)

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3.7 KC-number estimation 27

With all the statistical parameters estimated, the MPM for a 3 hour duration can be estimated assuming that the response is normal(or Gaussian) distributed and the wave spectrum is narrow-banded, using Equation 3.13 where t is the duration and logis the natural logarithm. The significant valueH1

3 of heave response can also be estimated by using Equations 3.14.

M P M =SD

√ 2∗log

( t T2

)

(3.13)

H¹

3 = 4√

M0 (3.14)

Now, Equation 3.7 gives a clear definition of the KC-number in a harmonically oscillatory flow, and the empiricalC_d-coefficient used in this thesis will be dependent on the KC-number. Despite the fact that a sea state of 3 hour duration is considered when calculating the MPM, the KC-number and hence the C_d-coefficient is held constant in the panel model analysis in Aqwa when obtaining RAOs, hence the C_d- coefficient is not dependent on time.

Resulting from the response in the Equation of Motion in Section 2.4, the vertical motion amplitude can be differentiated from the displacementx2:

x3(t) =ℜ{ ξ3e^iωt}

=⇒x˙3(t) =ℜ{

iωξ3e^iωt}

(3.15) The motion amplitudex˙3can then be implemented as the amplitude of the flow velocity oscillation of the structure. The KC-number can then be obtained by Equation 3.16.

KC_estimated= x˙3T p

D (3.16)

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3.8 C _d -coefficient estimation

In order to investigate how the heave motion of the pontoon is dependent on the Cd-coefficient, a range of Cd- coefficient had to be tested. In this thesis CFD analysis will not be performed as the focus for the thesis is optimization of the pontoon and analysis of response in irregular waves. For the pontoon configuration (hull+flange) investigated, there are no direct model test data available. It is worth mentioning that NPRA is performing model tests at Marintek for the bridge pontoon designs the summer of 2017. Given access to the test results would create many interesting Master project in the years to come.

Regardless, the available data when this thesis is written, are data for a circular hull with circular bottom flange, typically a FPSO or a Spar platform. So in this thesis Cd-coefficients for a cylindrical hull with circular bottom disc will be used. To have some clue of which range to stay in regarding the KC-number and Cd-coefficient, earlier literature is used. Tao and Cai(2004) investigated how a circular cylinder with bottom plate is dependent on the damping coefficient at different KC-number.

Figure 3.10 (2016) represents Tao’s findings with the 3 curves for KC-number 0.15, 0.44, 0.74. Also included in Figure 3.10 is the findings from Shao et al.(2016) where the Sevan 1000 FPSO geometry is tested for the same technique. The findings of Shao et al.(2016) are the three red points at ratio 1.24, and they are underestimating a bit on two of the curves, but it seem to give a good indication for the range of KC-number for the structure.

Figure 3.10: Behaviour of Damping Coefficient Z against diameter ratioD_d/D_c at different KC-numbers from paper of Shao et.al. (2016)

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3.8Cd-coefficient estimation 29

Adopting this method and testing it for the pontoon would be very interesting.

As can be seen from Figure 3.10 Tao and Cai has calculated the damping coefficient for a range of bottom plate diameter and hull diameter ratios, Dd/Dc, and the curves show how the damping coefficient is changing from a non-linear state to a linear state as the ratio is increasing. The pontoon should also be tested for this ratio.

In order to find the range ofCd-coefficients, the values from the curves in Figure 3.10 is adopted by using the Spars parameters presented in Table 3.3, the equations for added mass, ma, and damping coefficient, Z, of the Spar presented in Equation 3.17 and 3.20 respectively. The ratios that the damping coefficient will be estimated for is presented in Table 3.4, and the KC-numbers used are KC = [0.15, 0.44, 0.74].

Spar diameter (Dc) 39 [m]

Total draft (Td) 198.1 [m]

Disc thickness (td) 0.475 [m]

Table 3.3: Dim. Spar

Disc diameter (Dd) [m] Ratio (Dd/D_c)

39 1

42 1.077

45.1 1.16

48.4 1.24

53 1.359

57.72 1.48

65 1.66

70 1.8

Table 3.4: Disc diameter and Ratio.

ma =ρD_d³ 3 −

[πρD_c²

8 (Dd−D) +πρ(Dd−D)²(2Dd+D) 24

]

(3.17) where

D=

√

D²_d−D²_c (3.18)

The mass can be found using the geometrical parameters whereR_hullis the radius of the cylindrical hull of the Spar andR_{f lange}is the radius of the bottom plate:

m=ρ(πR²_hull(Td−tf) +πR²_{f lange}tf) (3.19)

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Z = 1 3π²

D_d²Dc

D_c²T_d+D²_dt_d

Cd

(m+m_a)/m∗KC (3.20)

Cd-coefficients can be found by rearranging Equation 3.20:

C_d= Z∗(3π²)(D_c²T_d+D²_dt_d)((m+m_a)/m)

KC∗(D_d²Dc) (3.21)

By using the three KC-numbers with corresponding damping-coefficients at a given ratio, polynomial fitting can be used to find theC_d-coefficients for any given KC-number between 0.15 and 0.74. It is chosen to evenly divide the KC-number range into 8 numbers with equal spacing ranging from 0.15 - 0.74.

Cd,i=p3+p2KCi+p1KC_i² (3.22) wherep1,p2and p3 can be estimated by a polyfit function in Matlab. Now that all of theseCdcoefficients are obtained for each ratio, Aqwa can be run by inputting theCd-coefficients on the Morison drag elements. The resulting RAO data will be used to calculate theM P M andH1

3 and a new set of KC-numbers can be calculated, by inserting them into Equation 3.7.

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CHAPTER 4 Results

In Chapter 2 and 3 the theory and methods used in this thesis is explained thoroughly.

The objective is to analyze the pontoon’s motion response when the flange is varying in width.

4.1 Shifting of Natural period

As mentioned earlier, adding the bottom will serve different purposes. Like raising the added mass and adding more damping in to the dynamic system. The pontoon has been simulated for 4 different flange sizes: Flange width,D_f = 0, 4, 6,8 [m].

Figure 4.1: Jonswap and Pierson-Moskowitz spectrum compared for the given metao- cean data.

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The Jonswap wave spectrum is shown in Figure 4.1 for the given properties in Section 3.6. The Jonswap spectrum has this extra peak factor γ to account for continuous growing waves. By using the Added Mass, mass and hydrostatic stiffness estimated by Aqwa together with the estimated structural stiffnessk_wb, the Natural Period for a structure can be calculated according to the theory stated in section 2.5 for a given range.

Figure 4.2: Added mass comparison for different flange width.

In Figure 4.2 the Added Mass is increasing with the increasing flange width as expected. The added mass of the structure is between 1.5 and 3 times the magnitude of the pontoon itself. The circtangle shape of the pontoon and bottom disc might have an impact on these large magnitudes.

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4.1 Shifting of Natural period 33

As can be seen from Figure 4.4 and 4.3, the Natural periods are compared in relation to the wave spectrum. Both natural period with and without bridge stiffness are calculated and shown in Figure 4.3 and 4.4 respectively. The natural period can be found using this graphic method. The wave spectrum is representing the energy in the waves at different periods, and the goal is to shift the natural period away from the wave spectrum as far as possible in order to avoid a resonance effect. The intersection point with the green intersection curve is representing the Natural period of the different structures. As can be seen there are large differences when adding the bridge stiffness. Studying the plot with the stiffness included, the pontoon with no flange is inside the wave spectrum, which is not ideal at all and risk of resonance is present. By adding the flange and increasing it, the Natural period is shifting away from the wave spectrum as predicted, and the risk of resonance is minimizing. For the plot where the bridge stiffness is excluded, all of the natural periods are safely out of range.

Figure 4.3: Natural period without bridge stiffness.

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Figure 4.4: Natural period with bridge stiffness included.

The resulting Natural Periods are listed in Table 4.1. These periods are resulting natural periods for head sea, hence sway, roll and yaw motion are not excited.

Flange width [m] Tn,3 incl. kwb [sec] Tn,3 excl. kwb [sec]

0 5,96 9,4

4 6,9 10,9

6 7,5 12

8 8,2 13,4

Table 4.1: Resulting natural period for different flange sizes

When studying the damping coefficient on the structure, there seem to be a negative side effect when adding a flange. As can be seen from Figure 4.5 the damping load is getting cancelled out at a certain period when adding the flange and adjust the width.

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4.1 Shifting of Natural period 35

Figure 4.5: Damping load comparison for different flange width.

For the pontoon with no flange, the damping load starts at zero and gives no cancellation effect along the increasing period. As the flange size increases the cancellation effect start to occur and a pattern can be seen where the cancellation period is increasing with the flange size. Without any damping loads the structure motion will loose this decaying motion effect from the damping, and the structure might be exposed to resonance effects at these certain wave periods. It is also worth noticing that for pontoons with 8 and 6 meter wide flange, the there is occurrence of negative damping load. As regular damping has a decaying effect on the body motion, negative damping will contribute to a resonance effect on the body. The occurrence for this negative damping effect might be due to errors in the mesh at this period for the pontoon. They should be expected to be >=0.

Numerical analysis of wave-induced responses of floating bridge pontoons with bilge boxes

Mas ter Thesis