Validation of a new hydroelastic code for flexible floating structures moored in waves

(1)

Date June 2009 Author Kessel, J.L.F. van

Addresu Delft University of Technology

Ship Hydromechanics Laboratory

Mekelweg 2, 2628 CD Delft

TU Deift

DeIft University of Technology

Validation of a new hydroelastic code for

flexible floating structures moored in waves

by

J.L.F. van Kessel

Report No 1625-P

₂₀₀₉

Proceedings of the ASME 2009 28th International

Confe-rence on Ocean, Offshore and Arctic Engineering, OMAE

2009, May 31June 5, Honolulu, Hawaii, USA, ISBN:

978-0-7918-3844-0, OMAE2009-79989)

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WELCOME FROM THE CONFERENCE CHAIRS

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OMAE2009: Welcome from the Conference Chairs

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ft Cengiz Ertekin H. Ronald Riggs Conference Co-Chair Conference Co-Chair

OMAE 2009 OMAE 2009

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On behalf of the OMAE 2009 Organizing Committee, it is a pleasure to welcome you to Honolulu,

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OMAE2009: Welcome from the Conference Chairs

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OMAE2009: Message from the Technical Program Chair

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MESSAGE FROM THE TECHNICAL PROGRAM CHAIR

-! Welcome to the 28th International Conference on Ocean, Offshore and Arctic

Engineering (OMAE 2009). This is the 28th conference in the OMAE series

guided by and influenced significantly by our friend and colleague, Subrata K.

Chakrabarti. It was a shock for me to learn that he had passed away so suddenly;

all involved with this conference express sincere condolence to his family, friends

and colleagues (the sentiments echoed by all of us are eloquently expressed in

the dedication included in this program). It is a great honor for me to have been

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leadership and friendship greatly. Although this series of conferences was

formally organized by ASME and the OOAE Division of the International

Petroleum Technology Institute (IPTI), it was Subrata's skill and dedication to this

Daniel T. valentine

_{division of ASME that made this series of conferences the success that it has}

Technical Program Chair

OMAE 2009

been and IS today.

The papers published in this CD were presented at OMAE2009 in thirteen

symposia. They are:

SYMP-1: Offshore Technology

SYMP-2: Structures, Safety and Reliability

SYMP-3: Materials Technology

SYMP-4: Pipeline and Riser Technology

SYMP-5: Ocean Space Utilization

SYMP-6: Ocean Engineering

SYMP-7: Polar and Arctic Sciences and Technology

SYMP-8: CFD and VIV

SYMP-9: C.C. Mei Symposium on Wave Mechanics and Hydrodynamics

SYMP-lO: Ocean Renewable Energy

SYMP-li: Offshore Measurement and Data Interpretation

SYMP-12: Offshore Geotechnics

SYMP-13: Petroleum Technology

The first eight symposia are the traditional symposia organized by the eight

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symposia organized and encouraged by members of the technical committees to

focus on topics of current interest. The 9th symposium was organized to

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offer papers in the areas of renewable energy, measurements and data

interpretation, geotechnical and petroleum technologies as they relate to ocean,

offshore and polar operations of industry, government and academia.

The first symposium, Symposium 1: Offshore Technology was always Subrata

Chakrabarti's project. It was typically the largest of the symposia at OMAE. His

exemplary work on this symposium provided the experience and guidance for

others to continue to develop the other symposia. Symposium 1 in conjunction

with the OMAE series of conferences is Subrata's legacy. The Executive

Committee has a most difficult yet honorable task of finding a successor to carry

on this important annual symposium in offshore engineering. We are all grateful

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OMAE2009: Message from the Tecirnical Program Chair

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for the inspiration and encouragement provided to all of us by Subrata.

Please enjoy the papers and presentations of OMAE2009.

Daniel 1. Valentine, Clarkson University, Potsdam, New York

OMAE2009 Technical Program Chair

(6)

OMAE2009: International Advisoiy Committee

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INTERNATIONAL ADVISORY COMMITTEE

R.V. Ahilan, Noble Denton, UK

R. Basu, ABS Americas, USA

R. (Bob) F. Beck, University of Michigan, USA

Pierre Besse, Bureau Veritas, France

Richard J. Brown, Consultant, Houston, USA

Gang Chen, Shanghai Jiao Tong University, China

Jen-hwa Chen, Chevron Energy Technology Company, USA

Yoo Sang Choo, National University of Singapore, Singapore

Weicheng C. Cui, CSSRC, Wuxi, China

Jan Inge Dalane, Statoil, Norway

R.G. Dean, University of Florida, USA

Mario Dogliani, Registro Italiano Navale, Italy

R. Eatock-Taylor, Oxford University, UK

George Z. Forristall, Shell Global Solutions, USA

Peter K. Gorf, BP, UK

Boo Cheong (B.C.) Khoo, National University of Singapore, Singapore

Yoshiaki Kodama, National Maritime Research Institute, Japan

Chun Fai (Collin) Leung, National University of Singapore, Singapore

Sehyuk Lee, Samsung Heavy Industries, Japan

Eike Lehmann, TU Hamburg-Harburg, Germany

Henrik 0. Madsen, Det Norske Veritas, Norway

Adi Maimun Technology University of Malaysia, Malaysia

T. Miyazaki, Japan Marine Sci. & Tech Centre, Japan

T. Moan, Norwegian University of Science and Technology, Norway

G. Moe, Norwegian University of Science and Technology, Norway

A.D. Papanikolaou, National Technical University of Athens, Greece

Hans Georg Payer, Germanischer Lloyd, Germany

Preben 1. Pedersen, Technical University of Demark, Denmark

George Rodenbusch, Shell lntl, USA

Joachim Schwarz, JS Consulting, Germany

Dennis Seidlitz, ConocoPhillips, USA

Kirsi Tikka, ABS Americas, USA

Chien Ming (CM) Wang, National University of Singapore, Singapore

Jaap-Harm Westhuis, Gusto/SBM Offshore, Netherlands

Ronald W. Yeung, University of California at Berkeley, USA

(7)

OMAE2009: Copyright Information

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Proceedings of the

ASME 2009 28th International Conference on Ocean, Offshore and Arctic

Engineering (OMAE2009)

May 31

- June 5, 2009

Honolulu, Hawaii, USA

Copyright © 2009 by ASME

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ISBN 978-0-7918-3844-0

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Proceedings of the ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering

OMAE2009

May 31 - June 5, 2009, Honolulu, Hawaii, USA

OMAE2009-79989

VALIDATION OF A NEW HYDROELASTIC CODE FOR FLEXIBLE FLOATING STRUCTURES MOORED

IN WAVES

).L.F. van Kessel

Offshore Engineering Department, Deift University of Technology,

Delft, The Netherlands

ABSTRACT

Natural periods of elastic modes can be in the range of

the wave spectrum for relatively long and slender floating

bodies. As a result elastic body deformations such as

vertical bending, horizontal bending and torsion may be

significant and need to be taken into account in the

hydrodynamic analysis of very large floating structures. The behavior of flexible floating bodies in waves has been studied at Delft University of Technology. For this purpose

the existing linear three dimensional diffraction code DELFRAC was modified to take into account the

fluid-structure interaction of deformable bodies at zero forward

speed in waves.

This paper focuses on the validation

of the new

hydroelastic code for flexible floating structures moored in

waves. Numerical results are validated by model

experiments of a flexible barge in waves from different

headings. In addition, the obtained results are compared with results from other existing hydroelastic programs. In

general it is shown that numerical results show good

agreement with experimental values. KEYWORDS

Ship hydroelasticity, Flexible barge, FEM, Seakeeping, Vertical Bending, Horizontal Bending, Torsion, Model tests, Numerical simulation.

INTRODUCTION

It is common practice in seakeeping theory to describe

the displacements of a floating body in the six rigid-body

modes surge, sway, heave, roll, pitch and yaw. This is

appropriate when the eigenfrequencies of elastic

deflections of a structure are significant higher than the

frequencies of the first order wave loads.

On the other hand, hydroelastic behaviour should be

taken into account when natural periods of elastic

deformations are in the range of the wave spectrum. This

may be the case for relatively long and slender floating

bodies like a Mobile Offshore Base, floating airports [3], large containerships [4], FPSOs [9] and large aircushion supported structures [10].

A general linear hydroelastic theory was presented by

Bishop and Price in 1979 [1]. Some adjustments and

extensions to this theory were presented by various other authors [2, 3, 5] in recent years.

The existing DELFRAC code was modified to take into

account displacements and deformations of flexible

floating structures at zero speed in waves. DELFRAC is a linear diffraction program developed at DeIft University of Technology. So far the prograni was used to compute the

motion behaviour of rigid structures, i.e. floating bodies with six degrees of freedom. The code has now been

extended to take into account elastic deformations which

are represented by an arbitrary number of degrees of

freedom.

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This paper is aimed to show the agreements and differences between the numerical results and results

obtained from model tests which were performed within the GIS-HYDRO organization as described by Remy et. al. [6].

NUMERICAL APPROACH

In the present approach a 3D Finite Element Method (FEM) is

coupled with a 3D hydrodynamic diffraction method in order

to calculate the motion behaviour of flexible floating

structures at zero speed in waves. First, a finite element method is used to compute the dry natural modes of the

structure in absence of external forces. The structure is modelled by a finite number of elements each having

constant mass and stiffness properties. Next, the fluid

actions associated with the distorting three-dimensional wet

structure are determined by a hydrodyriamic model involving translating pulsating sources.

Several other authors used a similar approach, see for

instance [1 - 91. However, the main difference between the

new program and the existing codes is the coupling between

flexible modes and possible aircushions underneath the

structure. The present paper discusses the results of the new

program and focuses on conventional floating structures in

which aircushions are not taken into account. For this reason aircushion theory is omitted in the next sections.

DRY FINITE ELEMENT APPROACH

The structure is modelled by a finite number of elements

each having constant mass and stiffness properties (m, E, G,

I). The nodes of the elements suffer generalized

displacements {U4 ={U,.U2...UN } where N denotes the

number of nodes of the element concerned, and e means shows that the quantity relates to an element. The matrix equation of motion for an element may be replaced by the

following general equation of motion:

[M]{U} + [B]{U}

+ [K]{U} = {F} + {Q} (1)

where:

[M]

= mass matrix

[B]

= damping matrix

[K] = stiffness matrix

U) = nodal displacement vector

F) = vector of external forces

(Q} = vector of external concentrated nodal loads The matrices [M},[B],[K] are real and symmetric since

they are associated with the dry structure. The fluid loading is represented by an external generalized force matrix {F" }.

When the damping and forcing term are ignored, the equation of motion reduces to:

[Mj{U}+[K]{U}=O (2)

The nodal displacements associated with the natural

frequencies of the structure are harmonic and may be

written as:

The real and positive eigenvalues co, (r=1,2...N) are the

dry natural frequencies, each associated with a characteristic dry eigenvector of the form:

D,}

(5)

In which r

is the principal mode and N represents the

number of degrees of freedom of the dry structure. The generalized displacement vector of the r -th principal mode at the j -th node is:

Drj } = !1,,v,, ,O,, , 0,, } (6)

The dry eigenmodes are orthogonal and therefore mass and stiffness matrices can be generalized:

[ni] = {D}T [M]{D} (7)

[k] {D}T [K]{D) ₍₈₎

Note that [m] and [k] are diagonal matrices of which the

first six modes represent rigid body modes,

WET HYDRODYNAMIC COUPLING

When considering floating bodies, it is customary to

determine the wave forces on the captive structure based on

the undisturbed wave potential 0,, , the solution of the

diffraction potential çb1 and the added mass and damping coefficients of the structure oscillating in any one of the

modes of motion in still water based on the motion potentials

0, . The motions of the structure are then determined by solving the equation of motion taking into account the wave forces, added mass and damping and restoring terms in all

degrees of freedom.

The contribution of the total potential due to the discrete pulsating source distributions over the structure may be

expressed as:

Ø () =

a ()G(, i)is

(9)

(4)

{U) =

{D}e'

(3)

(10)

= strength of a source on surface element s due to motion mode j

(x)

= potential in point due to

_j

-mode of

motion

The unknown source strengths a,, are determined based on

boundary conditions placed on the normal velocity of the

fluid at the centres of the panels:

= ,n=i,2,...,N, (10) The right hand side of the above equation depends on the case to be solved. If the source strengths for determination of the diffraction potential are required the normal velocity

vector becomes:

-

,

'1,,lN.l) - _a,,,

-The added mass and damping coupling coefficients are found by applying normal velocity requirements:

in which the panel index m covers the panels on the

structure and n,, are the general directional cosines for the panels. The directional cosines of the six rigid body modes

(1j6) are:

= cos(n,,,,x,) = cos(n,,,,x3) = cos(n,,x,) (13) = X,,,2 fl,,,3 - X,,,3 fl,,,2 = X,,,3fl,,,1-X,,,1fl,,,, =

in which X,,,, are the coordinates of the centre of a panel

relative to the body axes. De directional cosines of the

flexible modes

_(j

> 6) can be written as:

= fl,nuJ (14)

where a, represents the displacement of the panel in the / -th mode. From -the solutions of -the source streng-ths for all

in which:

= diffraction potential at panel 1t.

X,, wave force in the n-th mode

N,, = number of panels involved in the force in the ,, -mode.

/?,, = generalised directional cosine of panel k related

to the ,, -th mode.

area of panel i related to the force in the -th

mode.

Added mass and damping coefficients follow from:

= - Ic

_[p

,

0 ,k

(16)

b,,, = -

In1[Pw0i

'7,,.A

in which O,k is the motion potential on panel k obtained

from Eq. (9).

The restoring coefficients follow from:

where w is the displacement in vertical direction from eq.

(6). The total deflection and motion of the structure may be expressed as the sum of displacements in the principal

modes:

{U}={X,,}{D,,} (18)

where the vector {x, } with (r = 1,2...N) is a set of

principal coordinates. These coordinates can be subdivided

into rigid-body (.v,) and flexible-body (x1. ) contributions,

where:

XR{XI,X2 ...x6} (19)

XN} (20)

Substitution of eq. (18) in eq. (1), multiplying by {D}T and

making use of the results of eq. (15), (16) and (17) results in the general equation of motion:

in which: these cases the wave force vector A',, added mass a, and

N,,

x

= total number of panels of the structure

= X,X,X3= a field point

damping coupling coefficients b, can be obtained. The wave force follows from:

A = A, A,, A, = location of a source

G(X,A) = Green's function of a source in A relative ₍₁₅₎

to a field point

A',, =_pw2(Ok

+ 0,/k) fl,, k"I

= surface element of the body

aøj

j=1, N

(12)

c =-pgn,,,

S,,.k (17)

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2 ([ni] + [A]) {v} + w[B] {.v} +([k]+ [C]){x} = {X) (21) where:

= added mass matrix = damping matrix

= hydrostatic stiffness matrix

[X] = vector with wave forces

The deflections at any point of the floating body can be

expressed as:

{u} ={x,.}(u,}e"

(22)

COMPARISON OF NUMERICAL RESULTS WITH

EXPERIMENTAL RESULTS

This section describes the results of the new hydroelastic code for a flexible barge moored in waves. These results will be compared with experimental values described by

Remy et. al. [6]. In addition, computational results

presented by Tomasevic [8] and Senjanovic et. al. [7] will be used for comparison as well.

The flexible barge consists of twelve caissons which are connected by a steel beam at deck level. Figure 1 shows the geometry of the model. Caisson nr. 1 is referred to as

the bow section and the stern of the floating body is

located at the end of caisson nr. 12.

L-l- .l

r

Figure 1: Geometry of structure

C3 II llC5 C12

The bow caisson has a beveled shape as shown in Figure

2, while the other eleven caissons are rectangular with

the following characteristics:

Length = 190 mm Width = 600 mm Depth = 250 mm Draft = 120 mm Mass = 13.7 kg KG = 163mm K< = 225 mm

The KG-value is the distance above keel level and K, the roll radius of gyration. The bow caisson has a mass of 10 kg and a roll radius of gyration of 213 mm. According to

Remy et. al. [6] the centre of gravity of the bow caisson

above keel level is 87 mm. However this value seems unrealistic compared to the shape and KG-value of the

other caissons. For this reason the KG-values of the bow

caisson is set to 163 mm in the present computations

which is equal to the other caissons.

The length of the steel beam on top of the caissons is

2.445 m, which corresponds to the length of the structure since all caissons are separated by a gap of 15 mm. The beam is located 307 mm above keel level and has a square cross section of 1 cm2. The bending stiffness is

175 Nm2 and the torsional stiffness is 135 Nm2.

Fi'ure 3: Pane/model

The third column of Table 1 shows the amount of

damping as percentage of the critical damping which should be added to the potential calculations based on

decay tests [8]. The fourth column shows the added

damping used by Tomasevic and Senjanovic. The amount of damping added to the new computations can be found in the last column.

<I

>

Figure 2: Geometry of bow caisson

A beam consisting of 501 elements with concentrated

loads representing the floaters was modeled in NX

Nastran. The obtained dry mode shapes were used to

compute the wet displacements of the structure in waves as described in the previous section.

Figure 3 shows the panel model used by the diffraction

program. The same figure shows the panels below and

above the waterline. However, elements between the

caissons and above the waterline are excluded from the computations of added mass, damping and wave forces.

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Table 1: Additional modal damping as percentage

of the critical damping

DISCUSSION OF RESULTS

Figure 4 shows the absolute displacements of the flexible

barge in

head waves of

5.0 rad/s. Numerical and

experimental results for the flexible barge in a frequency

range of 2.5 - 10.5 rad/s in different wave headings is

shown in Figure 6 to Figure 17. The present section

discusses the motion behaviour of the flexible barge in

head waves and quartering waves of 120 deg.

LJ_I

0,0 01! 020 0. 0 014 043 0430*30!? 002 003 013 034 003 4, 024 044!02'!4 . (15300

04003*02001*333420043023052

l.'

?' "'0 033 043043 I'll OW 404

'II

Figure 4: Displacements and absolute RAOs of the

flexible barge in head waves of 5.20 rad/s

Figure 6 shows the heave RAOs of caisson 1, 3, 5, 7, 9

and 12 in head waves. The dots mark experimental

results

while the

blue and red lines correspond to

numerical results of Tomasevic and the new calculations

respectively. It can be seen that the maximum heave responses are largest at the ends of the structure and

decrease towards the centre of the structure. There is a

good agreement between the computations and the

experiments. However computations for caisson 7 tend to overestimate the heave responses of the experiment.

The overestimation by the computations occurs around 5.20 rad/s and

is mainly due to a

collision between

caissons in the centre of the structure. Although this

effect is not taken into account in the linear computations, it reduced the heave motions of caisson 7 during model

tests. Figure 5 shows the results of the three floaters in

the centre of the structure based on the new

comput:ations. This figure clearly shows that a collision between caissons 6, 7 an 8 must have occurred during

the model tests.

Figure 5: Collision between caissons in head waves of

5.20 rad/s

Figure 7 shows the mean heave responses which are the average values of the six measurement points on caissons

1, 3, 5, 7, 9 and 12. It is shown that the results of the new hydroelastic code are close to those of Tomasevic

and agree well with model tests.

Pitch R.AOs of the caissons in head waves are presented in Figure 8. The numerical results are slightly higher than experimental values at low wave frequencies. Pitch

responses at the end of the structure are again largest at the ends and decrease towards the centre of the structure. This shows the importance of the modal analysis, since a

rigid body approach will not show good agreement with experimental results.

The new pitch results agree well with those of Tomasevic.

Both numerical methods overestimate the results of model tests in case of caisson 7, which was also the case

for the heave RAOs.

Figure 9 shows a good agreement of the mean pitch RAO in head waves between numerical and experimental results. This is due to the fact that the computed pitch RAOs of the individual caissons correspond well with

5 _{Copyright © 2009 by ASME} 1 Surge 2 0 2 2 Sway 4 0 4 3 Heave 6 5 6 4 Roll 4 5 7 5 Pitch 6 5 7 6 Yaw 5 0 5 7 Vertical Bending 7.5 5 7.5 8 Horizontal Bending 2.5 9 8.5 9 Torsion 2.5 7 4.5 10 Mode 10 5 11 Mode 11 5 12 Mode 12 5 Decay Tornasevic /

Mode test Senjanovic Caic.

(13)

model tests and the contribution of caisson 7 to the mean

pitch RAO is small.

The vertical bending in head waves, which is the pitch

motion of caisson 1 minus the pitch motion of caisson 12, is presented in Figure 10.

Figure 11 shows the roll responses of six caissons in case the wave heading is 120 deg. It can be seen that there is

an excellent agreement between numerical and

experimental results. The mean heave RAO in Figure 12 also shows a good agreement, this means that there must be a good agreement with respect to phase angles in the roll motions of the individual caissons as well. The results

of Remy et. al. [6] and Senjanovic et.

al [7] are also

plotted in the same graph.

The mean heave and mean pitch transfer functions are

plotted in Figure 13 and Figure 14 respectively. It can be seen that experimental results and numerical results from the various authors correspond well with each other.

Figure 15, Figure 16 and

Figure 17 show a good

agreement between numerical and experimental results of

vertical bending, horizontal bending and torsion

respectively.

CONCLUSION

The present paper describes part of the validation of the

new hydroelastic code developed at Delft University of Technology. Model experiments of a flexible floating

barge moored in waves from different directions were

used to

validate the new hydroelastic program. In

addition, numerical results presented by Senjanovic, Tomasevic and Remy et. al. were used for comparison. The new computations indicated that collisions between

the caissons in the centre of the structure must have

occurred during the model tests in head waves.

In general it was shown that there is a good agreement between model tests and results obtained by the new

hydroelatic program.

As such the new program is a suitable tool to accurately predict the motion behaviour of flexible floating structures. Besides it is also possible to easily obtain displacements

and accelerations at any point at the flexible floating

structure in waves.

REFERENCES

Bishop, R.E.D. and Price, W.G. Hydroelasticity of

ships. Cambridge University Press, 1979.

Bishop, R.E.D., Price, W.G., and Wu Y., A General Linear Hydroe/ast/city

Theory of Floating

Structures

Moving in a Seaway. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and

Physical Sciences, 1986, pg. 375-426.

Malenica, S., Molin, B., Remy, F., and Senjanovic, I., Hydroe/astic response of a baige to impulsive and

non-impulsive wave loads. 3rd International Conference on

Hydroelasticity in Marine Technology, 2003.

Malenica, S., Senjanovi, I., Tomasevi, S., and

Stu mpf, E., Some aspects of hydroelastic issues in the design of ultra large container ships. 22nd IWWWFB,

2007, pg. 133-136.

Newman, 3. N., Wave effects on deformable bodies. Applied Ocean Research, 1994, vol 16, nr 1, pg. 47-59.

Remy F., Molin B., and Ledoux, A., Experimental and numerical study of the wave response of a flexible barge. Hydroelasticity in Marine Technology, 2006, pg. 255-264.

Senjanovic I., Malenica S., and Tomasevic, S.,

Investigation of shio hydroelasticity. Ocean Engineering, 2008, pg. 523-535.

Tomasevic, S.,

Hydroelastic model of dynamic

response of container shios in waves. PhD Dissertation, FSB Zagreb (In Croatian), 2007.

Ledoux, A., Mary, C., and Couty N., Modelling of

Springing and Whi'ping of FPSO's in a Time Domain Sea-keeping Tool. ISOPE 2004, pg 666-671.

[10j Van Kessel, J.L.F. and Pinkster, J.A., Wave-induced structural loads on different types of aircushion supported

structures. Proceedings of ISOPE, 2007,

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1.60 1.40 1.20 E 0.80 0,60 0.40 0.20

Heave RAO - Caisson I Heave RAO - Caisson 3 Wave headIng: 180 dog Wave heading: 180 deg

ExpOata -Tomasevic -CaIc E 0060 0 040 020 0.00 . - 000 2.00 3.00 4.00 5.00 6.00 7.00 300 900 lW ii 00 2.00 3.00 4.00 5.00 6.00 7 00 7300 9.00 10.00 11 00 1.00 0.80 0.60 0.70 ?0,6o O 0.50 o 0,40 0.30 0.20 0.10 0,00

Wave frequency (racEs]

Heave RAO - Caisson 5 Wave heading: 180 dog

2.00 300 400 500 600 7.00 800 900 10.00 11 00 Wave frequency fradls]

0.00 2.00 3.00 400 500 600 700 900 900 10.00 1100 I 00 090 080 070 O 060 O 0.50 o 040 0.30 020 0 10 000 1 40 1 20 100 .000 E 0060 040 020

Wave frequency (radls] Wave frequevcy [racEs]

Figure 6: Heave RAOs of caissons 1, 3, 5, 7, 9 and 12 in head waves

Moai9 Heave RAO Wave reading: 180 dey

200 3.00 4.00 500 600 700 800 9.00 10,00 1100 Wave frequency (radis]

Figure 7 Mean heave RAO of all caissons in head waves

Wave frequency [racEs)

Heave RAO - Caisson 7 Wave heading: 180 deg

300 400 5.00 600 700 000 900 1000 1100 Wave frequency (radls(

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0.14 0.12 0.10 '0,0.08 0.06 0 004 002 0.14 0.12 E 0.10 P 'Or 0.08 002 Fop Data - Torvasevic -Caic. 0.00 2.00 300 400 500 600 700 800 9.00 10.00 II 00 Wave frequency jradlsj

0.00 200 300 4,00 500 6,00 7.00 8.00 9)0) 10.50 1100 Fop Data -Tomasevic -Cab E p 0 0.14 0 12 5 0 10 008 006 -0 004 0.02 0.00 2.00 0 14 0 12 5 0 10 1008 006 0 004 0,02 000 2.00 3.00 4.00 000 6.00 7,00 800 9.00 10.00 1100 014 0.12 E 0.10 10.08 006 0 004 002

Wave frequency [radls] Wave frequency (radls]

Figure 8: Pitch RAOs of caissons 1, 3, 5, 7, 9 and 12 in head waves

Mean Pitch RAO Wave heading: 180 deg 007 0.06 6 005 04 003 0 0.02 001 000 2,00 3.00 4.00 500 600 700 800 9.00 1000 1100 Wave frequency IradlsJ

F,iure 9: Mean pitch /Z-40 of all caissons in head waves

Vertical Oending Wave heading: 180 deg

3,00 400 500 6,00 700 8,00 9,00 10,00 1100 Wave frequency (rad/s)

000 200 3.00 4.00 500 600 700 8.00 9.00 10.00 1100 0 18 0 16 0.14 0 12 0 10 000 006 004 0,02 000 2.00 3.00 4,00 500 600 7 00 800 900 10,00 11 00 Wave frequency (rad/s]

Figure 10: Vertical Bending £40 of caissons in head waves

Fop Data -Tomaseslo

-Cab,

Pitch RAO - Caisson 1 Pitch RAO - Caisson 3 Wave heading: 180 deg Wave heading: 180 dog

Pitch RAO - Caisson 5 Pitch RAO - Caisson 7 Wave heading: 180 dog Wave heading: 180 dog

Wave frequency (rad/sJ Wave frequency (rad/s] Pitch RAO - Caisson 9 Pitch RAO - Caisson 12 Wave heading: 180 deg Wave heading: 180 deg

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0.00 2.00 3.00 4.00 500 6.00 1.00 800 900 10.50 1100 000 200 300 400 500 600 700 800 9.00 1000 1100 0,00 2,00 3.00 4.00 5,00 6.00 700 600 9.00 1000 11,00 Wave frequency ]rad/sJ

025 020 Is o 1)10 005 030 020 020 - 0 75 0I0 005 2.00 3,00 400 500 600 700 8,00 900 10.00 1100 000 2.00 300 400 500 600 700 000 9,00 1000 1100 3.13) 400 500 600 700 800 900 Wave frequency [radts]

Fiqure 11: Roll RAOs of caissons 1, 3, 5, 7, 9 and 12 in quartering waves of 120 deg

Mean Roll Transfer Function Wave heading: 120 deg 030 Enp Data 025 Rnrrry 0.20 - Senjanovic_{- Caic.} 0 15 010 0.00 200 300 400 500 600 700 000 cc 1000 1100 Wave frequency ]radlsJ

Figure 12: Mean roll RAO of all caissons in quartering waves of 120 deg

1000 1100

Roll RAO - Caisson I Roll RAO - Caisson 3 Wave heading: 120 deg Wave heading: 120 deg

Wave frequnecy jradls] Wave frequency ]rad/s] Roll RAO - Caisson 5 Roll RAO - Caisson 7 Wave heading: 120 deg Wave heading: 120 deg

Wave frequency rudis] Wave frequency ]radls) Roll RAO - Caisson 9 Roll RAO - Caisson 12 Wave headng. 120 deg Wave heading: 120 deg

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0.35 0.30 O 025 020 0 15 0 0 10 005 3.00 400 5W 600 100 600 900 10.00 11 00 Wave frequency Irad/s)

Wave frequency Irad/s]

waves of 120 deg 0,30 0,25 020 015 010 Torsion Wave heading: 120 dug

0,00

2,00 300 4W 500 600 700 800 9,00 10,00 1100 Wane frequency (lad/SI

000

2,00 3,00 4,00 500 600 700 800 900 10,00 II 00 Wane frequency Irad/sI

F,'ure 13: Mean heave RAO of all caissons F,iure 14: Mean pitch RAO of all caissons in

0.45 0.40 030-g1025 020 015 010 005-000 - 0.00 2.00 3.00 4.00 5.00 600 7.00 000 9,00 10.00 1100 2,00 3.00 4.00 500 600 7,00 000 900 10.00 1100 Wave frequency Irad/si

F,iure 15: Vertical bending RAO in quartering Fiqure 16: Hor,onta/ bending RAO in quartering waves of 120 deg

F,ure 17: Torsion RAO in quartering waves of 120 deg

Mean Pitch Transfer Function Wave heading: 120 deg Mean Heave Transfer Function

Wave heading: 120 deg

quartering waves of 120 deg Horizontal Bending

Wave heading: 120 deg

in quartering waves of 120 deg Vertical Bending