energies
ArticleNumerical Study on Sectional Loads and Structural
Optimization of an Elastic Semi-Submersible Floating Platform
Yuliang Liu 1,2 and Takeshi Ishihara 2, *1

2*Changjiang Institute of Survey, Planning, Design and Research, No.1863 Jiefang Avenue,
Wuhan 430010, China; yuliang_liu@outlook.com
Department of Civil Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-8656, Japan
Correspondence: ishihara@bridge.t.u-tokyo.ac.jp

Abstract: This study investigates the sectional loads on an elastic semi-submersible platform for a
2 MW FOWT (floating offshore wind turbine) used in the Fukushima demonstration project. A watertank test is firstly carried out with an elastic model to study the dynamic responses and sectionalloads of the platform in regular and irregular waves. Numerical simulations are then performed using
multiple hydrodynamic bodies connected by elastic beams. The dynamic responses of the elastic
model are compared to those of a rigid model to clarify the influence of the structural stiffness on theplatform motion and mooring tension. The predicted sectional loads on the deck, brace and pontoon
by the proposed nonlinear hydrodynamic models show good agreement with the experimental dataobtained from the water tank test and a simplified formula is proposed to evaluate the distributionof the moments on the platform. Finally, the structural optimization of the elastic semi-submersible
platform is conducted. The sectional moments and fatigue loadings on the pontoons are significantly
reduced using the strut between the pontoons since the horizontal wave loads on the side column are
dominant and the vertical wave loads acting on the platform are relatively small due to the deep draft.
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Keywords: semi-submersible floating platform; elastic model; dynamic response; sectional loads;
structural optimization

Citation: Liu, Y.; Ishihara, T.
Numerical Study on Sectional Loads
and Structural Optimization of an
Elastic Semi-Submersible Floating
Platform. Energies 2021, 14, 182.
https://doi.org/10.3390/en14010182
Received: 26 November 2020
Accepted: 29 December 2020
Published: 31 December 2020
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Copyright: © 2020 by the authors. Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

1. Introduction
Floating offshore wind turbines (FOWTs) have been considered the best way toharvest wind energy in deep water regions. Compared to a bottom-fixed foundation,a floating platform has fewer constraints regarding water depth and soil conditions. For
FOWTs, however, the high costs associated with the design, construction, installation andmaintenance are problems that need to be solved. One of the strategies to reduce thelevelized cost of energy is increasing power generation [1] and decreasing the amountof steel used for the platform [2], which implies that a larger wind turbine and flexible
platform are designed. Thereafore, structure analysis and optimization for the platform are
critical for cost reduction and safe design.
Some aero-hydro-servo-elastic tools have been developed for the prediction of the
dynamic responses of platform and mooring lines [3,4]. The linear hydrodynamic force
coefficients of the platform have been obtained by a potential theory and presented as
the matrices of integrated hydrodynamic force coefficients for a rigid platform with six
degrees of freedom, which have been used in research. However, since the elastic responsesof the platform cannot be captured by a rigid body, the distributed loads and the elastic
models were proposed for structural analysis of the platform. Borg et al. [5,6] investigatedthe hydro-elastic interactions between the flexible floating platform and fluid by using
structural modal analysis and boundary element method (BEM). However, this study
was not validated by experiments. Silva de Souza and Bachynski [7] concluded that thehydroelasticity had negligible consequences for motion and load responses for a TLP

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https://www.mdpi.com/journal/energies

Energies 2021, 14, 182

2 of 22

FOWT. The sectional loads of the element of the platform can be easily assessed, as shown
in Ishihara et al. [8] and Suzuki [9–11] when the Morison’s equation was used to estimate
the hydrodynamic load on each element. Campos et al. [12] utilized Morison’s equation
to study the interaction between the wind turbine and the platform for a spar FOWT.
Morison’s equation performs well when the elements of the platform are slender, while it
has a limitation on the evaluation of diffraction force when the diameter of a cylinder isfive times larger than the wavelength, as mentioned by Faltinsen [13]. It is inapplicablefor some structures with large volumes where the diffraction loads are notable, such as
Ideol’s FOWT [14] and an advanced spar FOWT used in the Fukushima demonstration
project [15].
The elastic response is critical for the design of floating wind turbines and platforms,
especially when the wave frequencies are close to the eigenfrequencies, as pointed out by
Lamei and Hayatdavoodi [16]. Zhang and Ishihara [17] found that the natural frequencies
of a floating wind turbine with rigid and elastic platforms were different and the elasticity ofthe platform should be considered in the design of a wind turbine tower. To consider moreaccurate and efficient responses of the elastic platform, the distributed hydrodynamic loads
obtained from BEM were widely used without consideration of hydroelasticity [18–24].
In this approach, the platform was separated into multiple bodies connected by elastic
beam elements. The hydrostatic and hydrodynamic loading of each body is integrated
from pressure on the body surface by BEM and applied to the reference point of eachbody. Guignier et al. [18] compared the dynamic motion of a single body model anda multibody model based on the platform and observed little difference between the
motions of the two models. Kvittem and Moan [19] investigated the fatigue damage ofthe braces considering time history of structural response where Morison’s equation wasused to simulate loads on braces and BEM was applied for load prediction of columns.
Luan et al. [20–22] systematically studied the motion and sectional loads on a conceptsemi-submersible FOWT with the rectangular cylinders used as pontoons. They concluded
that the motion and sectional loads by the multibody model showed good agreement withexperiments in the combined wave and wind conditions. However, the distribution ofsectional load on the platform was not examined for the pontoon designed with variablecross sections of elements as shown in the Fukushima demonstration project [25]. Moreover,the nonlinear hydrodynamic coefficient models considering the effect of Reynolds andKeulegan–Carpenter (KC) numbers for the multibody have not been proposed yet.
The semi-submersible FOWT usually uses side columns to provide sufficient stability.
In the early stage of development for the semi-submersible platform, braces and decks
were widely used. In the Fukushima demonstration project [25], the 2 MW wind turbine“Fukushima Mirai” was a four-column semi-submersible FOWT and included three sidecolumns and a center column. The side columns are connected to the center column by
the pontoon, brace and deck to form an integrated structure. WindFloat [26] is anotherwell-known floating platform using braces and decks for connecting columns. Recently, asemi-submersible platform without braces and decks, in which columns are connected by
the pontoon only, was proposed to reduce costs and simplify the manufacturing process,such as the 7 MW FOWT “Fukushima Shimpuu” [27], which is a semi-submersible platform
with three square side columns connected by rectangular pontoons. However, cracks were
found on the pontoons during demonstration due to fatigue. This indicates that the elastic
response prediction and structural analysis on the element of the platform is significant forcost reduction and safe design.
This study investigates the sectional load on a semi-submersible platform used in the
Fukushima demonstration project. In Section 2, the geometry and structural information of
the platform are described. The hydrodynamic coefficients for the multiple hydrodynamic
bodies and the numerical model considering the elastic beams are developed. In Section
3, a water tank test is carried out with the elastic pontoon, brace and deck. The dynamic
response of platform motion and sectional loads on the pontoon, brace and deck areanalyzed and validated by the water tank test in regular and irregular waves. A formulaEnergies 2021, 14, 182

3 of 22is proposed to estimate the distribution of sectional loads on the elements. The sectionalloads on the pontoons are analyzed based on the four platforms, with and without struts
connecting the pontoons, for the structural optimization of the platform and fatigue load
reduction of the pontoon. Finally, the conclusions obtained from this study are presented
in Section 4.
2. Models
The dimension and mass properties of a 1:60 scale elastic semi-submersible platform,
which is used in the water tank tests and numerical simulations to investigate the dynamic
responses and the sectional loadings of the platform, are described in Section 2.1. Section
2.2 proposes the formulae to evaluate the integrated nonlinear hydrodynamic coefficientsfor the multibody. Finally, the numerical model for the dynamic analysis of the platform
and the hydrodynamic force models used in this study are summarized in Section 2.3.
2.1. Description of a 1:60 Scale Elastic Platform Model
To investigate the dynamic response of the floating platform and the sectional loadingson the elements, a water tank test was carried out based on an elastic platform model. Thisfloating platform corresponds to the 2 MW semi-submersible FOWT used in the Fukushima
demonstration project. The model is scaled down by Froude scaling law. For the stiffness,
it is calculated by Kmodel = K prototype /λ5 , in which Kmodel and K prototype are the stiffness ofmodel and prototype, and λ is the scaling factor and equals 60. The platform comprises a
center column and three side columns connected to the central column by pontoons, braces
and decks. The pontoons are rectangular cylinders with variable width. Braces and decks
are circular cylinders. The skirts are installed on the pontoons and heave plates to reduce
the heave motion as shown in Figure 1. The origin of the right-handed coordinate system
is defined at the center of the center column on the water plane with a positive X in the
direction of wave propagation and Z upward. The local x and y axes on the pontoon, braceand deck are presented in Figure 2. The local z axis is in the axial direction of the element.
Table 1 lists a summary of the dimensions of the laboratory scale platform.
Table 1. Dimension and hydrostatic properties of the semi-submersible platform.
Elements

Dimension (m)Draft of the platform
Height of central column (Hcc ) and side columns (Hsc )Diameter of central column (Dcc )Diameter of side columns (Dsc )Height of heave plate (HHp ) and pontoon (HHp )Diameter of heave plate (DHp )Width of pontoon
Length of pontoon (LPn )Diameter of brace (Dbrace )Length of brace (Lbrace )Height of deck
Width of deck
Center of gravity below still water levelRadius of gyration Kxx
Radius of gyration Kyy

0.2667
0.46670.08230.12500.06670.23330.05~0.10
0.28270.03750.38870.03750.0375~0.0608−0.0687
0.370
0.374

the pontoons and heave plates to reduce the heave motion as shown in Figure 1. The
origin of the right-handed coordinate system is defined at the center of the center column
on the water plane with a positive X in the direction of wave propagation and Z upward. The local x and y axes on the pontoon, brace and deck are presented in Figure
4 of 22
2. The local z axis is in the axial direction of the element. Table 1 lists a summary of thedimensions of the laboratory scale platform.

Energies 2021, 14, 182

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4 of 22

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4 of 22(a) Bird’s eye view(b) Top view(c) Side view(c) Sideview and global coordinate system.
Figure 1. Schematic of the semi-submersible
platform
Figure1. Schematicof ofthethesemi-submersible
andglobal
globalcoordinate
system.
Figure1. Schematicsemi-submersible platform
platform andcoordinate
system.

(a)

(a)

(b)

(b)

(c)

(c)

Figure 2. Local
systemfor thepontoon,(b)(b)brace(c) deck.
Figurecoordinate
2. Local coordinate
systemfor components
the componentsofof(a)(a) pontoon,
brace
and and(c) deck.

Figure 2. Local coordinate system for the components of (a) pontoon, (b) brace and (c) deck.
Two different numerical models are adopted for the platform in Orcaflex [28]. One
Table 1.
Dimensionand hydrostaticof thesemi-submersible
platform.
of themassumes
thehydrostatichull ofproperties
theproperties
platformis athefully
rigid structureand hydrodynamic
Table
1. Dimension
andofsemi-submersible
platform.
loads are given to a reference point. This model is hereafter named the rigid model andElements(m)
Elements
Dimensionis analyzed for the motion
andmooring tension of the platform. The otherDimension
model, based(m)on the multiple hydrodynamic
assumes the pontoon (Pn-2), brace (Br-2) and
deck
Draft
ofbodies,theplatform
0.2667Draftof theplatform
0.2667(Deck-2)
aselastic
beams,
modeledwithfiniteelements.
This
model
will
be
referred
Heightof central
columncc) andside
columns(H(H
sc)
0.4667Heightof central
column(Hcc(H) andsidecolumnssc)
0.4667toas an elastic model and is used to predict the sectional loads on the given sections andDiameterof central
column(Dcc)
0.0823
Diametercentralcolumncc)elastic
0.0823dynamic responses.
Theofstructuralelementsin(Dthemodel, except for Pn-2, Br-2
andDiameterofsidecolumns(D
sc)
0.1250Deck-2, are considered
Althoughan effect
Diameterasofrigid
sidebodies.
columns(Dsc)the stiffness of the platform has 0.1250

Height of heave plate (HHp) and pontoon (HHp)
Heightof heave plate (HHp) and pontoon (HHp)
Diameter of heave plate (DHp)
Diameter of heave plate (DHp)
Width of pontoon
Width of pontoon
Length of pontoon (LPn)

0.0667
0.06670.23330.23330.05~0.10
0.05~0.10
0.2827

Energies 2021, 14, 182

namic responses. The structural elements in the elastic model, except for Pn-2, Br-2 andDeck-2, are considered as rigid bodies. Although the stiffness of the platform has an ef22 andfect on the natural frequency of the tower as mentioned by Zhang and Ishihara5 of[17]
has an impact on the bending moment at the tower base as pointed out by Yamaguchi etal. [29], the elasticity of tower is neglected due to the high stiffness of the platform. In thison thefrequencyof the tower
mentionedby Zhang
Ishihara
and(Deck-2)
has an are
study,
thenatural
structural
responsesof theaspontoon(Pn-2),
braceand(Br-2)
and[17]deck
impactonsincethe bending
momentthe tower
pointed out
Yamaguchi
al. [29],
focusedon,the stiffnessof atcentralandbasesideascolumns
are bymuch
higher etthan
those oftheelasticity
of
tower
is
neglected
due
to
the
high
stiffnessoftheplatform.
In
thisstudy,
connecting elements.
thestructural
responses
the pontoon
brace (Br-2)
and deck (Deck-2)
are focused
Figure3 displays
theofelastic
beams(Pn-2),
and multiplehydrodynamic
bodiesused in theon, since the stiffness of central and side columns are much higher than those of connecting
water tank test and the numerical simulation. The Pn-2, Br-2 and Deck-2 connected to theelements.
central Figure
column(CC) andcolumn(SC-2)
are built
as the elastic
by the3 displays
thesideelastic
beams andmultiplehydrodynamic
bodieselements
used in thesquaresteel
beams,
while
the
other
parts
of
the
platformaremodeled
as
the
rigid
water tank test and the numerical simulation. The Pn-2, Br-2 and Deck-2 connected to thestructuresshown
as Figure3a sidein thewatertankare
test.
Inas
the
simulation,
the elastic
centralcolumn(CC) andcolumn(SC-2)
built
thenumericalelastic elements
by the square
steelofbeams,
while the isotherparts of theare modeled
theconnected
rigid structures
shown
model
the platform
separated
intoplatform
five bodies,
whichasareby threeelasticasFigure
3a
in
the
water
tank
test.
In
the
numerical
simulation,theelastic
model
of
the
beams as shown by the dashed lines in Figure 3b. Body 1 includes CC, SC-1, SC-3 andplatformbraces,
is separated
into
fivepontoons.
bodies, which
are connectedbeams Theas shown
respective
decks
and
Body-2consistsby
ofthree
SC-2elastic
and Hp-2.
points inbythedashedlines
inFigure3b.
Body
1
includes
CC,
SC-1,
SC-3
and
respective
braces,Figure 3b represent the reference point of each body. For each elastic beam, the bending
decks and pontoons. Body-2 consists of SC-2 and Hp-2. The points in Figure 3b represent
moments at the three sections as defined in Figure 3a are measured in the water tank test
the reference point of each body. For each elastic beam, the bending moments at the three
andsections
are predicted
ininthenumerical
simulation.
shows
the arestiffness
properties
as defined
Figure3a are measured
in theTable
water 2tanktest andpredicted
in the ofeachelastic
beam.
Table
3
lists
the
structuralproperties
of
each
body,
such
as
mass,
numerical simulation. Table 2 shows the stiffness properties of each elastic beam. Table
3 thecenterandmomentinertia.
The asoriginthelocalofcoordinate
listsofthegravity
structuralproperties
of of
each
body, suchmass,ofthe
centergravity and system
moment is locatedat the centerof gravityof eachbody. The
similarities
thecentermassofandmomentof inertia.
The originof the local
coordinate
systemis located forat thegravity
of of
each
body.
The
similarities
for
the
mass
and
momentofinertiaareachievedbyscaling
inertia are achieved by scaling dimensions of columns, pontoons, braces and decks anddimensionsof columns,
pontoons,
decks andadjusting
distribution.
adjustingweightdistribution.
Thebraces
windandturbines
andnacelleweight
are simplified
asThe
a rigidwind
turbines
andnacellearesimplified
as
a
rigid
pole
to
consider
their
mass
and
momentpole to consider their mass and moment of inertia as shown in Section 3.1.
of inertia as shown in Section 3.1.

(a) Elastic model in the water tank test(b) Elastic model in the numerical simulation

Figure3. Elastic
modelsof theplatform
tanktest
testandandthenumerical
simulation.
Figure3. Elastic
modelsof theplatformusedusedininthe
the waterwater tankthenumerical
simulation.
Table
2. Stiffness
properties
beamininthe
thelaboratory-scalemodel.
Table
2. Stiffness
propertiesofof thethe elastic
elastic beamlaboratory-scalemodel.

PontoonPontoonBrace
Brace
DeckDeck2) 2
EIxx EI(Nm
xx (Nm )193.00193.00
20.80
20.80
31.10
31.10

EIEI
yy (Nm2)2
yy (Nm )277.92277.92
20.84
20.84
33.10
33.10

EAEA(N)
(N)
7
2.32
×
10
2.32 × 107
66
6.95
6.95××1010
8.11××1010
66
8.11

GJ(Nm
(Nm
/rad)
2 /rad)GJ
2

213.08213.08
13.52
13.52
24.78
24.78

Energies 2021, 14, 182

6 of 22

Table 3. Structural properties of each body.
Mass (kg)
Body1
Body2
Body3 (Pn-2)
Body4 (Br-2)
Body4 (Deck-2)

14.972
4.239
1.732
0.271
0.420

Moment of Inertia (kg m2 )Center of Gravity (m)
X

Y

Z

Ixx

Iyy

Izz

Ixy

Iyz

Izx

−0.074
0.237
0.102
0.128
0.153

−0.147
0.411
0.177
0.221
0.265

−0.045−0.098−0.233
0.069
0.248

1.452
0.188
0.007
0.004
0.002

2.196
0.186
0.002
0.002
0.001

1.917
0.019
0.009
0.002
0.002

0.671
−0.002−0.004−0.001−0.001−0.040
0.000
0.000
0.001
0.000

−0.108
0.000
0.000
0.002
0.000

2.2. Hydrodynamic Coefficient Models for the Multibody
The linear hydrodynamic force, such as diffraction force and radiation damping force,for the multiple hydrodynamic bodies are estimated by the software AQWA [30] as 6 × 6
matrices with respect to the reference points, while the integrated added mass matrices foreach body from the added mass of each component are necessary, considering the effect ofKC number, Reynolds number and interaction as shown in Liu and Ishihara [31].
The hydrodynamic inertia and drag forces on the cylinder depend on the interaction
between cylinders, KC number and Reynolds number. The hydrodynamic coefficientmodels were proposed by Liu and Ishihara [31] and Ishihara and Liu [32] to take into
account the influence of these factors on the added mass and drag coefficients for eachcylinder and are expressed ask kk ki Ca ( β 0 , KC0 , η )= r Cak ( βk0 , KC0k ) × r γak × i ηak

(1)

k k
k k
i Cd ( β 0 , KC0 , η )= r Cdk ( βk0 , KC0k ) × r γdk × i ηdk(2)

where subscript i and r are the structure component i and referenced component r, superscript k represents the direction for the hydrodynamic coefficients, which can be expressedas the normal direction by n or the axial direction by t. i Cak ( βk0 , KC0k , η k ) and i Cdk ( βk0 , KC0k , η k )represent the added mass and drag coefficients of the component i at βk and KC k in the kdirection including the contribution of the interaction effect. r Cak ( βk0 , KC0k ) and r Cdk ( βk0 , KC0k )mean the representative Ca and Cd for the referenced component r at βk0 and KC0k in the
k direction. Correction factors i ηak and i ηdk are introduced to account for the interaction
between components. Correction factors r γak and r γdk are functions of KC and Reynolds
numbers.
The global added mass for a rigid body can be obtained by integrating the distributedhydrodynamic coefficients in the global coordinate system. In this study, the platform
is discretized to 79 elements and Ca of each element in the local coordinate system canbe computed according to Equation (1). The drag force by Morison’s equation is utilized
to account for the distributed viscous damping force induced by flow separation. The
integrated added mass coefficients for each body are expressed as 6 × 6 matrices [Ca ] andthe components in the global matrices of [Ca ] are presented as
Ca11
 Ca11
 Ca11[ Ca ] =  Ca11
 Ca51
Ca11


Ca11
Ca22
Ca11
Ca42
Ca11
Ca11

Ca11
Ca11
Ca33
Ca11
Ca11
Ca11

Ca11
Ca24
Ca11
Ca44
Ca11
Ca11

Ca15
Ca11
Ca11
Ca11
Ca55
Ca11


Ca11
Ca11 

Ca11 

Ca11 

Ca11 
Ca66(3)

The simplified formulas to integrate distributed added mass were derived from
Ishihara and Zhang [33]. The off-diagonal components in the matrix are neglected except forthe components Ca24 , Ca42 , Ca15 and Ca51 due to the symmetry of the platform. However,
the geometry of the multibody is not symmetrical and all terms in the added mass matrix

for the components Ca 24 , Ca 42 , Ca15 and Ca51 due to the symmetry of the pof 22
However, the geometry of the multibody is not symmetrical and all7 terms
in thmass matrix are necessary. The general formulas to integrate the distributed add
are derived in this study, which are applicable for any shape of the platform.
arenecessary.
The
generalformulas tointegratexyz
the distributed
addedmass are derived
located onan elementin the inglobal copresents
thelocalcoordinate
systemthis study, which are applicable for any shape of the platform. Figure 4 presents the local
system XYZ . x and y are normal to z axis of the element and z corres
coordinate system xyz located on an element in the global coordinate system XYZ. x and ytheaxialtodirection.
arenormalz axis of the element and z corresponds to the axial direction.

Energies 2021, 14, 182

Figure 4. Local and global coordinate system for an element.

Figure 4. Local and global coordinate system for an element.

The added mass matrix of the element in the local coordinate system can be expressed as
The added mass matrix of the element in the local coordinate system ca
i
i
i
C
C
C
axx
axy
axz
pressed as
 i

i
i
Cayy
Cayz
Cia =  Cayx
(4)

i i
i
i
i
i
Cazx CazyCaxx
Cazz CaxyCaxz 
i



i

i

i



Ca = iCayx Cayy Cayz 
i , Ciwhere the diagonal components Caxx
added mass coefficients inayy and Cazz representiiiCazx components
Cazy Cazz  in the added mass
the x, y and z directions, respectively. The off-diagonalmatrix account for the coupling between the motions in the two degrees of freedom. Theii the elementi platform is comprised ofoff-diagonalof the Cia are 0 sincethewhere thecomponents
diagonal components, Cayy andof C
Caxx
azz represent added mass coeff
the cylinders. In the global coordinate system, the matrix of added mass coefficients for an
the x, isy calculated
and z directions,
respectively. The off-diagonal components in the add
elementasmatrix account for the
coupling
betweenini the twodegrees of freed i the motions
i
i
i
i
Caxx
Caxy Caxz
Ca11 Ca12
Ci a130 sincethei element
ofi theis cooff-diagonalcomponentsi of the
Ca 1 are platform
−1
i
i
Cayy
Cayz
CiaXYZ = R × Cia × R−1 , Ca21
Ca22
Ca23  = R ×  Cayx
×R
i
i
i
i matrixii added mass coeffi
of the cylinders. In the global
system, theof
Ca31
Ccoordinate
Cazx
Cazy
Cazz
a32 Ca33
an element is calculated as(5)
where R is the rotation matrix of three Euler angles
i between thei local-1and global coordinates
CaXYZ = R × Ca × R ,and R−1 is the inverse rotation matrix. Each component
in the [Ca ] can be expressed as
k

i
Camn

i
i
i
 Cai 11 Cai 12k hCai 13 
Caxx

Ciaxy
Caxz
1
i
i
m, n = 1, 2, 3
=
C
y
−
C
z
m
=
4
n
=
6
Camn


∑
i
i  4, 5,-1a1(m−1)n ia(mi −2)n i i
R2i
Ca 21 Cai22=k 1h Ca 23  = R × Cayx Ci ayy Cayz  × Riim = 1, 2, 3 n = 4 CCamn= 1 i ∑ Ci i  z − Cai C
x mi = 5Cni = 1, 2, 3−2)n iCazy
azz  a 31 CRai 32= 1 Ca(am33−4)n i
 (mazx

Camn = ∑
i=1
i
k h
1
i
i
Camn = R ∑ Cam(n−1) yi − Camzi( n −2)
i=1
i
i
k h
k h
i
i
i
Camn = R1 ∑ Camz − Cam
x R
m =
1,the
2, 3 rotationn = 5 Camn
= R12 ofCai (m−4)Eulerz − Cangles
x m = 5 n = 4, 5, 6 (6)∑ three( n −4) i( n −2) in ia(m−2)n i between the local and glo
where
ismatrix
i=1
i=1
i
i
-1
k h
k h
dinatesandistheinverse1 rotation
1R
ii
Camn = R ∑ Cam(n−4) xi − Cam(n−5) yi
m = 1, 2, 3 n = 6 Camn = R ∑ Cai (m−4matrix.
x − Cai (Eachy component
m = 6 n = 1,in
2, 3 the [C a ] ca)n im −5) n i
i=1
i=1
i as
i
pressedk h
k hCamn = R1 ∑ Cai (m−1)n yi − Cai (m−2)n zi
m = 4 n = 1, 2, 3 Camn = R12 ∑ Cai (m−4)n xi − Cai (m−5)n yi m = 6 n = 4, 5, 6
i=1

k

i=1

1 k
Cai ( m −1) n yi − Cai ( m − 2) n zi  m = 4 n = 4,5,6

where xi , yi and zi indicate the location of thein the global coordinate system.
R 2 element
i =1
i =1
k is the number of elements. R is the characteristic klength of the platform.
k
1
1
i
i
 m = 1, 2,3 n = 4
Camn =  Cam
Camn =  Cai ( m − 4) n zi − Cai ( m − 2) n xi  m = 5 n = 1, 2,3( n −1) yi − Cam ( n − 2) zi 
R i =1R i =1k1
1 kii
 m = 1, 2,3 n = 5
Camn =  Cam
Camn = 2  Cai ( m − 4) n zi − Cai ( m − 2) n xi  m = 5 n = 4,5,6( n − 4) zi − Cam ( n − 2) xi 
R i =1R i =1k1
1 kii
 m = 1, 2,3 n = 6
Camn =  Cam
Camn =  Cai ( m − 4) n xi − Cai ( m − 5) n yi  m = 6 n = 1, 2,3( n − 4) xi − Cam ( n − 5) yi 
R i =1R i =1
k
1
1 k
i
i
i
i
i
Camn =  Camn

m, n = 1, 2,3Camn =

Energies 2021, 14, 182

8 of 222.3. Numerical Model for the Dynamic Analysis
The dynamic analysis for a floating platform is performed by solving the equation of
motion in the time domain as shown below.
 ..
.
M x + C x + K{x} = {FB } + {FR } + {FG } + {FM } + {FH } + {FW }(7)
 ..  .
where vector x , x and {x} denote the acceleration, velocity and displacement in thesix degrees of freedom, respectively. M represents the mass matrix. C is the damping
matrix and K is the stiffness matrix. The hydrostatic loads on the platform refer to thedisplaced water by the submerged body and its motion. This can be divided into a constantintegrated buoyancy force and a restoring force caused by the motion of the platform. On
the right side of Equation (7), {FB } presents the buoyancy force and {FG } indicates thegravitational force; {FR }, {FH } and {FM } denote the restoring force., hydrodynamic force
and mooring tension, respectively. The last term {FW } is the aerodynamic load, which
is neglected since the wind turbine is simplified in the water tank test. As mentioned by
Ishihara and Zhang [33], the dynamic model for mooring lines performs better than thequasi-static model for accurate prediction of mooring tension. The dynamic model using
lumped mass method is implemented, where the hydrodynamic loads of the mooring line
are computed by Morison’s equation, including hydrodynamic inertia force and drag force.
The buoyancy force is a vertical force and equals the gravity force of the displaced
water. It is calculated asT(8)0, 0, ρw g∀{ FB } =where ρw is density of water and ∀ is the volume of the submerged part of the platform.
The force that equilibrates the buoyancy force is the gravity force of the platform and is
obtained as shown in Equation (9), where m is the mass of the platform.

{FG } =



0,

0,

mg

T

(9)

The restoring force is calculated as the product of hydrostatic stiffness and displacement vector. The hydrostatic stiffness represents the static movement properties of theplatform when it is brought out of an equilibrium position by the external force or moment.
The translations in the surge and sway directions and the rotation in the yaw direction leadto no resultant hydrostatic force. So, there are no hydrostatic terms in the surge, sway andyaw directions. The vertical movement of the platform causes a change of buoyancy force,
which tends to return the structure back to the balance position. The restoring moments
arise with the pitch and roll motions since the center of gravity and center of buoyancy are
not situated on the same vertical line. The restoring force is calculated as Equation (10) andhydrostatic stiffness is described as Equation (11).

{FR } = KR {x}
0 0
 0 0

 0 0
KR = 
 0 0

 0 0
0 0


0
0
−ρw gAw
0
0
0

0
0
0
− FG × GMX
0
0

(10)
0
0
0
0
− FG × GMY
0


0
0 

0 

0 

0 
0

(11)

where GMX and GMY are the meta-centric heights of the platform about the x and y axes,
respectively.
The hydrodynamic force, which refers to the dynamic forces and moments due to thefluid on an oscillating platform in wave, can be written as(12){FH } = {Fm,a } + Fm,d + {Fw,F-K } + Fw,d + {Fd }Energies 2021, 14, 182

9 of 22

where{Fm,a } and {Fd } represent hydrodynamic inertia force and drag force, respectively,
Fm,d is the radiation damping force. {Fw,F-K } and Fw,d are the Froude–Krylov (F-K)
force and diffraction force caused by the wave excitation, respectively.{Fm,a } is computed as
 ..
(13)
{Fm,a } = ρw ∀[Ca ] x
where ρw is density of water and ∀ is the volume of displaced water.
The distributed drag force on each element is expressed as Equation (14).
Fdi =

.
.
1
ρω Cdi Ai (ui − xi ) ui − xi
2

(14)
.

where Cdi is the drag coefficient of the element in the corresponding direction, ui − xi means
the relative velocity of the element to water particle, Ai is the characteristic area of theelement. In the combined wave and current condition, the drag force can be estimated byan advanced hydrodynamic model proposed by Ishihara and Liu [32].
The elastic platform can be divided into several rigid bodies connected by elastic
beams. In the dynamic analysis of the platform, the wave-induced loads and radiation
damping force obtained by the potential theory are used for the multiple hydrodynamic
bodies and hydrodynamic interactions between each body are included in the hydrodynamic force matrices. In the dynamic analysis, hydrodynamic forces obtained from thepotential theory are attached to the reference points as shown in Figure 3b. Similar tothe single body model, the global added mass matrix of each body is integrated from theelemental added mass matrix Cia as shown in Equation (6). The drag force on each element
is computed as the distributed loads on the elements. Since Deck-2 is located above thewater surface, it does not experience any hydrodynamic load and only inertia load isconsidered. The structural response of the elastic beams is obtained by a finite element
model in the time domain. The elastic beam is divided into a series of segments, which arethen modeled by straight massless elements with a node at each end. The element only
considers the stiffness of the beam in the axial and normal directions. The mass, weight andbuoyancy are lumped to the nodes. The shear forces and moments of the beam are appliedat the nodes at the end of elements. No hydrodynamic loads act on the elastic beams.
3. Results and Discussion
The structural response of a semi-submersible platform is analyzed using the experimental and numerical elastic model. Section 3.1 describes the water tank test and environment conditions. In Section 3.2, the dynamic responses of the platform are predictedby the rigid and elastic models and are compared with the experiments. The predictedsectional loads on the pontoon, brace and deck by the elastic model are presented and are
validated by the water tank test in Section 3.3. A simplified method is also proposed to
predict the load distribution. Finally, the local stress on the pontoon of the original platform
is investigated and the structural optimization is performed by connecting pontoons by ashort strut in Section 3.4.
3.1. Water Tank Test with an Elastic Platform
Figure 5a shows an overview of the elastic model with elastic beams and Figure 5b
presents the water tank test. Three connection components are developed as the elastic
structure using steel beams as described in detail in Section 2.1. The pontoon, brace anddeck are segmented as illustrated in Figure 5c. The beams are connected with the hull of
the platform by the flanges and bolts as shown in Figure 5d. The strain on the four surfacesof the beam is monitored by the strain gauges. The sectional loads are then calculated
by M = ∆σI/y, where ∆σ is the different stress between the upper and lower surfaces,
I is the moment of inertia of the section about the neutral axis, y is the perpendiculardistance from the neutral axis to a point on the section. I/y is obtained by the experiments.

Energies 2021, 14, 182

structure using steel beams as described in detail in Section 2.1. The pontoon, brace anddeck are segmented as illustrated in Figure 5c. The beams are connected with the hull of
the platform by the flanges and bolts as shown in Figure 5d. The strain on the four surfaces of the beam is monitored by the strain gauges. The sectional loads are then calcu10 of 22
lated by 𝑀 Δ𝜎𝐼/𝑦, where Δ𝜎 is the different stress between the upper and lower surfaces, 𝐼 is the moment of inertia of the section about the neutral axis, 𝑦 is the perpendicular distance from the neutral axis to a point on the section. 𝐼/𝑦 is obtained by the
The sectionalTheloads
at the ends
and atmiddlelocationsthe beams
are measured
the areexperiments.
sectional
loads
the ends
andofmiddle
locationsof theforbeams
validation.
measured for the validation.

(a) Elastic model(b) Water tank test(c) Elastic beams(d) Location of strain gauges

Figure5. 5.Overview
modelandandthe
the
water
tank
FigureOverviewof
ofthe
the elastic
elastic model
water
tank
test.test.

Figure6 6illustrates
illustrates thethe configuration
platform
andandmooringlineslines
in theinwater
Figureconfigurationofofthe
theplatform
mooring
the water
tank
test.
The
platformisstationedby
fourcatenary
mooringlines,
wherethetank test. The platform is stationed by four catenary mooring lines, wheremooring
the mooringlines ML1 and ML2 connect the platform by the same fairlead. The length of the mooringlines ML1 and ML2 connect the platform by the same fairlead. The length of the mooringlines is 10.5 m. The arrangement of the mooring lines is different from the actual platform
lines
is 10.5 m. The arrangement of the mooring lines is different from the actual platform
due to the limitation of the water tank. The motion of the platform, mooring tension anddue
to the limitationof elastic
the watertank.
motiontoof
the platform,
mooring
tensionsectionalloads on thebeams
areThemeasured
capturethe dynamic
responseof andEnergies 2021, 14, x FOR PEER REVIEW
11 of 22sectionalloads Theon theelasticloads,
beams
areasmeasured
tomoments,
capture thethe platform.
sectional
such
the bending
are dynamic
calculatedresponseusing the of theplatform.
The
sectionalloads,
the bending
measuredstrainobtainedby thesuch
strainas
gaugespasted onmoments,
the beams. are calculated using themeasured strain obtained by the strain gauges pasted on the beams.
Table 4 lists the environmental conditions used in the experiments. Free decay tests
identify the natural periods of the platform in the three degrees of freedom (DOFs). In theregular wave cases, the response amplitude operators (RAOs) are measured at two wave
heights and several wave periods. In irregular wave cases, the waves are generated basedon the Pierson–Moskowitz spectrum (PM spectrum) with two significant wave heights of
0.05 m and 0.195 m and a peak wave period of 2.16 s. The wave elevations are measured
at a location 7.29 m away from the center of the platform in the upstream. All waves
propagate in the positive surge direction.
Figure 6.
ofofthetheplatform
andandmooringlines lines
in theinwatertank test.
Figure6. Configuration
Configuration
platform
mooring
the watertank test.

Table 4. Environmental conditions of the tests.

No.
1

Case
Free decay

Wave Height (m)-Wave Periods (s)-NoteStillwaterat a location 7.29 m away from the center of the platform in the upstream. All waves
propagate in the positive surge direction.

Energies 2021, 14, 182

11 of 22

Table 4 lists the environmental conditions used in the experiments. Free decay tests
identify the natural periods of the platform in the three degrees of freedom (DOFs). In theregular wave cases, the response amplitude operators (RAOs) are measured at two wave
heights and several wave periods. In irregular wave cases, the waves are generated basedon the Pierson–Moskowitz spectrum (PM spectrum) with two significant wave heights of
0.05 m and 0.195 m and a peak wave period of 2.16 s. The wave elevations are measured
Figureof away
the platform
andcentermooringlines
in the watertest.
at6.a Configuration
location 7.29 mfrom theof theplatformin thetank
upstream.
All wavespropagate in the positive surge direction.
Table 4. Environmental conditions of the tests.

No.
1
2
3

Table 4. Environmental conditions of the tests.

Case

Wave Height (m)Wave- Height (m)1Free decay
Regularwaves
0.05, 0.15
2Regular waves
0.05, 0.15
Irregular
waves waves 0.05,0.05,
0.195
3
Irregular
0.195
No.
Free decay Case

Wave Periods (s)Wave Periods(s)1.03~3.10
1.03~3.10
2.16
2.16

Note
Note
Stillwater
StillwaterPM
spectrumPM
spectrum3.2. Dynamic
Responseof thePlatform
and
3.2. Dynamic
Responseof thePlatformandMooring
Mooring Lines
Lines
The Thenaturalperiods
forforboth
models are
areevaluatedevaluatedmeans
naturalperiods
bothrigid
rigidandand elastic
elastic models
byby
means
of of
decaydecay
simulation
and
are
tothe
theexperimental
experimental
the surge,
heavesimulation
andarecompared
compared todatadata
in theinsurge,
heave andpitchandIn In
thethestillstillwater
condition,
the initialeach direction
is imposedisandthepitchdirections.
directions.
water
condition,
the offset
initialinoffset
in each direction
imposedplatformisthenreleasedto
decay
freely.
The
responses
in
the
first
five
periods
areused
and the platform is then released to decay freely. The responses in the first five periods
to evaluate
the natural
periods.
Figure Figure
7 shows7 thenaturalthe surge,
heave
are used
to evaluate
the natural
periods.
shows
the periods
natural of
periodsof thesurge,and pitch motions. Both elastic and rigid models show the same natural periods and agree
heave and pitch motions. Both elastic and rigid models show the same natural periods
well with those by the water tank test since the stiffness of platform is determined by theand agree
well
withby thetest
since
stiffness
platform
deterrestoring
force
andthosemooringlineswater
whichtank
are thesame
forthethe elastic
andofrigid
models.isTheminedby therestoringand mooringlines which
are the same
forthetheelastic
massesof elastic
andforce
rigid models
in the numericalsimulations
are keptsame,
andandrigidthus
models.
The
massesofelastic
andrigid
modelsinthenumerical
simulations
arethe natural periods of elastic and rigid platforms are the same.
kept the same, and thus the natural periods of elastic and rigid platforms are the same.

Figure 7. Natural periods of motion.

The dynamic responses of the platform in the regular waves are characterized by
RAOs, in which the amplitudes of platform motion and mooring tension are normalized
by the wave amplitudes as
A
RAOs =(15)
H/2
where A is the amplitude of dynamic response for the platform motion, mooring tension
and sectional loads, and H refers to the incident regular wave height.
The normalized time history of dynamic response of the platform is also expressed asx ∗ (t) =

x (t)
Hs /2

(16)

as
x* (t )=
Energies 2021, 14, 182

12 of 22

Heave (m/m)

Surge (m/m)

3Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. Rigid model(H=0.05m)
Cal. Rigid model(H=0.15m)
Cal. Elastic model(H=0.05m)
Cal. Elastic model(H=0.15m)

2

1

0

1

1.5

7.75

11.62

2

2.5

3

15.49

19.37

23.24

0

1

1.5

7.75

11.62

2

2.5

3

15.49

19.37

23.24

2

2.5

3

15.49

19.37

23.24

Periods (s)(a) Surge RAO(b) Heave RAO
15

Tension of ML4 (N/m)

Pitch (rad/m)

1

Periods (s)Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. Rigid model(H=0.05m)
Cal. Rigid model(H=0.15m)
Cal. Elastic model(H=0.05m)
Cal. Elastic model(H=0.15m)

2

1

0

Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. Rigid model(H=0.05m)
Cal. Rigid model(H=0.15m)
Cal. Elastic model(H=0.05m)
Cal. Elastic model(H=0.15m)

2

Full scale

3

Full scale(16)

where x(t ) is the time history of displacement, rotation, mooring tension, moment andstress. H s denotes the significant wave height in the irregular wave and the wave
heightof the regular wave, and x * (t ) is used to calculate the PSD (power spectral denwhere x (t) is the time history of displacement, rotation, mooring tension, moment andsity)of theresponse
in irregularwave
cases.
stress.
Hsdynamic
denotes thesignificant
wave heightin theirregular wave and the wave heightof Thethe regularandandx ∗ (twave) is used
to calculate
theregular
PSD (power
spectraldensity) offrom
the thewave wave,
heights
periodsof thewaves
are obtained
dynamic
responseinirregular
wave
cases.
measurements in the water tank tests. The simulations in the time-domain are perandwave
periodsof the regularwaves areobtained
fromthetheRAOs
formedThefor wave
100 s heights
to obtain
thedesiredsteady-state
responses.
Figure8 shows
in theandwatertankoftests.
simulations
in regular
the time-domain
are performedof measurements
the surge, heave
pitch
the Theplatformin thewave conditions,and thefor 100 s to obtain the desired steady-state responses. Figure 8 shows the RAOs of thetension of the mooring line ML4 at the fairlead. The numerical results show goodsurge, heave and pitch of the platform in the regular wave conditions, and the tension ofagreement
withexperimental
They indicate
the mooring
linetheML4
at the fairlead.data.
The numericalresults thatshowthegoodproposed
agreementnonlinearwith the hydrodynamic
model
works
well
for
the
multibodymodel.
The
predicted
responses
experimental data. They indicate that the proposed nonlinear hydrodynamic model worksby therigidelastic
modelsmodel.
showThelittledifference,as mentioned
Guignier
et al. [18],wellandfor themultibody
predicted
responses
by the rigidby
andelastic models
whichthat theaselasticity
ofby
themodelethasno effect
the platform
motion andshowimplies
little difference,
mentioned
Guignieral. [18],
whichonimpliesthat the elasticityof the model
has no effect on the platform motion and mooring tension.
mooring
tension.

3

Full scalex (t )
Hs / 2

1

1.5

7.75

11.62

2

2.5

3

15.49

19.37

23.24

Periods (s)(c) Pitch RAO

Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. Rigid model(H=0.05m)
Cal. Rigid model(H=0.15m)
Cal. Elastic model(H=0.05m)
Cal. Elastic model(H=0.15m)

10

5

0
Full scale

1

1.5

7.75

11.62

Periods (s)(d) Tension RAO of ML4

Figure 8. The predicted and measured RAOs (response amplitude operators) of the dynamic responses in the regular waves.

The dynamic responses of the rigid and elastic models in the irregular waves arealso predicted. The measured wave elevations are used as the inputs of the numericalFigure 8. The predicted and measured RAOs (response amplitude operators) of the dynamic responses in the regularwaves.
of 22 are
The dynamic responses of the rigid and elastic models in the irregular 13
waves
also predicted. The measured wave elevations are used as the inputs of the numericalsimulations. The simulation time is 99 s, which is the same as the experiment. The predicted
power spectral
density time
(PSD)
of s,
thewhichnormalized
platformand mooring
simulations.
The simulation
is 99
is the sameas the motion
experiment.
Thetensionare shownin Figure
9. Similartheregular wave
cases,
the andrigid
and elastic
predicted
power spectral
density(PSD) oftothe
normalized
platform
motionmooring
tensionaretheshown
in results
Figure 9.
to the regularwave
the rigid
and elastic
modelsgive
same
forSimilar
the platform
motionandcases,mooring
tension.
The lowermodels
give
the
same
resultsfortheplatform
motionandmooring
tension.
The
lower tofrequency peak of 0.134 Hz is observed in the PSD of surge motion, which corresponds
frequency
peak
of
0.134
Hz
is
observed
in
the
PSD
of
surge
motion,which
corresponds
tothe natural frequency of the surge motion. The peaks at 0.464 Hz observed in the PSD
ofthe natural frequency of the surge motion. The peaks at 0.464 Hz observed in the PSD of
platform motion are excited by the wave. Similarly, the first peak in the PSD of mooring
platform motion are excited by the wave. Similarly, the first peak in the PSD of mooring
tensioncorresponds to the low frequency of surge motion and the second peak is excitedtension corresponds to the low frequency of surge motion and the second peak is excitedby bythethewave,
respectively.
responsesshow
show
favorable
agreement
wave,
respectively. The
The predictedpredicted responses
favorable
agreementwith withthe theexperimental
data.
experimental
data.

Energies 2021, 14, 182

2

10

0

-1

10

-2

10

-3

10

-4

2

2

10

Full scale

6

10

5

10

4

10

3

10

2

10

1

10

0

0.5

1

1.5

0

0.0065

10

1

10

0

Exp. (H=0.05m)
Exp. (H=0.195m)
Cal. Rigid model (H=0.05m)
Cal. Rigid model (H=0.195m)
Cal. Elastic model (H=0.05m)
Cal. Elastic model (H=0.195m)

0.129

0.194

10

-1

10

-2

10

-3

10

-4

10-5
Full scale

0

0.5

1

1.5

0

0.0065

0.129

0.194

Frequency (Hz)Frequency (Hz)(a) PSD of surge motion(b) PSD of heave motion
10

Exp. (H=0.05m)
Exp. (H=0.195m)
Cal. Rigid model (H=0.05m)
Cal. Rigid model (H=0.195m)
Cal. Elastic model (H=0.05m)
Cal. Elastic model (H=0.195m)

4

10

3

10

2

10

1

10

0

Exp. (H=0.05m)
Exp. (H=0.195m)
Cal. Rigid model (H=0.05m)
Cal. Rigid model (H=0.195m)
Cal. Elastic model (H=0.05m)
Cal. Elastic model (H=0.195m)

2

2
2

2

2

10

0

Tension (N /m /Hz)

10-5

Pitch (deg /m /Hz)

10

2

1

Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. Rigid model (H=0.05m)
Cal. Rigid model (H=0.15m)
Cal. Elastic model (H=0.05m)
Cal. Elastic model (H=0.15m)

Heave (m /m /Hz)

10

2Surge (m /m /Hz)

10

10-1
Full scale

0

0.5

1

1.5

0

0.0065

0.129

0.194

10-1
Full scale

0

0.5

1

1.5

0

0.0065

0.129

0.194

Frequency (Hz)Frequency (Hz)(c) PSD of pitch motion(d) PSD of tension for ML4

Figure9. Normalized
dynamic
the platform
platformininthe
theirregular
waves.
Figure9. Normalized
dynamicresponse
response ofof theirregular
waves.

3.3. Sectional Loads on the Platform

3.3. Sectional Loads on the Platform

TheThesectionalloads on the elastic beams at the specified sections are computed bysectional loads on the elastic beams at the specified sections are computed bythethefiniteelement
betweenthe
the
RAOs
from
experiments
and thefiniteelementanalysis.
analysis.The
The comparison
comparison between
RAOs
from
experiments
and thenumericalmodel
ispresented
inFigure10.
Thebending
momentisinducedby theM
numerical model is presented in Figure 10. The bending moment My is yinduced by theEnergies 2021, 14, x FOR PEER REVIEW

14 of 22

Energies 2021, 14, 182

14 of 22

horizontal loads acting on the side column and pontoon, while the bending momentM x is caused
byacting
the verticalacting andon theheave whileplatestheandbending
pontoons.
The norhorizontalloads
on the loads
side column
pontoon,momentMx
M acting
malizedmoments
the beam
consistentforThedifferent
wave
M the
is causedbendingby the vertical
loads
heave
platesareandpontoons.
normalized
x and ony ofbendingsince
moments
Mx and M
consistentfor different
wave heights
since
y of theheights
the pontoon,
brace
andbeamdeck areresponselinearly.
The sectionalloads onthethe
pontoon,
brace
and
deck
responselinearly.
The
sectional
loads
on
the
pontoonaremuchpontoon are much larger than those on the brace and deck, which indicates that thelarger thanon thestructure
brace andtodeck,which
indicatesthatcolumns
the pontoon
is thecriticalpontoonis thosethe criticaltransferloads
betweenin thehorizontal
structuretransfer
betweenofcolumns
in thehorizontal
The cracksdemonon theplane.
Thetocracks
on loads
the pontoons
the 7 MW
FOWT
used inplane.
the Fukushimapontoonsof thewere
7 MWmainly
FOWTcaused
used inbythe
demonstration
project
weremainly.
Thebending
momentstrationproject
theFukushima
bending momentMy
caused by the bending moment My . The bending moment Mx for all the elastic elements
M x for all the elastic elements of the platform is smaller compared to the bending moof the platform is smaller compared to the bending moment My , which is attributed to thement
is attributed
to loads
the sidecolumns
to On
the thewave
loads
as showninM y , whichside columns
subject
to the wave
as shown
in subject
Figure 3.
other
hand,
the wave
Figure3. On onthetheother
hand, inthethewave
loadsdirection
acting ontheplatformsmall
in thedue
verticalloads acting
platform
verticalarerelatively
to thedirecdeep
tion
relatively
small
due to the loads
deep ondraft.
implieselementthat theofhorizontalloads
on thedraft.are
It implies
thatthe horizontal
theItslender
the platform
causedbyslenderelementoftheplatformcausedby
thesidecolumnshouldbepaidattentionto.
the side column should be paid attention to. The influence of stiffness of the element on thesectionalloadsofisstiffness
neglected
they areon
only
near theresonant frequenciesasThe
influenceof since
the element
the amplified
sectional loads
is neglected
since they are
mentionedby Ishihara
et resonant
al. [8]. The
resonant responsesof the byelements
areetnotonly
amplifiednear thefrequencies
as mentioned
Ishihara
al.observed[8]. Thein the fieldmeasurements
the watertanktests sinceof the platform
resonant
responsesof the andelements
are not
observedin thethe stiffness
field measurements
andusedthein the tankdemonstrationhighand theresonant
frequency
fardemonstration
away from theisfrequency
water
tests since isthestiffnessof theplatform
used inisthehigh andrange
of the wave
energy.
theresonant
frequency
is far away from the frequency range of the wave energy.
5050Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal. (H=0.15m)

40

Pn-2A My (Nm/m)

Pn-2A Mx (Nm/m)

40

30

20

10

0
Full scale

Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal. (H=0.15m)

30

20

10

1

1.5

7.75

11.62

2

2.5

3

15.49

19.37

23.24

0
Full scale

1

1.5

7.75

11.62

Periods (s)

2

2.5

3

15.49

19.37

23.24

Periods (s)(b) M y at section Pn-2A(a) M x at section Pn-2A
3

3
Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal. (H=0.15m)

Br-2A My (Nm/m)

Br-2A Mx (Nm/m)Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal. (H=0.15m)

2

1

0
Full scale

1

1.5

7.75

11.62

2

2.5

3

15.49

19.37

23.24

2

1

0
Full scale

Periods (s)

1

1.5

7.75

11.62

2

2.5

3

15.49

19.37

23.24

Periods (s)(d) M y at section Br-2A(c) M x at section Br-2A
Figure 10. Cont.

Energies 2021, 14, x FOR PEER REVIEW

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Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, 182

15 of 2215 of 2210

10
Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal.
(H=0.15m)
Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal. (H=0.15m)

7.5

10

Deck-2A
My (Nm/m)
Deck-2A
My (Nm/m)

Deck-2A
Mx (Nm/m)
Deck-2A
Mx (Nm/m)

10
7.5

5
5
2.5
2.5
0

Full scale

1

0 7.75
1
Full scale
7.75

1.5

2

2.5

3

11.62
1.5

15.49
2

19.37
2.5

23.24
3

15.49

19.37

23.24

11.62

Periods (s)Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal.
(H=0.15m)
Exp. (H=0.05m)
Exp. (H=0.15m)
Cal. (H=0.05m)
Cal. (H=0.15m)

7.5
7.5
5
5
2.5
2.5
0

Full scale

1

0 7.75
1
Full scale
7.75

1.5

2

2.5

3

11.62
1.5

15.49
2

19.37
2.5

23.24
3

15.49

19.37

23.24

11.62

Periods (s)Periods (s)Deck-2A(f) M y at section

Periods (s)(e) M x at section
Deck-2A

M y atatspecified
section Deck-2A(f) axis(e) M x10.atRAOs
sectionFigureof Deck-2A
moments about the local coordinate
sections.
Figure 10.
10. RAOs
RAOs ofof moments about
about the local
local coordinate axis
axis at
at specified
specified sections.
sections.
Figure
M at the sections Pn-2A and
Figuremoments
11 presents thethe PSDcoordinateof the normalized
y

4

10

Exp. (H=0.05m)
Exp.(H=0.195m)
Cal. (H=0.05m)
Cal.(H=0.195m)
Exp. (H=0.05m)

3

2

3
10
10

2

3

4
10
10

Exp.(H=0.195m)
Cal. (H=0.05m)
Cal.(H=0.195m)

2

My2((Nm)
Pn-2A Pn-2A
My ((Nm)/m2/Hz)/m /Hz)

4
10
10

1

2
10
10
0

1
10
10
-1

0
10
10
-2

-1
10
10

-3
-2
10
10 0
Full scale
10-3 00
Full scale
0

0.5

1

1.5

0.0065
0.5

0.129
1

0.194
1.5

0.0065

0.129

0.194

Frequency (Hz)Frequency (Hz)(a) PSD of M y at section Pn-2A2
2
Deck-2A
My2((Nm)
Deck-2A
My ((Nm)/m2/Hz)/m /Hz)

10

Deck-2A.
The
bending
moments
show favorablewith theFigure11predictedpresents the
PSD ofthe normalized
My at theagreement
sections Pn-2A
andmeasured
Deck-2A.
M y at the sections Pn-2A and
Figure 11 presents the PSD of the normalized
The predicted
momentsof
show
agreementwithM y favorable
at the section
Pn-2A
by the measured
numerical moments.
model ismoments.
The bending
underestimation
Deck-2A.
The predictedbending
moments
showbyfavorable
agreement withthe measured
The underestimationM
sectionThis
Pn-2A
theduenumerical
is observedin they at therange.
observedin the low of
frequency
may beto the model
second-orderhydrodyM y toatthe
thesecond-ordersection Pn-2A
by the numericalmodel
ismoments.
The range.
underestimationof duelowfrequency
This
may
be
hydrodynamic
loads,
whichisnamic loads, which is not considered in the numerical model.
not considered
the frequency
numerical model.
observedin theinlow
range. This may be due to the second-order hydrodynamic loads, which is not considered in the numericalmodel.
4

2

3
10
10
1

Exp. (H=0.05m)
Exp.(H=0.195m)
Cal. (H=0.05m)
Cal.(H=0.195m)
Exp. (H=0.05m)

Exp.(H=0.195m)
Cal. (H=0.05m)
Cal.(H=0.195m)

2
10
10
0

1
10
10
-1

0
10
10
-2

-1
10
10

-3
-2
10
10 0
Full scale
10-3 00
Full scale
0

0.5

1

1.5

0.0065
0.5

0.129
1

0.194
1.5

0.0065

0.129

0.194

Frequency (Hz)Frequency (Hz)(b) PSD of M y at section Deck-2AFigure11.
Normalized
moments ofof thethe pontoon
pontoon at
at section
sectionPn-2A
andDeck-2A.
M y at11.
M yDeck-2A.
(a) PSD ofFigure
sectionPn-2A moments(b)Pn-2A
PSD ofandat section Deck-2A
Normalized

The sectionalload at
specifiedatsections
can beandevaluatedthe multibody model.
Figure 11. Normalized
momentsof thethe pontoon
section Pn-2A
Deck-2A.by
The sectionalload atthespecified sections
can be evaluatedby the multibody modIt is possible to increase the number of bodies to calculate the sectional load at any section.
el. It is possible to increase the number of bodies to calculate the sectional load at any
However,the increaseof thebodiesmay leadto theproblemofthe numerical
TheHowever,sectional
load
atspecifiedsections
canconvergent
bethe
evaluatedby themultibodymodsection.
the increaseof bodies
may leadtoconvergent
problemof thenumodel.
Toestimatethedistributionofsectional
loads
on
a
beam,
the
bending
momentMyel.
It
ispossibletoincreasethenumberofbodiestocalculatethesectional
load
at
any
merical model. To estimate the distribution of sectional loads on a beam, the bendingon the beam
is assumed
as a functionof zmay
and leadis expressedassection.
However,thebeamincreaseof bodies
tooftheconvergent
problemmomentis assumed
as a functionz andis expressed
as of the nuM y on themerical model. To estimate the distribution of sectional
loads
on
a
beam,
the bending
My11 − M0y00moment M y on the beam is assumed
as M
a functionz andis expressed as−M
My ( z ) =+ z M y of
(17)
y
(17)
M y ( z ) = Myy0 + z
LL
M 1y − M y0
where z means the distance between
sectionthe( z) =
M ytheM y0 + zfor moment prediction and the end of(17)L moment My at the end sections ofbeam close to center column, My0 and My1 are the bendingwhere z means the distance between the section for moment prediction and the end
01
the beam close to center column, M y and M y are the bending moment 16Mofy 22at the eEnergies 2021, 14, 182

sections of the beam close to the center column and the side column, respectively, andis the length of the beam.
the beam close to the center column and the side column, respectively, and L is the length
Figure12 shows the bending moments on Pn-2, Br-2 and Deck-2. The horizonof the beam.
axes in Figure
12shows
denotedistancefromthe endbeams
to theFigure 12
thethebending
moments
on Pn-2,
Br-2 ofandtheDeck-2.
Theclosehorizontal
axescenter cumn inandare 12
normalized
by thefrom
lengthof thebeams.
is foundthat column
the distributions
Figuredenote the distancethe endof thebeams Itcloseto the centerandarenormalizedbythelengthofthebeams.
It
is
found
that
the
distributions
of
M
on
Pn-2,
momen
y
M y on Pn-2, Br-2 and Deck-2 are almost linear and the predicted bending
Br-2 and Deck-2 are almost linear and the predicted bending moments agree well withagreethosewellobtainedwith thosefromthe water tank test.
fromobtained
the water tanktest.
4
Exp.(T=1.29)
Exp.(T=2.19)
Cal.(T=1.29)
Cal.(T=2.19)
Eq.(17) (T=1.29)
Eq.(17) (T=2.19)

30

Br-2 My (Nm/m)

Pn-2 My (Nm/m)

40

20

10

0

0

0.25

0.5

0.75

3

2

1

0

1

Exp.(T=1.29)
Exp.(T=2.19)
Cal.(T=1.29)
Cal.(T=2.19)
Eq.(17) (T=1.29)
Eq.(17) (T=2.19)

0

0.25

0.5

Z

0.75

1

Z

(a) Moment distribution on Pn-2(b) Moment distribution on Br-2

Deck-2 My (Nm/m)8
Exp.(T=1.29)
Exp.(T=2.19)
Cal.(T=1.29)
Cal.(T=2.19)
Eq.(17) (T=1.29)
Eq.(17) (T=2.19)

6

4

2

0

0

0.25

0.5

0.75

1

Z

(c) Moment distribution on Deck-2
Figure12. Distributionof bending
momentsalong
along the
of of
Pn-2,
Br-2Br-2
and Deck-2(H = 0.05
Figure 12.
Distributionof bending
moments
thebeams
beams
Pn-2,
and Deck-2(Hm).
= 0.05 m).

3.4. Structural
Optimization
3.4. Structural
Optimization

The pontoon
the most
most important
structure
for connecting
the center thecolumnand theThe pontoon
is istheimportantstructure
for connecting
centercolumn asidecolumnasdiscussed
inSection
3.3.
Recently,
conceptsofaplatformwithoutdecks
the side column as discussed in Section 3.3. Recently, concepts of a platform witho
and braces have been proposed, as shown in the 7 MW FOWT used in the Fukushima
decksdemonstration
and braces havebeen proposed, as shown in the 7 MW FOWT used in the Fukproject. The deck and brace provide the stiffness of the platform, but theyshimaalso
demonstration
project.ofThedeck and especiallybrace provide
the stiffnessof the platfor
increase the complexity
manufacturing,for the weldingof the diagonal
but they
also
increasethe
thecomplexityof manufacturing,
especiallyfor the weldingof tbrace.
To optimize
platform,the sectionalloads on the pontoon
are investigated
based
on
the
four
models
as
shown
in
Table
5.
The
baselinemodel
removes
all
decks
and
bracesdiagonal brace. To optimize the platform, the sectional loads on the pontoon are inv
in the original model used in the Fukushima demonstration project. The struts connecting
tigatedbased on the four models as shown in Table 5. The baseline model removes
pontoons are installed in the optimized model, but the braces and decks are removed
decks and braces in the original model used in the Fukushima demonstration proje
The struts connecting pontoons are installed in the optimized model, but the braces adecks are removed from the original model. The full model is equipped with the dec
braces, and struts as shown in Figure 13. To keep the mass and external load for thefour models the same, the mass and hydrodynamic loads on the struts are neglected.

Energies 2021, 14, x FOR PEER REVIEW

17 of 22

Energies 2021, 14, 182

17 of 22

simulate the baseline model, but the mass and hydrodynamic load are included. Thisof 22means that the end nodes of the elastic deck and brace at the cross sections17Deck-2A
arefrom
thedisconnected
original model.
full model
is equipped with the decks, braces, and struts as
Br-2A
are
at Thethe central
column.

Energies 2021, 14, x FOR PEER REVIEW

shown in Figure 13. To keep the mass and external load for these four models the same,
simulatethe baseline
butloads
the
mass
and
hydrodynamic
load
This
the5.massand hydrodynamic
onfor
the
strutsare neglected.
In arethenumerical
models,
Table
Description
ofmodel,four models
usedthestructural
optimization
ofincluded.
the platform.
means
theandendbraces
nodesareof theelastic deckfrom
and thebrace
at thecolumn
cross sections
Deck-2A
arethethatdecks
disconnected
centralto simulate
the baseline
and column.
Br-2A
are disconnected
atandthehydrodynamic
centralmodel,
but the massDeckload are included. This means that the end nodes
Model
Strut
Description of Modelof the elastic deck andBrace
brace at the cross sections Deck-2A are Br-2A are disconnected at
Table
5. Descriptionof four models used for the structural optimization of the platform.
thecentralcolumn.
Baseline
model
Platform without deck, brace and strut Platform
Deck andOriginal model
with
deck
and brace, but without strut
Table 5. Description
structural
optimizationof theplatform.
Model of four models used for the
Strut
Descriptionof ModelModel
Baseline model
Original model
Optimized model
Full model

Optimized model BraceDeckand Brace Baseline
modelFull model

7
Original model#
Optimized model
7
Full model#

Strut









Platform with strut, but without deck and brace

Descriptionof ModelPlatformwithoutdeck,deck,brace
and strut
Platformwithbrace
and strut
Platformwithoutdeck,
brace
and strut
Platform with deck and brace, but without
strut
Platformwith butdeckwithout
and brace,
butandwithout
strut
Platformwith strut,
deck
brace
Platform with strut, but without deck and brace
Platformwith deck, brace and strut
Platform with deck, brace and strut

7
7
#

#

Figure 13. Overview of the platform with brace, deck and strut.
Figure 13. Overview of the platform with brace, deck and strut.
Figure 13. Overview of the platform with brace, deck and strut.

Figure 14 shows the predicted PSD of normalized platform motions and mooring
Figure 14 shows the predicted PSD of normalized platform motions and mooring
tensions
fourmodelsinintheirregular
wavewith thewave motions
heightofand
0.15
m. It is noticed
Figureof14
the predicted
PSD ofwave
normalized
tensionsof shows
four models
the irregularwith theplatform
wave heightof 0.15
m.mooring
It is noticedthatthepredicted
dynamic
responses
andmooring
tensionsbythenumerical
modelstensionsof fourmodelsdynamic
in the irregularwave
the waveheightm. It is noticed
that thepredicted
responses
andwithmooring
tensions
by of
the0.15
numerical
modelsare arealmostthethe
same.
This
that
the
changeofof
stiffnessdoes
notinfluence
thethat
the predicted
dynamic
responses
tensionsbythenumerical
modelsareplatform
almostsame.
Thisimplies
impliesthatandthe mooringchangestiffnessdoes
notinfluencethe platform
almosttheandsame.
This implies
that the change of stiffness does not influence the platform
motions
mooring
tensions.
motionsandmooring
tensions.
motions and mooring tensions.

2

10

2

2

2

-2

-2
10
10

-3

-3
10
10

-4
-4

10
10

-5

10-5
10
0
Full scale 0
Full scale
0
0

0.5

1

0.00650.129

0.5

0.0065Frequency (Hz)

1
0.129

10

0

1

10

0

Cal. Baseline model

10

-1

-1

10

-2

10

-2

10

-3

10

-3

10

-4

10

-4

10

2

-1

10 -1

2

2

0

10

10

2

10

Surge(m(m/m/m/Hz)/Hz)
Surge

Cal. Baseline model

Original
model
Cal.Cal.
Baseline
model
Optimized
Cal.Cal.
Originalmodel model
Full model
Cal.Cal.
Optimized
model
Cal. Full model

0

1

Heave (m /m /Hz)

10

10

1

Heave (m /m /Hz)

10

1

10-5

-5
0 10
1.5 Full scale
0
Full scale
0.194
0
0.194
0

1.5

Frequency (Hz)(a) surge

Original modelCal. BaselineCal.
model
Optimized modelCal. OriginalCal.
model
Cal.model
Full modelCal. Optimized
Cal. Full model

0.5
0.00651

0.5
0.00650.129

Frequency (Hz)

1
0.129

Frequency (Hz)(b) heave(a) surge(b) heave

Figure 14. Cont.

1.5
0.194

1.5
0.194

Energies 2021, 14, x FOR PEER REVIEW

10

4

18 of 2210
Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model

5
10 3

10

2

Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model

4

10102
3 1

2

2

3

10
3
10

2

10

1

10

18 of 22Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full modelCal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model

2

2

1010
2

10100
1

1010-1

Tension of ML4 (N /m /Hz)22
Tension of ML4 (N /m /Hz)

5

Pitch (deg /m /Hz)
2
2
Pitch (deg /m /Hz)

Energies 2021, 14, 18210

0Full scale
0
10 0

0.5

1

1.5

0.0065

0.129

0.194

10
1
10

0

10-1
0
0
10
Full scale
0

0.5

1

1.5

0.0065

0.129

0.194

Frequency (Hz)10-1
Full scale

Frequency (Hz)00.5(c) Pitch

1

1.5

0

0.0065

0.129

0.194

10-1

0

Full scale

0.51(d) Tensionof ML400.0065

1.5

0.129

0.194

Figure 14. Comparison of the normalized dynamic responses of three platforms in the irregular waves (H = 0.195 m, T =Frequency (Hz)Frequency (Hz)2.16 s).
(c) Pitch(d) Tension of ML4

Figure 15 illustrates the RAOs of bending moment M x and M y at the section

FigureComparisonofof thethe normalized
responsesof threeplatforms
in the inirregular
waves (H
= 0.195= 2.16
Figure14.14.
Comparison
normalizeddynamic
dynamic
responsesof threeplatforms
the irregularwaves(Hm,= T0.195m,s).T =Pn-2A for the three models in regular waves with a wave height of 0.15 m. It is found that
2.16 s).
Figure 15moments
illustrates theof baselinebending moment
Mx and Mmodels,Pn-2A infor Figurey at the sectionof theand optimized
as shown
the bending
M x RAOs
the three models in regular waves with a wave height of 0.15 m. It is found that the bendingM y toatthe
15 illustrates
RAOs
of bending
momentwhich
thesectionof
15a,Figure
are larger
than thosetheof theoriginal
and full models,is dueremovalM x and
moments Mx of the baseline and optimized models, as shown in Figure
15a, arelarger than
the
brace
and
deck.
In
the
original
andfullmodels,
the
deck,
brace
and
pontoonconstiPn-2Amodelsregularwhich
waves
withtoathe
wave
heightof 0.15
It isfoundthoseforof thethe threeoriginal
and fullinmodels,is dueremovalof thebracem.
anddeck.
In thattute
aoriginal
frame
structureprovidestiffnesstoresistthemodels,verticalThebendingofthe
thebaseline
andpontoon
optimized
asloads.
shownin
Figurethebending
moments
Mand
the
and
full models,
deck,
brace
and
constitutea framestructureandxM
momentsoftheoptimizedandfullmodels,as
shown
inFigure
15b,
are
much
providestiffnesstoresisttheverticalloads.
The
bending
moments
Mofthe
optimizedyy15a, are larger than those of the original and full models, which is due to the removalofandfullthanmodels,asIn
shown
in Figure15b,
aremuchmodels
smaller
than
those
of
the
baseline
smaller
thoseoftheoriginal
baselineandoriginaldue
totheandinstalled
strut.
This
the
brace
and
deck.
the
and
full
models,the deck,
brace
pontoonconstiand originalmodelsdueto the installed
strut.
This implies
that theinfluence
strut connecting
theimpliesthat structure
the
strut
connecting
pontoons
has atheslightthebending
bendingtute
a frameandprovidethe
stiffnessto resistvertical loads.on
Thepontoons has a slight influence on the bending moment Mx , but significantly reduces the
M y at the15b,
moment M
butsignificantly reduces
bending
momentsectionPn-2A.
Myx ,of
moments
theandfullthemodels,
asthe
shown
are
bending moment
Myoptimized
at the section
Pn-2A.
Oppositely,brace inandFiguredeck decrease
themuchOppositely,
the brace
andbaseline
deck decrease
the bending
moment. installed strut. This
smaller
than
those
ofx .theand originalmodelsdue to M
thebending
momentMx

implies that the strut connecting the pontoons has a slight influence on the bending
50
moment M x , but significantly reduces
the bending moment Cal.MBaseline
the section Pn-2A.
y at model50

Cal. Baseline model

Original model
Original model
Oppositely,Cal.
the
brace
and deck decrease the bending moment Cal.
.
M Optimized
Cal.
model
x
Cal.
Optimized
model

40

Cal. Full model50
Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model

30
40
20
30
10
20
0

Full scale
10

1

1.5

2

2.5

3

7.75

11.62

15.49

19.37

23.24

Pn-2A My (Nm/m)
Pn-2A My (Nm/m)

Pn-2A Mx (Nm/m)
Pn-2A Mx (Nm/m)40

Cal. Full model50
Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model

30
40
20
30
10
20
0

Full scale
10

1

1.5

2

2.5

3

7.75

11.62

15.49

19.37

23.24

Periods (s)0Full scale

1
7.75

Periods (s)

1.5

2

2.5

3

11.62

15.49

19.37

23.24

(a) Moment M x of Pn-2A0
Full scale

1
7.75

1.5

2

2.5

3

11.62

15.49

19.37

23.24

(b) Moment M y of Pn-2A

Figure 15. RAOs of bendingaround x and y axes at the section Pn-2A in theregular
Periodsmoments(s)Periods(s) waves (H = 0.15 m).

Figure 15. RAOs of bending moments around x and y axes at the section Pn-2A in the regular waves (H = 0.15 m).

Figure16 presents the PSD of the momentat the section
Pn-2A
in the irregular wave
M y ofPn-2A(b) Moment(a) Moment M x of
Pn-2A
presents
them.
PSD
the moment
the section
the irregular
withFigurea wave16heightof 0.15
Theof
conclusions
are atsimilarto thosePn-2A
for theinregularwaves, wave

with a wave height of 0.15 m. The conclusions are similar to those for the regular waves,Figure 15. RAOs of bending moments around x and y axes at the section Pn-2A in the regular waves (H = 0.15 m).

Figure 16 presents the PSD of the moment at the section Pn-2A in the irregular wave
with a wave height of 0.15 m. The conclusions are similar to those for the regular waves,that is, the bending moment M x decreases by the brace and deck, while the bending
19 of 22
19 of 22
moment M y is primarily reduced by the strut.

Energies 2021, 14, x FOR PEER REVIEW

4

10

2

4

10
that is, the bending
moment M x decreases
by the brace and deck, while the bending
Baseline modelthat is, the Cal.
bending
momentM
decreasesby the brace and deck, while the bendingx
Cal. Original model
M yMisCal.
momentprimarily
reducedbythe
thestrut.
Optimizedmodel
2 strut.
momentisprimarily
reduced byyCal. Full model

4

10 0
10
2

2

0

10-4
10
10-2-6100Full scale
-4
10 0

y

2

x

10

4

10 0
10
2

10-2
10

2

10-2
10

y

Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model2
2
Pn-2A M ((Nm)/m /Hz)Pn-2A
M ((Nm) /m /Hz)

10

2

x

2
2
Pn-2A M ((Nm)/m /Hz)Pn-2A
M ((Nm) /m /Hz)

Energies 2021, 14, 182

0.5

1

1.5

0.0065

0.129

0.194

0

10-4
10
10-2-6100Full scale-4
10 0Frequency (Hz)
10-6

0
0.5
Full scale (a) PSD of moment
00.0065Cal. Baseline modelCal. Original modelCal. Optimized modelCal. Full model

0.5
0.0065

1

1.5

Cal.
Baseline model 0.194
0.129
Cal. Original modelCal. Optimized model
Frequency (Hz)

1 Pn-2A
M x for

1.5

0.129

0.194

Cal. Full model

10-6
Full scale

0 PSD of moment
0.5
1 Pn-2A(b)
M y for
0

0.0065

0.129

1.5
0.194

Figure 16. Comparison of normalized
acting on three platform models inFrequency
the irregular
Frequencymoments(Hz)(Hz) waves (H = 0.195 m, T =2.16 s).
(b) PSD of moment M y for Pn-2A(a) PSD of moment M x for Pn-2A

For a given location on the cross section, axial stress can be calculated as

Figure 16. Comparison of normalized moments acting on three platform models in the irregular waves (H = 0.195 m,

Figure 16. Comparison of normalized moments acting on three platform models in the irregular waves (H = 0.195 m, T =T = 2.16 s).
F M y M x
2.16 s).
σ= z+ x + y

A axialI x stress
I y can be calculated as
For a given location on the cross section,(18)

For a given location on the cross section, axial stress can be calculated as
My x section, M x and M y denote thewhere Fz is the axial force, A is theFz areaMx ofy cross
σ =
+ M+
(18)
Myx
y
F
A
I
x
z
bending moments around x and σy =axes,
the+ x I x and+ Iy I y are the inertial moments of(18)A
Ix
Iyx force,crosssection,andaxialand A
y isindicate
thecrosscoordinate
coordinate
system. The
where
Fz is thethe area ofsection, inMxtheandlocalMy denotethe bending
momentsaround
x
and
y
axes,
I
and
I
are
the
inertial
moments
of
the
cross
section,andxystress atFzthe
A, Bforce,and CAinisthe
sectionPn-2A
shown in17 are calculated
MFigureis points
the axial
thearea ofcrossassection,
where
x and M y denote thexandyindicatethecoordinate
in
the
local
coordinate
system.
The
stressatthepointsA,
B
based onmoments
Equationaround(18).
x and y axes, I and I are the inertial moments of thebendingyand C in the section Pn-2A as shown in Figure 17x are calculated
based on Equation (18).

cross section, and x and y indicate the coordinate in the local coordinate system. The
stress at the points A, B and C in the section Pn-2A as shown in Figure 17 are calculated
based on Equation (18).

Figure 17. Local coordinate system of the connection components in the sections Pn-2.

Figure 17. Local coordinate system of the connection components in the sections Pn-2.
Figure 18 presents the standard deviation of the normalized stress, which represents
theFigurefatigue18
load
at the points
A, B and C
in the irregularwith a wave
heightof 0.15m.
presents
the standard
deviationof thewave
normalized
stress,which
represents
The stress in the cross section is caused by the axial force and moments. The stress at thethe fatigue load at the points A, B and C in the irregular wave with a wave height of 0.15
point
ALocalis obtained
from Fz and of
Mxthe, whilethe stressat the pointinCthe
is obtainedPn-2.
from Fz and
Figure17.
coordinate
connection
components
m. M
Thestressinthe crosssystem
section iscausedby theaxial force andsections
moments. The stress at
y . Point B is located on the corner of the upper and side edge, where Fz , Mx and My
Fz andM x deviation
thecontributepoint A tois theobtainedfrom, while thestressatoccurs
the point
is obtained
stress. The
largeststandardof thestressin theCbaseline
Figure 18 presents the standard deviation of the normalized stress, which represents
model,
since thewithout
the brace,deck
andandstrut.
In theoriginal
Fz andfrom. Point B isisdesigned
located on
the cornerof the
upper
side
edge,
where Fz
M y platform
themodel,fatigueload
at thepointsA, B andin thetheirregular
witha waveof 0.15
thevalueof stress
at point
C is C
almostsame as wave
stress at
point
B. Thisheight
implies, M
to the stress.
Theby
largeststandard
deviationof theThestressoccursMiny contribute
x and
m.
The
stressthecrosssectioniscaused
the
axial
force
and
moments.
stressatthe fatigue loads in the section are dominated by the horizontal loads. The stresses in thein thebaselinemodel,
since
theFplatform
designed
without
brace,
deck
strut.
In
Misvalue,
the
point
A is
obtained
from
and
the
stress theatthat
the
point
C and
is obtained
optimizedandfull models
show
small
which
indicates
adding
the
strut
is
z the
x , while
theoriginalmodel,
thevalueofstress
at
point
C
is
almostthesameasstress
at
point
more
effectivethanadding
the
deck
and
brace
to
decreasethefatigue
loads
on
the
pontoon.
from Fz and M y . Point B is located on the corner of the upper and side edge, where FB.
zTheimplies
steel materialof the optimized
model
can bereducedby 10%bycomparedto the originalThis
the fatigue
loads in thesectionaredominatedthe horizontal
loads. The, Mmodelandcontributetothestress.
Thelargeststandard
deviationofthestressoccursMxsincebraces andanddecks
are models
removed.show the small value, which indicates thatstressesintheytheoptimized
full

in the baseline model, since the platform is designed without the brace, deck and strut. In
the original model, the value of stress at point C is almost the same as stress at point B.
This implies the fatigue loads in the section are dominated by the horizontal loads. The
stresses in the optimized and full models show the small value, which indicates that

Energies 2021, 14, x FOR PEER REVIEW

20 of 22

adding the strut is more effective than adding the deck and brace to decrease the fatigue
loads on the pontoon. The steel material of the optimized model can be reduced by 10%
compared to the original model since the braces and decks are removed.

Standard deviation of stress (MPa/m)

Energies 2021, 14, 182

21 of 23

Figure 18. Standard deviation of normalized stress at the points A, B, C for the baseline, original,

600
optimized and full models in the irregular wave (H = 0.195 m, T = 2.16 s).
Baseline model
Original model
4. Conclusions
Optimized model
450
Full model
The dynamic responses
of an elastic semi-submersible platform were investigated,such as the platform motion, mooring tension and sectional loads on the element. The
conclusions are obtained as follows:

300

1.

The nonlinear hydrodynamic coefficient models for the multibody are proposedconsidering the effects of Reynolds and KC numbers on the added mass and dragcoefficients. The predicted dynamic response and mooring tension of the elastic
150
platform by the proposed model show good agreement with the experimental dataobtained from the water tank test. The influence of structural stiffness on the dynamic
response of the platform is negligible since the stiffness of the platform used in the0demonstration
Point
A
PointisBquite high.
Point C2.
The sectional loads on the deck, brace and pontoon were investigated using multiple
Figure 18. Standard deviation of normalized stress at the points A, B, C for the baseline, original,hydrodynamic bodies. The bending moments due to horizontal hydrodynamic loads
optimized and full models in the irregular wave (H = 0.195 m, T = 2.16 s).
are much larger than those caused by the vertical hydrodynamic loads. A linear
formula is proposed to predict the distribution of the sectional loads on the connection
4. Conclusions
element.
The
dynamic
responses
semi-submersible
wereinvestigated,
3.
The sectionalload ofon antheelastic
pontoon
was analyzed platformby the four
platforms
with andsuch as theplatform
motion,mooring
tensionandsectional
loads
on
the
element.
without the braces, decks and struts. The largest contribution to the fatigueTheload ofconclusions
obtained
as follows:thearepontooncomesfrom the horizontal loads and the struts significantly reduce the
1. The nonlinearhydrodynamic
coefficientmodels fortheproposedstress on the
pontoon, while
the contributions
fromthemultibody
decks and arebracesare limited.
considering the effects of Reynolds and KC numbers on the added mass and dragcoefficients.
The predicted
dynamic response
mooring
tension ofY.L.;
thevisualization,
elastic
AuthorContributions:Conceptualization,
T.I.; formal andanalysis,
Y.L.; investigation,
platformby the proposedmodel show
good agreement
theT.I.;
experimental
data
Y.L.;
writing—originaldraft preparation,
T.I.; writing—review
andwithediting,
project administration,
T.I.;
All authors
andtank
agreedto theversionof the manuscript.
obtainedfromhavethe read
water
test.
Thepublishedinfluenceof structural
stiffness on the dynamic responseof thereceived
platformis negligible
since the stiffness of the platform used
Funding:
This researchno externalfunding.
in the demonstration is quite high.
Institutional Review Board Statement: Not applicable.
2. The sectional loads on the deck, brace and pontoon were investigated using multiple
Informed
Consentbodies.
Statement:
applicable.
hydrodynamic
TheNotbending
moments due to horizontal hydrodynamic
Data
Availability
Statement:
Thedata
not publicly
loads are much larger than those arecausedby theavailable.
vertical hydrodynamic loads. A
linearformulais
proposedtopredict
thedistributionof Fukushima
the sectionalloadsoffshoreon thewind
floating
Acknowledgments: This research is carried out as a part of theconnection
element.
farm demonstration project funded by the Ministry of Economy, Trade and Industry. The authors
3. wishThe tosectional
loaddeepest
on thegratitude
pontoonanalyzedparties
by theplatformsduringwith thisandstudy.
express their
to wasthe concerned
for fourtheir assistancewithout the braces, decks and struts. The largest contribution to the fatigue load ofConflicts of Interest: The authors declare no conflict of interest.
the pontoon comes from the horizontal loads and the struts significantly reduce theEnergies 2021, 14, 182

21 of 22

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