Marine Structures 18 (2005) 149–168
Overview: Research on wave loading and
responses of VLFS
Shigeo Ohmatsu
Ocean and Ice Engineering Department, National Maritime Research Institute of Japan, Japan
Received 1 October 2004; received in revised form 19 July 2005; accepted 19 July 2005
Abstract
For the basis of the design and operation of very large floating structures (VLFS), the
comprehension of the hydroelastic behavior of VLFS is indispensable. Various methods have
been proposed in order to predict the hydroelastic responses of VLFS to waves and other
external loads during the Mega-float project in Japan. By virtue of these many studies, we can
now confidently estimate the hydroelastic responses with good accuracy.
This paper categorizes and presents a brief outline of these estimation methods. The
analytical considerations of hydroelastic waves are also provided and compared to the
numerical results.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: VLFS; Mega-float; Hydroelastic response; Mode-expansion method; Integral-equation
method; Eigenfunction expansion-matching method; Modified dispersion relation; Modified free surface
condition; Optics; Slowly varying wave drift force
1. Introduction
In Japan, very large floating structures (VLFS) have been intensively studied
for use as floating airports and floating cities. This Mega-float project was begun in
1995 and completed in 2000. During this project and subsequent to its completion,
various technologies involving VLFS have achieved great progress. Among these
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doi:10.1016/j.marstruc.2005.07.004
E-mail address: ohmatsu@nmri.go.jp.
technologies, one of the most important problems was to estimate the hydroelastic
response of VLFS from the viewpoint of hydrodynamics. The Mega-float project
allowed the basic characteristics of the hydroelastic response to become clear and
various analysis methods were developed.
This paper first outlines the development of various calculation methods of
hydroelastic responses in waves for both pontoon-type VLFS and semi-submersible
type VLFS and introduces the analytical features of hydroelastic responses in waves.
Then hydroelastic responses in other conditions is described. For example, in case
of airplane landings or take-offs, the response analysis in the time domain is pursued.
For the towing conditions of VLFS unit-structures, the effects of forward velocity
need to be included in the analysis. For the assembling conditions of VLFS, two or
three floating structures and their interactions need also to be considered.
Finally, the qualitative risk analysis of mooring systems is described. For the
safety of VLFS, the estimation of both the horizontal displacement of VLFS and
the reaction forces of mooring devices in designed environmental conditions is quite
important. Here, the estimation methods used to determine wind forces and slowly
varying wave drift forces will be shown.
2. Hydroelastic responses in waves
2.1. Categories of calculation methods
VLFS have two distinct hydrodynamical features. One feature is its huge
horizontal size. The wavelengths of practical interest are very small compared to the
horizontal size of a typical VLFS. Another feature is its small bending rigidity, such
that the hydroelastic responses become more important than rigid body motion.
For the estimation of such structural responses a huge computer memory and vast
computation time are needed, and conventional methods cannot be applied directly.
In order to overcome these difficulties, many studies were undertaken and many
calculation methods have been developed.
Here, the author reviews and categorizes the various calculation methods
developed so far. Many works involved pontoon-type VLFS, therefore, the review
of calculation methods for pontoon-type VLFS will initially be performed and
categorized by a representative method of elastic deformation.
One calculation method is the mode-expansion method. In this method, the elastic
motion is represented by a summation of many modes of motion, as shown in Fig. 1.
A second method is the mesh method. In this method, the elastic motion of a thin
plate is represented by the succession of vertical displacement of these substructures,
as shown in Fig. 1.
There are other methods, but all are primarily categorized using the above two
methods.
For treatment of hydrodynamic forces, there are also two methods (Table 1). One
is the Green function method or the integral-equation method. In this method, the
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S. Ohmatsu / Marine Structures 18 (2005) 149–168
150
velocity potential of flow fields is represented by the Green function distribution. The
second method is the eigenfunction expansion-matching method.
To solve the integral equation, the structure is discretized to a large number of
panels. Utsunomiya [1] employed the higher order boundary element method
(HOBEM) that utilizes quadratic 8-node panels in order to reduce the number of
unknown values. HOBEM can improve accuracy but still requires a large
computation time.
Kashiwagi [2] developed an alternative method that expresses the unknown
pressure distribution by a cubic B-spline function, used as a Galerkin scheme in
order to determine the coefficient of spline functions. In general, a Galerkin scheme
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Mesh method
Mode-expansion method
Fig. 1. Representation of elastic deformation.
Table 1
Categorization of hydrodynamic response calculation methods
Representation of elastic motions
Mode-expansion method
Mesh method
Treatment of hydrodynamic forces
Green function method Integral
equation method
Utsunomiyo [1]
Kashiwagi [2,12]
Yago and Endo [3]
Eigenfunction expansion-matching
method
Ohmatsu [4,5]
Seto and Ochi [6]
Murai et al. [7]
Modified free surface condition
Green function method Integral
equation method
Ohkusu and Namba [8]
Eigenfunction expansion-matching
method
Kim and Ertekin [10]
For semi-sub type VLFS
Iijima et al. [11]
Kashiwagi [2,12]
Murai et al. [7]
S. Ohmatsu / Marine Structures 18 (2005) 149–168
151
increases the computation time. However, when the structure is discretized to a panel
of equal size, the computation time can be drastically reduced by considering the
relative similarity relation in evaluation of the influence coefficient of matrix.
Therefore, Kashiwagi’s B-spline Galerkin scheme is recognized as one of the fastest
calculation schemes in practical use.
Yago [3] employed the mesh method. The pressure distribution is calculated using
the boundary element method (BEM), and the motion equation of the elastic plate is
solved using the finite element method (FEM). This method involves an ordinary
calculation and requires both a large memory and large computation time. Its
practical use, therefore, is limited to structures of approximately 1000m. This
program can handle different structural configurations, boundary conditions
between different panels, and variable rigidity of the structure.
Ohmatsu [4,5] employed both the mode-expansion method and the eigenfunction
expansion-matching method, which introduced the analytical representation of
solutions to the Dirichlet problem in the Helmholtz equation for rectangular regions.
The application is limited to a rectangular plate, but the surface integral for the
hydrodynamic force calculation can be performed using a line integral, greatly
reducing the computation time.
In this category there is also the Seto’s code [6]. For free surface flow, the hybrid
finite/infinite element scheme was introduced with special attention on the balancing
of structural analysis, such as NASTRAN. This code is very versatile; it needs both
computation size and time, but can be applied to both a complicated structure and
complicated sea area.
Murai and Kagemoto [7] developed a unique efficient method. They improved the
group body theory that treats the assembly of substructures as a single body and
introduces analytical coordinate transformation. This approach is called the
hierarchical interaction theory.
In addition to these methods, there is another line of approach. The presence of
the elastic plate is studied by the modification of free surface conditions. This
approach has been widely used in the study of elastic deformation in ice-covered
regions. In this category, there are also the Green function method and the
eigenfunction expansion-matching method.
Ohkusu and Namba [8,9] introduced the modified Green function that
corresponds to a modified free surface condition. They considered the infinitely
long plate initially and then the finite length, three-dimensional problem. This study
is applied to the basic understanding of hydroelastic behavior of pontoon-type
VLFS.
Kim and Ertekin [10] introduced the eigenfunction method in the region beneath
VLFS that satisfies the modified free surface condition. They also efficiently utilized
the representation of solutions of the Helmholtz equation for rectangular regions.
For the semi-sub type VLFS, there are not many recent studies. For these cases,
the treatment of hydrodynamic interactions among a great number of columns is
most important.
Iijima et al. [11] introduced the concept of the group body for hydrodynamic
analysis as well as the sub-structure method for structural analysis. This code is very
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152
versatile as it requires both computation size and time, but can be applied to a
complicated structure.
Kashiwagi [12,13] and Murai and Kagemoto [7] introduced the hierarchical
interaction theory independently. This method can be applied to an equally spaced
regular arrangement of semi-sub structures.
A summary is shown in Table 1. Among these methods, Ohkusu and Namba’ s
method was applied for a basic consideration of the characteristics of hydroelastic
responses.
Kashiwagi and Ohmatsu’ s code was used for the initial design stage of the Mega-
float project and Seto and Iijima’s code was used for the full design stage of both the
Mega-float project and the Haneda airport re-expansion and construction
competition.
In order to assess the validity of these calculation methods, model experiments in a
wave basin were carried out. The model experiment itself needed the development of
innovative technologies, such as the fabrication method that satisfies the law of
similarity on rigidity, and the measuring method that is used for very small
deformation at many points. Fig. 2 shows an example of an experiment of the
Haneda floating airport model using regular waves. In this case, the vertical
displacement was measured at 128 points, the structural strain at 32 points and
the mooring force at 4 points. The planned Haneda airport will be 3000 m long. The
model length was 15m, at a scale ratio of 1:200. Scale ratios of 1:100 or 1:200 are
normally used.
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Fig. 2. Model experiment of VLFS in wave basin.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
153
2.2. Analytical consideration of the hydroelastic response
Consider analytically the characteristics of the hydroelastic response of regular
waves. Here the problem is treated as simply as possible and the basic characteristics
are made clear.
For simplicity, let us consider the long uniform floating plate. The equation of
vertical displacement of a thin beam is given as
m
d2z
d
t2
ю
EI
d4z
d
x4
ю r
gz ¼ Аiorf.
(1)
Here zр
x;
tЮ is the vertical displacement,
m is the mass of the plate, and
EI is the
flexural rigidity.
From this equation and the body boundary condition,
df
d
z
¼
dz
d
t
at
z ¼ 0
(2)
we can derive the relationship
1 А
mo2
r
g
ю
EI
r
g
A2
df
d
z
¼
o2
g
f ¼
Kf.
(3)
This relationship is considered as a modified free surface condition. If
m ¼ 0 and
EI ¼ 0, the result becomes a free surface condition of the water surface. By the same
argument, from this modified free surface condition we can derive the dispersion
relationship of elastic waves of a thin plate:
1 А
o
o0
2
ю
k
kp
4
"
#
k tanh
kh ¼
o2
g
K,
(4)
where o0 ¼ r
g=
m,
k4
p ¼ r
g=
EI.
Eq. (4) is called a modified dispersion relation. o0 corresponds to heave mode
natural frequency, and
kp is the characteristic wave number.
From this modified dispersion relationship, the phase velocity of elastic waves of a
thin plate can be obtained as shown in Table 2.
In Table 2, the well-known phase velocity of water waves is also shown. When
wave number
k becomes very small, the phase velocity of elastic waves becomes the
same as that of water waves. For usual VLFS and for practical periodic waves, the
phase velocity of elastic waves is bigger than that of water waves except in the case of
periodic extremely long waves. Table 3 shows the group velocity. As is well known,
the group velocity of water waves is always smaller than the phase velocity, but in the
case of shallow water, we can see that the group velocity of elastic waves is greater
than the phase velocity.
Now let us consider the analogy of optics. Incident waves come from the a
direction, and the elastic waves will be refracted because of the difference in the
phase velocity as shown in Fig. 3.
Here
ca is the phase velocity of the incident water waves and
cb is the phase
velocity of the elastic wave. Usually, wave number
k is smaller than
k0, and
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S. Ohmatsu / Marine Structures 18 (2005) 149–168
154
subsequently,
cb is bigger than
ca. Snell’s law is shown as
sin a
sin b
¼
ca
cb
¼
1
n
.
(5)
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Table 3
Group velocity of elastic waves of thin plates and water waves
Dispersion relation
VLFS
Water waves
1 А o
o0
2
ю
k
kp
4!
k tanh
kh ¼ o2=
g K
k0 tanh
k0
h ¼ o2=
g K
Vg ¼
do=
dk јБББ
Vg ¼
do=
dk0 ¼ Ѕр1=2Ююр
kh= sinh 2
khЮЉ В
Vp
h ! 0
Vg ¼ ½1 ю р2р
k=
kpЮ4Ю=р1 ю р
k=
kpЮ4ЮЉ В
Vp
Vg ¼
ffiffiffiffiffi
gh
p
¼
Vp
h ! 1
Vg ¼ 1=2½o2
0=рo2
0 ю
gkЮюр4р
k=
kpЮ4Ю=р1 ю р
k=
kpЮ4Ю В
Vp
Vg ¼ 1=2
ffiffiffiffiffiffiffiffiffiffi
g=
k0
p
¼ р1=2Ю
Vp
water
VLFS
k0
ca
cb
αα
β
VLFS
∧∨
k
Fig. 3. Refraction at an interface.
Table 2
Phase velocity of elastic waves of thin plates and water waves
Dispersion relation
VLFS
Water waves
½1 А рo=o0Ю2 ю р
k=
kpЮ4Љ
k tanh
kh ¼ рo2=
gЮ
K
k0 tanh
k0
h ¼ рo2=
gЮ
K
Vp ¼ o=
k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
go2½1 ю р
k=
kpЮ4Љtanh
kh=р
kрo2
0 ю
gk tanh
khЮЮ
q
Vp ¼ o=
k0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g tanh
kh=
k0
p
h ! 0
Vp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ю р
k=
kpЮ4=р1 ю р
gk2
h=o2
0ЮЮ
q
В
ffiffiffiffiffi
gh
p
Vp ¼
ffiffiffiffiffi
gh
p
h ! 1
Vp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ю р
k=
kpЮ4=р1 ю р
gk=o2
0ЮЮ
q
В
ffiffiffiffiffiffiffiffi
g=
k
p
Vp ¼
ffiffiffiffiffiffiffiffiffiffi
g=
k0
p
S. Ohmatsu / Marine Structures 18 (2005) 149–168
155
Here
n is called the refraction coefficient. This equation shows that b is larger than
a, and b can be calculated by this equation. Fig. 4 shows an example of the
relationship between incident angle a and refraction angle b, estimated for a given
structure, water depth, and wave period.
From this figure we can see the various characteristics of elastic waves. The longer
waves can easily propagate near the incident wave angle, but the shorter waves are
strongly refracted. When b ¼ 901, the corresponding a is called the perfect reflection
angle.
Fig. 5 shows an example of an exact numerical result that corresponds to a wave
period of
T ¼ 7:7s and a ¼ 151. The propagation angle (e.g., β
E 481) agrees very
well with the estimated value in Fig. 4.
Next let us consider the magnitude of elastic deformation. When the floating
elastic plate deforms as a series of progressive waves, the corresponding fluid velocity
potential can be expressed by
f ¼ z
a
io
k
cosh
kр
z ю
hЮ
sinh
kh
exp½iрo
t А
kxЮЉ.
(6)
Here, z
a is the amplitude of deformation. The fluid pressure and horizontal
velocity are then given by
p ¼ rz
a
o2
k
cosh
kр
z ю
hЮ
sinh
kh
exp½iрo
t А
kxЮЉ,
(7)
u ¼ z
ao
cosh
kр
z ю
hЮ
sinh
kh
exp½iрo
t А
kxЮЉ.
(8)
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T = 3sec
T = 5sec
T = 10sec
T = 12sec
T = 16sec
T = 5sec
T = 7.7sec
T = 12sec
T = 7.7sec
0
90
80
70
60
50
40
30
20
10
0
5
10
15
20
25
30
35
40
45
Incident Angle (deg)
Refraction Angle
(deg)
Fig. 4. Relation of incident wave angle a and refraction angle b.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
156
From these equations, the relationship between fluid pressure and horizontal
velocity can be derived in
p ¼ r
o
k
u ¼ r
cu.
(9)
Here,
c is the phase velocity of the sinusoidal elastic waves. This relationship is
valid at any depth
z, but not near the boundary of the end part of the elastic plate
because the wave number differs in open water. In such cases, there will be an infinite
complex wave number at the ends of the boundary area. We can then consider the
balance of flux and pressure of the incident wave, reflected wave, and transmitted
wave using Eqs. (6)–(9).
Assuming that the mean pressure is equal at the boundary, then we obtain
r
caр
ua ю
ucЮ ¼ r
cbub.
(10)
Here,
uc is the velocity of the reflected wave component. The mean velocity
perpendicular to the boundary on both sides of the boundary should be equal, then
we obtain
ua cos a А
uc cos a ¼
ub cos b.
(11)
From these two equations, both the transmission and reflection coefficients of
mean velocity can be obtained:
UT ¼
ub
ua
¼
2
m cos a
cos a ю
m cos b
,
(12)
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Incident Waves
= 75°
0
4770
x (m)
y (m)
1714
0
Bottom = -1.0977,
Peak = -0.6378
Fig. 5. Vertical displacement amplitude distribution for the incident wave angle a ¼ 151.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
157
UR ¼
uc
ua
¼
cos a А
m cos b
cos a ю
m cos b
,
(13)
where
m ¼ р
ca=
cbЮјр1=
nЮ.
By the integration of Eq. (8), the relationship between mean velocity and
amplitude is written as
z
a ¼
hu
c
.
(14)
The amplitude transmission coefficient of flexural wave (i.e., ratio of amplitude
operator (RAO)) can be derived as follows:
RAO ¼
z
a
Z
a
¼
ub
cb
ca
ua
¼
mUT ¼
2
m2 cos a
cos a ю
m cos b
.
(15)
When the incident wave angle a ¼ b ¼ 0, RAO is simply written as
RAO ¼
2
m2
1 ю
m
.
(16)
Fig. 6 shows an example of a comparison between this analytically estimated
amplitude (solid line) and the numerical result (marks). The agreement is fairly good,
indicating that the numerical result and theoretical considerations are both reliable.
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0
2
4
6
8
10
12
14
16
18
20
Wave Period (sec)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
= 0°
= 90°
Eq.(10).(11)
a /
Fig. 6. Estimated and calculated response function for the incident wave angle a ¼ 0.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
158
3. Hydroelastic responses in various conditions
3.1. Time-domain analysis
For the treatment of the response of VLFS in irregular waves or under a moving
load, such as airplane landings/take-offs, the analysis in a time-domain is required. It
is well-known that the linear problem of the frequency-domain formulation can be
transformed into a time-domain formulation and vice versa by a Fourier
transformation technique. Ohmatsu [14] compared both the numerical and
experimental results of response of VLFS in irregular waves. Fig. 7 shows a time
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Displacement at Point A
Displacement at Point B
Displacement at Point C
Measured
Estimated
Measured
Estimated
Measured
Estimated
T (sec)
T (sec)
T (sec)
4
3
2
1
0
-1
-2
-3
-4
-5
1.5
1
0.5
-0.5
-1.5
-1
0
15
20
25
30
35
40
45
15
20
25
30
35
40
45
15
20
25
30
35
40
45
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3
-2
-1
0
1
2
Disp. (mm)
Disp. (mm)
Disp. (mm)
Fig. 7. Time series of vertical displacement at three points.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
159
series of vertical displacement at three points of structure. The solid line represents a
measured value and a dotted line represents an estimated value, using the RAO in
frequency domain. The agreement was good, indicating that linear superposition can
be applied.
On the other hand, Kashiwagi [15] developed a direct method, using the time-
domain mode-expansion method. Special attention was paid to the accurate
calculation of so-called memory effect functions for hydrodynamic forces.
Additional drag forces acting on the airplane, due to elastic deformation of the
runway, were estimated from numerical results.
3.2. Towing condition
VLFS are too big to construct in its entirety at a shipyard. The floating unit
structure is constructed at a shipyard and towed to a construction site. During
towing, the floating unit structure should be safeguarded against waves. Hara [16]
conducted an at-sea experiment of a Mega-float unit. He measured the structural
strain at various points during towing. Fig. 8 shows an example of both a measured
bending moment and an estimated moment.
For an exact estimation, the response analysis requires the inclusion of the effects
of forward speed. Watanabe and Utsunomiya [17] developed a wave response
analysis method of an elastic floating plate in a weak current using the perturbation
expansion of velocity potential and the Green function with current velocity,
U. This
method can be applied to the towing problem. Fig. 9 shows the deformation
distribution in head sea and following sea conditions, as well as stopping conditions.
The differences were not large and could be roughly estimated by considering the
changes in the wave period that were encountered.
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Cal. ( = 30°)
Exp.
G_2
0.01
0.005
0
0
0.5
1.5
2.5
3
2
1
M/pgl
2
Bha
e (rad./sec)
Fig. 8. Bending moment measured by at-sea experiment and calculated bending moment.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
160
3.3. Assembling condition
VLFS should be assembled at construction sites in an open sea. Under such
conditions, the responses and connecting loads of multiple floating units in waves
can be estimated. For this purpose, a calculation method that incorporates the mesh
method can be applied more easily. Fig. 10 shows an example of the numerical
results and measured values in both freely floating and pin-joint conditions.
4. Risk analysis of the VLFS mooring system
Usually, VLFS are moored by a number of mooring devices. The safety
assessment of mooring devices is the most important parameter for safety of VLFS.
Kato et al. [18] conducted comprehensive studies for risk analyses of mooring
systems. For external loads, we consider wind, waves, and current. For both wind
and current load estimations, we have to consider the friction drag component
because VLFS has a vast horizontal plane. As for the wave load, we have to consider
the slowly varying wave drift force,
½
Mij ю
mijр1ЮЉ €
Xр
tЮ ю
FV р _
XЮ
ю
X
n
j¼1
Z
t
А1
_
xjрtЮ
Lijр
t А tЮ dt ю
FMр
X; _
XЮ
¼
Fwind ю
F1 ю
F2.
р17Ю
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1.4
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
x/a
w(x,0)/A
Fig. 9. Amplitude distribution of vertical displacement in head sea, following sea and stopping condition.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
161
Eq. (17) describes the motion in a horizontal plane.
FV is damping due to viscosity,
FM is the nonlinear mooring reaction force, and the right-hand side of the equation
represents the external forces.
The power spectrum of fluctuating wind forces can be calculated using the wind
spectrum:
SFF р
f Ю ¼ r2
a
ZZ
A
Cdiр
f Ю
Cdjр
f Ю
UiUj Re½
Rijр
f ЮЉ
В
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Siр
f Ю
Sjр
f Ю
p
d
Ai d
Aj,
р18Ю
where
Rijр
f Ю ¼ exp А
k1
f j
yi А
yjj
ffiffiffiffiffiffiffiffiffiffiffi
UiUj
p
!
exp Аi
k2
f р
xi А
xjЮ
ffiffiffiffiffiffiffiffiffiffiffi
UiUj
p
!
.
Cd is the pressure drag coefficient for sidewalls or the friction drag coefficient for
horizontal planes,
U the average wind velocity at points
i and
j,
Rij the spatial
correlation function,
x the coordinate in the mainstream direction,
y the coordinate
at right angles to the mainstream direction, and
k the spatial correlation coefficient.
For the wind spectrum on the ocean, Ochi-Shin’s spectrum is used usually. By the
spatial integration of Eq. (18), the fluctuating component of wind force is greatly
reduced.
The wave force,
F1, shows a linear wave force, and
F2 shows a slowly varying wave
drift force. The estimation of the
F2 term is quite important because it can cause a
long period of resonance in the horizontal motion of VLFS. The second-order wave
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Free condition
Pin-joints
2.0
1.5
1.0
0.5
0.0
−1.0
− 0.5
0.0
0.5
1.0
z a/ζ
a
2.0
1.5
1.0
0.5
0.0
z a/ζ
a
x/(L T /2)
−1.0
− 0.5
0.0
0.5
1.0
x/(L T /2)
λ∞
/ L = 0.4
λ∞
/ L = 0.4
Fig. 10. Measured and calculated vertical displacement distribution in freely floating condition and pin-
joint condition.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
162
force for shallow draft structures can be obtained as follows:
F2р
tЮ ¼ r
ZZ
SB
d
S Fр1Ю
t
z¼0
r
wр1Ю ю
1
2
r
g
I
C
d
Cр
wр1ЮЮ
2~
nр0Ю
А
1
2
r
g
I
C
d
Cрxр1ЮЮ
2~
nр0Ю.
р19Ю
Here,
w is the vertical displacement and x is the relative water surface elevation at the
sidewall.
If we consider the equation of motion of the floating thin plate,
D
d2
d
x2
ю
d2
d
y2
2
wр1Ю ¼ Аr
gwр1Ю А rF
р1Ю
t
z ¼ 0.
(20)
Eq. (19) can be written as
F2р
tЮј А
D
ZZ
SB
d
S
@2
@
x2
ю
@2
@
y2
2
r
wр1Ю
А
1
2
r
g
I
C
d
Cрxр1ЮЮ
2~
nр0Ю:
р21Ю
This first term represents the drift force due to the inclination of the bottom of the
VLFS. This term can be changed to a line integral using the free end condition of the
plate.
F2р
tЮј А
Dр1 А nЮ
I
C
d
C
d2
wр1Ю
d
ndt
2
~
nр0Ю
А
D
2
р1 А n2Ю
I
C
d
C
d2
wр1Ю
dt2
2
~
nр0Ю
А
1
2
r
g
I
C
d
Cрxр1ЮЮ2~
nр0Ю.
р22Ю
Let us compare the magnitude of the steady components of these first and second
terms and the relative water elevation term for a typical VLFS structure.
Fig. 11 shows a typical example in the case of a periodic wave with
T ¼ 10 s. The
horizontal axis is the rigidity of VLFS and the vertical axis is the force in both the
longitudinal and transverse directions. These upper figures show the summation of
the first and second terms. The lower figure shows the third term. It is clear that the
first and second terms are very small and can be neglected. The drifting force for
flexible structures is small, compared to that of rigid structures.
Now we consider only the last term of Eq. (22). The time series of incident wave
heights and relative water surface elevations for multi-directional irregular waves can
be represented as follows:
Z
I р~
x;
tЮ ¼ Re
X
i
X
j
Z
ijeiр
kij
!
~
xАo
itЮ
"
#
,
(23)
xр~
x;
tЮ ¼ Re
X
i
X
j
Z
ijx
ijр~
xЮeАio
it
"
#
.
(24)
ARTICLE IN PRESS
S. Ohmatsu / Marine Structures 18 (2005) 149–168
163
ARTICLE IN PRESS
-9
-8
-7
-6
-5
-4
-3
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
(Fx
1+Fx
2)/(ρ
ga
2 B/2)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Fx
3
/(ρ
ga
2 B/2)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Fx
3
/(ρ
ga
2 B/2)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
(Fx
1+Fx
2)/(ρ
ga
2 B/2)
log10(EI/ρBL4)
-9
-8
-7
-6
-5
-4
-3
log10(EI/ρBL4)
-9
-8
-7
-6
-5
-4
-3
log10(EI/ρBL4)
-9
-8
-7
-6
-5
-4
-3
log10(EI/ρBL4)
0°
15
30
45
60
75
15
30
45
60
75
90
Fig. 11. Steady wave drift force due to (1st+2nd terms) and 3rd term of Eq. (22),
T ¼ 10 s.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
164
Here, i depicts the wave period and
j represents the wave direction. x
ij is the RAO
of the relative water surface elevation, calculated in the frequency domain. The time
series of the slowly varying wave drift force can be calculated as follows:
F2р
tЮ ¼ Re
X
i
X
j
X
k
X
l
FА
ijkleАiрo
iАo
kЮ
t
"
#
,
(25)
where
FА
ijkl ¼ А
1
4
r
gZ
ijZГ
kl
I
C
d
Cx
ijxГ
kl~
n.
Here, the asterisk shows complex conjugate value. If we prepare the x
ij in the
frequency domain, we can obtain the time series of the slowly varying wave drift
force.
For the wave drift force evaluation, Shimada [19] developed a rather simple
method. He assumed that the incident wave is perfectly reflected and that the drift
force can be evaluated by the change of momentum in unit time (Fig. 12).
The drift force in
y direction can then be expressed as
f y ¼ r
g
X
i
X
j
Z
ijSi sin y
j cosрo
it ю
ij А
Kix cos y
jЮ
(
)2
¼
1
2
r
g
X
i
X
j
X
k
X
l
Z
ijZ
klSiSk sin y
j sin y
l
В cos рo
i А o
kЮ
t ю
ij А
kl А
aijklx
И
Й
,
р26Ю
where
aijkl Ki cos y
j А
Kk cos by
l,
S
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ю 2
Kh= sinhр2
KhЮ
p
.
Here,
S is the shallow effect coefficient. The total
Fy, after integration, can then be
shown as follows:
Fy ¼
Z
L
0
f y d
x
¼
1
2
r
gL
X
i
X
j
X
k
X
l
Z
ijZ
klSiSk sin y
j sin y
l
В
P cosfрo
i А o
kЮ
t ю
ij А
kl А a
ijklg.
р27Ю
ARTICLE IN PRESS
Fig. 12. Coordinate system.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
165
where
a
ijkl tanА1 1 А cos
aijklL
sin
aijklL
;
P
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 А 2 cos
aijklL
p
aijklL
.
Shimada calls
P the phase difference effect coefficient.
P becomes small with a
large
L. P shows that the fluctuation can be reduced by the phase difference at
different points. Fig. 13 shows an example of a time series of drift forces, calculated
by Hsu’s method, Pinkster’s method, and Shimada’s method. It is clear that the
fluctuating component is very small for very large VLFS.
By introducing these external forces into Eq. (17), we repeat the numerical
simulation more than 10,000 times for the same environmental conditions, and thus
ARTICLE IN PRESS
0
0
100
200
300
Time (s)
400
500
0.1
0.2
0.3
0.4
0.5
Fy/ρ
gLH
1/3
2
Eq. (3)
Hsu, et al.
Pinkster
Fig. 13. Phase-difference effect on slowly varying wave drift force on 1000m long VLFS in long-crested
irregular waves (y ¼ 601).
U = 50m/sec
U = 55m/sec
U = 52.5m/sec
U = 53.5m/sec
U = 57.5m/sec
U = 60m/sec
U = 65m/sec
U = 70m/sec
U = 80m/sec
Design load
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
Maximum mooring force / final dolphin strength
1.0000
0.1000
0.0100
0.0010
0.0001
Exceeding Probability
Fig. 14. Exceeding probability of maximum mooring force in various mean wind speeds.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
166
we can obtain the probability of exceedance of maximum reaction forces acting on
mooring devices for various environmental conditions, as shown in Fig. 14.
Here, the horizontal axis shows a maximum reaction force, and the vertical axis
shows the probability of exceedance for each mean wind speed. Using these
calculated values and the natural environmental conditions of specific sites, we can
estimate the final failure probability of mooring devices, as shown in Fig. 15.
5. Concluding remarks
This paper reviews and categorizes the various estimation methods of hydroelastic
responses of VLFS in waves and other load conditions. This article also outlines the
risk analysis scheme of VLFS mooring system specifying evaluation method of
second order slowly varying wave drift force. It is indispensable for the safety
evaluation of VLFS.
Finally the author is grateful to Professor Torgeir Moan who gave him the chance
to make this review article.
References
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ARTICLE IN PRESS
1.E-08
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
10
15
20
25
30
35
40
45
50
Number of mooring dolphin on the long side
Failure probability
χ=120°
EXP[-3.74763*(N-3.49707)0.413556]
χ=90°
EXP[-2.9*(N-3.49707)0.413556]
Fig. 15. Variation of failure probability to a number of mooring device.
S. Ohmatsu / Marine Structures 18 (2005) 149–168
167
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