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UFL/COEL-TR/099
DESIGN FORCES FOR CONNECTING ELEMENTAL
UNITS OF VERY LARGE FLOATING STRUCTURES
by
M.K. Ochi
and
S.B. Malakar
November 1993
DESIGN FORCES FOR CONNECTING ELEMENTAL UNITS
OF VERY LARGE FLOATING STRUCTURES
by
M.K. Ochi and S.B. Malakar
National Science Foundation
Contract BCS-8920838
November 1993
ABSTRACT
This paper presents results of a study that analytically evaluates the
magnitude of design extreme forces and moments in random seas required to
rigidly connect adjacent units of a very large floating structure comprised of
a large number of identical elemental units. The effect of all units
contained in the entire platform (instead of just the neighboring units) is
considered for connecting any two specific units. Numerical computation is
carried out for a platform consisting of 60 elemental units (6 units in the
X-direction and 10 units in the Y-direction) in a sea of significant wave
height of 10 meters, and extreme connecting forces and moments required in a
one-hour operation in this sea with a risk parameter of 0.01 are evaluated.
KEYWORDS
Very large floating structure, Multi-unit offshore structure,
Wave-induced loads, Extreme design forces
TABLE OF CONTENTS
Page
INTRODUCTION 1
FORCES AND MOMENTS REQUIRED FOR CONNECTION OF UNITS
IN ONE DIRECTION 3
Platform Connected in Y-Direction 3
Platform Connected in X-Direction 8
FORCES AND MOMENTS REQUIRED FOR TWO-DIMENSIONAL RIGID
CONNECTION OF MULTI-UNIT PLATFORM 11
EVALUATION OF CONNECTING FORCES IN RANDOM SEAS 12
EXAMPLE OF COMPUTATION 14
CONCLUSIONS 17
ACKNOWLEDGEMENTS 18
NOMENCLATURE 19
APPENDIX: EQUATIONS OF MOTION OF A PLATFORM IN WAVES 22
REFERENCES 25
INTRODUCTION
The concept of a very large floating offshore structure has become the
object of attention for commercial use. Such a large platform could serve as
a floating city, as an airport for a heavily populated coastal city, as an
isolated locale for a nuclear facility, for a waste disposal plant, etc. For
design of such a very large floating structure, it appears most logical and
practical to consider the structure to consist of many elemental units which
can be constructed and transported separately, and replaced, if necessary.
The problem involved with this kind of structure is that of safe
connection of the elemental units at their boundaries. Hence, irrespective of
whether the connection mode is rigid or hinged, the significance of the
estimation of wave-induced forces and moments (including extreme values
expected in a lifetime) acting on the boundary between units cannot be
overemphasized for design of very large floating structures.
Many interesting and useful studies have been carried out on very large
floating structures. Among others, dynamic responses of a structure comprised
of several elemental units connected in a straight line was studied by Riggs,
Che and Ertekin1, Che et al.2, Riggs and Ertekin3, Wu et al.4, etc., and
responses of a structure comprised of many units two-dimensionaly connected by
hinges were studied by Paulling and Tyagi5. Response characteristics of a
very large ring-shape floating structure supported by eight lower hull-type
floating bodies in waves were studied by Goo and Yoshida6, and Yoshida
et al.7, while extensive studies on hydrodynamic interactions associated with
multi-legs of a very large floating structure were carried out by Kagemoto8,
Kagemoto and Yu9-11. Experimental and theoretical studies on the heaving
force and pitching moment required to connect three units in a straight line
in head seas were made by Maeda et al.12 In this study, the effect of other
elemental units contained in the structure on the connection of any two
specific units was considered. A method for estimating forces and moments
required to rigidly connect the neighboring units of a multi-unit structure
was developed by Ochi and Vuolol3. In this study, estimation was made by
evaluating the difference in the forces (or moments) between two units having
a common boundary.
The present paper presents an analytical development of a method for
estimating the magnitude of extreme forces and moments in random seas required
to rigidly connect adjacent units of a rectangular-shape large structure
comprised of a large number of identical elemental units as shown in Figure 1.
The effect of all units contained in the entire platform (instead of just the
neighboring units) is considered in estimating forces and moments for
connecting any two specific units. The platform is not moored but it is
assumed that no drift movement due to waves and/or current takes place.
FORCES AND MOMENTS REQUIRED FOR CONNECTION OF UNITS IN ONE DIRECTION
Prior to developing a method for evaluating forces and moments to
rigidly join adjacent units of a floating structure consisting of a very large
number of units having the same dimensions, it may be well to develop a method
for evaluating forces and moments of a floating structure consisting of
multiple element units in a straight line along either the X or Y-axis. A
right-hand coordinate system with the Z-axis positive upward is adopted in the
present study.
Platform Connected in Y-Direction
Let us consider a floating structure consisting of N-units in a straight
line along the Y-axis as shown in Figure 2. Let Q be any unit in the platform
and let aq be a vector representing forces in three directions and moments
about the three axes of Unit Q required for the unit to be rigidly connected
with Unit (Q+l) at its deck boundary. The boundary is located at a distance
L/2 from and a height H above the center of gravity of the unit as shown in
Figure 3. We may write the six elements of the restraint vector at the deck
edge of the Unit Q, denoted as aq, as follows:
a 0 0
qx
a 0 0
qy
a L 0 0
a qz + + H
q 2
q aq -a q-a
qz qy
aq0 0 aqx
a a 0
q# qxqx
q_ aq qx_
L
a + a + Ha
q* 2 qL qH
where aq* is the vector representing forces and moments at the center of
gravity of the Unit Q, and aqL, aqH are those vectors necessary to convert
from the center of gravity to boundary with Unit (Q+1). These are given by,
aq*- a qx, aqy, aqz, aqg,' qO, acq',]
aqL- 0 0 0 -aqz 0 aqx]
aqH- [ 0 0 -.qy, aqx, 0
In evaluating the restraining vector aq, we consider wave-induced forces
and moments as well as the resulting motions of all units comprising the
structure. For this we first consider the equation of motion of an arbitrary
Unit J in waves coming from direction p. Since the motions of Unit J are
constrained by aj at one end and aj_- at the other end, the equation becomes
[s.] F + aj aj- (2)
where [s ] Ajs + Bjs + Cjs
s vector representing six-degrees of motion of the unit
I- 1 y, z, z #, '
A Bj, C coefficient matrix of acceleration, velocity and displacement
terms, respectively, in the equation of motions of Unit J (see
Appendix)
F wave-induced forces and moments (vector) on Unit j
L
a -a. + a + Ha
j j* + jL + H jH
a a- L + Ha
j (j-l)* (j-l)L + (j-1)H
4
It is noted that aj_- is evaluated on Unit J-1 as a vector of
constraining Unit J at the boundary between them which is the opposite end of
the Unit J where aj is evaluated. Hence, the signs of a(jl)L and ajL are
different. We may write Eq.(2) as follows:
[S I] + I aj* a-* + I L + a(j-)L 1+ H IaJH a(j-)H } (3:
sj (Jj-1)j-1)* 2 jL Q -1)
By accumulating the forces and moments given in Eq.(3) from j 1 to
j q, we have
- [I]
j-1
- al* + (al* a2*) + (a2* a3*) + ...... + (a(q-l)* aq*)
L
SI1L
-H alH
+ (a1L + a2L)
+ (a2H alH)
+ ............... + (a(q-l)L + aqL)
+ ...............+ (qH (q-l)H)
q* q
j-1
1H a
jL 2 qL qH
Similarly, by accumulating from j q+l to j N, we have
N N-1
F [j] L jL + qL +H aqH
S-q+ j-q+qL q
j-q+l j-q+l
Then, by multiplying Eq.(4) by (N-q) and Eq.(5) by q, and by subtracting
the former from the latter, we can obtain aq* as
N q q N-I
S [s [s L L L- L H qH
Lq* N Ij d ;a C 2N q H 0qH
j-1 j-1 j-1 j-1
(6)
aq* thusly derived represents a restraint vector at the center of
gravity of Unit Q required for rigid connection of the unit with Unit (Q+l)
taking into account the effect of all other units contained in the platform.
Since the connection boundary of the unit is located at a distance L/2 from
and a height H above the center of gravity, the moments about the center of
gravity of Unit Q may be converted to forces at the boundary of the unit. In
combining forces and moments, it is necessary, in general, to consider the
phases between them. However, for safety considerations, it may be best to
consider the severest situation; namely, all phases are zero between the
forces and moments. Thus, the forces in three directions at the boundary of
Unit Q required for rigid connection with Unit (Q+l) becomes,
X-direction aq*x + aq* /H + aq* /(L/2)
Y-direction aqy aq* /H (7)
Z-direction aq*z a /(L/2)
In evaluating the accumulation of forces and moments associated with
motions, 2[sj] in Eq.(6), we may separate sj into two parts: So, a vector
representing that of the entire platform and Asj, a vector representing that
of the difference in the motions between Unit J and the entire platform. For
the latter, rolling 0o and yawing o0 of the entire platform affect the surging
and heaving motions of Unit J. By letting Ij be the distance between the
center of gravities of the entire platform and Unit J (see Figure 2), the
affects on the surging and heaving motions become -Ijo and tfjo,
respectively.
Hence, we may write sj as follows:
sj s + As.
where s x j, y zj <, 8 i,
S x, yo, z 0
Asj [- 0o, j0 0, 0, I
Furthermore, wi
N
S[sj] -
j-1
N
J-1
N
- 2
j-1
N
J-
j-1
e may write [sj] as follows:
(As. + B s + Csj
'A (s + As) + B (so + As ) + C (s + As
A.Sj + B + C + A.Aj + j + Cj ,j
j-1
and
j ]- A ol+ B oS Co ) + A As.+ B.As.+ C.As.
j-1 j-1 j-1
(10)
Note that the coefficient matrices Aj and Cj of all units are the same
if the under-water configuration as well as the dimension of the units are the
same. However, the coefficient matrix of velocity Bj which is associated with
the damping term of the motion is different for each unit, since some elements
of Bj include nonlinear terms whose magnitude depends on the location of the
unit.
For a platform consisting of N units as shown in Figure 2, we have (N-l)
connections, and for each connection we must determine 6 unknown restraints
(3 forces and 3 moments); hence, there is a total of 6(N-1) unknowns. On the
other hand, we have 6 equations of motions for each unit; hence, it appears
that we have a total of 6N equations of motions. However, this is not the
case. The number of equations is actually 6(N-1) for the following reasons:
For Unit #1 shown in Figure 2, we can determine the unknown al from
[sl] F1 al. Then, from a knowledge of al, the unknown a2 can be obtained
from the equation of motion for Unit #2, [s2] F2 a2 al. By applying
this procedure to succeeding unit, the equation of motion for the last unit
becomes [sN] FN aN aN-1, where aN 0 because of the free edge, and
aN_- is already obtained from the equation of motion for Unit (N-l). This
implies that the equation of motion for Unit N is unnecessary and thereby the
total number of equations is 6(N-1) which agrees with the number of unknowns
to be evaluated.
Platform Connected in X-Direction
The same method developed for evaluating the restraining forces and
moments necessary to connect units which are arranged in a straight line along
the Y-axis can also be applied for connecting M units arranged in a straight
line along the X-axis as shown in Figure 4. That is, let pp (vector) be the
constraining forces and moments required for rigid connection of Unit P with
Unit (P+1) at their deck edges. Let the distance between the edge and the
center of gravity of Unit P be B/2, and the height above the center of gravity
be H as shown in Figure 3. We may write pp in a form similar to that given in
Eq.(l). That is,
Ppx
ppy
ppz
p4,
0
0
0
0
Ppz
-qx_
O
--^
B
- p* + 2 PpB + H 0pH
where
p* [Ppx' ,py' Ppz' Pp' P p' pI P
IpB- 0 0 0 0pz' -py]'
pH 0 0 0 -py, px 0]'
Let Unit I be an arbitrary unit with constraints Pi at one end with
Unit I+1, and 6i-i at the other end with Unit I-1. The equation of motion in
waves coming from direction A can be written as
[si] Fi + i pi-1
where
(12)
Fi wave-induced forces and moments (vector) on Unit I
B
pi fi* + B2 iB + H PiH
Pi-1 (i-1)* 2 B(i-1)B + H 'iH'
0
0
0
-ppy
Ppx
0
(11)
,p-
and thereby [s.] may be written as
[SB] F + j + | + + H (H /(-) } (13)
[Si Fi + IJi* (i-)* 2 iB+ (i-)B+ H IiH (i-)H (13)
Using the same procedure considered for derivation of Eq.(6), we can
derive a vector representing forces and moments required for rigid connection
of Unit P in the platform with Unit P+1 as follows:
M M-1
Pp* F f I- [si] F- [si] I- B iBH H iB- 1pB }- H BpH
i-i i-1 i-1 i-1
(14)
Furthermore, by separating si into two parts: so, a vector representing
that of the entire platform and si, a vector representing that of the
difference in the motions between Unit I and the entire platform, [si] can be
written as
M M o M A ,
where C[si] Aso+ Bi+ Co +B C A+ si+ B i+As CiAsi (15)
i-I i-i i-1
pp p o
and [si] ( io+ Bi o+ Cio + A AiAsi+ BiAsi+ CiAsi (16)
i-1 i-1 i-1
By converting the moments about the center of gravity of Unit I to
forces at the edge of the unit, we have the forces in three directions at the
edge of Unit I required to be rigidly connected with Unit (I+1). With the
same assumption as considered in Eq.(7), the forces become as follows:
X-direction
Y-direction q*y Pq*. /H 0p /(B/2) (17)
Z-direction p*z + Pp* /(B/2) .
FORCES AND MOMENTS REQUIRED FOR TWO-DIMENSIONAL RIGID CONNECTION
OF MULTI-UNIT PLATFORMS
In this section, we consider a very large rectangular platform
consisting of M x N units as shown in Figure 1, and evaluate forces and
moments required for rigid connection of an elemental unit with adjacent units
at its boundaries. Let us denote constraining forces and moments (vector) on
the four boundaries of Unit PQ as ap,q, ap,q-1, Pp,q and Pp-l,q as shown in
the figure. Here a represents the restraining vector for connecting units in
the Y-direction, while 8 represents that in the X-direction.
The method for evaluating restraining forces and moments developed in
the previous section cannot be directly applied for evaluating restraints
required for each unit in the two-dimensional X-Y plane. This is because the
number of unknowns in this case exceeds by far the number of equations.
However, the concept developed on the previous section can be applied for
evaluating restraints required to form a large platform consisting of M x N
units as follows:
We first connect M units in the shorter side of the platform
(X-direction in the example shown in Figure 5) by applying Eq.(17). This
forms a rigidly connected column unit. Next, connect N column units in the
Pp*x + fp*O /H
Y-direction. The resulting forces in three directions at the boundary between
the column units Q and (Q+l) can be evaluated by applying Eq.(7), and these
forces in each of three directions are equally distributed over M units. That
is,
X-directions (Q*x + aQ* /H + a /(L/2)} / M
Y-directions (aQy aQ* /H} / M (18)
Z-directions (aQ*z a /(L/2) / M
where M is the number of units comprising the column unit. Note that
computation of a should be carried out for column.units.
EVALUATION OF CONNECTING FORCES IN RANDOM SEAS
The forces and moments discussed thus far are those in regular waves.
By using the results obtained in regular waves, the connecting forces in three
directions on the four boundaries of an arbitrary elemental unit, including
extreme values for design consideration, can be evaluated in random seas by
the following procedure:
(1) Let a platform consist of M x N units. First, compute fp* and aq*
by Eqs.(14) and (6), respectively, for the unit wave height yielding the
frequency response functions. aq* should be evaluated for column units
comprising M elemental units.
(2) Consider the six-parameter wave spectral family for a sea having a
specified significant wave height. The six-parameter wave spectral family was
developed from analysis of 800 spectra obtained from measured data in the
X-direction
Y-direction q*y Pq*. /H 0p /(B/2) (17)
Z-direction p*z + Pp* /(B/2) .
FORCES AND MOMENTS REQUIRED FOR TWO-DIMENSIONAL RIGID CONNECTION
OF MULTI-UNIT PLATFORMS
In this section, we consider a very large rectangular platform
consisting of M x N units as shown in Figure 1, and evaluate forces and
moments required for rigid connection of an elemental unit with adjacent units
at its boundaries. Let us denote constraining forces and moments (vector) on
the four boundaries of Unit PQ as ap,q, ap,q-1, Pp,q and Pp-l,q as shown in
the figure. Here a represents the restraining vector for connecting units in
the Y-direction, while 8 represents that in the X-direction.
The method for evaluating restraining forces and moments developed in
the previous section cannot be directly applied for evaluating restraints
required for each unit in the two-dimensional X-Y plane. This is because the
number of unknowns in this case exceeds by far the number of equations.
However, the concept developed on the previous section can be applied for
evaluating restraints required to form a large platform consisting of M x N
units as follows:
We first connect M units in the shorter side of the platform
(X-direction in the example shown in Figure 5) by applying Eq.(17). This
forms a rigidly connected column unit. Next, connect N column units in the
Pp*x + fp*O /H
Y-direction. The resulting forces in three directions at the boundary between
the column units Q and (Q+l) can be evaluated by applying Eq.(7), and these
forces in each of three directions are equally distributed over M units. That
is,
X-directions (Q*x + aQ* /H + a /(L/2)} / M
Y-directions (aQy aQ* /H} / M (18)
Z-directions (aQ*z a /(L/2) / M
where M is the number of units comprising the column unit. Note that
computation of a should be carried out for column.units.
EVALUATION OF CONNECTING FORCES IN RANDOM SEAS
The forces and moments discussed thus far are those in regular waves.
By using the results obtained in regular waves, the connecting forces in three
directions on the four boundaries of an arbitrary elemental unit, including
extreme values for design consideration, can be evaluated in random seas by
the following procedure:
(1) Let a platform consist of M x N units. First, compute fp* and aq*
by Eqs.(14) and (6), respectively, for the unit wave height yielding the
frequency response functions. aq* should be evaluated for column units
comprising M elemental units.
(2) Consider the six-parameter wave spectral family for a sea having a
specified significant wave height. The six-parameter wave spectral family was
developed from analysis of 800 spectra obtained from measured data in the
North Atlantic (Ochi and Hubblel4). The formulation yields 11 different
spectra including spectra with double peaks. The advantage of using a family
of spectra for design is that the upper and lower bounds of responses in
various sea states cover the variation of marine systems responses computed
using the measured spectra in various locations throughout the world. Thus,
it may be safely concluded that the upper bound of the response can be used
for the design.
(3) Assuming a linear system, evaluate response spectra ?p* and aq* in a
specified sea from Items (2) and (3). Evaluate imo of each element of 3p*
and aq*, where mo is the area under the spectrum.
(4) Using the V/mo -value obtained in Item (3), evaluate forces for a
specified elementary unit PQ. That is, evaluate ap,q by Eq.(18), Pp,q by
Eq.(17), ap,q-l by Eq.(18) with q q-1 and Pp-l,q by Eq.(17) with p p-l.
Choose the largest value in 11 responses of ap,q, Pp,q, etc.
(5) Obtain the 2nd moment of the spectrum, m2, for the three forces in
the X, Y and Z directions in Item (3), and evaluate
Sn (602T m
2 I 2 ;;a m '
o
where T time in hour, a risk parameter.
This is the coefficient for evaluating the design extreme value in T-hours
operation of the system with the assurance (1 a) (see Ref. 15).
(6) Evaluate the design extreme value from Items (4) and (5).
(7) Carry out the same procedure for various wave directions.
EXAMPLE OF COMPUTATION
As an example of application of the method developed in the preceding
sections for estimating connecting forces of elemental units, we may consider
a large floating platform comprised of 60 elemental units (6 units in the
X-direction and 10 units in the Y-direction). Each elemental unit is 50 m
x 100 m, displacement of 14,800 tons, and has eight floats of the same
configuration with 25 m separation between them as shown in Figure 6. Each
float consists of vertical circular cylinders of two different diameters; the
upper portion is 6 m in diameter, while the bottom portion is 12.5 m in
diameter. This configuration is chosen to minimize the wave-induced vertical
force following Motora and Koyama16 who found that the wave-induced vertical
force of a marine structure with small water-plane area becomes almost zero at
certain wave frequencies resulting in a substantial reduction of the vertical
force in random seas.
Computations of design extreme forces necessary for rigid connection of
elemental units are made in a sea of significant wave height of 10 meters.
The extreme forces are those expected to occur in one-hour with a risk
parameter of a 0.01.
Figure 7 shows the six-parameter wave spectral family for a significant
wave height of 10 m. The spectrum identified by the heavy line is the most
probable spectrum expected in this sea, while the other ten spectra are
estimated based on the statistical analysis of measured data with a confidence
coefficient of 0.95. The weighting factor for the most probable spectrum is
0.50, while that for each of the other spectra is 0.05. Using this wave
spectral family, we have eleven response spectra for each force and moment,
and therefrom eleven design extreme values can be evaluated following the
procedure given in the previous section. The largest of these eleven design
extreme values is considered for design.
Computations of the extreme forces are made for the wave directional
angle, p, from 0 to 90 degrees. The extreme forces for other angles can be
obtained from those computed for p 0 to 90 degrees as will be shown later.
Figure 8 shows the design extreme forces in X, Y and Z directions for
connecting six elemental units in the X-direction. The forces are the maxima
computed for the eleven wave spectra for various angles from p 0 to 90
degrees. As can be seen in the figure, the forces in the X-direction are much
larger than those in the other two directions for all five connecting
boarders, and they become maximum at the center connection; namely between
Units 3 and 4.
Figure 9 shows the extreme forces computed by Eq.(18) in the three
directions for connecting elemental units in the Y-direction. Again, the
extreme forces shown in the figure are the maxima computed for the eleven
spectra for various angles from p 0 to 90 degrees. The figure indicates
that the forces in the Y-direction about the quarter length from the edge of
the structure become maximum in this case.
From the results obtained in Figures 8 and 9, we can evaluate the design
forces in the X, Y and Z directions at the four boundaries of each unit of the
structure. Figure 10 shows an example of these forces at six arbitrarily
chosen elemental units for wave directions from 0 to 90 degrees. In the
figure, the first digit of the unit represents the row number of the
multi-unit structure, while the second digit is that for the column number.
Although Figure 10 shows the connecting forces for wave directions from
0 to 90 degrees, it represents forces for all wave directions. This is
because the connecting forces on a unit shown in the figure are equal to those
of the unit located symmetrically to it with respect to the origin. Although
not shown here, the forces on any unit are symmetric with respect to X and Y
axes. For example, the forces on Unit 1.1 for wave direction from 90 to 180
degrees are the same as those on Unit 6.1 computed for wave directions from 0
to 90 degrees but forces along the Y-axis are interchanged, which is
essentially equal to the forces on Unit 1.1 for wave direction from 0 to 90
degrees. Thus, the forces obtained on Unit PQ in a very large floating
structure consisting of M x N units shown in Figure 10 can be equally applied
on Units (M+1-P)Q, P(N+1-Q) and (M+1-P)(N+1-Q) by interchanging the forces
along the X and/or Y axis depending on the location.
CONCLUSIONS
A method for estimating the magnitude of design extreme forces required
to rigidly connect adjacent units of very large floating structures consisting
of many identical units, called elemental units, is analytically developed.
In estimating forces and moments for connecting any two specific units, the
effect of other units contained in the entire structure (instead of just the
neighboring units) is considered.
The frequency response functions of wave-induced forces and moments are
first calculated through a solution of the equations of motions in six degrees
of freedom. Then, the response spectra of forces and moments in a specified
sea state are obtained, and therefrom the design extreme values are evaluated
by applying extreme value statistics. The moments above the center of gravity
of the elemental units are converted to forces at the boundary of the units,
thus the connection forces in three directions (X, Y and Z directions) on four
boundaries of each elemental unit are evaluated.
As an example of numerical computations, connecting forces are evaluated
for a floating platform comprised of 60 elemental units (6 units in the
X-direction and 10 units in the Y-direction) in a sea of significant wave
height of 10 meters. Each elemental unit is 50 m x 100 m, has a displacement
of 14,800 tons, and has 8 floats of the same configuration. The design
extreme forces on four boundaries of each elemental unit expected to occur in
a one-hour operation in the sea are estimated with the risk parameter of 0.01.
ACKNOWLEDGEMENTS
This study was sponsored by National Science Foundation, Environmental
and Ocean Systems Program, through contract BCS-8920838 to the University of
Florida. The authors would like to express their appreciation to Mr. Norman
Caplan, Program Director, for his valuable technical advice received during
the course of this project. The authors also wish to express their thanks to
Ms. Laura Dickinson for her patient typing of the manuscript with its
intricate equations and tables.
NOMENCLATURE
A
aij
A
B
bij
B
C
cij
Cdx Cdyl Cdz
D x, D Dz
x y z
D D ,y D
x y z
dl, d2
F
G
(gi)
g
H
(hi)
H
k
L
11, 12
M
mj
mx*, my*
N x, Ny Nz
x y z
- coefficient matrix of acceleration terms in equation of motion
- (i,j)th element of A
- projected area of float in XY-plane
- coefficient matrix of velocity terms in equation of motion
- (i,j)th element of B
- breadth of elemental unit
- coefficient matrix of displacement terms in equation of motion
- (i,j)th element of C
- eddy-making drag coefficient of float in X, Y, and Z directions,
respectively
- frictional drag coefficient of float
- added-mass coefficient
- drag coefficients of float in X, Y, and Z directions,
respectively
- linearized drag coefficients of float in X, Y, and Z directions,
respectively
- diameter of upper and lower floats
- wave-induced forces and moments (vector) on elemental unit J
- vector coefficient of cosine component of forcing function
- ith element of G
- gravity constant
- vector coefficient of sine component of forcing function
- ith element of H
- vertical distance between deck and center of gravity of
elemental unit
- 2/g
- length of elemental unit
- wetted length of upper and lower float, respectively
- mass of platform
- j-th moment of spectrum
- added mass of float in X, Y, and Z directions, respectively
- added mass associated with incident waves of float in X and Y
directions
- wave-generating damping coefficient of float in X, Y, and Z
directions, respectively
s vector representing six-degree of motion
[x, y, z 0, #]'
s s of the entire platform
(x Y z 0 0 00 ]'
[Xo' 0o, 0' o, o '
s. s of unit J
As. s- s
J j o
s distance from the center of gravity of elemental unit to
free-water surface
s(w) spectral density function
T duration time in hours
ul, u2 distance from free water surface to geometric center of
underwater configuration of upper and lower part of float,
respectively
W wetted surface area of float
(X, Y, Z) coordinate of entire structure
(x, y, z) coordinate of elemental unit
(xi, yi) location of j-th float in reference to the center of gravity of
elemental unit
(xj, y ) location of j-th float in reference to the center of gravity of
entire structure
x surge motion
y sway motion
z heave motion
a vector representing forces and moments at the boundary of Unit Q
required to be connected with Unit (Q+1) in a platform connected
in the Y-direction
[aqx, a qy, aqz, aq aqO, a q]'
(aqx' ,qy' aqz' oLqO' Oq,' cq, It
a vector representing forces and moments at the center of gravity
of Unit Q in a platform connected in the Y-direction
aqH vector representing forces of a which contribute to moments in
a at the boundary vertical distance H from the center of
gravity
[0, 0, 0, -aqz 0, aqx
a -vector representing forces of a which contribute to moments in
qL a at the boundary distance L/2 qrom the center of gravity
q
[0, 0, 0, -a qy aqx 0]'
8p vector representing forces and moments at the boundary of Unit P
Required to be connected with Unit (P+l) in a platform connected
in the X-direction
[[x px py' pz' Pp' Pp' P pp]
p. vector representing forces and moments at the center of gravity
of Unit P in a platform connected in the X-direction
SpB- vector representing forces of 6. which contribute to moments in
Bp* at the boundary distance B/2 from the center of gravity
[0, 0, 0, 0, pz py]
ppH vector representing forces of _. which contribute to moments in
f6 at the boundary vertical distance H from the center of
gravity
[0, 0, 0, -p -0px, 0]'
wave elevation
o surface wave amplitude
0 pitch motion
p angle of wave incidence
v kinematic viscosity of water
p fluid mass density
T k(x. cos p yj sin p)
roll motion
XxjX X amplitude of displacement of j-float relative to waves in X,
Syj zJ Y and Z directions, respectively
w circular frequency in radian per second
APPENDIX: EQUATIONS OF MOTION OF A PLATFORM IN WAVES
The equations of motions of a floating platform consisting of many
identical rectangular-shaped units, each supported by multiple vertical
floats, in regular waves are outlined in the following.
Figure A-1 shows a right-hand coordinate system with the Z-axis upward
and the origin at the center of gravity of the entire platform, denoted by 00.
Let us consider Unit j whose center of gravity, denoted by 0, located at (bj,
lj), in the coordinate system (x, y, z). Two coordinate systems (X, Y, Z) and
(x, y, z) are necessary for evaluating wave-induced moments (rolling moment
for example) of a unit and those of the platform separately. Let the location
of a given float of a unit be (xj, yj) in the (X, Y, Z) coordinate system and
(xi, yi) in the (x, y, z) system. Here, we have x. bj +xi and yj lj +
yi. The equations of motion discussed below refer to those of the entire
platform.
The wave profile f encountered by the float located at (xj, yj) is
written as
((t) Re (o exp[ il xj cos p yj sin i) Wt j]J, (A.1)
where 0o surface wave amplitude
k w2/g
S- circular frequency rps
p angle of wave incidence.
We may assume that (a) floats are symmetrically arranged with respect to
the origin 0, (b) there is no modification of the wave profile due to the
presence of the floats, (c) the hydrodynamic interaction between floats is
negligible and (d) water is incompressible and water particle motion is
irrotational. In developing the equations of motion, two different types of
damping are considered; one is that due to energy loss through generation of
surface waves (designated as N), and the other is that associated with
generation of turbulent eddies around the float (designated as D). The latter
is proportional to the square of the velocity. Thus, the damping force in the
Z-directions, as an example, becomes,
Nz+ =C I IN
..A.. 1 'Iz
Nz + Dz zz| N z + p(C A + C W)zz ,
z z z 2 dz z f
(A.2)
where p
N
z
D
z
Cdz
A
z
W
Cf
- fluid mass density
- wave-generating damping coefficient of a float in the Z-direction
- drag coefficient of a float in the Z-direction
- eddy-making drag coefficient of a float
- projected area of a float in the XY-plane
- wetted surface area of a float
- frictional drag coefficient of float.
The nonlinear damping given in Eq(A-2) may be linearized following the
technique known as equivalent linearization, and the linearized damping
coefficient, Dzj, can be written as
- 8 4
Dzj ~~ oX Dz PX CdzAz + C
(A.3)
where Xzj is the amplitude of vertical displacement of the j-th float relative
to waves (Ochi and Vuolol3). Taking these assumptions into consideration, the
equations of motions of the platform in 6-degrees of freedom can be written
as,
S+ B + C G cos t + H sin t,
A s + B s + C s G cos ut + H sin ut,
(A.4)
where
s [x, y, z, 0, 0, ] vector representing surge, sway, heave, roll, pitch
and yaw motions, respectively.
A coefficient matrix
0 0
a22
0
U a42
a51 0
0 0
coefficient
b1 0
0 b22
0 0
0 b42
b61 b62
Coefficient
0 0
0 0
0 0
0 0
0 0
0 0
0
a33
0
0
0
matrix
0
0
b33
b43
b53
0
matrix
0
0
C33
0
0
0
o
o
of acceleration terms
0 a15 0
a24 0 0
0 0 0
a44 0 0
0 a55 0
0 0 a66
of damping
0 b15
b24 0
b34 b35
b b45
b44 b45
b54 b55
b64 b65
of restoring
0 0
0 0
0 0
C44 0
0 C55
0 0
terms
b16
b26
0
b46
b56
b66
terms
0
0
0
0
0
0
B-
C-
G Vector coefficient of cosine component of forcing function
gl' g2' g3' g4' g5' g61
H Vector coefficient of sine component of forcing function
[h, h2, h3, h4, h5, h 61
Elements of A, B and C matrices are given in Table A-1, and elements of
G and H vectors are given in Table A-2. Figure A-2 shows the under-water
configuration of float.
REFERENCES
1. Riggs, R., Che, X. and Ertekin, R.C., "Hydroelastic Response of Very
Large Floating Structures" 10th Int. Conf. on Offshore Mechanics &
Arctic Eng., OAME '91, ASME, 1991.
2. Che, X.L., et al., "Two-Dimensional Hydroelastic Analysis of Very Large
Floating Structures", Marine Technology, Vol.29, No.l, 1992, pp.13-24.
3. Riggs, H.R. and Ertekin, R.C., "Approximate Methods for Dynamic Response
of Multi-Module Floating Structures", Marine Structures, Vol.6, No.2/3,
1993, pp.117-141.
4. Wu. Y., et al., "Composite Singularity Distribution Method with
Application to Hydroelasticity", Marine Structures, Vol.6, No.2/3, 1993,
pp.143-163.
5. Paulling, J.R. and Tyagi, S., "Multi-Module Floating Ocean Structures",
Marine Technology, Vol.6, No.2/3, 1993, pp.187-205.
6. Goo, J.S. and Yoshida, K., "A Numerical Method for Huge Semisubmersible
Response in Waves", Trans. Soc. Naval Arch. & Mar. Eng., Vol.98, 1990,
pp.365-387.
7. Yoshida, et al., "A Conceptual Design of a Huge Ring-Like Semi-
submersible", Proc. 1st, Int. Workshop on Very Large Float. Structures,
Hawaii, 1991, pp.81-96.
8. Kagemoto, H., "Minimization of Wave Forces on an Array of Floating
Bodies The Inverse Hydrodynamic Interaction Theory", Appl. Ocean Res.,
Vol.14, 1992, pp.83-92.
9. Kagemoto, H. and Yue, D.K.P., "Wave Forces on Multiple Leg Platforms",
Proc. 4th Int. Conf. on Behaviour of Offshore Structures, 1985,
pp.751-762.
10. Kagemoto, H. and Yue, D.K.P., "Wave Forces on a Platform Supported on a
Large Number of Floating Legs", Proc. 5th Int. Conf. on Offshore
Mechanics & Arctic Engineering, Vol.1, 1986, pp.206-211.
11. Kagemoto, H. and Yue, D.K.P., "Hydrodynamic Interaction Analyses of Very
Large Floating Structures", Marine Structures, Vol.6, No.2/3, 1993,
pp.295-322.
12. Maeda, et al., "On the Motions of a Floating Structure which Consists of
Two or Three Blocks with Rigid or Pin Joints", Journal Soc. Nav. Arch.
of Japan, Vol.145, 1979, pp.71-78.
13. Ochi, M.K. and Vuolo, R.M., "Seakeeping Characteristics of a Multi-Unit
Ocean Platform", Spring Meeting, Soc. Nav. Arch. & Mar. Eng., 1971.
14. Ochi, M.K. and Hubble, E.N., "On Six-Parameter Wave Spectra", Proc. 15th
Int. Conf. Coastal Eng., Vol.1, 1976, pp.301-328.
15. Ochi, M.K., "Principle of Extreme Value Statistics and Their
Application", Proc. Ext. Loads Response Symp., Soc. Nav. Arch. & Mar.
Eng., 1981, pp.15-30.
16. Motora, S. and Koyama, T., "Wave Excitationless Ship Forms" Sixth Symp.
Naval Hydrodynamics, ACR-136, 1966.
G Vector coefficient of cosine component of forcing function
gl' g2' g3' g4' g5' g61
H Vector coefficient of sine component of forcing function
[h, h2, h3, h4, h5, h 61
Elements of A, B and C matrices are given in Table A-1, and elements of
G and H vectors are given in Table A-2. Figure A-2 shows the under-water
configuration of float.
REFERENCES
1. Riggs, R., Che, X. and Ertekin, R.C., "Hydroelastic Response of Very
Large Floating Structures" 10th Int. Conf. on Offshore Mechanics &
Arctic Eng., OAME '91, ASME, 1991.
2. Che, X.L., et al., "Two-Dimensional Hydroelastic Analysis of Very Large
Floating Structures", Marine Technology, Vol.29, No.l, 1992, pp.13-24.
3. Riggs, H.R. and Ertekin, R.C., "Approximate Methods for Dynamic Response
of Multi-Module Floating Structures", Marine Structures, Vol.6, No.2/3,
1993, pp.117-141.
4. Wu. Y., et al., "Composite Singularity Distribution Method with
Application to Hydroelasticity", Marine Structures, Vol.6, No.2/3, 1993,
pp.143-163.
5. Paulling, J.R. and Tyagi, S., "Multi-Module Floating Ocean Structures",
Marine Technology, Vol.6, No.2/3, 1993, pp.187-205.
Table A-i: Elements of Matrices A, B and C
a b c
11 M+ n m n Nx + xj 0
where, mx where Dx Dxj + Dx
pCM (d1/2)2 1
+ (d2/2)2 9
12 0 0 0
13 0 0 0
14 0 0 0
15 n Imxl(s+u ) (nNx + Dxlj) (s+ul) + Dx2j (s+u2) 0
+ x2 (s+u2)
(- 1
16 0 ( 6Yx ) 0
21 0 0 0
22 +n m nNy y + Eyj 0
23 0 0 0
24 n ( my(s+u) nNyj(S+ul) + Dylj (s+u) + Ey2j (s+U2) 0
+ my2(s+u2)
25 0 0 0
26 0 E(fyjxj) 0
0
0
M+n m
There m ml+ mz2
0
Same as a14
Same as a24
Same as a34
Jx + m ,2
+ n Imyl(s+u" )
+ my2(s+u22 2
0
0
0
n N + Ezj
SDzjyj
- E Dzjj
0
1'
Same as bl4
Same as b24
Same as b34
Nz1( + 5yiDzjy +( nNl + ylDgJ (s+ul)2
+ I( By2j (s+u2)2
( Dzjxjyj
I "yljxj) (s+ul) + I Ey2jxjl (s+u2)
0
0
n( 1pgd
0
0
0
Same as cl4
Same as c24
Same as c34
4pgird( j 2
0
11
1
Same as a15
Same as a25
Same as a35
Same as a45
Jy + m' Ex2
Jy
+ n (mx(s+)2
+ mx2(s+U)2
0
Same
Same
Same
Same
b15
b25
b35
b 45
nNxl+: xlj (s+ul)
+ ( Ex2j (s+u2)
I ( x1jYjl (s+ul) + (I x2jYj (s+u2)
-II 4
Same as a16
Same as a26
Same as a36
Same as a46
Same as a56
Jz + mx Yj
+ m x
Same as b16
Same as b26
Same as b236
Same as b46
Same as b56
N ) j) +( xj Nyl x() +( yj 2
I I
Same as c15
Same as c25
Same as c35
Same as c45
ZEpgxd] CX I
0
Same
Same
Same
Same
Same
as c16
as c26
as c36
as c46
as c56
_
zj Ex2 +I Fix +
Table A-2: Elements of Vectors G and H
g h
1 o cosA [-2 Im Esinrj + I Nie -kul IcostJ) *o cos -- 2aM cosrJ) + Ne-kul ( sinrJ)
(1-kul 1 l12 1 -kul 1 -ku2 1
+ I Ixljcosj e + (lI D" j ) -ku2 + DI xljin1 e + DI kx2jsinurjle
where, mx* pxCH (dl/2)21te-kul + (d2/2)2t2e-kU2}
2 f2 I II 2) 1 1-kul
2 o sinip I-u yi sinrj) wu Nyle icostjj eo' sin( u2I [ cosrj) u ( ye ( sinrj)
+ ( LD cost) e + ( cDy2jCosrj) e 2 I + D( ylsinr e + Dy2jsi)nr'j) k2 u
where, my mx
S2 -kul -ku2 2 -kul -ku21 I.i..j
3 o -, ( rl' + z2u ( os 1 -o ." (ezlU~e + z2er sin r)
+ ( Nzle-kul ( lsin) + ( I nr e -ku2 1 I Nze -kul I E casj) + ( Ee ICO) -ku2
2 -ku l 2 -kul
+ pgxd2e 1 Cl + 4 pgwdi e E
4 ro [2 1azle m + mz2e -k2)( yj Cor) To 2 mzlek + mz2e L( yjsinrj)
-ku -ku2 -ku -ku2
sini myle (s+ul) + my2( e (s+U2)} osinrj) t + sinr (mZ e (s+ul) + mz2e (s+u2)}( cosr()
+ u (N- kue ( yjslnrj) + Dz2jYjainrj) ku l Nz1 Eyjsinrj) + ( Ez2jYjcosrj I
+ -kuW ir+ i W I (T.
a sin Nye-kul (s+ul)( cosri a sin Nyle-ku (s+ul)( asinrj)
+ e-ku D ycosrj) (s+u1) + e-ku Dy2jcosrj) (s+u2) + e-ku E yljsinrj) (s+ul) + e-ku I y2jSinrjl (s+u2)}
+ ( pgd2 I e1 Lyjcosj + ( g pgir di ekul L yj sinrj)
5 o 2 { (mzle kul + mz2e )(, xjcosoj) oz, [ (leu l + mz2e- ( xjsintj)
-kul -ku2 [_kul -ku2
I ~ -l -ku11 f "2 11 l1
cosA exle (s+ul) + mx2e -u(s+u2)) ) Fdsinrij + cos., xle l (s+ul) + mx2e (s+u2) ) EcosrJ
Nzle -kul( xjsinr) + I z2 rjx I + u Nzle-kul( xjcosrj + E 2jXjcosT)
cosA Nxl-kul(s+ul) ( cosrJ) w cosA IN kul(s+ul) snrj)
+ e-kul xjcosr (s+u) + e ( +s+u) + e-kl) Dl isianr J (s+ul) -+ e -ku2 2jsinrjl (s+u2)J
pgnd2 ) ek xjcosTrj + ( pgirdi )e -kul xj sinj)
6 -o "2 ( cos mx* ( yjsinrj) + sinp y* ( Exjsinrj) {I o [2 cosA ( Yjcosrj) + sint My* ( Eyjsinrj I
-W cos Nj N -kule ( i cosr -- cos I Nxleku ( Eysinrj)
+ -ekul xlfjyjE" + e -ku2( dx2jY Cos I + e -kul xljj sin + e -kuI dx2jxsinjl I
+ s in Nyle ( xj cost) + sin N1e- ul( Ixjsinrj)
+ ku( 5Dyljxjcosrj + e-ku2 ( D2jxjcos rJJ + e-ku x j sinr)t + -ku2l +xjxjslnrj
WAVES
\ ~x
1 2 ---------- J---.-- Q ---- ------------- ------------ N
\I
\ i l
\ I
PI q p,q
IPP_
Figure 1: Very large foating structure comprised of Mx N elemental
units.
WAVES
\.x
1 2 --------- J
---a---
----N
Figure 2: Floating structure with elemental unit arrangement
in a straight line along the Y-axis.
1
2
Y P
M
Y -1!-1 140. 1KI-IA
I--L/2 I-- B/2-t
HH I
Figure 3: Longitudinal and transverse cross sections of
elemental unit.
x
WAVES
\ 2 _
\I
\ I
'I
I
I
-pp
M
Figure 4: Floating structure with
elemental unit arrangement
in a straight line along
the X-axis.
Y 4- -- -- -
P IPQ
M _
Pp-, q
ap, q-1 Op, q
Op,q
Figure 5: Definition of connecting forces and moments
(vectors) on four boundaries of Unit PQ.
Figure 6: Dimension of elemental unit used for
computation.
35
30 1
S 25- I
20 \
.0 \I
s i\ \
10 -, j
uI \
5-
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
FREQUENCY IN RADIAN/SEC
Figure 7: Family of six-parameter wave spectra for significant wave
height of 10 meters.
250
200
0
I-.
_2 .oo-------- ---- --'--i---.--
z 50
0
U.- 100
50
y
1 2 3 4 5
UNIT
Figure 8: Connecting forces in X,Y and Z-directions of
structure with unit arrangement in a straight
line along the X-axis.
UNIT
Figure 9: Connecting forces in X,Y and Z-directions on elemental unit
of each column arrangement in a straight line along the
Y-axis.
11 1.10
S5*8
6*1 *10
< 1,000 >
1*1
3*5
II I
I 46
I
S610
*I
X DIRECTION
-- Y DIRECTION
------- Z-DIRECTION
0 2 4 6TON/M
I I I I
Figure 10: Example of connecting forces in X,Y and Z-directions for
design consideration of elemental units of very large
floating structure. Significant wave height of 10 meters.
T
300
_y
WAVES X
YN
x1 bl
Y 00
Figure A-1: Right-hand coordinate systems (X,Y,Z)
and (x,y,z).
#z Z
YI
^ 4 Y
X, iz
-r------- --
S1 100
d I
2
Figure A-2: Under-water configuration of float.
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