• TABLE OF CONTENTS
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 Front Cover
 Title Page
 Abstract
 Table of Contents
 Introduction
 Forces and moments required for...
 Forces and moments required for...
 Evaluation of connecting forces...
 Example of computation
 Conclusions
 Acknowledgement
 Nomenclature
 Appendix: Equations of motion of...
 References
 Tables and Figures






Group Title: Technical report – University of Florida. Coastal and Oceanographic Engineering Program ; 99
Title: Design forces for connecting elemental units of very large floating structure
CITATION PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00075016/00001
 Material Information
Title: Design forces for connecting elemental units of very large floating structure
Series Title: Technical report – University of Florida. Coastal and Oceanographic Engineering Program ; 99
Physical Description: Book
Creator: Ochi, Michel K.
Malakar, S. B.
Affiliation: University of Florida -- Gainesville -- College of Engineering -- Department of Civil and Coastal Engineering -- Coastal and Oceanographic Program
Publisher: Dept. of Coastal and Oceanographic Engineering, University of Florida
Publication Date: 1993
 Subjects
Subject: Coastal Engineering
Hydraulic engineering   ( lcsh )
University of Florida.   ( lcsh )
Spatial Coverage: North America -- United States of America -- Florida
 Notes
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
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Bibliographic ID: UF00075016
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Abstract
        Abstract
    Table of Contents
        Table of Contents
    Introduction
        Page 1
        Page 2
    Forces and moments required for connection of units in one direction
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    Forces and moments required for two-dimensional rigid connection of multi-unit platform
        Page 12
        Page 11
    Evaluation of connecting forces in random seas
        Page 12
        Page 13
    Example of computation
        Page 14
        Page 15
        Page 16
    Conclusions
        Page 17
    Acknowledgement
        Page 18
    Nomenclature
        Page 19
        Page 20
        Page 21
    Appendix: Equations of motion of a platform in waves
        Page 22
        Page 23
        Page 24
        Page 25
    References
        Page 26
        Page 27
        Page 25
    Tables and Figures
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
Full Text





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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