Diffraction theory and statistical methods to predict wave induced motions and loads for large structures

OFFSHORE TECHNOLOGY CONFERENCE 6200 North Central Expressway

Dallas, Texas 75206

PAPER

NUMBER

OTO 2502

Diffraction Theory and Statistical Methods

to Predict Wave Induced Motions and Loads

for Large Structures

By

Arne E. LOken and Odd A. Olsen, Det Norske Veritas

THIS PAPER IS SUBJECT TO CORRECTION ©Copyright 1976

Offshore Technology Conference on behalf of the American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. (Society of Mining Engineers, The Metallurgical Society and Society of Petroleum Engineers), -American Association of Petroleum Geologists, American Institute of Chemical Engineers, American Society of Civil Engineers, American Society of Mechanical Engineers, Institute of Electrical and Electronics En-gineers, Marine Technology Society, Society of Exploration Geophysicists, and Society of Naval Architects and Marine Engineers.

This paper was prepared for presentation at the Eighth Annual Offshore Technology Conference, Houston, Tex., May 3-6, 1976 Permissionto copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. Such use of an abStract should contain conspicuous acknowledgment of where and by whom the paper is presented.

ABSTRACT

This paper presents results from com-putations of wave loads on one fixed and one floating large volume structure using three-dimensional sink-source technique (diffraction theory). The motions and mean drift forces are also included for the latter: Theoretical calculaticns are compared with the results from model experiments. Generally, flood agreement was found between computer programme results and experiments.

-Three-dimensional diffraction theory was further used to predict the effect a large volume caisson may have on the Incident wave and to estimate the added Mass pressure in vibration.

The design wave concept and the use of statistical methods in connection with diffraction theory are discussed on the basis of different wave spectra forms. The application of.two-dimensionalsink-source method (strip theory) compared to the three-dimensional method is

References and illustrations at end of paper.

discussed with reference to sample

cal-culations.

INTRODUCTION

The basic contributions to the applica-tion of three-dimensional diffracapplica-tion theory are the papers of Lebreton and Cormault (1969) /1/ and Garrison and

Seetharama Rao (1971) /2/. Since their presentations this method has increasing-ly become the predominant technique for dealing With wave induced loads on large offshore structures of arbitrarY form. However, strip theories .based on

two-dimensional (diffraction) theories have for some time been applied in ship. design to determine responses inregular waves.- The linear superposition tech-nique-proposed by St. Denis and Pierson /3/ is then normally used to determine the-responses in irregular seas.

-Different computer programmes have been developed on the basis of the method proposed by Lebreton and Cormault.

Computations of short-term responses in irregular seas using different wave spectrum formulations are-also included. This procedure has not yet gaineda real break-through in the design of large offshore structures although it has been widely used in ship design for years.

THEORETICAL FORMULATION

A review of the basic equations used in the diffraction theory program is

presented in the following. The equations of motions

The motions of a floating structure are derived from the condition of dynamic equilibrium betweenexcitation,and restoring, damping and inertial reac-tions of the system. Assuming that the system is linear-and-harmonic, the six coupled linear differential equations -of motions may be written:

ym.1.21.)i.14:33.T!.c.ri

3k 3k k 3k k 3k k

=F.e-iwt(1)

k=1

Here M.k is the component of the gene-ralize

a

mass matrix (j=k=1,2-6) and A.k and Bikare the added mass and dampina coefficients respectively. The restoring _coefficients are. denoted C. the

-cOmpiex amplitude of the ekeiting forces and:mbMents P.. It is understood that the real part3should be used in all expressions when e-lwt is involved. The -frequency of wave encounter which is thefrequency

same as the frequency of the response is further denoted w. Displacements are represented by n. Dots are used to indicate time derivatives.

A right-handed coordinate system x,y,z fixed with respect to the mean position of the structure with origin in the plane of the free undisturbed surface is used. The coordinate z points ver-tically upwards through the centre of gravity and the coordinate x in the direction of backward motion. It is assumed that the structure is symmetric about the x-z plane and the centre of gravity is located in (0,0,zc).

The generalized mass matrix is written as:

Here M is the mass of the structure, I. is the moment of inertia, and Ijk is J

the product of inertia.

The added mass or damping coefficients are:

(orB.=A11 0

A13 0 A15-0'-(3) Ajk

M

_{A22 0} A24 0 _A26

.A310

_{A33 0 A35 0}

0

A42'0 A44 0 A46

A510

A530

A550

-0

A60

A640

A66 For a structure floating in the free surface the only non-zero restoring coefficients are:

C. C t1C # (4)

331 C44' 55 an 5 C.35

Different methods of taking into account non-linearities of restoring and damping coefficients exist, but these will not be dealt with here.

Hydrodynamic boundary value problem

The problem is formulated in terms of linear potential flow theory in steady state condition.

The total velocity potential is written as:

0(x,y,zit)

_{Re(p(XtY,Z) e)}

(5)

This velocity potential is conveniently broken down into the following parts:

6

= ₀+4) +

_7.

_j

_j

(6)

.3=1

Here c..7

et

is the diffraction poten-tial for the restrained body while qb., j#1,6 are the velocity potential due3 to the-different modes of motions. The velocity potential of the incident wave is written as:

_iwt g.ca cosh k(z+h)

o e w cosh k.h

i(kx cos(3+ky sinfi-wt) (7)

e

Here is the wave amplitude,- h is the

(2)

one of these programmes together with 0 M 0 -Mzc 0 0

information gained from experiment 0 0 M 0 0 0

results, see also Faltinsen and Michel- 0 -Mzc 0 14 0 -146 sen (1974) /4/ and Faltinsen et.al. Mzc 0 0 0 Is 0

(1975) /5/. o o -14:6 0

6

-DIFFRACTION THEORY AND STATISTICAL METHODS TO PREDICT

798 _{WAVE INDUCED MOTIONS AND LOADS FOR LARGE STRUCTURES} OTC 2502

ARNE E. LOKEN

ODD . A. OLSEN 799

OTC 2502

water depth and p, is the direction of propagation of incident waves. The wave number is related to the frequency

of

waves by the dispersion relationship:

2

= k tanh k.h

§

The different contributions to the po-tential have to satisfy the Laplace equation, i.e.:

-2

v 0, = 0 in the fluid domain (9)

-w 02 + g - 0 on z = 0 .(10)

az

The free surface condition Eq. (110) may be rewritten in'the.case w co to give

the high frequency Condition:

= 0 (11)

714,

j=1,6 and0, e-iwt

aJradiation condition and condition at the sea bed,

q) = 0 on z The terms 0. must satisfy the boundary i.e.:

44

on z = -h (12) a2

The following body boundary condition On the average position of the wetted surface of the body will apply:

a0,

7r12 nj

j

= 1,6

47

-40

an

Here a/an is the normal derivative in the direction;of the outward normal veqtor n.

The generalized normal n., is . defined by: j, n = (n n n-) l' 2' 3

r

n = (n_{A' 5}nein-) 6

The position vector is defined by:

= Ici

+ 173 + z15. (17)

The solution of 0, (j=1,7) can be written as:

c.=: II

Q.(,11,)G(x,Y,zi,7140ds

(18)

S

-

7

Here Q. is the unknown source

density

fundtign

and G(k.17,z;Eirl,C) is the

Green's function. The latter can be

'Written in

three different ways

accor-ding to Wehausen and Laitone /6/ and

(8)

Garrison /7/p i.e.:

2 2

d(xtlr,z4,11,C) 12c1(1.1)..

cosh k(c+h)(Y.(kr1 )-iJ0 (kr ))

22

cc-k+N). / 2 2-k=1, ukh+v- h7v cOs(11

(C+h))X0(pIcr

) or: q.(x,Y,z; = 1. R1 +2PV cc I ' -ph e coshp(c+h)coshp(z+h)J (pr psinhph - vcoshph 27(k2-v2)coshk(h)coshk(z+h) 0-+ i J (kr ) 2 -2 k h - v h+v (20 ) or: G(x1Y,z4Eirl,) = op 1 1 + [(-1)nR2 + R ) n=1 n 4n cos(p 1?+10) (19) + 1-1)n-/--(R3 1 R5 ) + cosh k(z+h) ) p (21) n

In Eq. (19) pk is the solutions of the

equation: Pktan p h + v = 0 (22) and 2 v = ₍₂₃₎ -=

/4(k--E)

+(Y-11)2+(z"t)2* :(24) = +(y-n)-2+(z+2h+t) (25) = 2+ (y-h). (26)

2+(1.02

R/ 1 -.(27) (28) 2 1

R2=

/ r1 +(z-2nh-c

Added mass in vibration

The three-dimensional diffraction theory problem was formulated in the previous section.to obtainthepressure in phase with rigid body accelerations and thus the resulting added mass.

In vibration analyses of elastic struc-tures in a fluid it is necessary to describe the effect of fluid in the problem as well as the stiffness pro-perties of the body. Several methods exist to take into account the virtual mass effect.

In global vibration analyses the struc-ture is usually approximated by lumped masses, and two-dimensional sectional added mass coefficients are used. The

latter may be based either on Lewis form technique (conformal mapping) or Frank close fit method (two-dimensional

sink-source method), and correction factors (J-factors) are introduced to account for three-dimensional effects. During recent years highly effective

fluid finite element methods have been developed on the basis of potential theory

formulation, see Zienkiewicz (1967) /9/, Matsumoto (1972) /10/ and Gallagher

et.al. (1975) /11/.

These methods may also be used in the calculations of added mass in the fre-quency range of wave-induced rigid body motion. In the vibration range

of frequencies these methods may either be used to produce reliable J-factors

to be applied in global vibration analyses or applied in detailed ana-lyses where a complete interaction between elasticity and the effect of

the surrounding fluid is to be achieved. To attain the interaction between local elastic properties of body and sur-rounding fluid, the added mass effect has to be known as an influence matrix before complete solution of the dynamic problem.- This is due to the fact that all displacements at the interacting surface are unknown and determined in the final solution from the combined effect of elasticity and fluid, /12, 13/. Blaker (1975) /13/ gives a com-plete derivation of how the added mass effect is included in one of the

structural finite element programs of SESAM-69. The added mass effect at a certain stage of computation is repre-sented as pressures on the immersed surface in phase with the, accelerations DIFFRACTION THEORY AND STATISTICAL METHODS TO PREDICT

800 WAVE INDUCED MOTIONS AND LOADS FOR LARGE STRUCTURES OTC 2502

R4=

i(r 2+(z+2nhC)2% 3n = i/r 2+(z+2nh+C) (29) (30) / 2

R5=

ri +(z-2nh+c)2 (31)

PV means the principal value integral in Eq. (20). For practical numerical evaluation of Green's function Eq. (19)

is used for kr > 0.1, Eq. (20) when kr.1 < 0.1 andiEq. (21) when w co;

i.e. in the high frequency range. The source densities Q. Eq. (18) are

found from compliance

4ith

the boundary conditions in Eqs. (13) and (14) which result in the following integral

equa-tion: a

27ryk,y,z)+JVQ.(E,n,OTT

s

3 (G(x,17,z;E,n,c))ds n. when j = 1,6 7 9n when j = ' (32)

This is similar to the infinite fluid problem formulated by Hess and Smith

/8/. In Eq. (32) one has to exclude

the integration of the source part of the Green's function over the immediate neighbourhood of each point

(E,T1,0 =

(x,y,z)

on the surface S where the

integral is evaluated. The contribu-tion to the normal derivative from the immediate neighbourhood of (x,y,z) is

takencareofbytheterm-27Q.(x,y,z).

The total hydrodynamic pressures are found from the linearized Bernoulli's equation:

30

p(x,y,z;t) = -p (33)

and the hydrodynamic forces and moments: =-Lrp(x,y,z;t)n .₃ ds (34) The added mass and damping coefficients Ajk and Bjk are as follows:

Ajk = p Re(ff (Pi nkds) (35)

Bjk =

pwIm(PI

cp.

.K

n,ds) (36)

S

where k=1,2-6 and j=1,2-6 and Re and 1m are the real and imaginary parts of cp. respectively.

and given as an influence matrix for each degree of freedom; i.e.:

1,.=[ps] for j = 1,2,3. (37)

7

The influence matrix gives the column-wise pressure at the interacting

sur-face nodes for unit acceleration at the same nodes.

However, the fluid finite element method may have some practical disad-vantages. The integral equation over the immersed surface of the body with a relatively complicated kernel using three-dimensional diffraction theory is replaced by a system of equations in the F.E.M. (finite element method) over a much larger domain throughout

the fluid.

An alternative method to the applica-tion of fluid finite elements in vibration obtained by extending the

sink-source method described in the previous-section is presented in the

following.

A velocity potential 08 is then intro-duced which has to satisfy. the condi-tions expressed by Eqs. (9), (10), (11)

and (12) as well as the following linearized body boundary condition on the average position of the wetted surface (j = 8):

48

_{= (nl}

_{dx +}

_{n2 dy + n3 dz)}

(38)

Here dx, dy and dz are the x, y and z components of the oscillating displace-ment from the mean position.

The boundary condition in vertical sinusoidal flexural vibration will for example be:

,,

- n3cost27m3c)

----an

where nl is the unit normal vector in z-direcfion, m is the mode of vibration, x is the coordinate of wetted surface and L is the total length of the body. The Green's function in Eq. (21) may be used for the vibration problem. The potential 08 is found using the equations presented in a preceding section.

The boundary condition on the form as presented in Eq. (39) is suitable for investigation of J-factors both as function of mode shape and geometry

3n

(39)

of bodies.

The authors propose to use the three-dimensional sink-source method in structural finite element programmes to take into accourit the added mass effects. The procedure is formulated in principle and is intended to be used to compute the influence matrix L. in Eq. (37) instead of using fluid finite elements.

The influence matrix is obtained through successive application- of unit motion

(acceleration) in x; y and z directions at each different element on the sub-merged boundary of the body.

For numerical evaluation Eq. (32) is

rewritten on matrix form:

[(314] [Q} = [a] (40)

The matrix IGNIcontains derivatives of Green's function for all elements on the body for one degree of freedom. The column vector {Q} contains source densities and lal is a diagonal matrix representing unit displacement (accele-ration) at each element on the sub-merged surface. Eq. (40) is to be solved for each right hand side repre-sented by a column of the matrix

lad.

The potential is computed Using Eq.

(18).

{O81 =

{Q} iG] (41)

The matrix containscontains velocity

poten-tials for eact element.

The added mass pressure is found from:

= Pl(P.81

By following this procedure the in-fluence matrix Eq. (37) may be

estab-lished.

Nave drift forces

Based on the complete Bernoulli's equation the pressure can be obtained from the velocity potential 0 which will,include a linear term proportional to 7* and 4 quadratic term proportional to I4rad0)`.

The first term has been dealt With in a previous section. The last texia will produce a second order force which will comprise a second order harmonic part

(42)

ARNE E. LOKEN

plus a steady component independant of

time.

Although the magnitude of the steady force Will be of second order compared to linear wave induced forces, this force is of vital Importance for the design of positioning systems for a structure at Sea.

The values Of the steady second order

term in regular sea may in turn be used to

calculate the mean drift force in ir-regular seas and also the slowly varying force /14/.

Newman /15/ has derived an expression for the horizontal drift force and moment in regular waves assuming in-finite water depth. Faltinsen and Michelsen /4/ generalized Newman's expressions to finite water depth. According to Newman:

P =-f f[pcose+pV (V cose-V sine)]rdedz

x S. r r (43) P ,,1 f[psine+pv (V sine+Vecose)lrdedz y S.

(40

'

M =-pf _{IVr Ve2 r dedz}

z S.

A cylindrical polar coordinate system

(r,*e,z) is used. The surface of a

cylinder with a large radius extending from the mean free surface to the sea bed is denoted S. while V, and V, are the radial and tangential velocity .

_components and p the dynamic pressure.

Px,

_{components of the horisontal drift}F are the time average x and y

R

force and the mean drift moment about z'axii. Eqs. (43), (44) and (45)

are approximated according to Faltinsen and Michelsen /4/ using the first order velocity potentials and retaining the contributions of second order.'

DIFFRACTION THEORY COMPUTER PROGRAM NV459

A computer program NV459 based on the three-dimensional sink-source techni-que has been developed at Det norske Veritas /4, 16/. The program evaluates the integral Eq. (32) by-approximating the immersed surface by plane quadri-lateral elements. The derivative of the Green's function in the direction of the outward normal vector is written for numerical purposes as follows:

3G G 3G 1. DG .3G

Dn N nl*Dx n2.Dy ' n3 Dz

The source density is constant over each element. The integral equation

is thus transformed into a set of

simultaneous linear algebraic equations:

[G] (Q1 = {Bc} (47)

This procedure is in principle the same as used by Hess and Smith /8/ for the

infinite fluid.problem. The only dif-ference is that in NV459 the Green's function and its derivatives are eva-luated in the centroid of each element, while Hess and-Smith used the point where the velocity component in the plane of the surface element due to the source distribution of that ele-ment is zero.

The different parts of Green's function and its derivatives are evaluated ac-cording to the method presented by Faltinsen and Michelsen /4/ and Hess and Smith /8/.

The linear algebraic equations that are established to find the unknown source densities are solved using a solution procedure which may handle equation-systems independent of the core memory of the computer. The source densities are found using Gauss elimination by Crout's method /17/. For practical purposes a limit of 500 elements is used in the computer pro-gramme. However, this can easily be extended.

The numerical work in the computer programme is considerably reduced by taking advantage of the symmetric properties of the source densities. If the body has one plane of symmetry

(which is the x-z plane) the source density has to be symmetric about this plane when j = 1, 3 and 5. Further, when j = 2, 4 and 6 the source density is asymmetric about the same plane. When j = 7 the source density can be split into a symmetric and an asymmetric part. When the body has both the x-z plane and the y-z plane as planes of symmetry, then the source density will be symmetric about the x-z plane and asymmetric about the y-z plane for j = 1 and 5. Further, when j = 3 the source density is symmetric about both the x-z plane and y-z plane. When j =-2 and 4 the source density is sym-metric about the y-z plane and asym-metric about the x-z plane. When j = 6 the source density is asymmetric about DIFFRACTION THEORY AND STATISTICAL METHODS TO PREDICT

ARNE E. LOKEN

orre 2502 ODD A. OLSEN 803

both the x-z plane and the y-z plane. Finally, for j = 7 we can split the source density into four parts in the same way as mentioned above. Thus, when there is only one plane of sym-metry it is only necessary to satisfy

the integral equation for positive y-.values of the body surface S. When

both the x-z plane and y-z plane are planes of symmetry it is only necessary

to satisfy the integral equations for positive x- and y-values on S.

It can easily be seen by examining the formulas of Green's function and deri-vatives that the Green's function it-self (and derivatives) contains sym-metrical properties when it is evaluated over the wetted surface._ Thus taking this symmetry into account the com-puting time is further reduced.

The geometry of the immersed surfaceof a body as input to programis described by adopting the procedure given by Hess and Smith /8/.

The output from the program of time dependant variables is presented in the form of amplitudes andphase angles. The variables are written on the time dependant form:

a(t) = /A/ sin (wt+c) (48)

The corresponding incoming wave as a function of time is written:

= C_a sin(wt)

,

(x=y=2=0)

The mainfeatures of the computer program NV459are summarized in the

following:

Fixed structures

-1 - Total linear dynamic forces and moments on the body.

2 - Linear dynamic pressure distribu-tion over the wetted surface of the body.

3 - Horizontal mean non-linear drift forces and moments.

4 - Linear dynamic velocity, accelera-tion and pressure for any point in the surrounding fluid.

-

_Facatijag_s

truc tur es

1 - Added mass and'damping coefficients.

(49)

2 - Wave 'excitation forces and moments 3 - Linear dynamic motions in six

de-grees of freedom.

4 - Linear dynamic pressure distribu-tion for fixed points on the body. 5 - Horizontal mean non-linear drift

forces and moments.

6 - Linear dynamic velocities, accele-rations and pressures for any point in the surrounding fluid. Both absolute and relative velocities and accelerations are caloulated. RESPONSE IN IRREGULAR WAVES

Research carried out several years ago has revealed that the irregular wave pattern could be described by linearly superposing regular waves of different frequencies. In this way it was made possible to take into account all the wave components present in a realistic

sea /3/.

The normal Procedure followed when cal-culating the response in an irregular wave system is to work in the frequency

domain) to utilize the linear superpo-sition principle and obtain a response spectrum as the product of a wave spectrum and a hydrodynamic transfer function, i.e.:

Sx = S(w) Y2(w)

Here S(w) and S(w) denote the power spectral density of the waves and the response respectively and Y(w) is the transfer function for the response in question.

Any wave-induced response which is linear with respect to wave height and can be represented as a function of frequency, will have its own transfer function defining the response in regular sinusoidal waves.

The parameter describing the statisti-cal distribution of the amplitudes of the response is directly related to the area under the response spectrum. As the wave spectrum is described by

parameters related to reaorded or ob-served statistical wave parameters, the response amplitude is calculated as a function of the same parameters. Know-ledge of the wave alimate for-a parti-cular ocean area will then in turn provide sufficient information to

-

establish long-term extreme value distributions of the responses of interest /18/.

Although working in the frequency do-main is preferable in many respects,

one serious drawback cannot be

over-come. Phase information available

together with the. transfer functions is lost when deriving the response spectrum and applying statistics. This makes it difficult to Utilize. predicted loads in

a

structural ana-lyses procedure as the force system is not in equilibrium. The only way to solve this problem is to work in the time domain again and utilize a conventional simulation procedure. So far, only long-crested waves with different frequencies have been con-sidered, i.e. they have infinitely long parallel crests. However, in reality the waves are short-crested. This effect is obtained by adding irregular wave systems from different directions, each associated with a certain percentage of the total energy. The energy distribution is represented by

a

directionality function;

normally a cosine squared function. Also the response is calculated for-waves from various directions and the

effect of short-crestedness may conse-quently be taken care of in this

proce-dure.

The wave energy spectrum

All linear statistical properties of the wave field (i.e. wave height,

velocity and aoceleration spectra) are today considered to-be adequately de-scribed by means of the wave energy spectrum.

By integrating the wave spectrum over all frequencies one arrives at the

total variance of the sea surface,i.e.:

s2 = 7S(w)dw (51)

When hydrodynathic transfer functions are applied to determine hydrodynamic statistical response of a structure, the actual shape of the wave spectrum will completely define the response characteristics.

During recent years a number of varying spectral shapes have been proposed. Most of these proposals can be written in the form:

S(w) pxp(-C2!w-n) (52)

where w is the circular frequency, R.

and n are empirical coefficients and C1 and Cl are constants with the wind speed pakameter incorporated in C,. The Neumann spectrum has 9., = 6 and

n = 2 while the original Pierson-Mosko-witz spectrum has 2, = 5 and n = 4 /19/. A new spectrum not yet applied much by engineers is the Jonswap spectrum which is a result of the Joint North Sea Wave obseniation project - a comprehensive international project undertaken off the Island of Sylt /20/.- The shape of the Jonswap spectrum is very different from that of Neumann and Pierson-Mosko-witz. The main difference is that the Jonswap spectrum is more peaked, i.e. a peak enhancement factor has been multiplied to theoriginal P.M. spectrum Written in the-original notation, the Jonswap spectrum may be expressed as:

4 5 f -4 S(f) = ag2(27)-f-5exp{-1(y-) } 2 (f-f ) exp 2 2 . y 20 fP a_a for

f

_{- p}

< f C a for f > fp

Here Phillips' constant which is equal to 0.0081 in the P.M. spectrum is de-noted a and y, aa and ab define the

shape of the spectral peak. The term y is the ratio between the maximum energy level of Jonswap spectrum and that of the P.M. spectrum. The physi-cal significance of the parameters is shown in Fig. 25.

The average Jonswap spectrum has: Y = 3?,

However, it is Often Convenient to express the wave spectrum with the significant wave height as an additional parameter so that the energy Content may vary independently of the location

along the frequency axis. Such a for-mulation is furthermore a requirement When long-term distributions of

wave-induced motions and loads are computed by means of the method outlined by

-NordenstrOm /18/.

DIFFRACTION THEORY AND STATISTICAL METHODS TO PREDICT

804 WAVE INDUCED MOTIONS AND LOADS FOR LARGE STRUCTURES OTC 2502

(53)

(54a) (60)

OTC 250

The parameterized P.M. spectrum having both the significant wave height and

the average zero-uperossing period as parameters may be written /18/:

H 2.T wT -5 wT -4

872 27r' 7 271

(55)

The average zero-uperossing wave period may be expressed in terms of the

spectral moments as:

mo Tz = 27(R-)

2

The relationship between the period T of maximum energy and the period dEfined above will depend on the actual spectral shape. For both the P.M. spectra we have:

T = 1.408-T

Analytical parameterization of the Jonswap spectrum is difficult to carry

out due to the large number of para-meters and the complex mathematical

expression. However, numerical inte-gration is used to introduce the

signi-ficant wave height as a parameter. It should also be emphasized that the Jonswap spectrum may be parameterized in terms of the non-dimensional fetch

length.

The question of which spectrum should be applied for engineering purposes and which parameters should be used to describe the spectral

characteris-tics isat present a matter of contro-versy. Besides the spectra mentioned earlier, other spectrum formulations such as those according to Darbyshire, Scott and Scott-Wiegel have been of-fered for practical applications. Extrapolation of response amplitudes The statistical distribution of wave heights or response amplitudes during stationary conditions is assumed to be adequately described by the Rayleigh distribution function. This assumption it strictly valid for narrow-band

spectra only. However, waves and wave-induced respahses are normally consi-dered to be sufficiently narrow-banded to support this approximation.

The most probable largest wave height in N cycles is then related to the

ARNE E. LOKEN

ODD A. OLSEN 805

significant wave height through:

HMAX = H1/3 0.5 inN (58)

The significant wave height may also be expressed as a function of the

-standard deviation of the wave process:

H113= A s =

--w

where s is the standard deviation of the wave process and Ew is the para-meter of the Rayleigh aistribution. The cumulative Rayleigh distribution of wave-induced response amplitude X may then be written:

-X2, P(X) = 1 - exp(E--,

X

The standard deviation sx of a random response time history may then be derived as the area under the response spectrum:

s = _{7sX (w)dw}

X 0

Furthermore, the standard deviation is related to the parameter E through:

2

Ex = 2s

X

-The most probable largest response amplitude, for instance motions or loads, on any probability level may now be derived. The following equa-tion yields:

XMAX = -i'N'In(t/TzX)

where t is the time period and average zero-uperossing period response normally chosen equal average zero-uperossing period waves.

A more detailed derivation of the mathematical theory may be found else-where /18/.

NUMERICAL COMPUTATIONS

The computer program NV459 /16/ has been applied to different hydrodynamic problems and the results have been

compared with model experiments conduc-ted at The Norwegian Ship Model Tank,

see also ref. /4, 5/.

From the results of forces and motions per unit wave amplitude in regular

T , the oE'the to the of the

806

DIFFRACTION THEORY AND. STATISTICAL METHODS TO PREDICT

WAVE INDUCED MOTIONS AND LOADS FOR .LARGE STRUCTURES_

_OTC 2502

Waves and ,wave spectral forms, the short-term responses in irregular waves have been computed.

Hydrodynamic loads on a gravity structure

The wave-induced loads for a box-shaped structure (caisson of 90 m x 90 m x 40 m) with a vertical column 20 m in diameter positioned in centre

of

the caisson top and piercing the sur-face, have been computed. The water depth used was 80 m. The results have been compared with model experiments which were performed in scale 1:100.

Three different wave heights were then used to check the linearity of the

response.

The results of the total hydrodynamic forces and moments per unit wave amplitude on the caisson and local pressure distributions are displayed

in Figs. 1 -

5.,

Values obtained by spline interpolation to be used in the computation of short-term responses later are also included in Figs. 1 -

4.

The experimental results are compared to theoretical calculations of forces

and moments on the column in Fig.

6.

The theoretical results are based on the calculations of accelerations at five equidistant positions by NV459

along the centreline of column (column not included in computer model). The accelerations are in turn used in Morison's equation

/21/

withaddedmass

coefficient equal to onewhen computing total forces and moments. As can be seen, the experimental data cluster about the computed values.

"Caisson effects"

The ,pressure in any point in the fluid

outside the caisson may also be calcu-lated by NV459. If the pressure is calculated in the still water level it can easily be converted to yield crest elevation. The crest increase eleva-tion above the caisson is expressed as percentage increase compared to the

crest of the incident wave ("caisson effect"). Olsen, O.A. and Hessen, F.

/22/

investigated the effect of

dif-ferent caisson geometries on the crest elevation. The crest elevation in-crease envelope for a typical caisson shape at different wave conditions is shown in Fig.

7.

As may be observed the "caisson effect" is an Important factor in determining the necessary

clearance between wave crest and lower .deck of fixed

pletfdrm,

Hydrodynamic coefficients and motions of a free floating body

The hydrodynamic coefficients (added mass and damping), drift forces and

wave induced motions were computed for a free floating box

(90

m x 90 m), at - draft 40 m and infinite water depth.

The mass moments of inertia of the box

14 expressed as squared radius of gration in m times mass of the body were:

I

4

= 33

.042-M

I5 =

32.092.M

16 =

32.922-M

The centre of gravity was located at

zc =

10.62

m.

The computer programmes

NV459 /16/

and

NV417 /23/

were applied in the.compu-tations.

The strip theory computer programme

NV417 is

based on the method of

Salve-- sen et.al.

/24/

and calculates ship

motions and wave loads in regular waves. The two-dimensional velocity potential used in

NV417 is

calculated using either Lewis form technique or the Frank close fit method. The strips

in NV417 were placed in the lengthwise direction for 13 = 0°.

The results from the computations have been compared with model experiments in scale 1:100. The model was oscil-lated in heave, surge, pitch and yaw amplitudes of 3 am and

0.05

rad.

respectively. Some of the tests were repeated using 6 am and 0.1 rad.

exciting amplitudes to check the linearity of results.

Added mass and damping coefficients are presented in Figs. 8 -

13.

The theo-retical predictions based on the three-dimensional sink-source method agree quite well with the experimental values. The most pronounced dif-ferences occurred at rather small wave periods. This may be due to difficul-ties in obtaining a pure one degree of freedom motion of the model. It may be noticed that great differences ap-peared between the values obtained by

representation of the velocity poten-tial seems to be inadequate for the present problem.

The hydrodynamic exciting forces in surge and heave are displayed in Figs. 14 - 16 and drift forces on the box are further presented in Figs. 17

-18. It may be noticed that the heave resonance at approximately T = 16.3 s have a large effect on the drift force. However, comparisons with experimental results are in close agreement.

The motion responses in surge and heave are plotted per unit wave amplitude of the incident wave in Figs. 19 -21. The results from these computa-tions reveal that the use of strip theories may in some cases give

reason-, able results in modes of motions not

subjected to any resonance effects (Fig. 19). The difference in the hydrodynamic coefficients obtained from NV459 and NV417 (Fig. 9) result in different resonance periods in heave (Fig. 21).

Added mass pressure in vibration

The three-dimensional sink-source method seems to be suitable in dealing with added mass effects in Vibration.

The results from application Of NV459 to calculate added mass pressure dis-tribution in translational motion

(zero node) and 2-node vertical flexu-ral sinusoidal vibration is shown in Fig. 22. The results from application of the Lewis form technique is also included.

/P/ P

-p.w2/a(x)/.1°

Here /a(x)/ is the absolute value of deflection as function of position given by the right hand side of Eq.

(39), and b is equal to B/2.

The results from the hydrodynamic pressure distribution may be used to obtain the correction factor J accoun-ting for three-dimensional effects if strip theories are used in vibration analysis.

Short-term response amplitudes

The short-term response of wave-induced motions and loads are calculated ac-cording to the procedure described in the previous section using computer

program NV406 which has been developed at- Det' norske Veritas /25/.

Short-term' 'surge and heave motions of a floating box in an irregular short-crested wave system are presented in Fig. 23. These results are based on the hydrodynamic transfer functions shown in Figs. 19 and 21. Total forces and moments on a fixed caisson in an irregular long-crested wave system are quite similarly presented in Fig. 24. Corresponding transfer functions are shown in Figs. 1, 3 and 4.

The computations were based on 21 dif-ferent frequencies covering the range of significant energy density of the

response spectra. The transfer func-tions were established on the basis of diffraction theory computations for

6 frequencies and asymptotic values. The remaining frequencies were quite well estimated using a cubic spline

interpolation procedure.

All results are_presented in terms of the parameter /8E\L/E113 as a function Of the period T correkponding to peak energy density 8f the wave spectrum. The parameter in the Rayleigh distri-bution of response amplitudes

ic

is non-dimensionalized in the following way:

Surge and heave motion:

Ifflirc/L

Force: VELVE3c/p-g-v

moment:

Two differentwave,spectra, the mean .

Jonswap numerically parameterized and the modified PiersonMoskowitz, were used in the computations. The results do not differ significantly if the total energy in the wave systertits kept constant, i.e. the-areas under the spectra are equal... However, relative comparisons will depend on the shape of the.. ,transfer function in question as well as the location of the wave

spectrum along the-frequency axis. These-effects are particularly well illustrated

in

the short-term-distri-bution of heave motion for the floating box and the overturning moment on the caisson.

The short-term distributions are all presented as a function of the period of spectral peak. This is theonlyway to compare the consequences in short-term load predictions resulting from ARNE E. LOKEN

different spectral shapes as the posi-tion of this peak relative to the transfer

function

in question, will often be of main importance for the final result. However, the average. zero-uperos sing period of the Pierson-Moskowitz spectrum may easily be deri-ved from Eq. (57). The relationship between Tz and T for the Jonswap spectrum on the &her hand, will be a function of the energy content, i.e. the level of significant wave height. Numerical integration of the spectrum with application of Eqs. (53) and (56) should then provide a sufficiently close approximation of T.

Wave-induced surge and heave motion of the floating box and wave loads on the fixed caisson are furthermore presented in Table 1 and 2 respective-ly. Both irregular and regular sinus-oidal waves were considered. The results for regular waves were derived from the transfer functions presented previously.

The irregular wave system was defined as:

=

H1/3

8m

= 12s

The corresponding regular wave was chosen to have a period of T=T =12 s and a crest-to-trough height equal to the most probable largest wave height resulting from the wave system given

above. Significant parameters of the

Rayleigh distribution were derived from Figs. 23 and 24 using the scaling factors presented previously to main-tain the proper dimension. The most probable largest response amplitude in 3 and 6 hours were calculated according to Eq. (63) assuming the average zero-uperossing period of the response to be

8 sec. This was also considered to be

a sufficiently close estimate of the average zero-upckossing wave period when deriving the most probable largest wave height.

These two methods of Calculation repre-sent the following different design procedures:

Design wave approach

Spectral analysis approach

The correlation of the results derived

according to these two procedures of analysis are seen to be rather good as far

as

the surge motion of the floating box is concerned. However, the heave motion differs significantly. Studying corresponding transfer functions, Figs. 19 and 21, it is found that whereas surge motion varies approximately linearly with period, heave motion increases exponentially towards a resonance peak.

Corresponding results for loads on the caisson are in close agreement. The transfer functions range of interest do not incorporate any extreme peaks. Whereas the deterministic approach is questionable when the transfer func-tions in question undergo large varia-tions within limited frequendy bands, it is considered to be a good approxi-mation in the case of relatively steady transfer functions. Total wave loads on concrete gravity structures are typical examples which in most cases fulfil the requirements. The results will then normally be on the conserva-tive side, provided the period of the deterministic wave is chosen equal to the period of spectral peak and the time duration is Chosen equal for the waves and fot the responses to these waves.

The effect of the wave spectrum also appears from Tables 1 and 2. However, the influence is seen to be of minor importance in this connection; less than approximately 10%. Extreme peaked transfer functions will, on the other hand, increase this influence, see for instance heave motion of floating box in Table 1.

CONCLUSIONS

The basic underlying assumptions, equa-tions and methods in a three-dimensio-nal diffraction theory programme deve-loped by Det norske Veritas have been reviewed. A procedure to utilize the three-dimensional sink-source method in estimating the added mass effect in vibration has been proposed.

The computer programme based on three-dimensional diffraction theory has been applied to solve the hydrodynamic

boundary value problem for a fixed and floating box. The results from compu-tatibns show good agreement with model tests.

DIFFRACTION THEORY, AND STATISTICAL METHODS TO PREDICT

The results from computations reveal that the "caisson effect" is an Iffipor-taht factor in determining the necessary clearance between wave crest and lower deck of a fixed platform.

Two-dimensional representation of the total velocity potential for the fluid surrounding the floating box appeared to be inadequate.

The short-term responses of motions and loads have been calculated using a modified Pierson-Moskowitz spectrum and a numerically parameterized mean

Jonswap spectrum. The results for the two spectra are not found to differ significantly.

The difference in the results between a design wave approach and the spectral

analysis approach of wave load calcula-tion may be insignificant for certain shapes of transfer function.

NOMENCLATURE

Diagonal matrix containing unit displacement (in x,y or

z-direction) at each element (node) of immersed surface of ,a vibrating body multiplied

with corresponding direction cosines

a(t) General response variable as function of time

/A/ Absolute value of general response variable in NV459 Added mass coefficients

(j,k = 1.2...6) Beam width

Matrix of right hand side of linear algebraic equation used to find source densities

Damping coefficients

Hydrostatic restoring coef-ficients

ds Element of area

dx,xy,dz x,y and z coordinates of oscil-lating displacement of immersed boundary of a body in vibra-tion

Parameter of the Rayleigh distribution [a] B. Cjk P-F1 Frequency

Frequency of spectral .peak Complex amplitude of exciting

force

/Fi/

_/vi

.1'.

3

*

'Complex conjugate of Fj

Magnitude of second order steady

state force, drift force in x-direction

Gravitational constant Green's function

Matrix containing values of Green's functions evaluated at

each element (node)

Coefficient matrix containing derivatives of Green's function in the direction of the outward normal vector evaluated at each element (node) at the wetted surface

Water depth

H1/3 Significant wave height HMAX Most probable largest wave

height

Moment of inertia

Product of inertia

Bessel function of first kind and zero order

Wave number

Modified Bessel function of second kind and zero order

Characteristic length of object

Spectral moments Mass of body

Unit normal into the fluid

Generalized normal (j=1,2... )

Number of cycles Dynamic pressure -ARNE E. LOKEN

OTC 250.2 ODD A. OLSEN 809

3

I.

Jo

n4

DIFFRACTION THEORY AND STATISTICAL METHODS TO PREDICT

810 _{WAVE INDUCED MOTIONS AND LOADS FOR LARGE STRUCTURES OTC 2502}

Sco

Standard deviation

Average wetted surface of the body

Surface of ayertical circular cylinder

of

large radius

S( ) Spectral density

Time variable Wave period

Period of spectral peak Average zero-uperossing wave period

Displaced volume of water V

r,Ve ,V Fluid velocity components in

cylindrical polar coordinate system

x,y,z Coordinate system fixed to mean position of structure

X Response amplitude

XMAX Most probable largest response amplitude

Transfer function

Bessel function of second kind and zero order

z-coordinate of centre of gravity

Direction of propagation of incident waves (=0 means propagation along positive x-axis)

Phase angle Of transfer function

Peakedness parameter

438

Complex COnjugate, Of (# u /g = ktanh k'h)

Coordinates of a point on the immersed surface of object Mass density of water

a Spectral width parameter

(I)(x,y,z,t) Total velocity potential Complex potential aMplitude

of incident wave,

Complex amplitude of diffrac-tion potential

Oscillatory potential for unit motion in j th mode

Velocity potential used in the vibration problem

CirCular frequency of wave,

ACKNOWLEDGEMENT

The authors wish to thank colleagues in DnV for their assistance and valu-able advice in-the preparation of this manuscript.

REFERENCES

1. Lebreton, J.C. and Cormault, P.: "Wave Action on Slightly Immersed Structures. Some Theoretical and Experimental Considerations", Proc. Symp. Research on Wave Action 1969

4.

2.- Garrison, C.J. and SeetharaMa Rao, V.: "Interactions of Waves with Submerged Objects",-J. Waterways, Harbours coastal Engineering Div., ASCE 1971 97. P( Qj ) Cumulative 'probability Source densities (j=1,2...7,8) Position vector

(j=1,2...6 refer to surge, sway, heave,

roll,

pitch and yaw respectively)

r.

[p Influence matrix of added mass pressure along wetted surface for each degree of freedom

Wave amplitude of incident waves m

(x,y,z)

for successive unit

motion at-each node on the

Wave length

wetted surface Ti Complex amplitude of motions

t T_ Tz V Y( ) Yo zc a

ARNE E. LOKEN

OTC 2502

ODD A OLSEN

811

St. Denis, M. and Pierson, W.J.: "On the Motions of Ship in Confused

Seas". Trans. SNAME, 1953, 61.

Faltirisen, 0.M. and Michelsen, F.: "Motions of Large Structures in Waves at Zero Froude Number". Proceedings of International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, University Col-lege London, 1974.

Faltinsen, O.M. et.al.:- "Wave Loads on Gravity Platforms". Proc. POAC Conference, Alaska 1975.

Wehausen, J.V. and Laitone, E.E.: "Surface Waves", Handbuch der Physik, 9. 1969, Springer-Verlag Berlin,'1969. Garrison, C.J. and Berklite,'R.B.: "Hydrodynamic Loads Induced by Earthquakes", OTC '1554, 1972.

Hess, J.L. and Smith, A.M.O.: "Cal-culation of Non-Lifting Potential Flow about Arbitrary Three-Dimensio-nal Bodies". Rep. No. E.S. 40622, Douglas Aircraft Division, Long Beach California, 1962.

Zienkiewicz, 0.C.: "The Finite Ele-ment Method in Structural and Conti-nuum Mechanics", McGraw-Hill, London,

1967.

Matsumoto, K.: "Application of Fini-te Element Method to Added Virtual Mass of Ship Hull Vibration".

Jour-nal of the Society of Naval Archi-tects of Japan 1970 vol. 127.

Gallagher, R.H., Oden, J.T., Taylor, C. and Zienkiewicz, 0.c.: "Finite Elements in Fluids", John Wiley & Sons 1975.

Araldsen, P.O. and ROren, E.M.Q.: "The Finite Element Method using Superelements - The SESAM-69 System". Presented at the University of

Cali-fornia Berkeley 1970.

Blaker, B.: "Vibrations of Submerged Structures as Computed by the Finite

' Element Method", COMputas Report .

No. 75-8, Oslo 1975.

Hsu, F. and Blenkarn, K.A.: "Analy-sis of Peak Mooring Force Caused by Slow vessel Drift Oscillations in Random Seas", OTC 1970, Paper No.

1159.

Newman, J.N.: "The Drift Force and Moment on Ships in Waves", J. Ship. Res. 1967.

Faltinsen, O.M. and LOken, A.E.:

"NV459. Wave Forces on Large

Ob-jects of Arbitrary Form. User's Manual". Pet norske Veritas Report No. 74-13-S, Oslo 1974.

Forsythe, G. and Mpler, C.B.: "Com-puter Solutions of Linear Algebraic Systems". Prentice Hall 1967. NordenstrOm, N.: "A Method to Pre-dict Long-Term Distributions of Waves and Wave-Induced Motions and Loads on Ships and Other Floating Struc-tures". Det norske Veritas Publi-cation No. 81, Oslo 1973.

Pierson, W.I. and Moskowitz, L.: "A proposed Spectral Form for Fully Developed Wind Seas Based on

Simi-larity Theory of S.A.

Kitaigorod-skii". Journal of Geophys. Res. 69

(24), 1964.

Hasselmann, K. et.al.: "Measure-ments

of

Wind Wave Growth and Swell during the Joint North Sea Wave Project". Deutsches Hydrographi-sches Zeitschrift, Nr. 12, 1973. Morison, J.R., O'Brien, M.P.,

Johnson, J.W. and Shaaf, S.A.: "The Forces Exerted by Surface Waves on Piles". Trans. Am. Petrol. Inst.

1950, 189.

Olsen, 0.A. and Hessen, F.: "In-fluence of Large Volume Body on the Crest Elevation of a Regular Wave of 1.order". Det norSke Veritas Report No. 7469-S, Oslo 1974.

Faltinsen, 0.M.: "Computer Program Specification NV417. Second Edition. Wave Induced Ship Motions and Loads. Six Degrees of freedom". Det norske Veritas Report No. 74-19-S, Oslo 1974.

Salvesen, N., Tuck, E.O. and Faltin-sen, 0.M.: "The Ship Motions and Sea Loads". Trans. SNAME 1970, 78. LOken, A.E.: "Computer Specifica-tions NV406. Second Edition. Short and Long-Term Response of Wave

Induced Motions and Loads. Volume I and II". Det norske Veritas Report No. 74-22-S, Oslo 1974.

TABLE I MOTIONS OF FLOATING BOX PARAMETER (SHORt-CRESTED IRREGULAR WAVE , SYSTEM) H =8 in T =12 s SURGE ( (3=0) HEAVE (=0) P.M. JONSWAP P.M. jONSWAP

IgL

. 0.27 0.30 0.64 0. 5 4 H1/3 rE-.1 . 0.85.10 0.94-102 -

0.20-101

- 0.17.101--X (.,m) 0

.77

0.85 1.81 1.53 'I' =8 s zX -XMAX t=3 hours (m), 2.10

2.30

- 4.80 4.10

T =8s

zX XMAX' t=6 hours (111) 2"20 2:40 5 10 4 30 PARAMETER (REGULAR SINUS-OIDAL WAVE H=1.1AX

M T= =1T2 s) P SURGE (13=-0°) HEAVE (3=0°) . TRANSFER FUNCTION -X/c

t=3 hours

0.37

- 0.20 HmAx=1.90 3

C =7.6 m

_a

X(M)

t=6 hours

- 2.80 1.50. HmAx=1.99

H3

. Ca=7.95 m X(m) , 2.90

1.60

_

TABLE 2 - FORCES AND MOMENTS ON -CAISSON PARAMETER (LONG-CRESTED IRREGULAR WAVE SYSTEM) H =8 m T =12 1/3 p s) HORIZONTAL FORCE

(a=e)

VERTICAL FORCE (=0°) OVERTURNING MOMENT ($=00)

P.M. JONSWAP P.M. JONSWAP P.M. JONSWAP

i

AP-I,

0-50 0.56 0.70 0.78 0.045 ' 0.050 H 1/3 , 0.57.10-2 1.76-10-2 2.20-10-2 2.4-51O 1.57.1 -2 5.08-103 5.70-103 7.12;103

7.954O.

4.57-104 ir =8 s (t) (t) (t) (t) (tm) (tin) XMAX t=3 hours 1.37-104

1.54104

1.92104

2.24.104 1.10.105 1.25-105 (t) (t) (t) (t) (tin) (tin) T

=8s

zX X_{MAX t=6 hours} 1.43.104 1.60.104 2.00-104 2.23-104 1.15-105 1.28-105 (t) (t) (t) (t) (tm) (tin) PARAMETER (REGULAR SINUS-OIDAL WAVE H=HM AX T=T =12 s) A HORIZONTAL FORCE (13=0°) VERTICAL FORCE (3=0°) OVERTURNING MOMENT (3=0o) TRANSFER FUNCTION FORCE.L/pgV- a 0.58 0.80 0.054 - -MOMENT/pgV.ca0L t=3 hours HmAx=1.90 HI/3 c_=7.6 m

_{a x}

1.60104

2.20.104 1.30.105 t=6 hours (t) (t) (tin)

H=1.99 H

MAX 1/3 ...=7.95 m 1.70.104 2.30.104 1.40'105 " X ( t) ( t) (tin)

r=1: NV 459 108 ELEMENTS MEASURED VALUES X: SPLINE INTERPOLATION 1.2 Q8 0.4 X X 10 14 18 22 26 PERIOD T1s)

Fig. 3 - Vertical force on caisson

(LxBxH=90mx90mx40m). XX 80m 40m x WAVE Y--0.4 NV 459 108 ELEMENTS : MEASURED VALUES X: SPLINE INTERPOLATION 0.02 12 0.8 NV 459 108 ELEMENTS : MEASURED VALUES X: SPLINE INTERPOLATION 0.06 80mii

Fig. 2 - Horizontal force on caisson

(LxBxH=90mx40mx40m).

1 140m

X

PERIOD Tm

Fig. 4 - Overturning moment on caisson (LxBxH=90mx90mx40m). . ,

lilliiill

_

frg-v-La , X X X ca WAVE x X cm 10 , 14 18 22 26 m: NV 459 108 ELEMENTS : MEASURED VALUES 80m 40m p-g-v-La -X: SPLINE INTERPOLATION 0.06 x WAVE 10 14 18 - 22- 26 - PERIOD T(s)

Fig. 1 -'Horizontal force on caisson

(LxBxH=90mx90mx40m). 10 14 18 ,22 26 PERIOD I(S) 14 ' 18 22 26 PERIOD I(S) -10 6.62 X X 1.6 0.8 80m 40m

HEADING ANGLE 0° 2=-40m Ellx! q. 2=-80m VIEW A-A HEADING ANGLE 450 01. 0 I 2=-40m Q4 WAVE PERIOD T=15(s, :MEASURED VALUES =I :NV 459 108 ELEMENTS 0.8 WAVE PROPAGATION 12 WAVE CREST INCREASE V./0 8 150 c. 2=-80m VIEW A-A H=47mi I I 50 45°

1425m

11:VE

WA 0° -50 CV

Fig. 7 - Crest increase envelope.

0.4 Q8

Fig. 5 - Dynamic pressure distributions along caisson (Laxli=90mx90mx40m).

V

LL=100m

J

0.8

-150

:NV 459 ACCELERATIONS AT 5 LOCATIONS ALONG COLUMN USED IN MORISON'S EQ. : MEASURED VALUES

pv

0.4

10 14 18 22 26

PERIOD Its)

Fig. 6 - Forces and moment on column wave height eight m.

10 14 18 PERIOD Tis, :NV 459 108 ELEMENTS

--

: NV 417 8 OFFSET POINT

0

:NV 459 48 ELEMENTS :1 EXPERIMENT VALUES (DIFFERENT EXCITATION AMPLITUDES)

Fig. 8 - Surge added mass coefficients for floating box

(LxBxd=90mx90mx40m).

00

_.

_.s. II. A

0

/

Ailleb

".

i

Z

/

:a X p=00

---

P=45° ..."'"

,

,

\

T=17sec TO 5sec 1 ID! .9. 08 04 [tonnes] 105 [trra 105 1.6 Q8 FORCES: MOMENTS:

---pv 0.8 0.4 0.04 0.02 :NV 459 106 ELEMENTS

=---A :NV 417

8 OFFSET POINT

:NV 459 ,48 ELEMENTS

:1 EXPERIMENT VALUES

(DIFFERENT EXCITATION AMPLITUDES)

-Fig. 9 - Heave added mass coefficients for floating box

(Lx5xd=90mx90mX40m)., NO

6--10 14 - 18 PERIOD ts) cl.:.NV 459 108 ELEMENTS : NV 417 8 OFFSET POINT :NV 459. 48 ELEMENTS :1 EXPERIMENT VALUES 11- (DIFFERENT EXCITATION AMPLITUDES)

Fig. 11 - Yaw added mass

-.coefficients for floating box (LxBx&90mx90mx40m): 14 18 PERIOD I(S) 0.04 OW. 1.0 0.6 Q2 c3::1IV 459 108 ELEMENTS : NV 417 8 OFFSET. POINT 0 :NV 459. 48 ELEMENTS :1EXPERilv1ENT VALUES (DIFFERENT EXCITATION AMPLITUDES)

Fig. 10 - Pitch.added mass coefficients fir floating

box

(LXBXCMOMX90MX4011).

:NV 459 108 ELEMENTS

----A : NV. 4)7

8 OFFSET POINT

,C) :NV 459 48 ELEMENTS

:1 EXPERIMENT VALUES

(DIFFERENT EXCITATION AMPLITUDES)

Fig. 12 - Surge damping coefficient for floating

box (LxBxd=90mx90mx40m). 10 14 18 PERIOD Ts) 10 14 18 PERIOD Ts) 1.2

4!--Ar

Ass pve 0.08 Ae6 pVt2 0.06

laV15712 /F1/ pg-VCa/L 1.6 0.8 180 120 60 0.4 0.2 0 10 14 18 PERIOD T(s) NV 459 .108 ELEMENTS

_--A

:NV 417 8 OFFSET POINT

0

:NV 1.59 48 ELEMENTS

63 EXPERIMENT VALUES (DIFFERENT EXCITATION

AMPLITUDES)

Fig.. 13 Heave damping. coefficient for floating

box (littxd=90mx90mx4(0), ABSOLUTE VALUES HEADING ANGLE=45° 10 14 18 PERIOD T HEADING -ANGLE =45° 10 14 18 PERIOD TIS; NV 459 108 ELEMENTS :NV 417 8 OFFSET POINT

0

:NV 459 48 ELEMENTS

Fig. 15 - Exciting force and phase on floating box

-(LxBxd=90mx90mx40m).

Tivp:

1.6 0.8 Ui LU cc CD 180 LU 120 J

2 60

w 0 tri a. 10 14 18 PERIOD Tis) HEADING ANGLE=0° 10 14 18 PERIOD T(s) I= :NV 459 108.. ELEMENTS

---A : NV.417. "

.8 .OFFSET POINT

0 NV. 459 48 ELEMENTS

Fig. 14 - Exciting force and

phase on floating box

(LxBxd=9°Mx90Mx40m)..-10 14 ' PERIOD Tml

\`254,

HEADING ANGLE =0° -'1 1--"©-1 1-06-E)--1=1-10 14 18 PERIOD I'M NV 459 108 ELEMENTS NV 417 8 OFFSET- POINT (D :NV 459 48 ELEMENTS

Fig. 16 - _Exciting force and .

phase on floating _box (txhd=90mx90mx40m), ABSOLUTE VALUES HEADING ANGLE =0° Z5. -ABSOLUTE VALUES HEADING ANGLE-= 00 JFI/ g.V. E 1.6 a8 w 180 Ili 120 CD

0.6 0.4 0.2 10 14 18 PERIOD Tm NV-459 108 ELEMENTS,

EFFECT OF MOTION INCLUDED NV-459 108 ELEMENTS

EFFECT OF MOT ION NOT INCLUDED 0: NV-459 48 ELEMENTS,

EFFECT OF MOT ION INCLUDED

O

NV-459 48 ELEMENTS,

EFFECT OF MOT ION NOT INCLUDED

: EXPERIMENT VALUES

-Fig. 17 - Drift force on floating box (LxBxd=90mx90mx40m). 10 14 18 PERIOD Ts) 10 14 18 PERIOD I(S) 1:3 "NV 459 108 ELEMENTS :NV 417 8 OFFSET POINT

0

:NV 459 48 ELEMENTS :SPLINE INTERPOLATION Fig. 19- Motions of floating box (LxBxd=90mx90mx40m)

amplitudes and phases.

ITJtI

10 14 18

PERIOD T(s)

--1=1: NV-459 108 ELEMENTS,

EFFECT OF MOTION INCLUDED

: NV-459 108 ELEMENTS

EFFECT OF MOTION NOT INCLUDED 0: NV-459 48 ELEMENTS,

EFFECT OF MOT ION INCLUDED 0: NV-459 48 ELEMENTS,

EFFECT OF MOTION NOT INCLUDED

: EXPERIMENT VALUES

Fig. 18 - Drift force on floating box (LxBxd=90mx90mx40m). 0.8 0.4 HEADING ANGLE =45° ABSOLUTE VALUES HEADING ANGLE =45° 10 14 18 PERIOD I(S) 10 14 18 PERIOD Ts ci:NV459 108 ELEMENTS 0:NV459 48 ELEMENTS X :SPLINE INTERPOLATION Fig. 20 - Motions of floating box

amplitudes and phases

(LxBxd=90mx90mx40m). HEADING ANGLE =0°

rl

1/44)

_

₃

i1

_i I :-....

5

- .0

a

r, . -AS . co ABSOLUTE HEADING VALUES ANGLE =0° -47 ...-107.-."

-al ,m 40A Fx 0.6 0.4 0.2 ITJ C 0.8 0.4

10 14 18 PERIOD 1.(s) HEADING ANGLE=0° 10 14 18 : NV 417 8 OFFSET POINT rzl :NV 459 108 ELEMENTS :NV 459 48 ELEMENTS EXPERIMENT VALUES X :SPLINE INTERPOLATION Fig. 21. - Motions of floating box

amplitudes and phases (LxBxd=90mx90mx40m). Q2 1.5 1.0 0.5 0-NODE NV459 2-NODE NV459

---- STRIP THEORY LEWIS FORM

/Pi p0./a(x)/.b SURGE-MOTION MEAN JONSWAP PIERSON MOSKOWITZ (NV-406) 10 14 Tp 18

Fig. 23 - Short term responses of motions floating box

(LxBxd=90mx90mx40m).

10 20 30 40 50 m x'

Fig. 22 - Added mass pressure in zero and two-node vertical vibration for barge (LxBxd=100mx10mx5m).

HEADING ANGLE OF SHORT CRESTED WAVE SYSTEM: 0°

'AFL H113 Q6 HEAVE -MOTION MEAN JONSWAP //PIERSON MOSKOWITZ (NV-406) 6 10 14 Tp H1/3 1.6 Q8

HEADING ANGLE OF LONG CRESTED WAVE SY6TEM=0° - , 0.8 FORCES: MEAN JONSWAP

oN

PIERSON-MOSKOWITZ (NV-406) 10

14 -

18 PIERSON-MOSKOWITZ INV,406)__

Fi_g. 24 -,.5hort term responses

. of loads caisson . (LxBxH=90000mx40m).:-JONSWAP

S

MAX ONSWAP

Sj

_MAX PM MAX PM MAX

ag2

_f

-5

(2n)

fp

FREQUENCY

Fig. 25 - Parameters of the jonswap wave spectrum.

*TA_

H113 1.6