Modified theoretical techniques for motion predictions of offshore structures

Lab. y. Scheepsbouwkunde

Technische Hogeschool

WEMT SYMPOS M N

ADVANCES

INIORE

TECHNOLOGY

Amsterdam, 25-27 November 1986

MODIFIED THEORETICAL TECHNIQUES FOR MOTION PREDICTIONS OF OFFSHORE STRUCTURES

Xiong-Jian Wu, Ship Hydrodynamics Laboratory, Shanghai Jiao-Tong University,

China presently at Department of Mechanical Engineering, Brunel, The University

of West London, Uxbridge, Middlesex, UB8 3PH, U.K.

W.G. Price, Department of Mechanical Engineering, Brunel, The University of West

12 APR,

₁₉₈₈

ARCHEF

AUTHORS: Xiong-Jian Wu & W.G. Price Page No. 1

London, Uxbridge, Middlesex, UB8 3FB, U.K.

SUMMARY

General and modified motion prediction theories are briefly reviewed and their application to a wide selection of offshore structures illustrated. The

approaches involve two and three dimensional shallow draft formulations, a

modified integral equation together with a multiple Green's function expansion, and a hybrid three dimensional-strip theory. The resulting theories are

validated by comparing theoretical predictions with available experimental data and in general good agreement is obtained.

1. INTRODUCTION

Developments in offshore engineering have spawned offshore structures of various shapes and sizes, operating in ways which are partly or totally different

from those of ships. Thus the motion and sea load prediction techniques

originally developed in naval architecture for conventional ships may be no longer necessarily directly applicable to offshore structures and therefore to be of practical use may be suitably modified.

In general, conventional ships (mono-hulls and catamarans) have slender hulls and travel with forward speed in the seaway. Based on a hull slenderness parameter various strip theory procedures _{have been developed to predict ship motions and} in severe weather conditions additional motions caused by sla

_ing2

_{, deck wetness,}

flexibility of the ship3 _{etc all play important roles in determining the safe}

operational envelope of the ship.

Offshore structures range from very simple barges to complicated

semi-submersible constructions composed of one or more submerged floaters or pontoons and multiple columns or struts. These may keep station at a sea site by using

mooring and/or dynamic positioning systems as well as being able to travel at low forward speed between sites. Thus the developed analysis procedures are required

to describe all the important aspects of the problem and these extend to

PageNo

2

will not be discussed in any detail in this paper.

Semi-submersibles are extensively used in a wide range of offshore

opera-tions. Usually a semi-submersible consists of pontoons and columns of simple

sectional geometries, i.e. rectangular or circular. When the sub-structures are

of small characteristic diameters relative to the wavelength, Hoofts method6

based on Morison's equation produces reasonable motion predictions. When the

pontoons have large or intermediate dimensions, an approach combining a two-dimensional strip theory for the pontoons and a Morison equation description for

the columns7 may be successfully applied. Furthermore when three dimensional effects have to be included a practical approach may be developed combining a

three dimensional diffraction theory together with additional viscous damping

contributions8.

However when the struts of a semi-submersible are not short,

it has been proposed that a simplified formulation based on an extension of strip (9)

theory may be adopted

Transportation barges, crane barges, ocean production platforms, jack-up rigs in transit, semi-submersibles and TLPs in transit or under tow etc. all possess a common shallow draft feature. For these structures the flat ship

approach0)

and

pulsating potential method (11) can only describe the heave, pitch and roll motions.

However, by extending the approach of

Kim0),

a three dimensional shallow draft theory has been developed (12,13) capable of predicting the six rigid body motions of the structure. From this model a two dimensional shallow draft theory4 has

been derived capable of predicting the motions of shallow draft offshore

structures possessing a large or intermediate length to beam ratio value. For large offshore structures three dimensional theoretical models are appropriate but these might not be feasible in a preliminary design exercise because of the large computing costs and human effort in data handling.

Nevertheless the main hull of most offshore structures eg. the pontoons of a

semi-submersible) possess a large parallel midbody section with little or no variation in the sectional properties along its length and at the ends there usually exists a small region having large curvature or abrupt geometric changes. These

features have stimulated the development of a hybrid 3D-2D strip theory

producing hydrodynamic information in all six degrees of motion including the surge mode.

Naturally if the offshore structure is slender then two dimensional theories are again applicable. However, the general linear wave theory approach requires

some

modification4

before it becomes totally suitable for general offshore

predictions of flexible

structures3

are performed at higher frequencies than usually considered in a rigid body study. The singularity method may now

encounter serious numerical failure due to the presence of irregular frequencies7.

Fortunately their occurrence may be estimated using empirical based formulae or analytical solutions and they can be removed from the mathematical model

using a multiple Green's function expansion9.

In this paper, these theoretical approaches are discussed and modified motion prediction techniques developed to analyse the behaviour of large offshore

structures excited by a seaway.

2. GENERAL MATHEMATICAL MODELS

The general equation of motion describing the six rigid body motions

(i

= 1,2,...

_{,6) of a floating offshore structure excited by regular sinusoidal}

waves of amplitude

a

and frequency w may be expressed as

j=1

i

ii

vii

j

ii

i

i

E [(M+A)5(t)+(BB

6

_{)X(t)+C X(t)j=Fe}

-iwt

(1)

where the jth motion displacement of amplitude Xja is given by

X.(t) = X. e-iwt

j ja

In these expressions subscripts jor i=l denotes surge motion; i=2, sway; 13, heave;

i=4, roll; i5, pitch and i = 6, yaw motion. The constant M.. denotes the generalised mass and the frequency dependent hydrodynainic coefficients _A.., describe the fluid added mass or inertia and fluid damping respectively. The

remaining terms C.., F. describe the restoring coefficient and ith wave exciting force respectively whilst, to this usual linear model is included the additional viscous damping coefficient B . describing contributions arising from drag

(20)

v13

influences and this may be estimated by the Morison damping term.

Assuming an ideal fluid and a linear wave theory mathematical model, the coefficients A. ., B. ., F. can be determined using a three-dimensional source or

lj lj i

source-dipole distribution method. For example, for the latter the Green's

function (CC )) integral equation describing the unknown velocity potential amplitude of the jth motion is given by

2 (p) -dS =

-

J

v(Q) G(P,Q)

ds

(2)

s

_w s w

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: 3

where P(x,y,z) and Q(,fl,) are two points on the mean wetted body surface S and the normal velocity components

n - i w n. J

PageNo

: 4 for j = 1,2,... ,6 for

j=7

(3)

In this expression the unit normal = (n1, n2, n3) and

- X n=(n4,n5,n6)

where r = (x,y,z) and rG defines the position of the centre of gravity G of the

structure relative to the chosen origin O of the Cartesian coordinate axes Oxyz, with plane Oxy lying in the calm water surface and Oz is positive upwards. The velocity potentials °, describe the incident sinusoidal wave and diffraction wave respectively.

From this formulation it may be shown that the hydrodynamic coefficients are represented by the expressions

= -

_Im(b)n.

dS w i 'J - J

Re () n.

dS B. . = p i S w

and the ith wave exciting force by

F. =

- iwp

J )

n, dS

For an offshore structure floating upright, with port-starboard geometric symmetry, equation (2) can be modified to

2r J(p ) _{- J} J(Q[ BG(P,Q1) BG(P,Q2)

_]dS

SW1 Ql Q2 =

-

J

v(Q

[G(P,Q1)± G(P,Q2)] dS Sw 1 for y-symmetric i [y_antisymmetricj modes (4)

where S denotes one half (i.e. port side say) of the mean wetted body surface

s , P (x,y,z) , Q1(,fl,) E , Q (,-n,) is the symmetric point to Ql.

w.

Equation (4) describes the ideal floating position of the offshore

structure and the required computing time necessary to solve the problem is

reduced significantly. However, in general for an offshore structure inclined

at an angle of heel, produced by the effects of wind, current, waves and their

combinations, symmetry no longer exists and the general expression given in equation

(2) is required. It has been

shown2°

that depending on the magnitude, a steady angle of heel modifies the form of the hydrodynamic coefficients, couples all six motions, alters the motion behaviour of the offshore structure and greatly increases

computing effort.

When the structure has a twin or multi-hulled configuration, equation ₍₂₎

may be recast into the form

2îr

J(p)

I G (P,Q) -

i=l

_J dS =

-

I y 1(Q) G(P,Q) dS M M

Si

i=l

_si

n (5) w w M

where S = S and S is the mean wetted surface of the ith hull in

w w w

i=l

the group of M hulls which are not necessarily of identical form.

When two or more individual bodies are involved, theoretical formulations

have been described by oortmerssen(2U . In the case of two structures, the integral equation may be redefined as

2 2 (m)j

2îr fflfl(p) I ffl(Q) =

-

I

y

n (Q) G(P.Q) dS (6)

i=l aflQ 1=1

si

Si

w w

form = 1,2, the normal velocity

_inm)

_i=m

on

S1

for j = 1,2,...

6

(7) O

im

and (m)7 7 y

=v =--

onS

n n n w

In this formulation, the hydrodynamic coefficients for the ith body in the rth direction due to the jth oscillatory motion of the mth body are

si

w

and the rth wave exciting force on the ith body is

F(1) =

_-

_iwp

_{(° +}

) n dS

r _r

Si

w

for r = 1,2,.. .,6 = j; i = 1,2 _{m. Thus, the motion equations may be redefined as}

(i) -iwt

2 6

(m)

_C(1)

[

(M(i)

+A

(im)) . .(m)X.

+s

(im) = F e ₍₁₁₎

rj im rj j rj j rj im j r

l j=l

for i = 1,2, where . = 1 for i = m or 0 for i m. im

Another modification to the general expression (i.e. equation (2)) may be introduced when a portion of the structure has zero draft. It can be

shown4

that in this case the integral equation takes the form

4 ) (P) -

J

()

'ldS +

J

(Q2)G(P,Q2) dS i 2Tr s Qi

s2

w w

2'

vG(P,Q)

dS for

PES

_J w j n

?pss1

w S w

Xiong-Jian Wu & W.G. Price

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: 6

(12)

where

V =

2/g, s

= s

1 + s 2 and s 1, s

2

denote the non-zero draft and zero draft components of the mean wetted body surface S

_s2.

The general expression defined in equation (2) and the variations subsequently

present are capable of describing most of the applications expected to arise in the operations of offshore structures. However, because of the geometric properties of individual structures these mathematical models may be further refined and simplified

to produce tools suitable to provide practical engineering solutions to particular problems as will now be discussed.

A (fm) rj p J f Im

(mh)

_n r dS (8)

si

w 3(im) rj = _

Re(a)

n_r dS (9)

3. MODIFIED SHALLOW DRAFT THEORIES

3.1 Three dimensional formulation

For shallow draft structures possessing

a small draft, i.e. large beam (B) to draft (h) ratio (B/h » 1) a flat or nearly flat bottom, and

of small or intermediate length L (L/B < 3, say)

a three dimensional shallow draft theory has been developed to estimate all the

motions of a rigid or even flexible offshore structure2'13 . The approach relies

on the assumption that the Froude-Kriloff forces can be evaluated on the mean wetted body surface S _{whereas the radiation and diffraction forces are estimated on the}

approximate surface S located on the still water surface at z = O. Such reasoning allows the simplified relationships

=

n =--=-VG,

Bn Bz 3 Bz

BG BG BG

(13)

to be valid on S. _{Consequently, the integral equation (12 or 2) simplifies to}

4 (p) + V

J

(Q) G(P,Q)

dS =

-

J

v1(Q)

G(P,Q)

dS (14)

S S

o o

with normal velocity

-

i W

n.

j= 1,2,...

,6

o ₍₁₅₎

Bn

j

=7

for i = 1,2,... ,6 =

j.

Xiong-Jian Wu & W.G. Price

PageNo:

7

and fluid actions

p z =-h A. . =

-J

Im (1) _n. i dS S o ₍₁₆₎ B..

1]

= -p f

Re (J) n.i dS S o F. = - i w p ni dS - i w p ni dS (17)

Hence, the equations of motion in equation (1) reduce to for i = 1,2,6 and ip j. o X. = n. dS ia WM i ii S w AUTHORS

PageNo:

8 (18) 5 F e (19) E (M + A ) 3 (t) + B (t) + C X (t) 1 jWt j=3 ii i ii j ii _i J i

for i = 3,4,5 provided that B . . = O.

vi j

Extensive applications of this formulation to practical offshore structures

reveal that

the integral equation is greatly simplified by the removal of the derivative

of the Green's function,

the replacement of the mean wetted body surface S by an equivalent flat plate approximation S allows a crude panel discretisation to be used in the

numerical analysis without penalising the convergence or accuracy of solution, analytical expressions may be derived for the horizontal motions defined by equation (18) if the lateral walls of the shallow draft structure are

composed of simple vertical plates in a piecewise manner.

Figures (1-3) illustrate a selection of offshore structures (and computed

results) which have been analysed by the described approach.

For the rectangular barge model (L = 2.4m, B = O.8m, h = 0.105m) in Figure 1

the predicted pitch response is compared with experimental data and a more

conventional three dimensional calcu1ation22 . As can be seen, the agreement is favourable but achieved by a very large reduction in computing effort.

Figure 2 illustrates the predicted roll motion of a semi-submersible in transit and this again correlates well with the experimental data of Takaki et

al23

Finally for the triangular jack-up rig deck (L = 108m, B = 124.6m, h = 3.OSm) shown in Figure 3, the analytically predicted surge motion shows remarkable agreement

with the experimental data of

Chacrabarti24.

More extensive comparisons covering all motions can be found in the litera-ture(12ul3) _{and in all cases it is sho that this simplified shallow draft theory}

3.2 Two dimensional formulation

When L/B > 3, the previous theory may be further modified to a two

dimensional version4'25 . For example, for the cross-section at x with contour C (x) and contour C (x) at z = O, the integral equation can be shown to

w o

reduce to the form

2 _{1(p) -} (q)

G(p,q) d

=

v(q) G(p,q) d

(20)

where p,q are two points on the contour and G( ) denotes a two dimensional Green's function.

This formulation allows the sectional added mass, damping coefficients and

PageNo

9

where dZ denotes an elemental length on the contours C and C . When

o w

integrated along the length of the offshore structure these terms yield their corresponding global coefficients and the equations of motion defined by equations

(18) and (19) remain unchanged.

This approach is used to create the results displayed in Figure 4 for a

rectangular barge model (L = 3.Om, B =0.75m, h = O.0159m) similar to that illustrated in Figure 1. It can be seen that the two dimensional predictions for surge, heave and roll responses are in reasonable agreement with Nojiri's experimental data _as quoted in Ref. 12.

4. A MODIFIED TWO DIMENSIONAL INTEGRAL EQUATION METHOD

When an offshore structure or its main hull is of large or intermediate aspect ratio (say L/B ? 3) and isof finite draft (i.e. h _{O), a two dimensional strip} theory may be developed. Analogous to equation (2) , the equivalent two dimensional

Green's function integral equation is of the form

wave forces to be defined

1J

(A) C b.. = - p f. = -

iwp

i

o C

f

o C as

Im () n.

Re () n1

n. dZ

i

J

o w d .

iwp

Co

J

n (21) (22)

G(p,q)

_r îr

J(p)

+ J

--d = J

v(q) G(p,q) d

q C w w

where again G ( ) denotes the two dimensional Green's function. o

It is well known that at high frequencies, singularity methods break down

at an infinite number of irregular frequencies7. _{It was found that these}

frequency values can be adequately predicted from an empirically derived formula

based on an equivalent rectangle approximation. That is the mth irregular frequency occurs at

w_m = {

gk

coth (k h )_e (24)

k =

B e

for m=l,2 ... In this expression the equivalent beam B and draft

_he

are given by Be =

(C)

3

Xiong-Jian Wu & W.G. Price Page No : 10

h =A/B

, C

A/Bh

e s e _s

where A, B, h are the sectional area, beam _{at the waterline and draft measured}

from the mid-point of the beam, the correction coefficient ct

(l+nm)/8.

For wave frequencies below the first irregular frequency w1 in equation (24) the integral equation (23) produces unique solutions; otherwise, mathematical failures can occur. However for the development of singularity techniques in the evaluation of second order wave forces acting on offshore structures, this

irregular frequency restriction must be eliminated. A successful method has been

proposed9

, based on the modified integral equation

(23)

(p) + (q) d

=

J

v(q)

G* d

Cw q

and this incorporates the multiple Green's function expression

* M -G =

G(p,q)

+ G. (p,q,p.) 1=1 J

p)

C

sgn(yy)

_+c

< r _{3G(p,q) )}

G(p,q)

G.(p,q,.) = G(p,

jl - j2 z=O

y=i

z0

where M is the number of free surface piercing sub-hulls of the structure,

is an arbitrary point on the interior free surface of the jth free surface piercing sub-hull and C.1, C.2 are constants.

Extensive numerical tests have been performed _{to verify the present} formulations given in equations (25-27). In all applications with mono, twin or

multi-hulled two dimensional sections the irregular frequency problem has been eliminated without causing any significant deviation in the numerical predictions

at other frequencies. That is, the approach is also effective to the frequency at which

the numerical failure occurs. To illustrate this achievement a selection of results are presented for a mono, a twin and a four hull section associated with different offshore structures.

For a mono-hull triangular section of beam B, draft h = 0.5E the irregular

frequencies predicted analytically (see Appendix) (or by equation (24)) are

w1v'B/2g =1.54(1.53) andw2i/B/2g =1.98 (1.94) . Precisely at these two frequencies the heave (at w1) , _{sway and roll (at w2) hydrodynainic coefficients evaluated by}

equation (2) exhibit abrupt variations in their forms _{as shown by the solid line} in Figure 5. As discussed previously these changes are due to mathematical

inconsistencies. _{However, when the present modified method is applied the} predic-tions (as denoted by circular points in Figure 5) _{show no abrupt fluctuations and} the irregular frequency problem no longer exists.

For a twin-hulled structure at 15° heel, Figure 6 illustrates the calculated sway, heave and roll hydrodynamic coefficients determined using equations _(25-27) At w2B/2g _{1.52 a sharp variation occurs in the data but this frequency relates to} the physical existence of the first resonant wave excited between the inner span

(B.) of the two hulls. _{This resonant frequency can be satisfactory evaluated} from the approximation27 u n =

(gk)½

_{k = nn/B,} i

Xiong-Jian Wu & W.G. Price Page No : 11

(28)

for n=l,2 ...Thus the variation in these predictions is due to a real physical

phenomenumat w2E/2g _{l.52 and no irregular frequencies arise in this calculation.}

The final example in this section involves the floating four circular structure shown in Figure 7. _{This consists of circular cylinders of radius a = l.Om, with a} central distance between adjacent cylinders of 3.Om. _{The results derived using} equation (2) are denoted by the solid line in Figure 7 whilst those produced by

Page No : 12

A mathematical failure occurs in the ordinary method at frequency w2a/g 1.87

but this is eliminated by the modified mathematical model. As can be seen from Figure 7 the predictions derived by the two theories agree very well except in

the vicinity of the irregular frequency and since equation (2) yields unique and accurate solutions below the first irregular frequency the modified theory

produces results which

coincide with the original integral method results (i.e. derived from

equation (2)) at low frequencies,

remove the irregular frequencies from the calculations associated with

mono, twin and multi-hulled offshore structures oscillating at high

frequencies in calm water.

Therefore it is shown that the proposed modified theory presented in this section

is effective (and also computationally efficient) throughout the whole frequency

range.

5. A THREE DIMENSIONAL-STRIP APPROACH

Although two dimensional strip theories are widely used in offshore engineer-ing applications they suffer from many deficiencies. Namely,

they are unable to describe accurately the fluid actions existing at the bow and stern,

they are unable to calculate surge related hydrodynamic coefficients and

motion responses, and

at low frequencies the calculated heave added mass tends to an infinite value whereas in reality it is finite valued.

Before discussing a hybrid approach combining three (3D) and two (2D) dimensional mathematical models it is interesting to note the usual geometric

features exhibited by a full ship hull or the main hull of a large offshore

structure. In general, the structure has a long parallel midbody with little or

no variation in sectional geometry along its longitudinal length and

comparatively small regions at the bow and stern where large and/or abrupt geometric

variations exist. Wu5 proposed that for such structures the midbody region can

be modelled by a 2D strip-like distribution and the end regions by 3D panel

distributions of sufficient detail to describe the local geometric variations. These two mathematical models may then be combined using integral equations from which the fluid action data may be derived. _{Such an approach eliminates the stated}

deficiencies, increases the accuracy of the predictions over conventional 2D

strip theories and reduces the computational effort (without reducing accuracy) required by an elaborate 3D panel method.

5.1 Mathematical model

Figure 8 illustrates the form of a typical structure under discussion. It

has a long parallel midbody section between coordinates x _{x < Xb and it is}

assumed that in this region the unit normal component in the Ox direction is given by with O n1 = E: («1)

1=

f

X s Xb for

x() G(P,Q) d

no Page No 13 small longitudinal sectional variations.

The mean wetted body surface area S = S + S + S with S , S and S denoting

w s m b s m b

the surface area components of the stern, midbody and bow respectively.

A slowly varying midbody section implies slowly changing physical quantities along the length so it is assumed that all velocity potential _{amplitudes can be}

expressed by polynomial expansions of the form

(x,y,z) = i(y,z) x(x) ₍₃₀₎ where N x(x) = n n n=O n n=O fl X

and C denotes a constant defined by the midbody geometric form between _coordinates x to X

s b

Substituting equation (30) into equation (2) yields the following 3D-2D strip integral equation

(P

2)

2 J(p) -

J

(Q) dS - J (q) an

S+S

sb

C w =

-J

v(Q) G(P,Q) dS

-

J

Xb

v(Q) G(P,O)

d]

d (32)

S+S

C X s b w s (29) (31) (33)

Xiong-Jian Wu & W.G. Price Page No: 14

Nowadays the Green's function G(P,Q) can be very efficiently evaluated28

and there remain no difficulties in calculating the integrals in expressions (32-33) These now contain few unknowns and therefore require- much less computing effort to solve the subsequent numerical matrix equations.

The chosen value for N in equation (31) depends _{on the range of wave} frequencies under consideration. _{It has been found that in the low frequency}

range satisfactory hydrodynamic coefficient data were derived for a fore-aft symmetric structure when N=2 and at higher frequencies N=3 may be adequate.

5.3 Numerical examples

For a submerged pontoon of length L = 117m, beam B = 45m, draft h = 15m

and submergence H = _{15m the calculated heave and roll hydrodynamic coefficients}

are illustrated in Figure 9. _{The circular points denote data derived from}

equation (2) and an 88 panel discretisation, whilst _{the solid line denotes data} evaluated by the present hybrid theoretical approach, i.e. equations (30-33), and a 16 panel discretisation in the bow and stern regions. _{It can be seen that the}

latter calculations correlate well with the experimental _{evidence of Ohkawa29} and are in agreement with the conventional 3D calculation data.

The second exaple5 involves a series of floating

_{rectangular cylinders}

of dimensions 120 x 40 x 20 (L X B x h), 80 x 40

x

20 and 40 x 40 x 20m. The discretisation of the mean wetted body surface areas adopted in the 3D and hybrid calculations are given in Figures lO and 11 respectively. _{Figure 12 illustrates}

the computed surge hydrodynamic coefficients and it is _{clearly seen that a} similar degree of accuracy is derived by both approaches.

6. CONCLUSIONS

Based on a general Green's function integral equation _{(i.e. equation (2))} modified integral equations are developed capable of _{analysing the motion behaviour} of various offshore structures excited by waves. _{The modifications and}

simplifica-tions to the theory are suggested by the configuration of the offshore structure i.e. shallow draft, slender, etc. _{Three variations of the general motion theory}

are summarised in the paper and from the findings presented it is _{concluded that}

(i) _{for shallow draft structures, both}

the three dimensional (L/B<3) _{and two}

dimensional (L/B _{3) shallow draft theories}

allow accurate predictions to be made of all the bodily motions of the structure. _{By combining} analytical and numerical solutions a significant reduction in effort

Xiong-Jian Wu & W.G. Price Page No :1_5

(computer time, costs, etc) is achieved in evaluating the responses of

this type of structure.

the modified integral equation with multiple Green's function expansion

(i.e. equations (25-27)) provides a useful two dimensional strip theory able

to determine the motion responses and wave loads of mono, twin and multi-hulled

structures over a wide range of frequencies. Since this approach

eliminates the occurrence of 'irregular frequencies', it may be of particular use in the frequency domain analysis of second order wave

forces and also in hydroelasticity

theory3

the hybrid 3D-strip theory developed in section 5 is able to evaluate

accurately all six bodily motions of a large ship (such as a tanker) and

a large offshore structure possessing a midbody section exhibiting only small variations in sectional geometry along its length.

In all cases considered, the comparisons between theoretical predictions

and available experimental data show good agreement, validating the proposed

theoretical approaches developed and discussed in the paper.

ACKNOWLEDGEMENT

We are grateful to the Marine Technology Directorate, Science and

Engineering Research Council for their financial support during the preparation

of this paper.

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Shipbuilding Progress, November 1986.

Kim, W.D. "On the forced oscillations of shallow draft ships". J. Ship Research, Vol. 7, October 1963, pp. 7-18.

Yamashita, S. "Motions of a box-shaped floating structure in regular waves", IHI Eng. Review (Japan) , Vol. 14, No.2, April 1981,

pp. 21-30.

Wu, Xiong-Jian and Price, W.G. "Motion Predictions of sea-going barges in offshore operations", Offshore Operations Symp. (ASME) , New

Orleans, February 1986, pp.83-88.

Wu, Xiong-Jian and Price, W.G. "A method to analyse shallow draft offshore structures in six modes of motion", 5th OMAE Symp., Tokyo, April 1986, Vol. 1, pp. 476-482.

Wu, Xiong-Jian. "A two-dimensional source-dipole method for seakeeping

analysis of ships and offshore structures", mt. Conf. Computer

Added Design, Manufacture and Operation in the Marine and Offshore Industries, Washington DC, September 1986.

Wu, Xiong-Jian. "A hybrid 3D-strip method for evaluating surge coefficients of full shaped ships", Boundary Elements VII, Vol.2, Editor:

C.C.Brebbia, September 1985, pp.9.3-9.12, Springer-Verlag.

Miao, G.P. and Liu, Y.Z. "The theoretical study on the second order wave forces for two-dimensional bodies", 5th OMAE, Tokyo, April 1986,

pp. 330-336.

Page No : 17

Wu, Xiong-t3ian and Price, W.G. "Appearance and disappearance of irregular

frequencies in wave-structure interaction problems", 1st mt.

Workshop on Water Waves and Floating Bodies, MIT, February 1986,

pp. 193-200.

Wu, Xiong-Jian and Price, W.G. "An equivalent box approximation to predict

irregular frequencies in arbitrarily-shaped three-dimensional marine structures". To appear in J. Applied Ocean Research, October 1986.

Wu, Xiong-Jian and Price, W.G. "A multiple Green's function expression

for the hydrodynamic analysis of multi-hull structures". To appear in J. Applied Ocean Research, January 1987.

Shin, C. "On the motions of inclined ships in transverse waves", Trans.

west-Japan Soc. Naval Architects, No.63, March 1981, pp. 79-95 (iii west-Japanese)

Oortmerssen, G.V. "Hydrodynamic interaction between two structures,

floating in waves", BOSS 1979, paper 26, pp. 339-356.

Brown, D.T., Eatock Taylor, R. and Patel, M.H. "Barge motions in random seas - a comparison of theory and experiment", J. Fluid Mech.,

Vol. 129, 1983, pp. 385-407.

Takaki, M., Arakawa, H. and Tasai, F. "On the oscillations of

semi-submersible catamaran hull at shallow draft", Trans. West-Japan Soc. Naval Archi., No. 42, May 1971, pp. 115-130 (in Japanese).

Chakrabarti, S.K. "Wave interaction with a triangular barge", 5th OMAE, Tokyo, April 1986, pp. 455-460.

Wu, Xiong-Jian and Price, W.G. _{"Behaviour of Offshore Structures and}

service vessels of shallow draft in deeper water". To appear in

RINA mt. Symp. on Development in Deeper Water, October 1986.

Wu, Xiong-Jian and Price, W.G. "Irregular frequencies associated with singularity distribution techniques". Internal Report, Brunel Univ.

Wu, Xiong-Jian and Price, W.G. "Resonant waves in fluid structure

interaction problems involving a free-surface", mt. Conf. Vibration

Problem, Xi'an, June 1986, pp. 439-446.

Newman, J.N. "The evaluation of free-surface Green functions", mt. Conf.

Ohkawa, Y. "On the hydrodynamic forces acting on submerged hexahedrons", Report of Ship Research Institute (Japan) , Vol. 17, No.2, March 1980,

pp. 133149 (in Japanese)

-John, F. "On the motion of floating bodies - Part II",

Commun. on

Pure and Applied Math., Vol.3, 1950, pp. 45-101.

APPENDIX

Irregular frequencies occurring in a triangular section

Irregular frequencies associated with a source or source-dipole distribution method analysis for a free surface piercing body are defined by the set of

(30)

equations

=

o

=0

For a triangular section of beam B and draft h = as illustrated in Figure 5, the interior velocity potential satisfying (Al) can be derived as

with corresponding irregular frequencies at

g k cotan (kh) I (antisymmetrc)

= (A4)

g k

tan (k h) I

(symmetric)

AUTHORS

Page No 18

in the interior domain

on the interior free surface (z=O)

on the sectional contour C

w

(Ai)

sinh(ky)sin(kz') -

sin(ky)sinh(kz') (antisymmetric)

(A2)

=

'

cosh(ky)cos(kz') - cos(ky)cosh(kz') (symmetric)

for - z' y z' and O z' h with z' = Z + h. To satisfy the free surface

condition at z' = h (i.e. z = 0) it requires

tan(kh) = tanh(kh) (antisymmetric)

(A3)

Approximate solutions to equation (A4) may be shown to be

(m + ¼) T (anti symmetric)

kh

_(A5)

(m - ¼) 'lT (symmetric)

allowing the irregular frequencies to be simply given as

g (m + ¼) /h w m

01

for m = 1,2... g (m - ¼) /h

_(setric),

Xiong-Jian Wu & W.G. Price Page No : 19

(antisyrnmetric)

ZOPO

-: !ZOTY

¿250

Fig. 2

Roll response of a

semi-submersible in transit,

U

denotes the 3D shallow draft

theory calculation.

1.4 1.2 0.8 0.4

00

X / a a SURGE RESPONSE IN HEAl) SE AS .12 .1 0 .0 8 .0 6 .0 4 .0 2 O / - a a Pace No 20 1.0 0.5

Pitch amplitude operator (deglmm)

in head seas

- rounded keel-edge

sharp keel-edge exp.Brown et al sharp keel-edge, 3f) cal.

G the present shallow draft theory

Frequency (rad/s)

Fig. i

Pitch response of a barge (L2.4m, B=O.8m, hO.iO5m).

o

denotes the 3D shallow draft theory calculation.

ROLL itt beam seasS/

.

0 1 2 3 4 5 6 X/B X exp., Ref. 23

e-- present method

I t j

i exp.

Chakrabarti

- diffractions

theory cal.

present shallow draft theory

WAVE PERIOD

6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

Fig. 3

Surge response of a triangular platform (L=108m, B=l24.6m,

h=3.05m).

denotes the 3D shallow draft theory calculation.

¡kZ0

520 Deck

R

UI

II

Il

0 IO IS

2 r 1.0 0.5 0.0 i .0 0.0 o SWAS o OoO 2 r A/L 0.0 O

Xiong-Jian Wu & W.G. Price Page No : 21 z / =90° ß=120° a a HEAVE o exp. 3D cal4Nojir - - - _{2D cal.} D 20 shallow draft j___ a

-s--'

J

z a a -e.r 2 /5

Fig. 4

Surge, heave and roil

responses of a

rectarigu-]ar barge (L=3.Orn,

B=O.75rn, h=O.0159m).

Fig. 5

Sway, heave and roll coefficients for a triangular section

(h=B/2) calculated by the ordinary 2D method

_{(i.e. eq.(2))}

and by the multiple Green function method (i.e.

eqs. (25-27))

StAVE *44pVCp2J' ROLL

ordinary method (20) --e.-- multiple Greer

fnçticn method (20) G.5' (ØI 2 .W /kç a a ROLL ß=90° o B= 180° exp. ca1) Nojori D 20 shallow draft theory 1.0 0.5

a.

f 22"

B22/pVw

Fig. 6

_{Sway, heave and roll coefficients of}

a twin-hull in 15° of

heel computed by the multiple Green

_{function method.}

SWAY

,wö

a-1 .0

a

SWAY _B33/plku

K FA VE

AUTHORS : Xiong-Jian Wu & W.G. Price

Page No : 22

ROLL

P.4 e. i.? l. 2.

Fig. 7

Comparisons between the results by the ordinary 2D method

(i.e. eq.(2)) and by the multiple Green function method

(i.e. eqs. (25-27)).

ordinary method (2D)

--e--- multiple Green

function method 20)

3D calculation.

AUTHORS :

Xionq-Jiafl Wu & W.G. Price

Page No

: 23

Fig. 8

The 3D-strip discretisation

of the mean wetted surface

of a full-shaped marine

structure.

0.4 0.2 0.0 0.3

Fig. 9

Heave and roll coefficients of a submerged pontoon

(L117m, B45m, h15m and submerged depth H=15m)

denotes the present 3D-strip method calculation.

Fig.

10

DiscretisatiOn for

Fig.

il

Discretisation for

3D-strip calculation.

a p,, 6 6 2 30-STRIP METKOIJ 3D, 88 PANELS TOTAL EXP., OFIKAWA(1960) < s s A A 00 0.5 1.0 1.5 2.0 0 0 0.5 1.0 1.5 2.0 0) 1.0 _{30-STRIP METi0O} t 30, 88 PANELS TOTAL 0.6 EXP., OHKAA(1980) 0.6

Xiong-Jian Wu & W.G. Price

Page No 24

Fig. 12

Surge added mass and

damp-ing coefficients computed

by the present 3D-strip

method for a series of

rectangular cylinder of

B = 40m, h = 20m and

L/B = 3.0, 2.0 and 1.0

respectively (quoted from

Ref. 15)

0.0 0. D. a 06 -' 0.4 o 0.2 0.0 30-STRtP METHW 3D, BO PANELS TOTAL 0 0 0.5 .0 1.5 2.0 D. a o0 0.5 1.0 1.5 2.0 1.0 1.5 - 0.5 o 1.5 1.0 0.5 0.0 0.0 0 0 0.5 1.5 1.5 00 s- 3D-SIPIP P(TU) 30, 136 PAIELS TOTAL

a. Surging added mass and

damping for a barge 120x40x20m.

s 3D-STRIP ET}W 30, 48 PA1_3 TOTAL

2.0

b. Surging added mass and C. Surging added mass and

damping for a barge 80x40x20m. _{damping for a barge 40x40x20m.} 0.0 0.5 0.4

w..

00 0.5 1.0 1.5 2.0 D. a o 0.3 o 0.2 0.1 0.0 0.5 1.0 1.5 2.0 0.5 0.4 0.3 0.2 0. 1 w 00 1.0 1.5 2.0 1.0 0.6 0.4 0.2