A comparison of calculation methods for wave loads on twin pontoon semi-submersibles

1. INTRODUCTION

The twin hull semisubmersible platform having two parallel submersible pontoons has become the most common type of floating drilling rig. In recent years this type of structure has also been applied to other duties, such as; heavy lift crane platforms, dredging, hotel platforms, and multifunction support vessels with diving, fire fighting and crane facilities.

Wave-induced hydrodynamic loads on such structures may be obtained from model tests or by calculation. Model tests are most useful for validation

of calculation methods and verification of final

designs, while calculations are convenient for comparison of alternative designs and for detailed load distributions. During the past decade considerable progress has been made in the development of calcula-tion methods for twin pontoon sernisubmersibles; however, general agreement has not been reached oT the most suitable method for the calculation of hydro-dynamic loads on these structures. in some cases, increased importance is now attached to rapid transit, introducing a requirement to sustain a reasonable speed in surface (transit) condition in moderately rough weather. In other cases, narrow spacing between the pontoons increases the hydrodynamic interaction. The recent generation of twin pontoon semisubmersib-les thus introduces additional requirements to the methods used to determine hydrodynamic loads.

The present paper reviews available methods for the calculation of hydrodynamic loads on twin pontoon semisubmersibles. A comparison of numerical results for some of the methods is included based on three vessels with quite different forms, viz; (a) a Multifunction Support Vessel (MSV), designed by

Norwegian Maritime Research No. ¡/1982

26

Submarine Engineering and Seaforth Maritime (Fig. 1), (b) a semisubmersible, selfelevating, walking cutter dredger, SIMON STEVIN (Fig. 2), (c) a small waterplane area twin hull (SWATH) vessel. The MSV has a fairly conventional underwater form, while the SIMON STEVEN has unusually flat and wide

pontoons for shallow water work (Fig.

3). The

SWATH has a

coritinous surface-piercing strut

between each underwater pontoon and the deck

structure.

J. Mathisen

Keywords: offshore strucu res wave loads, calculation methods, Diffraction, Problens

Short list of abbreviations given at the end of the paper.

* Presented to limited audience at a Symposium on Ocean Engineering and Ship Handling 1980, Swedish Maritime Research Centre, SSPA, in Gothenberg.

C. A. Car/sen

Fig. ¡ MSV Mu/i/function Support Vessel.

2. HYDRODYNAMIC LOADS

Semisubmersibles are exposed to several types of environmental action. Waves ar the most important with respect to structural loads,and only wave-induced

A comparison of cacuiaion methods or wave 'oads oi twin

pontoon semisubmrs bies

vor

$Cheepshymj

By Ârchief

Meke!wg 2, 2628

_{CD D3ft}

J. Mathisen, C. A. Car/sen - P _oi -781

Det norske Ver/tas

ABSTRACT

Methoth available for calculation of wave-induced motions and loads on twi n pontoon semisubmersibles are reviewed, and problems associated willi evaluation of the distribution of the diffraction forces are discussed. Namerical results are compared for Morison's formula, strip theory, and 3-dimensional sink-source approaches.

-. -r4-.. .;

r..

Fig. 2 SIMON STE VIN - Semisubinersible Dredger.

AIT

SI.iON S1EVN

t

(A + M) + + F e jWt (1)

where A is the added-mass matrix, M is the generalised mass matrix, B is the linear damping matrix and C is the restoring matrix, including both hydrostatic and linear anchor force coefficients. !l is the body motion vector, and F is the exciting force vector. Further notations arc explained in the notation section chapter6. Fig. 3 shows the coordinate

axes. Matrices A, B, C, and M are typically quite sparse when the sernisubmersible has a normal amount of structural symmetry, (1). Added-mass, damping, and excitation coeffïcicnts are generally frequency dependent, and assumption of small ampiliude waves and motions allow all the coefficients to be considered otherwise constant.

lt is usual practiceto spilt the exciting force into two parts:

D (2)

27

The force fI) due to the pressure distribution of the undisturbed, incident wave is

referred to as the

Froude-Kriloff force. The diffraction

force FD

corresponds to the scattering of the incident waves by the structure; i. e. it is the force required to generate the scattered waves. Note that both Froudc-Kriloff and diffraction forces are calculated without considering motion of the structure. Hydrodynanìic effects of the m'otions are taken care of by the added-mass and damping term. The energy dissipation indicated by the damping term includes both drag effects and the energy content of the waves radiated by the structure due to the motions. Thus, the equations of motion actually implythe iinear superposition of three regular wave trains with the same frequency, namely: incident waves, diffracted (scattered) waves, and radiated waves due to the motions.

As might be expected, the greatest effort is expended in determination of the added-mass, damping and diffraction terms. Section 2.4 discusses the different methods that may be applied. The restoring matrix and Froude-Kriloff force are quite easily calculated. Once all the coefficients are known, it is straightforward to solve the equations of motion for a series of frequen-cies and heading angles (f3) with respect to the waves. 2.2 Element loads

When the overall motions of the semisubmcrsihlc have been determined the loads (P()) on the individual structural elements can easily be calculated:

(3) = F()e

j't

- (A(1) + M(1)) _{t -} B(1)

_{t -}

_C(jyt

or loads willbe considered here. Furthermore, regular sinusoidal waves (Airy waves), zero speed of advance and deep water conditions will be assumed.

The calculation of wave-induced loads on semisub-mersibles can basically be considered to consist of three steps:

I. Determination of the coefficients of motion for the semisubmersible considered as a rigid body.

Solution of the equations of motion for six

degrees of freedom.

Calculation of distributed loads on all the struc-tural elements of the semisubmersible.

2.1 Equations of motion.

F 2 A

The equations of motion for a semisubmersible can

- O) -

(i) + I w ( +M0)) + iw

B() - C() I

(4)

conveniently be written in matrix form:

The directions of the clement loads are here defined relative to the axis system OX1X2X3shown in Fig. 3. Matrices A(i), B(s), C() and M(i) are the contributions of Lh element to the correspondning coefficient mat rices for

the whole

semisubmersible. When applying these forces toast ructural model, it is usually convenient to transform them to forces (Q(i)) in plat-form aligned directions by adding in the components of static weight and buoyancy due to roll and pitch:

Norwegian Afariti,ne Research No. 1/1982

Id]tl

t-

I

r

B

Fig. 3 Pontoon proportions of Simon Stevin (Supper-most,) and MSV ('1ower, and orientation of coordinate axes.

X3 XX

0.4

0.3

where p is the density of the fluid, Cm and Cd are inertia and drag coefficients, A is the displaced volume, a is the area normal to the flow, and and 0.2

are velocity and acceleration of water particles in the incoming wave at the centre of the cylinde:. Applying this formulation to cylindrical elements of a floating platform, the contributions to the coefficients of equation (2.1) are indicated by the following

expressions:

A(i) 4p (Cm

-

I) A

_(i)

B(1) p Cd a(1) u

F() e _{p Cn A(1) + Á Cda(j)u *}

Drag and inertia coefficients must be selected

appropriate to the form of the cylinder and the

Norwegian Maritime Research No. 1/1982

28

direction of flow. A suitable, constant velocity (u*) is introduced in order to lincarise the drag force.

Particle velocities and accelerations calculated at the centre of the cylindrical elements are applied in Morison's formula instead of integrating the corresponding pressures over the element surface. This approximation is accurate only for wavelengths greater than 5 times the cylinder diameter.

Part of the excitation (p A(1) ) is equivalent to the Frouce-Kriloff force, and part (p (Cm- 1) A(i) ) is equivalent to the component of diffraction force in phase with the particle acceleration. The diffraction force component in phase with the particle velocity is omitted from this formulation. Fig. 4 shows that this component may be significant for structural members fairly close to the water surface, at short wavelengths. This figure also indicates that the drag force is signifi-cant only in relatively long waves.

Note that Morison's formula is unsuitable when interaction effects between neighbouring elements are

significant.

A simple expression for the horizontal force tending to separate two horizontal cylinders can be derived from the inertia terms of Morison's formula:

Q2 Cm Aw2

0.ks

sin (kp/2) (11)

where A is the displacement volume of one cylinder,s is the depth of the cylinder centres, p is the distance between the centerlines, and a is the wave amplitude.

--

.--.. s/Drl.O s/D r 0.75/

Fig.4 _{Ratio of diffraction force component in phase}

with particle velocity to total excitation force for a horizontal cylinder. (Based on data from Lee's contribution to discussion of ref (29)).

0(i) I = P()1 + (M()33 - A (i)) g 7)5 = P(1)2 _(M(i)33 - P A) g 14

(5) Q(03 =

Corresponding components of static anchor forces must also be included when present.

2.3 Sectional loads

Sectional loads on any specifIed part of the semisub-mersible may be obtained by summation of the element loads on one side of that section. When the structure is symmetrical about the centreline, then much sinipli-fled formulae for the centreline sectional loads can be obtained as shown in references (2, 3).

Centreline sectional loads are very useful at an early design stage in evaluation of the overall strength on a semisubmersible. Transfer functions for sectional loads are also useful in the selection of design wave parameters, cf. ref. (4).

2.4 Added-,nass, damping and excitation

2.4.1 Morison 's formula

This formula is widely used for the determination of hydrodynamic forces on cylindrical members of offshore structures. Lundgren (5), gives a survey of recent work concerning its application. In the terms of Morison 's formula, the hydrodynamic force on a fixed cylinder is expressed as the sum of an inertia force (Fa)

and a drag force (Fd):

FpCA

(6) Fd

_/PCda

(7) lo 15 20 o 5 'r-. -'r s/IJ r 1.5 -r-r- -r-DRAG FORCE / X r 1/14 C0 rQ7 stO = 0.75

The results presented in section 3.4 show that this expression is useful under certain conditions.

Only forces normal to the cylinder axis are given by Morison's formula. Axial forces acting at the exposed ends of cylindrical niembers and correction forces at the intersection of members must also be included. It is usually acceptable io calculate these forces on the basis of the pressure due to the incoming wave.

2.4.2 Strip theory

This method is widely used for calculating wave-induced motions and loads for ordinary ships. The «strips» are obtained by splitting the hull into a number of sections. Each strip has approximately constant cross-section, so that two-dimensional potential theory can be applied to determine added-mass, damping and excitation forces on the individual strips (1).

Strip theory has been extended to catamarans and twin pontoon semisubmersibles by making each strip include opposite sections of both port and starboard hulls (2, 3, 6, 7). Interaction between the hulls can then

be taken into account when calculating the added-mass, damping and exciting forces. This approach generally gives good results for the wave-induced motions. However, the two-dimensio-nal approach overestimates the effect of the standing wave-induced between surface-piercing hulls, when the incoming wavelength is twice the distance between the hulls. This wavelength is usually so short that accurate prediction of the motion transfer functions (for this wavelength) is of minor interest.

Problems are encountered when attempting to use strip theory to determine the distribution of hydrody-namic loads over the structure. The distributions of the Froude-Kriloff force, and the added-mass and

damping forces are given by the

incident wave potential and the forced-motion (or radiation) poten-tial respectively. However, the diffraction potenpoten-tial is not normally calculated in strip theories. Instead, an integral theorem, known as Green's second identity

(basis for the Haskind relations),

is invoked to determine the total diffraction forces on the structure from the known incident wave and forced-motion potentials. This approach s described in detail in the

appendix (A.lA.4).

It is applicable for the

diffraction force integrated over the whole wetted surface of the structure, hut the theory does not apply to the forces on parts of the structure. Lacking an appropriate theory to solve this problem, it is natural to try either of the following assumptions: (a) the diffraction force is

evenly dstributed over

the structure, or (b) the diffraction force is distributed as indicated by the expression for the total diffraction force on the structure. Assumption (a) was applied by Nordenstrom, Faltinsen and Pedersen (2), and both approaches are applied in the present paper.

§A.5 of the appendix describes an alternative

technique for the determination of the diffraction

force acting on one pontoon of a twin-pontoon

semisubmersible. Tb is approach involves the calculat-ion of an additcalculat-ional radiatcalculat-ion potential; however, this potential is very similar to the forced-motion potential

29

already calculated and is obtainable by the same type of numerical method. Proper division of the forces and moments between the two pontoons should, in many cases, provide a sufficiently detailed load distribution

for structural analysis.

An adequate distribution of the diffraction force is important for a reasonably accurate structural evalua-tion. Many publications presenting strip theory load calculations for catamarans and semisubniersibies make only a brief or unclear reference to this question. Nordenstroni et al. (2) assume the total diffraction

forces to be evenly distributed between the two

pontoons.

Lee eì al. (3) refer to Ogilvie's suggestion (8) for the division of the diffraction force between the two pontoons, but do not mention the necessity of determi-ning an additional radiation potential when using this method. Paulling and Hong (7) refer to Lee's report (9), which only applies the Haskind relations for the ovrali diffraction force. Kim (6) refers to his earlier report (10), where the 2-dimensional diffraction problem is solved directly for beam seas, with the application of the 2-dimensional Laplace equation and sink-source technique. This approach is valid for beam

seas, but is likely to present problems for other

headings; cfr. §2.4.4.

2.4.3 Three-dimensional sink-source theory

This type of theory was primarily developed for

application to large offshore structures, when Morison's formula is inadequate. The forced-motion and diffraction potentials can be determined by means of three-dimensional source techniques (11, 12). Since

the diffraction potential itself

is calculated, the problem of the distribution of thc diffraction forces is overcome. However, very much more numerical work is required than for a strip theory approach.

2.4.4 Slender body diffraction theory

This theory is in many ways similar to strip theory, but

a more stringent formulation of the problem

is generally employed, and the diffraction forces are determined directly from 2-dimensional diffraction potentials. Less numerical work is required than for 3-dimensional sink-source theory.

The slender body assumption requires the vessel beam and draught to be small compared tc the length, and the shape and dimensions of the cross-section to

vary slowly along the length. A high frequency

assumption is also employed, requiring wavelength comparable to the beam. In order to satisfy the body

boundary condition (eqt. A3), the diffraction potential

must have a longitudinal variation similar to the

incoming wave. This reasoning leads to a diffraction potential formulated as:

øDexp jkx1cosß }1ì (x1,x2,x3)

(12) where 1i is a series of 2-dimensional potential functions with parameter xl. Compare the exponent above with the incident potential in eqt. AS. Substituting this

Norwegian Mariti,ne Research No. 1/1982

expression in the 3-dimensional Laplace equation, gives a 2-dimensional Helnìholtz equation:

a21

+

_k2cosß

O (13)

ax22 òx32

where longitudinal derivatives of l' are neglected. This is the governing equation that must be solved in order to obtain the 2-dimensional diffraction potentials. Only in the beam seas case does this simplify to the 2-dimensional Laplace equation.

References (13, 14, 15, 16, 17) give applications of slender body theory. Troesch (14) includes an extensive comparison with excitation forces on a monohull calculated using the Haskind relations. Good agreement is exhibited for the overall forces, while the sectional forces show some significant deviations.

Choo (15) has applied slender-body theory to a

catamaran, indicating that this theory is promising for application to twin-pontoon senìisubmersibles. Liapis and Faltinsen (17) have recently developed a solution valid for all wave headings, employing a new, fast, integral representation method for the Helmholtz equation problem.

2.5 Non-linear effects

The analysis above assumes infinitely small wave amplitudes. For semisubmersibles in transit condition the freeboard of the pontoons is typically 0.5-1 .0 m, and for finite wave heights the pontoons will be partly

submerged for some part of a cycle, resulting in

non-linear variation of buoyancy force and in

additio-nal damping due to flow of water on top of the

pontoons.

Unpublished comparisons by Det norske VERITAS between analysis and model tests of a structure similar to Fig. I show experimental pitch and roll angles at the resonance peak about 40_5Olo lower than calculated, while the results agree well outside the resonant region. The heave motion agrees well. Similar discrepancies were observed for a semisubmersible loading station by Suhara et al. (18) for roll (40Vo) while larger devia-tions were found for pitch and heave (60-70o).

The problem of partly submerged circular cylinders was studied by Dixon et al. (19), including the variable buoyancy force in a modified Morison equation. Fór rectangular sections, however, the problem is more complex due to the abrupt change of water-plane area, and further theoretical and experimental studies are necessary. At the moment empirical correction factors

are applied at VERITAS for the transit motion

analysis.

Semisubmersibles in survival condition generally

have a constant waterplane area for quite large

excursions from the design level. The type of non-linearities discussed above for the transit condition will not occur so easily in the survival condition. Kistler and Nash (20) have carried out a compherensive series of model tests on a typical two-pontoon semisubmer-sible at drilling and survival draughts. Their results show significant non-linearities in heave response only

Norwegian Maritime Research No. 1/1982

30

for wave periods greater than 20 seconds. Such long period waves will usually be of only minor importance. Katayarna et al. (21) have also presented model test results showing that non-linearities in response become significant only for long period waves. 2.6 Forward speed effects

The proper inclusion of forward speed effects for semisubmersibles in transit condition or for catamarans poses .some problems. At zero speed, interaction effects exist between the hulls, but with forward speed there will be a tendency to leave these effects behind.

Fortunately, model experiments (22, 23) show that the loads on the structure between the hulls are greater at zero speed than with forward speed. Thus, useful predictions of the loads can be made without solving the forward speed problem.

3. COMPARISON OF NUMERICAL RESULTS Motions and centreline sectional loads have been calculated for a SWATH, the Simon Stevin in transit condition, and the NISV in both transit and survival conditions (Fig. 1--3). Some details of the vessels are given in table I.

Table 1

Principal dimensions of vessels used in numerical calculitions.

3.1 Numerical methods applied

Numerical results have been obtained using both strip theory and three-dimensional sink-source theory. The strip theory results were calculated with computer program NV407, presented in reference (24), and

containing the strip theory from reference (2). A

combination of Morison and strip theory approaches is used to handle the various structural elements of semisubmersibles. A modification of the program enables the calculation of the distributed diffraction force based on assumtpion (b) of section 2.4.2. This is referred to in the figures as the «odd diffraction force» because the even part of the diffraction force (having the same direction on both sides of the section) does not contribute to the sectional force. For the MSV at survival draught, calculations have also been carried out using Morison's formula only. The three-dimen-sional sink-source theory results were calculated with computer program NV459, based on reference (12).

This program has been throuhly tested and very

SIMON STE VIN MSV (TRANSIT) MSV (SURVIVAL) SWATH Length L lOOn, 89m 89m 80m Draugh d 6.1 n, On, (m

-Beam of one pontoon b/d 410 (.65 .04 0.667 Distance bctween

pontoon centrelines p/b (.69 3.96 3.87 2.25

Height of centre of

gravity above keel 1(81/4 2.86 2.25 1.20 0.539

Roll radius of

gyration RRG/(p+b) 0.874 0.416 0.398 0.373

Pit/b radios of

dely used. The authors therefore believe that the

results of NV459 can be used as a basis for the

evaluation of the other methods of calculation.

However, the limitations of linear potential theory must be borne in mind.

3.2 Modelling of structures

Care was taken in the mathematical modelling of the structures to ensure identical geometry and weight distributions when applying the different theories. When applying Morison's formula, no particular

effort was macle to adjust the drag and inertia

coefficients in order to obtain better agreement with the other methods. Appropriate frequency-independ-ent coefficifrequency-independ-ents vere simply extracted from ref. (25). The models employed for the 3-dimensional sinksource approach were fairly coarse, in order to use minimum computer time, while still providing reasonable accuracy.

3.3 Motions

Figures 5 to 12 show comparisons of the transfer functions for motions calculated by the different theories. The agreement is generally good, perhaps being weakest for the SWATH. Results not presented here how better agreement in roll and pitch for the SWATH. The results for

the MSV in

survival condition show unusually good agreement, confirming that useful results for the motions of semisubmersibles

in submerged condition can be obtained with Morison's formula.

Some additional results for motions of the MSV and the Simon Stevin in transit condition are given in ref. (4). Significant 3-dimensional effects are observed in the pitch motion of the Simon Stevin, probably due to the unusually large, closely-spaced pontoons.

150loo

-5

0-0

Fig. 6 Transfer function for pitch - MS V (transit). z O' 0 3-D 0IFFR. THEORY STRIP THEORY r90° O 3-D DIFFR THEORY - STRIP THEORY o

Fig. 5 Transfer function for heave MSV (transit). Fig. 7 Transfer function for sway - Simon Stevin(transi t).

31

fz.,Tjegjan Maritime Research No. 1/1982 025 05 10 20 40 80 025 05 10 20 40 80 z 90' O 3-D DIFFR THEORY STRIP THEORY '3 to 05-o o o

io o

50

o

0.6

p 9o

3-O OIFFR THEORY

STRIP THEORY

40 80

AIL

Fig. 8 Transfer function for roll- Simon Stevin (t,-an-sit).

:90°

O 3-D. DIFFR. THEORY

STRIP THEORY

Norwegian Maritime Research No. ¡/1982 112

a

1.0 0.5 o n3 0 25 1.5 3.0-0.5-rn o 0.6 p 90, o 3 -0. OIFFR. THEORY - SITUP THEORY 10 2.0 4.0 66 A/L

Fig. IO Transfer function fo, heave- SWA 7H.

go. O 3-D O!FFR. THEORY STRIP THEORY -. - MORISONS FORMULA I I 0.5 LO 2.0 /4.0 8.0 XiL

Fig. 9 Transfer function for sway - SWATH. Fig. 11 Transferfunctionforsway - MSV (survival).

32 1.0 20 0.5 2.0 .0 4.0 66 AIL

6.0 .fl

a/A

2.0 -0.0 0 25 13 O 3-D OIFFR THEORY - STRIP THEORY - MORISONS FORMULA 0.5 1.0

Fig. 12 Transfer function for roll - MSV (survival). 3.4 Hj'drodynarnic loads

Transfer functions for forces and moments at the centreline sections are shown in Figures 13 to 20. The transverse force will load the connecting structure between the hulls in tension and compression, while the transverse torsional morflent may be considered to apply a torsional load to the connecting structure. The curves indicated to be «siinplified Morison», arc

obtai-ned using eqt:

(li).

Since the pontoons are surface-piercing in the transit condition, mirroring of the cross-section about the water surface is considered when selecting the inertia coefficients. Morison's formula has not been applied to the SWATH.

Neither of the two assumptions used with the strip theory provides loads in consistent agreement with the three-dimensional diffraction theory. Reasonable agreement occurs only for long wavelengths, when a fair estimate of the transverse force can be made by Morison's formula. Peaks in the «simplified Morison» curves in Figs. 13 and 15 at short wavelengths should be disregarded, since the condition that the wavelength be greater than 5 times the pontoon dimension is not fulfilled.

The standing wave between the pontoons of the SWATH occurs at a wavelength

X ¡L =

0.6, corresponding to the peaks in the strip theory results in Figs. 17 and 18. At a wavelength X IL = 1.8, 3-dimen-sional sink-source results show a resonant vertical motion of the water between the pontoons, out of phase with the incoming wave. This will provide large pressure differences between the inside and outside surfaces of the pontoons, leading to large transverse forces.

This resonant frequency can be roughly

predicted by considering vertical oscillation of the water mass enclosed between the hulls.

33 07 pgp L a 0 75- 050- 025-¡ I

...\

o

/

I

ii

° I

¡ \

1/o

OS pOpL a o 025 05 TO 2.0 4.0 8 0 AI L

Fig. 13 Transfer function for transverse force at

centre/inc section - MS V (transit).

0

06-004-.

002-8 45

O 3-D. DIFFR. THEORY

STRIP THEORY WITHOUT

ODD OIFFR FORCE

STRIP THEORY INCLUDING

ODO OIFFR FORCE ßr 90' O 3-O DIFFR THEORY STRIP THEORY WITHOUT ODD DIFFR FORCE

- -

STRIP THEORY INCLUDING 000 DIFFR FORCE - . - SIMPLIFIED MOR ISON O 025 05 1.0 20 40 80 >II..

Fig. 14 Transfer.function for transverse torsional mo-nent at centre/me section - MSV (transit).

Norwegian Maritime Research

0.6 pgpLa 04 02 o 0.06 OS pgp L2a 0.04 0.02 o os 90 O 3-D OIFFR THEORY

STRIP THEORY WITHOUT

ODO OIFFR FORCE

STRIP THEORY INCLUDING

000 OIFFR FORCE

SIMPLIFIED MOR ISON

\

r'

'i

Norwegian Maritime Research

No. 1/1982 A A A A OS LO 20 4.0 80 AIL

Fig. 15 Transfer function for transverse force at cen-trelijie section - Simon Sievin (transit).

s 60

0 3 - O OIFFR THEORY

STRIP THEORY WITHOUT

.000 OIFFR FORCE

STRIP THEORY INCLUOING

ODD DIFFR FORCE

4.0 80 AIL

Fig. 16 Transfer function for transverse torsional moment at centre/me section - Simon Stevin (transit). 34 20 03 pgBLa T.5 O o : 90 O 3-D DIFFR THEORY STRIP THEORY WITHOUT ODD I DIFFS FORCE T O - STRIP THEORY t _{INCLUDING 'ODO} I 0IFFR FORCE' 0,5 A A A o O 06 10 2.0 40 6.6 AIL

Fig. ¡7 Transfer function for transverse force at

centre/we section - SWATH.

s6Q

O 3-D. DIFPR. THEORY

STRIP THEORY WITHOUT

'ODD DIFFR FORCE

STRIP THEORY INCLUDING

ODD OIFFR FORCE

A/L

Fig. _{18 Transfer function for transverse torsional}

moment at centreline section - S WA TH.

0.3 Q?

pgpL

0.2 OEl 0.0 0.1 os pgp L2a 0.05 I.

i

I' I

'

_{-. MORI SONS} III

'I

'I

Fig.

19 Transfer function for transverse force at

centre/me section - MSV 1tsurvivctl,).

O _{3-O Elf FR, THEORY}

STRIP THEORY WITHOUT

000 DIFFR FONCE _F2

STRIP THEORY INCL

ODD OIFFR FORCE

pgpL

3 r45

o 3-0 DIFFR THEORY

STRIP THEORY WITHOUT ODD DIFFR FORCE

STRIP IHEORY NCL.

ODD DIFFR. FORCE

--MORISONS FORMULA ß

FORMULA

Fig. 20 Tran.sferfunction for transverse torsional ¡no-ment at centre/inc section - AIS!" (sufrival,1.

35 0.1.

-

030.2 -0.1 r 9O o 3-D OIFFR. THECRY STRIP THEORY 0.01 0.25 0.5 MORISON S - FORMULA 1.0 2.0 4.0 80 X/L

Fig. 21 Total sway exciting force - MSV (survival). Despite close agreement in the motions, the loads

predicted for the MSV at survival draught using

Morison's formula do not agree very well with the 3-dimensional sink-source results. Fig. 21 shows the total transverse excitation force, and table 2 gives added mass coefficients illustrating that considerable difference in the force components_{may coincide with}

close agreement in the motions. The differences

between Morison's formula and 3-dimensional sink-source technique seem somewhat less for the sectional loads than for the force components. This comparison is primarily given for one wavelength, thus providing a limited basis for drawing conclusions.

Table 2

Added mass coefficients for MSVat survival draught, X/L= 1.23.

Norwegian Maritime Research No. 1/1982 MORISON'S FORMULA STRIP THEORY

3D

DIFFR. A22 _{0.86 1.16 1.51} A33 1.59 3.15 2.06 A44 _{0.144 0.0283 0.0467} A5ç 0.132 0.261 0.0715 10 2.0 025 0.5 ₄₀ 80 Xi L

4. CONCLUSION

Various theories available for the calculation of

wave-induced loads on twin pontoon semisubmersib-les have been briefly revicwcd. Difficulties associated with determination ofthe wave load component due to diffraction have been pointed out, and confirmed by numerical results. lt is recommended that two alterna-tive approaches for the calculation of wave-induced loads on twin pontoon semisubmersibles be investiga-ted further:

Strip theory combined with additional forced-motion potentials, for proper subdivision of the diffraction loads between the pontoons ref. (8). Slender body diffraction theory.

Comparison of motions calculated by strip theory and 3-dimensional sink-source theory gave good aoreement for the transit _{condition. At survival} draught Morison's formula also provided motions in good agreement.

5. ACKNOWLEDGEMENTS

The present study was partly sponsored by the Royal Norwegian Council for Scientific and Industrial Research.

The authors express their appreciation to Volker Stevin for releasing the study on the Simon Stevin, and to the designers of the MSV, Seaforth Maritime and Submarine Engineering, as well as the client Shell UK Exploration and Production, for permission to present the MSV study.

Discùssions with colleagues at Det norske VERITAS have also been invaluable in the preparation of this article.

6. NOTATION

Norwegian Maritime Research

No. 1/1982

36

wave amplitude

wave direction (3=00 corresponds to OX1, head seas) volume displacement

motion vector

713 are translations in X1 X2, X3 directions of Fig. 3 and

5, 6 are rotations about X1, X2,X3 axes. wavelength

v.'ater particle displacement in incoming wave density of sea water

total potential potential component

2-dimensional diffraction potential circular frequency (w2 = 2irg/X) added mass matrix

area of structural element normal to flow a overall breadth of structure

damping matrix breadth of one pontoon

restoring coefficient matrix Morison drag force coefficient

Morison inertia force coefficient cylinder diameter

draught

excitation force p

excitation force component gravitational acceleration excitation force component

complex unit lt is understood that the real part is to be taken in complex expressions w wave number (k = 2îr/X)

length of structure generalized mass matrix

2-dimensional normal components

unit normal vector, out of fluid, into vessel

P total load on structural element in X1, X2 X3 directions

P pressure

p distance between pontoon centrelines

Q total load on structural element in platform-aligned directions

R large distance from the vessel

r position vector

s, s surface area for integration

depth of immersion of cylinder centre

t i me

US _{velocity used to linearise drag coefficient}

V, V volume for integration

X1, X2X3 coordinate axes in Fig. 3 OX1 X2 plane lies in the water plane,

OX1 lies along the centreline, and OX3 passes through the centre of gravity

Supersc prit s:

P radiation potential for moving port pontoon in presence of fixed starboard pontoon

dots indicate derivation with respect to time

Subscripts:

D diffraction

d Morison drag force

incident vave or Froude-Kriloff direction number i

(i) pertaining to clement number i

ni Morison inertia force

R radiation

indicates time variable (q'),

as opposed to complex amplitude () 1,2, 3 indicate directions OX1, OX2, OX3 4, 5, 6 indicate rotations about OX1 OX2 OX3

Greek letters: A a B B b C Cd Cm D d F G g k L M N2, N3 n

7. REFERENCES

Salvesen, N., Tuck, E. O., and Faltinsen, O.: «Ship Motions and Sea Loads», Trans. SNAME Vol. 78, 1970, pp. 250-287.

Nordenstrøm, N., Faltinsen, O., Pedersen, B.: «Prediction of Wave-Lnduced Motions and Loads for Catamarans», _Offshore Technology Conference, Houston, Texas, 1971, paper No. OTC 1418.

Lee, C.M., Jones, H. D., Curphey, R.M.: «Prediction of Motion and Hydrodynamic Loads of Catamarans», Marine Technology, Vol. 10, No. 4, Oct. 1973, pp. 392-405.

Carlsen, C. A. and Mathiseri, J.:

«Hydrodyriarnic Loading for Structural Analysis of Twin Hull Semisubmersibles», ASME \Vinter Meeting, Nov. 1980.

Lundgren, H.:

«Wave Forces on Cylinders in the Drag/Inertia Regi-me>), Second WEGEMT Graduate School, Advanced Aspects of Offshore Engineering, Vol. 3, Norwegian Institute of Technology, Trondheim, 1979.

Kim, C.H.:

«Motions and Loads of a Catamaran Ship of Arbitrary Shape in a Seaway», Journal of 1-Jydronautics, Vol. 10, No. 1, Jan. 1976, pp. 8-17.

Paulling, J. R., Hong, Y. S.:

«Structural Loads on Twin-hull Semisubmersible Platforms», _Offshore Technology Conference, Houston, Texas, 1978, paper No. OTC 3246.

Ogilvie, T. F.:

«On the Computation of Wave-Induced Bending and Torsion Moments». Journal of Ship Research, Sept.

1971 pp. 217-220. Lee, C. M.:

«Theoretical Prediction of Motion of Small-Waterpla-ne-Area, Twin-Hull (S\VATH) Ships in Waves. DTNSRDC Report-76-0046, Dec. 1976.

Kim, C. H.:

«The Hydrodynamical Interaction between Two Cylindrical Bodies Floating in Beam Seas». Stevens Inst. of Tech. Rep. No. SIT-OE-72-10, October 1972. (Il), Hogben, N. and Standing, R. G.:

«Wave Loads on Large Bodies», Intgernational

Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London, April 1974.

Faltinsen, O. M . and Michelsedn, F. C.:

«Motions of Large Structures in Waves of Zero

Froude Number», International Symposium on the Dynamics of Marine Vehicles and Structures in Wa ves, London, April 1974.

Faltinsen, O. M.:

«Wave Forces on a Restrained Ship in Head-Sea Waves», Ninth Sy,np. on Naval Hydrodynamics,

Paris, Aug. 1972. Troesch, A. W.:

«The Diffraction Forces for a Ship Moving in Oblique Seas», Journ. of Ship Rsch., Vol. 23, No.2, June 1979 pp. 127-139.

Choo, K. Y.:

«Exciting Forces and Pressure Distribution on a Ship in Oblique Waves», Dissertation, Dept. of Ocean Engineering, Massachusetts Institute of Technology, Feb. 1975, Cambridge, Massachusetts.

Skjørdal, S. O. and Faltinsen, O. M.:

37

«A linear Theory of Springing», Jour,i. of Ship Rsch., Vol. 24, No. 2, June 1980, pp. 74-84.

Liapis, N. and Faltinsen, O. M.:

«Diffraction of Waves Around a Ship», to be publjs hed, based on a Doctoral thesis from the Norwegian Institute of Technology, Trondheim, Norway.

Suhara, T., Tasai, F. and Mitsuyasu, H.: «A Study of Motion and Strength of Floating Marine Structures in Waves», Offshore Technology Confer-ence, Houston, Texas, 1974, paper No. OTC 2068.

Dixon, A. G., Greated, C. A. and Salter, S.H.: «\Vave Forces on Partially Submerged Cylinders». ASCE Journal of the Waterway, Port, Coastal and

Ocean Division, Nov. 1979.

Kistler, E. L. and Nash, J. M.:

«Influence of Draft on Wave Steepness Effect in the Dynamics of Semisubmersibles», Offshore Technolo-gy Conference, Houston, Texas 1975, paper No. OTC

2367.

Katayama, M ., Unoki, K., Hatakenaka, K. and Uhihara, T.:

«Qn Structural Response Analysis of Scm isubmersible Offshore Structure in Waves», Japan Shipbuilding and Marine Engineering, Vol. 12, No. 3, 1978, pp.

11-20.

Wahab, R., Pritchett, C. and Ruth, L. C.: «On the Behaviour of the ASR Catamaran in Waves», Marine Technology, July 1971, pp. 334-360.

Dinsenbacher, A. L., Andrews, J. N. and Pincus, «Model Test Determination

of Sea Loads

on Catamaran Cross Structure», Naval Ship Research and Development Centre, Report No. 2378, May 1967. Pedersen, B., Egeland, O. and Langfeldt, J. N.: «Calculation of Long Term Values for Motions and Structural

Response of Mobile

Drilling Rigs», Offshore Technology Conference, Houston, Texas 1973, paper No. OTC 1881.

(25). «Rules for the Design, Construction and Inspection of Offshore Structures 1977. Appencix B Loads». Det norske VERITAS 1977.

Wehausen, J. \'. and Laitone, E. V.:

«Surface Waves, Handbuch der Physik», edited by S. Fluegge, Vol. 9, Fluid Dynamics 3, Springerverlag, Berlin 1960.

Frank, \V.:

«Oscillation of Cylinders in or below the Free Surface of Deep Fluids». NSRDC report No. 2375, Oct. 1967.

Potash, R. L.:

«Second Order Theory of Oscillating Cylinders». Univ. of Ca!ifornia report NA 70-3.

Hooft, J. P.:

<A Mathematical Method of Determining Hydrody-namically Induced Forces on a Semisubmersible».

Transaction SNAME, Vol. 79, pp. 28-70, 1971. APPENDIX A

CALCULATION OF EXCITING FORCES BY STRIP THEORY

A.l

This presentation is based on the derivations in refs. (1 &2), but includes some more detail in the derivation of

Norwegian Mari! i,ne Research No. 1/1982

the diffraction force, and in splitting the excitation into odd and even components.

We consider a rigid vessel floating at zero speed in regular sinusoidal waves. The resulting motions are assumed linear and harmonic. The coordinate axes shown in Fig. 3 arc employed. Viscous effects are ignored, allowing the fluid

flow to be assumed

irrotational, and potential flow theory may be applied. The total velocity potential (J

(xl, x2, x3; t) must

satisfy the Laplace condition in the fluid domain and appropriate boundary conditions. It is convenient to decompose the total potential as follows:

6

(I+D +

iøRi)e_Jwt

i=1

where 11is the potential due to the incoming wave, _D is the diffraction potential and 1Ri is the radiation potential due to motion in direction i.

A 2 BOUNDARY CONDITIONS

At the free surface kinematic and dynamic considerations lead to the classical linear free surface boundary condition which must be satisfied by all potential components:

=0

at2 ax3

On the surface of the vessel, the incident and

diffracted potentials must satisfy a kinematic boundary condition giving zero velocity through the surface at mean position:

a1

aøD

+ =0

an òn

a

-

(iiRi) =

JLfl7 fl

where the generalized normal ni is defined by: (n1,n2, n3 )=n

(A 5) (n4, n5, n6) = rx n

with n the unit normal vector and r the position vector. At intinite depth no disturbance occurs, giving the condition:

Norwegian Maritime Research No. 1/1982

(Al)

(A3)

38

Pt = p

where n is the unit normal vector out of the fluid into

the vessel surface, and - indicates the derivative in this

P jwp

direction. an

The radiation potentials must satisfy a kinematic boundary condition on the mean position of the vessel surface, giving a normal fluid velocity equal to the velocity of the hull surface due to the motions:

-(A9)

(A 10)

The corresponding forces are found by integrating the pressures over the vessel surface (SV)at the mean position: F.= s-i i

Pnds

SV =JWP

f

_nds

SV ¡

The Froude-Kriloff forces due to the incident

potential (AS) are thus given by:

(A 12)

Fj=pga51 exp fk (jx1 cosß+jx2sjnß+x3)}n.ds

Considering only the starboard side hull (SS) ofa

òX3

-.

asx

_(A6)

At large distances (R = \/x12 + x22') from the

vesse!, the forced-motion and diffraction potentials must fulfil a radiation condition for outgoing waves (26):

hrn R 1/2

(.. -jk)=O

_{(A 7)}

R = -' oo

where is the velocity normal to the control surface aR

in the outward direction.

A.3 DETERMINATION OF POTENTIALS The incident wave potential may be written:

-jga

-

_{exp {k(jx1 cosß+ jx2 sin ß + x3)} (A8)}

The radiation potentials are generally determined by two-dimensional sink-source methods such as refs. (27 & 28) within strip theory.

The diffraction potential is not determined directly, but is eliminated from the expressions for the diffract-ion forces.

A. 4 EXPRESSIONS FOR FORCES

The dynamic pressures

_{are determined from the}

potential components using the linearized Bernoulli equation:

0nx3 =0

(A2)

Hi _{is even for i = 2,4,6 and odd for i = 1, 3,5.}

Even components have the same sign for both port

and starboard hulls, while odd components have

opposite signs for the two hulls. When calculating the total forces on the structure, it is only necessary to take (twice) the even components into account, since the odd components cancel. However, the odd components must be included when considering loads on parts of the structure.

The expression for the diffraction forces starts from the same basis (All):

FDi'jwp f

_{(A 15)}

SV

Substituting the radiation potential for the unit normal by way of the body boundary condition (A4):

FDi p

- ds (A16)

v da

The region cf integration is indicated in fig. 22. The surface (SC) enclosing the volume (V) includes the vessel surface (SV = SS + SP), the free surface (SF), distant control surfaces (SR) and the sea bottom (SB). The identity is manipulated by making the potentials (A and ø) fulfil conditions relevant to the present problem. By stipulating Laplace's equation for the fluid domain, the left-hand side of (A 17) is eliminated. The bottom boundary condition (A6) shows that the surface integral over the bottom (SB) vanishes. Using

the harmonic function (eJ ()t) in the free surface

condition (A2) gives

39

SR

FREE SURFACE

SF

Fig. 22 Region of integration Green 's second identity.

= g ax3 FORT PONTOON SP SB SEA

f (A

-

ds = O

A result obtainable from Green's second identity is s n n now needed. This identity relates volume and surface

integrals over a closed region

for two arbitrary

potential functions (A' GB):

(A 17)

2B

B

2A)=

SCA

STAR BOARD PONT 00M SS BOTTOM

for

SR DISTANT CONI POL SURFACE application of

By employing this condition, we find that the

surface integral

over the

free surface (SF) also vanishes. Similarly, employing the radiation condition

(A7) with _

-

at the distant control surface, shows

dR dii

that the integral over this surface (SR) also vanishes. The only part of equation (A 17) remaining is then the surface integral over the vessel:

Setting 5A

_-

and = Ri finally gives:

v

a1

'dscRi

ds

Substituting in the expression for diffraction force (A 16):

Noni'cgiwi Maritime Research No. ¡/1982

symmetrical two-pontoon vessel (x2 is positive), the Froude-Kriloff forces on that hull may be split into even and odd components.

(A 13) = pga _{exp {jkx1 cos + kX3 } cos (2 sin ß)}

ni ds (A 14) U1 =iPga5exPUkx1 cosß +x3}sin(kx2 sine) nids

is even for i = 1, 3, 5 and odd for i = 2,4,6.

FDÍ=p

R.ds

(A21)

Applying

the body boundary condition

(A3) eliminates the diffraction potential:

F

Di 0Ri ds (A22)

SV

When using radiation potentials determined by 2-dimensional theory in this expression, it is consistent to apply the same strip theory justification for using the 2-dimensional derivative of the incident potential; i.e. for a long and slender hull the derivative in the longitudinal direction on the hull surface may be neglected.

(jN2 sinß + N3) k1 (23)

where N2 and N3 are the components of the unit

normal in the x2-x3 plane. Note that the precise

interpretation of fds must be adjusted in expressions involving N1 and N2.

lt

is convenient to split the evaluation of the

diffraction forces into two parts which need only be integrated over the starboard hull (SS).

GDj Ja

f

I - N3 cos(kx2 sin (3) +N2 sin3 sin (kx2 sin (3)

SS

expkxcos(3+ tO(3}Rj

ds

HDi = apw f [N3 sin ( °2 s1 ¡3) + N2 sin (3cos (2 '° ß)J

SS

exp jkx1cosß +kx3} _Ri ds

GDj is even for i = 3,5 and odd for i = 2, 4, 6 HDi is even for i = 2,4, 6 and odd for = 3,5

and KDI may not be evaluated since a 2-dimen-sional assumption has been applied.

FDj 2GDi i = 3,5 (A26)

FDÎ = 2Di i = 2,4,6 (A27)

The odd components of GDÌ and HDi need not be evaluated to find the total diffraction force on the vessel. Neither is use of the odd components in evalua-tion of loads ori part of the structure theoretically justified, because the integral relationship (A20) employed applies only for integration over the whole vessel.

Collecting the Froude-Kriloff and diffraction forces we obtain the total exciting forces for a symmetrical two-pontoon vessel as:

F1 = 2G1

i

F. =2(G11 +GDj)

Norwegian Maritime Research No. 1/1982

(A26)

Finally, the boundary condition (A3) is again

= 3, 5 (A27) _{employed to eliminate the diffraction potential:}

40

F1 =2(I1i +HDI)

i=2,4,6

(A28)

A. 5 DIFFRACTION FORCES ON ONE PONTOON

A slight variation on the procedure presented in §A.4 makes it possible to properly determine the diffraction force on one pontoon only, but an additional potential function must then be calculated. This is similar to the procedure suggested by Ogilvie (8).

Proceeding from equation (A19) we set = as

before, but introduce an alternative potential _Ri for This new potential is the radiation potential for forced motion of the port pontoon in the presence of the starboard pontoon which is held fixed. The body boundary condition is then reformulated as:

P

an

Ri _{= { ìwni}

_}

(A29)

Equation (A20) takes the altered form: P

SV D a

Ri

_{ds =}

f

P 3Ç _{ds (A30)} Ri

The diffraction force on the starboard pontoon (FS _{) is written as the difference between the force on}

Di

the whole vessel and the force on the port pontoon: FS

= FDi

f

ds

Di (A31)

The diffraction potential is eliminated from this integral in a similar way to that used in §A.4. First, the boundary condition (A29) is applied to replace n1 by

the derivative of the radiation potential, and the

integral over the port pontoon is replaced by the

integral over the whole vessel minus the integral over the starboard pontoon.

(A32)

P P

F8

_{= FDj + p ( f}

a Ri

_ds-f

D3 ds)

SV SS

The integral over the starboard hull vanishes due to the boundary condition (A29), and equation (A30) is applied to the integral over the whole vessel.

FS

_{= FD1 + p f}

- ds

_(A33)

P F

= FDj - p f

--ds

Di R n

SV

lt

should be fairly straightforward

to modify existing procedures for.calculat ion of the

2-dimensio-nal radiation potential

in order to determine the

1'

potential ç

R on the vessel surface (IO). When this potential is available equation (A34) may be evaluated in the manner applied previously for equation (A22). (A34)

41

Norwegian Mari ti/ne Research No. 1/1982