Hydrodynamic characteristics of prismatic barges

19 JUNI 1973

ARCHEF

OFFSHORE TEC}OLOGY CO1FEEÑcE 6200 North Central ExpresSway Dallas, Texa.s 75206

Lab.

y.

_{Scheepsbouwktmde}

HoschooI

lu

CTION PAPER

NUNBEROTC 1417

Hydrodynamic Characteristics of-Prismatic Barges

I'

By

¿

C. H. Kirn, C. J. Henry and F. Chou, Stevens Institute of Tecbnoior3 .

<r.

i

© Copyright 1971

Off shore Tecbnolo,r Conference on behalf of American Institute of Mining, Metailurg-ical, and Petroleum Engineers, Inc., The American Association of Petroleum Geologists, American Institute of Chemical Engineers, American Society of Civil Engineers, The Ameridan Society of Mechanical

Engineers, The Institute of Electrical and Electronics Engineers, IÌic., Marine Technolor Society, Society of Exploration Geophysicists, and Society of Naval Architects' & Marine Engineers.

This paper was prepared for presentation at the Third Annual Offshore. Techxiolor Conference to be held in Houston, Tex.,, April 19-21, 1971. Permission to copy is restrict ed to an abstract of not more than 300 words. Illustrations may nqt be copied. Such use of an abstract should

contain conspicuous acknowledgment of where and by whom the-paper is presented.

ABSTRACT

The.forces and moments acting on pris-matic hulls are presented, för both wave excitation and motions. Using these charts. a platform designer can easily calculate the motionsof a barge in head seas or beam seas. An example is given t illustrate, the

proced-urefor calculating motions. In addition,

predicted mtions for several barge. conf igu-rations agree well with corresponding model experiments.

INTRODUCTION

During the past several years, a number of ocean platforms have failed structurally and in many-cases utilization of platforms has had to be curtailed because of loads or motions induced by ocean waves. Under the sponsorship of the Sea Grant Program, three years of'study at Stevens Institute has led to the development of a reliable añalytical method for predicting motions in the design stage of océan platform development. This report describes the analytical 'technique briefly, and presents results with which

References and illustrations at end of paper.

platform designers can easily predict motions' of' prismatic barges without resorting to lengthy digital computer programs.'- It is hoped thereby that use of these methods in the design stage can reduce operational delays and failures due to wave induced platform mo-tions. In addition, by comparing motions of several alternate configurations, an optimum operational quality can be achieved.

In the representation of the hydrodynamic forces and moments, use has been made here of some of the most recent theoretical develop-ments ¡n marine vehicle motions analysis. The

recent historical development of these analyt-ical techniques i's briefly described as fol-lows. The vertical wave-exciting forces on -Lewis cylindrical forms (two-dimensional body with a ship-like cross-section) restrained in

beam seas were calculated by Grim. This method is based on the assumption that the

disturbance of an incident wave caused by the ship's hull can be represented by the poten-tial used in describing the flow around the same hull oscillating in.calm'water. This disturbance' potential plus the incident wave potential are supérposed with the disturbance' potential strength adjusted to satisfy the

kinematic boundary condition on the restrained

hull. This approach was exténdé

byTamura2-to the calculation of sway

ardMll exciting

forces and mòments on Lewis form cyltnd restrained in beam seas. The fundamentàl proach developed by Grim was e,ktended by K1n13 to ¡nc lude a much greater variety-ofcross---sections through the use of the two-dimension-al "close fit" disturbance potentitwo-dimension-al developed by Frank. This method can be used to evalu-ate the wave-exciting forces on the widely. varying conf iguratiöns of ocean platforms. In addition, the development presented by Frank

is used to evaluate the hydrodynamic forces induced by the motions of the vehicle itself. These two hydrodynamic force systems are then combined in the equations of motion to arrive at a procedure for predicting wave induced

platform mot iòns.

The techniques described thus far are for analysis of motions response to sinu-soidal waves at one frequency. But the ocean surface is a complicated superposition-of many waves of different frequencies, with no

apparent fundamental periodicity. The genéral techniques for analyzing responses to such random input signals have been developed in communication, network analysis and signal processing literature. lhes,e techniques have been applied specifically to marine vehicle motions in random seas by St. Denis and Pierson.5 The application of these techniqdes

is fully described there and will not be re-peated here.

With the results presented in this paper, the motions of prismatic barges can be

ana-lyzed for the cases of head -seas and beam seas which have been shown to be the most severe headings.3 The basic assumptions and

underly-ing theory are described briefly, then,a cursory discussion of several, applications of. the theory is given. An example of the use of the design charts is described and illus-trated in detail.

Methods for predicting ocean platform motions areunder further development in-this continuing program. Many shapes -other than prismatic hulls can be handled. Cylindrical

legs can be added at various depths

of

sub-mergence. The mean drifting force ¡n the direction of wave propagation can be.p're-dicted. Presently under development are ocei platform motions prediction techniques that

¡nclude mooring cable forces as'.well as pre-dicting longitudinal stress variations in the. cables themselves due to the motions. Also, the -effect of large plates or thick footings at the lower end of.cyìindrical legs are

be-ing studied. Most

of

the a'bove prediction techniques are analytical. In addition to.

-theoretical analyses, model test techniques have been studied at Stevens Institute of Technology for a wide range of marine vehicles

:i'ncl'iiding'oceaìi platforms, for many years. With an.appropriate combination of

analyt-¡cal and experimental studies during the de-_sign stagèsófdevelopment, the operational

limitationsand the chances of structuia1 ---fai-lure of--an-ocean platform dueto wave

in-duced motions can be minimized..

DESCRIPTION OF THEORETICAL APPROACH

In order to describe the sea and barge

geometry, let Oxyz be a right-handed

rec-tangular coordinate system with the plane z0 at the calm water surface, and let.the center of gravity of the barge be on the z-axis taken positive upward. This geome-try is

shown in Figure la. The y-axis is taken in

the direction of wave propagation, and the x-axis parallel to the wave crests. Then for the case of a sinusoidal wave propagating in the positive y-direction, the free-surface elevation above calm water is given by

h = a cos(wt-y) [1]

where a = wave ampi itude

-= wave number (2/g for deep water waves) w = wave frequency

Two barge headings are considered in this report, viz., head seas and beam seas, with three degrees of freedom -in each case. For the barge in head seas, the motions studied here are surge, pitch and heave while in beam seas, they are sway, roll and heave. The barge is assumed to be doubly symetric (i.e., port and starboard as well as fore and aft) so that yaw motions in ei-ther case are negligible.

- A cursory description of the theoretical

background

follows

only for the case of beam seas since the head.seas case is handled in exactly the same way.

Newton's Laws for rigid body motions are invoked for describing the surge, roll and heave motions of the barge in beam seas where, the inertial responses to the accelerated mo-tions are equated to the appliedhydrodynamic forces and moments. In analyzing the motions of any-marine vehicle, the representation of

these hydrodynamic forces and moments must be carefully outlined. It.is assumed in this analysis that the incident wave amplitude and

the resulting barge motions are sufficiently small so that the response amplitudes are

-linearly dependent upon the wave amplitude. The resulting theory, so called linearized theory, is tractable and has been shown to be valid for a wide class of marine vehicles and for a wide Êange of realistic sea conditions.

Based on the small ampi itude assumption,

4

T

OTC.lkl7

._C.._H.:KIM

C.._.J..HENRY and .E..CI-101J

it follows that the applied hydrodynamic forces and moments can be resolved into those due to the barge motions. in calm water plus those due to the incident wave while the barge

i.s held fixed at. its mean position. Further-more, if the incident wave is sinusoidalat

frequency w , then the barge motions will be als6. In this way,a linear relationship is.

attained between input wave amplitude and out-put motions amplitudes in sway., roll and heave so that linear spectral analysis techniques can be applied to analyze more complicated waves and responses.

'in

arriVin' at the final form of thé equations of-motion, a further resolution has been found .seful for the hydrodynamic forces and moments due to barge motions in calm water, which includes no further

approxima-tions thañ those aireadyméntioned. It is

found that these forces can be divided into three parts: the added inertial component which exhibits an explicit dependence on a damping component due to radiating waves

caused by the body motion which has an ex-plicit dépendence On ¡W (i-J-l), anda

re-storing component independent of w , due to changes in buoyancy distribution caused by the barge motion.

With these assumptions. and developments, the équations for the sway .fl ,. roll c , and

heave motions of the barge in beam seas described by Eq. [1] become

+

(-w2M+iwN)c =

(-w2M+iwN)

+ (-w2M

+iwN+B)c= F

[2]

(_uMcc+iwN+B)

where M = virtuál inertial force or moment per unit acceleration

N = damping fo.rce or moment per, unit velocity

B = restoriñg force or moment per unit

displacement

-and where the first subscript on .M, N or B

designates the motion and the scond denotes

the component. For example, (.) denotes a.

roll-induced, sway force. The assumed sym-metry of the barge has been used in deriving

Eq. [2] which results in uncoupling of the heave equation from sway and roll. It is

further stipulated that buoyant restoring forces and moments result only from roll and heave displacements. The.bars over some of

the symbôl.s 'in Eq. [2]. are Fntroduced to de-note complex amplitudes, which represent phase differences 'between each quantity and

the incident wave given' in Eq. 11].

The coefficients in Eqs. [2] as well as the wave exciting forces and moment F, FC

and can be evaluated for a prismatic barge

using the information contained in this paper, plus several additional inertial and geometric parameters. In order to evaluate, the hydro-dynamic contributions to the terms, in Eqs.[2], a two-dimensional representation of the barge

has been introduced. The pressure and veloc-ity fields have been assumed independent of, x, the coordinate parallel to the wave crests. Then the hydrodynamic force and moment can be

calculated for each segment of the barge be-tween x and x .+ .. This was accomplished

by replacing the barge elements by concen-trated two-dimensional sources with unknpwn strength where each source potential sati.sfies the appropriate free-surface boundary condi-tion. The source strengths are then deter-mined from the kinematic boundary condition on. the barge, hull.

Sumin

over all segments

then gives, the total force. and moment on the barge. The wave exciting forces and moments are treated by a technique similar to that for the motion induced components as.was originally des,cribed by Grim.1 A more de-tailed description, of this technique is given by Kim and Chou.3

Table 1 shows the relation between the hydrodynamic quantities evaluatéd and reported

in this study and the coeffiçients in Eqs. [2].

Those .hydrodynami.c quant.i.ties evaluated in the attached figures are given as functions of the ratio of beam to twice the draft B/2T and the wave frequency parameter 'VB/2 = w2B/2g = (or UB./X).

The frequency response functions for the barge motions are then evaluated from Eqs.[2]. Letting = _uP + i wN , Q = -W2M + iWN ,

=(B- w2ti)

iWN,

CC

w2M)+

[wN ,

[]

tAen Eqs; [2] become a set. of simultaneous linear algebraic equations of the form

:P'T1/va

'+

cp/va=1/va

, .

,a

+

i/va '=

/va

-C/a = /a

I-918 _{HYDRODYNAMIC CHARACTERI,STICS OF PRISMATIC BARGES OTC 1417}

=

L1+J

The response ratios ii/a, ¿/va, /a are complex valued functiOns of wave frequency, w, which are called the frequency response

functions. The magnitude of the frequency response function gives the motion response amplitude due to a unit amplitude sinusoidal input wave at frequency w and the argument of the frequency response function described the phase. In this way, the frequency re-sponse functions can be evaluated from the information ¡n this report for the sway, r011 and heave motions of a prismatic barge ¡n beam seas. Furthermore, simply by inter-changing beam and length, the same information can be obtained for head seas.

DISCUSSION OF RESULTS

The theoretical approach described in the previous section of this report was ap-plied to the prediction of the motions of two rectangular barges and the results were com-pared with corresponding meásurements from scale model tests. The measured and pre-dicted motions in head seas are compared in Figure 1 for the case of a prismatic barge with length-beam ratio LIB = 1.5' where, it

is seen that 'good agreement is obtained. Another example for the cäse of a similar

barge in head seas but with length-beam ratio 2.0 demonstrated good agreement be-tween measured and predicted motion (not

shown in figures). Some questions arise as

to the upper limit of length-beam ratio for' which the present approach is valid in head seas since for conventional ship hulls

(length-beam ratio 6 Or 7), a different representation is used; viz., two-dimensional strips running ¡n the thwartship direction as opposed to fore, and aft as used here for head seas. Thwartship segments were attempted for the barge (L/B = 1.5) in head seas but' were discarded because fore and aft strips were found more accurate in comparison to model test results. The critical value of

length-beam ratio at which the conventional hull representation becomes more accurate

could not be determined for-lack of data ¡n the intermediate range, 2< L/B < 6. However, as state above, the thwartship segment has proven more accurate, at least up to L/B2.0,

seas, the approach described here is identical to that used for. conventional ship hulls and therefore is considered valid for all values of length-beam ratio.

Another limitation of the theoretical approach presented here is due to the use of

linearized theory for the representation of the hydrodynamic forces. However, good agree-ment between theory and model tests has been obtained for the barge with L/B = 1.5 for wave conditions up t'hrough sea state 4. (equiv-alent full-scale sea state assuming barge

length of 250 feet-). Again, further tests are required to defi'ne the u,pper limit of validity for linearized 'theory but the authors feel that sea state 5 is within the range of valid application of these results.

Consequently,. the results presented here can be used for easily predicting three de-gree of freedom motions of prismatic barges in head 'seas or beam seas for length-beam ratios at least up to 2.5 and, for sea states

at least up to 5. .

-NOMENCLATURE

A = waterplane area w

a = wave amplitude

-B = beam or restoring coefficient F = force or moment

G = center of gravity or source potential g = gravitational constant

I = moment of ¡nertia of the hull about an axis through the center of gravity L = length of hull

2 = hydrodynamic moment arm

M = moment or inertial coefficient

-M' three-dimensional added mass

m"= two-dimensional added mass

N = two-dimensional or three-dimensional damping coefficient

O = origin of the coordinate system

T= draft

t time

W = weight

x,y,z body coordinates.

H = half beam-to-draft ratio, or indicating heave in two-dimensional hydrodynamics h = wave elevation, or suffix designating

wave

i = ¡-1, or suffix indicating the imaginary

part or damping part

R = suffix indicating roll in two-dimensional hydrodynamics

amplitude then gives

/\'a - /va

.)

and the authors are thereby encouraged to sug-gest the use of this approach up to LIB = 2.5 with reliable' results. Calculated motions at this value still appear 'reasonable in compari-son with values ant ic'ipated by model, test engineers. Further experimental results ¡n

-the intermediate range of length-beam ratio are required to clarify this point. In beam

a

flTC 14 7 C. H. KIM. C. J. HENRY and F. CHOUV APPENDIX

r = suffix indicating the real, part or

inertial part

S = suffix indicating swaying motion in

two-dimensional

hydrodynamics-sway, or suffix designätirig swäying mo-tion orswaying-exc.iting force

= radius of gyration

X = wave length

V volume displacement

= wave number

w = circular frequency.

cp = velocity potential or roll .ör suffix designating rolling motion or roll-èxcfting môment

= pitch, or suffix designating pitching motion or pitch-exciting moment

p = water density

heave Òì suffix designating heaving motiön or heave-exciting force

ACKNOWLEDGEMENTS

The wrk reported here tias sponsored by the Sea Grant Program Office of the National Oceanic and Atmospheric Agency (previàusly of the National Science Foundation) and by Stevens Institute of Technology. The compu-tations were carriéd out at the Stévens Computer Center which i partially supported by the National Science Foundation.

REFERENCESV . .

1. GRIM, O.: "Eine Methode fUr eine genauere

Berechnung der Tauch- and Stampfbewegun-gen in glattem Wasser und in Wellen," HSVA-Bericht Nr. 1217, June 1960.

2. TAMURA, K.: "The Calculation of Hydro-dynamical Forces and Moments Acting on the Two-Dimensional Body - According to Grim's Theory," Journ. SZK., No. 26, Sept.. 1963.

KIM, C.H. and CHOU, F.: "Prediction of Motions of Ocean-Platforms in Oblique Seas," Stevens Inst. of Tech., Ocean Engineering Dept. Repôrt 0E 70-1, May

.1970.

FRANK, W.: V "On the Oscillation of Cy.

linders

in or Below the Free-Surface

of Deep Fluids," Naval Ship Research &

Developmept Center Report 2375, Oct.. 1967.

ST. bENNIS, M. and PI.ERSON,W.J., Jr.: "On theMötions of Ships in Confused Seas," Transactions.., Society of Naval Architects & Marine Engineers, Vol. 61,

1953. . . V

Definitions of Coefficients in Eqs. [2]

M " 1]l M " CC M°" cpP M°" M°" M 4xp M 1n

M,

B B FC where = m L = m . L Ô', = mR = M°"/ .CPP _Rr = M " . £ lTh Sr =

M+ 2

.

.

M L

= Mj

+ M

:M;+.

M N .

N. =VÑH

L N°

cpR

N° L N°

N°/L

cpp R. N° N N

N° +.M

,1l Cpu "

-N°

+

N -, uP unì

=pV+M"

M = pV + M

=1

+M."

Cf Cp =

Mj

= _Ls

N4- 2

. pgA = pgV = L

_FHV..

=

+0G

F11 'Sr i-919

._Note: .R denotes Real Part; I denotes imaginary Part

TABLE 2 - SOLUTIONS OF UNCOUPLED MOTIONS OF JACKUP MODEL

L B

= length - .

= beam

i heave-exciting force per unit length

= roll exciting moment per unit length

T

=draft

. . .

.- ., .abQut.the axis. through the origin O

V A W = displacement volume = waterplane area .

Ls = sway inertial moment arm about the

r

_originO.

. .

£ = sway damping moment arm about the

s

P = water density . origin o

. -

-= mass moment of iñertia about the axis _{L = roll inertial moméntarm abòut the}

through the center of gravity R

originÖ

-= rretacentric height -, - .,

- .

-= ron damping moment arm. about the

' origin Q. .

m = sway added mass per unit length

= heave added mass per unit length

TABLE i - PARTICULARS OF JACKUP RIG MODEL - m" = roll added massfloment of inertia per

unit length about the axis through Len th L

ft - -

2558

g

the origin O . . Beam, B

, ft

.:

1.70k

N5 = sway damping force coefficient per Depth;.D ft .. . 0.21+8

unit length _{Draft, T}

, ft 0 16

NH = heave damping force coefficient per Weight, W , lb

43 52

unit length .,, .

-.

-V.C.G. above watrliñe,OG,ft 0.41+03

NR = roll damping moment per unit length . . -

,-about the axis through the origin O.: Pitch Gyradius, , ft

li877

= sway-exciting force per unit length Pitch Metacentric Height, ft

2 887

X/L 10.471 R I 0.8 3.927 R I 1.2 2.618 R I 1.3 2.417 R I. l.7l+5 :1 2.3

-

1.366 R

I..

4.8 0.6545 R I -12.87 .301 _-34.30 3.200 -50.01 6.416 -5.67 7.272 -71.71 11.026 -88.97 14.29 -171.29 24.20 92 63 1 42+1 41 87 11 01 8 41 17 31+6 - 224 18 255 -1+2 52 20 71 -85 z6 19 85 -318 05 7 825 179.64 85.14 84.06 140.73 10.65 163.35 -10.71 164.53 -111.87 169.10 -220.24 164.21 .-&+3.5 100.60 101+ 11 ¿+37 -1 813 22 790 -7 143 26 536 -8 975 25 ¿485 -18 15 2! 72 -25 ¿14 12 856 6 53 -2488

i/va

-1.014 178.29 -111.447 84.31+ 5.624 83.2 -70.81 132.20 -:

''

15;00 .45.78 136.0 14.74 6.12 41.86 132.10 18.63 .49.54 ._2260 102i9' 17.79 80.29 -8.995 59.15 -1.121+ 121Ô4 4.824, -53.40 .0128 - .888 .1139 -.653 .207 -.5040 .2274 -.41+4 .2928 -.2580. .3014 -.u961 -.0574 .1371 i/va -.0297 -1.2027 -.2903 _1.62+8 1.8327 -1.6634 -2.3024 .7793 -.5634 .2571 .2212 :051+0 ;0O39 -.0151 /a .9921 -.0007 .9526 -.022 .8350 -:0371+ .7971 ,.089! .558! -.0762 .3629 :0020 -.0215 .0617 i1i/a .8884 .6636 .5145 - .4989 .3901 .316j .1487 -89.2 .8012 -67.62 -62.88 -41.37 -17.68. 112.7

lI/va

l.2Ò3 1.641 2.475 -.. .2.43 .62 .228 .0156 -91.41 .100.2 -137.8 -161.3 155.5 166.3 -75.1+7

lt/Va

.9921 .9529 .8358 .8021 ;5633 .3629 .0644 -.0421 _I.331+ -2.564 -6.77 -7.779 .3164 109.5 1L92Ö

s

y 'a

"3f

WAVE

PS

Fig. la - Coordinate system.

CALCULATED lOo

si

THEORY TEST

-o 60 w 4

--X 80 -' 2.0 1.5 1.0 0.5 CALCULATED 'J 5 _X/L 240 200 I60 D D a I2O 2.5 80 120 4 w w 4 -j 80 4.01 351-i

T-THEORY TEST O va

/0

0

D

3.01-k,,

D 'I s-CALCULATED si

Fig. lc - Heaving motion of' jackup rig model in head

seas.

Fig. ld - Pitching motion of jackup rig model in

head seas.

Fig. lb - Surging motion of jackup rig model in

head seas.

BOW

D

Fig. 2a - Sway added mass coefficient.

I I . - I

-I ---I---

I

I.e 2.3 2.8

- 1TB/X

-3.3 3.8 4:3 4.8

Fig. 2c - Roll added mass coefficient.

PV X

Fig. 2b - Heáve added mass.

I I -I

0.3 0.8 113. I8 - 2:3 2.8

- lrB/

X

4.3 4.8

Fig.

3a-

Sway damping coefficièrit.

'ra

VS

pgvB À

0.3 0.8 .3 1.8 2.3 2.8 3.3 38 4.3 4.8

Fig.

3e -

_{Roll damping coefficient.}

0,3 0.8 .3 .8 23 2.8 .3. 3.8 4.3 4.8 -..7TB/X

6 4 2 -2 -4 -6 '8 -12 -14 -I6

Fig. #a Sway inertial mpment arm.

o -6 -8 -Io -12 -14 -'6 -18 -20 -22 Hr2 -_ H.3 24 -Fig. 0.3 0.8 1.3 1.8 2.3 2.8 33 3.8 4.3 4.8 irB vs T X

4

0.3 1.6

F,,

2!:.!

pQvaLBT

Fig. 5a - Sway-exciting force.

1.2 1.0 Q8 0.6 0.4 0.2 o

2

H 6 H. 7 H8 Hr9 Hr IO H2 4.3

14

F,1 vs2T-H4 PgIJOLBT À H 5 I I 0.3 0.8 1.3 1.8 2.3 2.8

-

ira/A

Fig. 5b - Sway-exciting force.

Fig. 5c - Heave-exciting force.

0.6 --a 0.2 -o

-2

-0.4 -0.8 -1.0 -1.2 -1.4 FC I _vs7TB pgOBL X

Fig. 5d. - Heave-exciting force.

Fig. f - Roll-èicciti±ig moment.