Hydrodynamic characteristics for the design of rectangular barges in restricted water depth

HYDRODYNAMIC CHARACTERISTICS FOR THE

DESIGN OF RECTANGULAR BARGES IN

RESTRICTED WATER DEPTH

K KOKKINOWRACHCS and L BARDIS

Technical University Aachen, Fed. Republic of Germany

ABS T RACT

This paper deals with the results of a systematic hydromechanic analysis of rectangular barges in restricted water depths. The numerical methods used are briefly reviewed and evaluated with respect to their accuracy and efficiency for design purposes.

In the examples presented in the paper the relevant parameters, i.e. the ratios length-to-beam, beam-to-draft and water depth-to-draft, have been

variied.

Exciting forces, hydrodynamic coefficients, shear force, bending and torsional moments are presented.

The effect of the restricted water depth on the hydrodynamic characteristics of barges is demonstrated. Furthermore, some conclusions are drawn con-cerning the applicability of the strip theory and the necessity for a three-dimensional analysis. Some results of calculations of drift forces on barges in restricted water depth are also presented.

NOTAT ION

x,y,z cartesian coordinates

sea surface elevation

d water depth

H wave height

A wave length

T wave period

k wave number

circular wave frequency velocity potential

p water density

1 half length of a barge

a half beam of a barge

h draft of a barge

s wetted surface of a body

V displacement

m nass

I moment of inertia

a. heading angle

sj motion component in j-th direction

(=1,2,3

for translations,

j=4,5,6 for rotations)

sjo amplitude of the motion component

Fk force in the k-th direction

Fk0 amplitude of the force Fk

FkR, Fki real and imaginary part of the force Fk

fk dimensionless force TECHNISCHE tJPd

Laboratorium voor

Scheepshy&omJpJ

zchIef Mekeiweg

_2,2628

_{CD DeHt}

TeL: 015- 788873- Fax 015-781838

Second International Symposium on Ocean Engineering and Ship Handling 1983,

Swedish Maritime Research Centre SSPA,

P.O. Box 24001, S-40022 Gothenburg

ak bk1 akj. bkj Ck j Fk2) f ko MB Q MT INTRODUCTION

Barges find a wide applicability in offshore activities. These floating units are used for the transportation of large components of offshore structures, for the storage of oil and of liquified gas and often as a working area in the open sea, especially during offshore installations. In

the last years, the suitability of barges as floating carriers for chemical process plants, for example, for desalination, LPG, LNG or methanol as well as for electric or nuclear power stations has been evaluated.

For almost all the applications a reliable prediction of the seakeeping characteristics of the barge is necessary. On the one hand the designer needs sufficient information about wave loads and wave-induced motions, on the other hand in many cases, especially for floating plants, the motions of the barge have to fulfill the requirements dictated by the chemical process. Thus, the behaviour of the barge in waves can decidedly influence the technical and economical efficiency of the entire system.

For the hydrodynamic analysis of these rather simple floating bodies several methods can be applied which are widely approved in the naval and offshore

field.

However, the barges used offshore show some particularities which can strongly affect their behaviour in the sea. Generally, these barges can be considered as rectangular boxes, but with a relatively low length-to-beam ratio, large beam and large draft. Barges operate often in regions with restricted water depth. The water depth-to-draft ratio can become small.

The effect of a small water depth-to-draft ratio was the subject of several investigations, mainly for ships in shallow waters, in the last fifteen years ([1] - [10)

In most cases a strip theory has been applied, the hydrodynamic coefficients and the exciting forces have been determined for the two-dimensional cross-section. A rather limited number of experimental data for restricted water depth is available.

The two-dimensional investigations have clearly shown that the influence of the shallow water on the motion of a floating body can be significant. Some remarkable conclusions concerning the hydrodynamic quantities could be drawn by these studies.

As the barges used offshore are often short and wide, the strip theory associated with slender bodies is not always applicable and can lead to greater inaccuracies. Hence, a three-dimensional theory for the hydrodynamic analysis is required. Particularly in the case of a body of large draft in shallow water the small underkeel clearance advocates for a

three-dimensional theory, since the water flows partly underneath the body and partly around the ends. However, the three-dimensional analysis of a barge using a computer program based on a sink-source technique or a finite-element approach is an extensive undertaking, not always possible in the project phase or in the earlier state of the design of such a system. With the aim to make the results of complicated numerical methods easily accessible and to provide practising engineers with useful information for

the hydrodynamic design of barges a research program has been carried out at the Division of Ocean Engineering (LMT) of the Technical University Aachen in closed co-operation with the industry. In extensive systematic

added mass and damping coefficient

dimensionless added mass and damping coefficient restoring coefficient

mean drift force in k-th direction dimensionless drift force

bending moment shear force torsional moment

calculations the relevant parameters have been variied widely in order to cover the most important areas of offshore applications.

Two- and three-dimensional computations have been performed. For the two-dimensional analysis a new approach based on the idealization of the flow around the cross-section using macroelements has been applied. For the three-dimensional calculations a sink-source technique has been used. Amongst the large amount of data obtained in the above mentioned research program, only a restricted number of characteristic results can be pre-sented in this paper, after a short overview of the theories applied. REVIEW OF THE THEORETICAL METHODS

The barge is considered as a rigid body in regular waves. The fluid motion is assumed to be irrotational and incompressible. A right-handed co-ordinate system (x,y,z) fixed with respect to the mean position of the body is

used, with positive z vertically upwards (Fig.1).

It is assumed that the amplitude of the wave as well as the amplitude of the response are small in comparison with the wave length, i.e. the linearized boundary-value problem is considered.

The formulation of the dynamic equilibrium between inertia, excitation, hydrodynamic reaction and restoring forces on the body leads to the six coupled linear equations of motion:

J1

= Fk(t)

where Sj is the motion component in j-direction, mkj is the component of the generalized mass matrix, akj and bkj are the added masses and damping coefficients, respectively, ckj the restoring coefficients and Fk(t) the exciting forces or moments.

The first-order velocity potential of the flow caused by the motion of a rigid body in a harmonic wave can be expressed in the form:

0

= (2)

where is the velocity potential of the incident wave, the

dif-fraction potential for the body restrained in the wave, ₄ (j1,...,6) the radiation potential resulting from the unit motion of the body in the jth mode in otherwise still water and jo the complex velocity amplitude of the body motion in the jth mode.

The exciting forces or moments are obtained from the solution of the dif-fraction problem which is unequivocally described by the velocity potential

= o7

(3)

The hydrodynamic coefficients, i.e. the added masses and the damping coefficients, are determined by calculation of the hydrodynamic reaction

forces of the radiation problem.

The results of the two-dimensional calculations presented in this paper are obtained by means of a new approach which can be called a

macroelement-technique.

This method has been developed for an arbitrary cross-section, the basic approach for the simple case of a rectangular cross-section is demonstrated in Fig. 2. The fluid around the section is subdivided in three

macro-elements with the sides of the rectangle, the sea surface and the bottom

as boundaries (Fig.2). The elements I and III extend in horizontal direction to infinity, whereas the element II is bounded in both directions.

Starting with the method of separation of variables for the integration of the Laplace equation for the flow field, appropriate expressions for the velocity potentials for the diffraction and radiation in each element can be established. These expressions in the form of Fourier series are

se-lected in such a way, that the kinematic boundary condition on the top of element II, i.e. on the bottom of the body, the linearized dynamic condition at the sea surface, the kinematic one on the sea bed and the radiation

condition at infinity are fulfilled. The kinematic boundary condition at the vertical wall as well as the requirement for continuity of the potential and its first derivate at the boundary of the neighbouring elements remain to be satisfied. Using for that the Galerkin method the unknown Fourier coefficients are obtained.

The macroelement-method has been presented in detail in previous publi-cations ( [10]- [14] ) . The general case of a cross-section of arbitrary shape

has been treated on the basis of the approximation of the contour by a step curve and by definition of macroelements bounded through the steps. Here, by wayof example, only the expressions for the velocity potential

47 of the diffraction problem are given.

Consider an incident wave with the profile

(x,t) =

and with the velocity potential

gH coshk(z+d)

_eWt

0(x,zt)

_{- 'J cosh(kd)}

The velocity potential ₄ can be expressed in the form:

7(x.z,t)

= -iw-W7(x.z)e

Element II

-iwt

For the three macroelements the functions are defined as follows: Element I -W(x,z) ₌

X-Q

_F*X+Q

+

-- '4J(xz) = F7

2a

+

o 2a

+ 2>1 [F7

sinhs(xa)

_{* sinhs(x+a)]}

+ F

_icoss(d-z)

nrl L n

sinh 2os

7n

sirih 2as

J Element III

i

=

>F e°

d

Zz)

where -1/2 Z

= N

cosa(z+d)

N -

[i+ sin(2.d)]

-

2ad

with a the roots of the equation

atan(a.d)+-- = O

the imaginary root o. = -ik being considered as the first, and

nit

- d-h

For a = -1k the dispersion equation

w2 =

gktanh(kd)

follows from eq. (12).

The advantage of the expressions (7) , (8) and (9) is that they lead to simple unharmonic Fourier series at the vertical boundaries of the macro-elements (x = a and x = -a)

The exciting forces, added masses and damping coefficients for the cross-section being known from the solution of the diffraction and the radiation problem, the corresponding quantities for the floating body are obtained by integration along its length.

In the context of the strip theory two coupled equations for pitch and heave and three coupled equations for the sway, roll and yaw modes of

motion are established. In the present investigation a conventional concept following Grim's formulation of the strip theory E15] has been used.

The three-dimensional calculations dìscussed in this paper are based on a sink-source technique.

The time-independent parts of the velocity potentials can be represented by a continuous distribution of single sources on the wetted surface of the body in the integral form:

'Px.y,z)

--

ffq

_{G(x.y,z,g,) dS}

₍₁₅₎

(j

= 1,...,6,7)

where the unknown source (or sink) density at the position (,11,) on the wetted surface S of the body and

_{G(x,y,z,,q»)}

_{the Green's}

function, singular in x,y,z.

A Green's function can be chosen satisfying the Laplace equation, the boundary conditions in the free surface, on the sea bottom and at infinity,

so that only the kinematic condition on the wetted surface of the body remains to be satisfied. This Green's function can be expressed in two ways using an integral or a series formulation, each of them being advantageous

for the numerical computations, depending on the values of the variables ( [i o] ,

[i

6])

-After satisfying the boundary condition on the wetted surface of the body using the formulation of eq.(15) the following two-dimensional Fredholm

integral equation of the second kind over the surface S can be obtained:

-q(x,y.z)

i

ac (x,y,z,)

dS =

an

n

when j= i...6

(nj: generalized direction cosine)

To solve eq. (16) numerically, the wetted surface S is subdivided into a

finite number of plane elements, over each of which the source density is assumed constant. This transforms the integral equation into a set of linear algebraic equations with the values of the source density on the elements as unkncwns.

This type of sirun-source technique has been widely applied in the last ten years for the hyrodynamic analysis of large offshore structures of

arbitrary shape. he details of this numerical method are widely documented

in the technical literature ([9] , [i 0] , [i 7] , [i 8])

NUMERICAL RESULTE

As mentioned in che introduction, systematic calculations of rectangular barges have been carried out at the Ocean Engineering Division of the Technical Univerrity Aachen [19]

The aim of these investigations was to extend the very usefull data for barges published 1971 by Kim et.al., [20], in order to cover broader areas of offshore appLications.

Attention was pa_-d to two additional aspects, namely the effects of the restricted water depth and of the three-dimensional flow around short and wide barges. Conerning the evaluation of results obtained from the strip theory by means three-dimensional computations only limited information has been published, for example in [18] and [21] for infinite water depth.

Two computer prc.'rams developed at the Technical University Aachen have been used for the caLrulations.

The code DIFRAC- is based on the macroelement approach for the two-dimen-sional case. Extensive tests have shown that the macroelement technique is very effective wth regard to the computer time and the storage needed and

also free from r.xaerical irregularities ( [1 2],[1 4]

The three-dimensonal calculations have been carried out with the computer program SING-A founded on a sink-source technique. Results of tests of this program in comparison to other theoretically and experimentally obtained

data have been pblished in [i 2]

In Figures 3 and 4 the effect of restricted water depth on the hydrodynamic coefficients of rectangular cross-section is demonstrated. The strong dependance of thse quantities as well as of the exciting forces on the water depth has E considerable effect on the motions of the body.

Special attentic'n was paid to the added mass and damping coefficient for heave in the rare of small frequencies, i.e. for long wave lengths.

Whereas in infinte water depth the added mass for heave tends to infinity as the frequency tends to zero, there is a finite limit of the added mass

in the case of rEstricted water depth ([6], [22] - [26] ). The existence of

this limit can b-e shown analytically.

In infinite water depth the damping coefficient varies ultimately as the frequency tends o zero, in restricted water depth this quantity varies ultimately as tle second power of the frequency ( w2 ) . These results which

are usefull in cecking numerical methods in long wave length have been confirmed using he macroelement approach.

In Fig.4 results obtained with the computer program DIFRC-K are compared with data from o-her numerical methods ( [26J - [28] ) . In Figures 5 and 6 the

results calcula.ed with the code SING-A can demonstrate the effect of the

ön

when j

three-dimensionality of the flow around the barge. A barge of 120x40x5 m has been analysed using both the strip theory and the three-dimensional

method, [29]

En Fig.5 the transfer function of pitch in head sea is plotted and compared with experimental data.A closer agreement with the experimental data is

achieved with the three-dimensional approach.

A greater discrepancy has been found in the midship bending moment in head sea the amplitude of which is shown in Fig.6. In the range of the maximum the strip theory leads to higher values, only the three-dimensional analysis provides results in good agreement with the experiment.

In Fig.7 the effect of the restricted water depth on the motion of a barge of 60x40x5,4 m is demonstrated. For this body (1/a = 1,5), only the three-dimensional analysis is reliable. By way of example, the drift forces and moments on this barge have also been calculated using the direct integration

method, ( [30] , [31] ) . It is evident from the results presented in Fig.8

that, mainly due to the water depth effect on the motion, considerable changes in the drift forces can result in shallow water. This fact has to be taken into account when mooring systems are designed for barges in restricted water depth.

In the research work, the main results of which are reported in [19] , the

complete hydrodynamic analysis of a great number of barges in restricted water depth has been carried out. In Table 1 the values of the parameters

1/a, h/a and d/a of the barges investigated in [19] are given. Table 1: Parameter of the barges investigated in [19]

In this paper only a limited number of selected results can be presented. Hereby, the length-to-beam ratios 1/a = 1,2 and 3 are chosen, the half beam-to-draft ratio a/h = 4 has been kept the same in all cases. For the

depth-to-draft ratio the values d/h = 1,5, 2, 3, 4, 6, 8 and 12 are considered. 1/u

0.375 0.5

0.75

1.0 1.5

2.0

3.0

5.0

1.0

0.25

05

S

0.75

5

.

S S 1.0

.

.

0.25

2.00.5

0.75

.

S S S

0.25

3Q0.5

S S

i

5

0.75

1.0 5 5 5 0.125

0.25

5

0.5

0.75

s s

Results of the barges with 1/a 1 are given in the charts of Fig.9. Here

the transfer functions of the exciting forces for the incident-wave

di-rection Q = 00 and the hydrodynarnic coefficients are given.

The charts of Fig.10 present results for barges with 1/a = 2 for the

incident-wave directions a = 0°, 45° and 90°. The real and imaginary parts of the exciting forces are separately plotted.

The added mass in heave increases considerably with decreasing water depth--to draft ratio. The behaviour of this added mass obtained from three-dimensional computations seems to be different from that of the

correspon-ding cross-section in the region of low frequencies. Numerically no finite limit for the added mass in heave could be detected from the three-dimensional calculations. This difference appears to arise from the

different mathematical nature of the radiation problem in the three- and two-dimensional case. Also the heave damping coefficient increases with decreasing water depth.

In Fig.11 some results for barges with 1/a = 3 are presented, by way of

example.

In Fig.12 the midship bending moment for a = 0° and 45° is plotted for a barge with 1/a = 3, h/a=0,25 and four values of the water depth-to-draft ratio. Similarly for the example of Fig.6, the three-dimensional theory leads to a smaller maximum bending moment than the strip theory. The same remark applies to the vertical shear force (Fig.13)

The amplitude of the torsional moment for a = 45° is plotted in Fig.14. The results from the three-dimensional calculation are slightly greater than those from the strip theory.

Finally, the drift forces in x-direction for the headings a = 0° and 45° are presented in Figures 15 and 16 for two barges and four water depths. The drift forces have been calculated according to the momentum approach,

[18] . As it has been demonstrated by the example of Fig.8, the effect of

the restricted water depth on the drift forces can be considerable.

CONCLUSIONS

The numerical methods, results of which have been presented in this paper, can be considered as valuable design tools for the hydrodynamic analysis of barges used in offshore activities.

Both methods, i.e. the macroelement-technique for the two-dimensional and the sink-source technique for the three-dimensional analysis, have been sufficiently verified by means of experiments.

The results demonstrate the large influence of the restricted water depth on the motion characteristics of floating structures. These effects have to be taken into account for the design of the barge itself as well as of the mooring system.

The comparison between the two-dimensional strip theory and the consider-ably more complicate three-dimensional method shows that for barges with relatively small length-to-beam, beam-to-draft and depth-to-draft ratios the two-dimensional analysis can lead to somewhat less reliable results. ACKNOWLEDGEMENT

Essential parts of this paper have been compiled in connection with a research project sponsored by the Federal Ministry for Research and Tech-nology, Bonn. This support is greatly appreciated.

REFERENCES

Sluija, M.F. van and Tan Seng Oie: The Effect of Water Depth on Ship Motions. Appendix 7 of Seakeeping Committee Report. 14th Inter-national Towing Tank Conference, Ottawa, 1975

Tuck, E.O. and Taylor, R.J.: Shallow Water Problems in Ship Hydro-dynamics. 8th Symposium on Naval Hydrodynamics, Pasadena, 1970 Tuck, E.O.: Ship Motions in Shallow Water, Journal of Ship Research, Vol.14, 1970

Kim, C.H. Hydrodynamic Forces and Moments for Swaying, Heaving and Rolling Cylinders on Water of Finite Depth, Journal of Ship Research, Vol.13, 1969

Yu, Y.S. and Ursell, F.: Surface Waves Generated by an Oscillating Circular Cylinder on Water of Finite Depth, Journal of Fluid Mechanics, Vol.11, 1961

Keil, H.: Die hydrodynamischen Kräfte bei der periodischen Bewegung zweidimensionaler Körper an der Oberfläche flacher Gewässer. Report 305, Institut für Schiffbau der Universität Hamburg, 1974

Tasai, F. et.al.: Ship Motions in Restricted Waters, Part _{I - Tank} Tests, Res. Inst. of Appi. Mech., Kyushu University, Vol.XXVI, No.81,

1978

Keuning, J. A. and Beukelman, L.: Hydrodynamic Coefficients of Rectangular Barges in Shallow Water, Conference on the Behaviour of Offshore Structures, BOSS '79, Imperial College, London, 1979 Oortmerssen, G. van: The Motions of a Moored Ship in Waves. NSMB Publication No.510, 1976

[io] Kokkinowrachos, K.: Hydrodynamik der Seebauwerke, Handbuch der Werften, Vol.15, Hamburg, 1980

[ii] _{Kokkinowrachos, K.: Einige Anwendungen der Potentialtheorie bei der}

hydrodynamischen Analyse meerestechnischer Konstruktionen, _Interocean '76, Düsseldorf, 1976

Kokkinowrachos, K., Asorakos, S. and Mavrakos, S.: Belastungen und Bewegungen großvolumiger Seebauwerke durch Wellen, Research _Report of Rhine-Westphalia No. 2905, Westdeutscher Verlag, Opladen, ₁₉₈₀

Kokkinowrachos, K.: Hydrodynarnic Analysis of Large Offshore _Structures, International Ocean Development Conference, Tokyo, 1978

Asorakos, S.: Ein potentialtheoretisches Verfahren zur Erfassung der Wechselwirkung zwischen Elementarwelle, starrem Körper _und

elasti-schem porösem Boden, Dr.-Ing. Dissertation, Technical _University Aachen, 1981

Grim, O.: Bewegungen und Belastungen dea Schiffes im Seegang, Lecture Notes No.3, Institut für Schiffbau, Universität Hamburg, 1973

Wel-jausen, J.W. and Laitone, E.V.: Surface _{Waves. Handbuch der Physik,}

Vol.IX, Springer Verlag, 1960

Garrison, C.J.: Hydrodynamics of Large Objects in the Sea, Journal of Hydronautics, Vol.8, 1974

Faltinsen, O.M. and Michelsen, F.: Motions of Large Structures in Waves at Zero Froude Number. Proc. Interntl. Symposium _{on the Dynamics}

of Marine Vehicles and Structures in Waves, University College, London, 1974

Kokkinowrachos, K. and Bardis, L.: Hydrodynamic Characteristics of Rectangular Barges in Shallow Waters. LMT Report SR _{80-02, Ocean} Engineering Division, Techn. Univ. Aachen, 1980

Kim, C.H., Henry, C.J. and Chou, F.: Hydrodynamic Characteristics of Prismatic Barges. Offshore Technology Conference, _{OTC 1417, Houston,}

1971

Migliore, H.J. and Palo, P.: Analysis of Barge Motion Using Strip and Three Dimensional Theories, Offshore Technology Conference, OTC 3558, Houston, 1979

Ursell, F.: On the Virtual-Mass and Damping Coefficients in Water of Finite Depth, Journal of Fluid Mechanics, _{Vol.76, 1976}

Sayer, P. and Ursell, F.: On the Virtual _{Mass, at Long Wave-lengths,} of a Half-Immersed Circular Cylinder Heaving on Water of Finite Depth. XI Symp. Naval Hydrodynamics, London, 1976

Kim, C.H.: Effect of Mesh Size on the Accuracy of Finite-Water Added Mass. Hydronautics, Vol.9, 1975

125] _{Keil, H.: Hydrodynamische Masse und Dämpfungskonstante tauchender}

Zylinder auf flachem Wasser, Schiffstechnik, _{Vol.23, 1976}

Bai, K.J. and Yeung, R.W.: Numerical _{Solutions to Free-Surface Flow} Problems, X Symp. Naval Hydrodynamics, M.I.T., Cambridge, 1974

Bai, K.J.: A Variational Method in Potential _{Flows with a Free Surface.} Univ. of Calif. Berkeley. College of _{Engineering, Report NA 72-2, 1972} Lebreton, J.C. and Margnac, M.A.: _{Traitment sur Ordinateur des Quelques} Problèmes Concernant l'Action de la Houle sur les Corps Flottants _en Theorie Bidimensionelle. Bulletin du Centre de Recherches et dEssaies de Chatou, No.18, 1966

Kokkinowrachos, K.: Charakteristika des Seeverhaltens einiger schwimmender Offshore-Konstruktionen, Yearbook of the Schiffbau-technische Gesellschaft, Vol.76, 1982

Pinkster, J.A. and Oortmerssen, G. van: Computation of the First and Second Order Wave Forces on Bodies Oscillating in Regular Waves, 2nd Intern. Confer, on Numerical Ship Hydrodynamics, Univ. of Cal., Berkeley, 1977

Kokkinowrachos, K., Bardis, L. and Mavrakos, S.: Drift Forces on One- and Two-Body Structures in Waves, Conference on the Behaviour of Offshore Structures, BOSS '82, M.I.T., Cambridge, 1982

22 pV a33 pV 5.0 4.0 3.0 2.0 1.0 5.0 4.0 3.0 2.0 1.0 o O

t

X

V

L

/// ///////////////////////////////////7/////////////////////

Fig.2: Macroelernent-Idealization of the flow around a barge

I

0.5 1.0 15 w Erad sec1l

z

b22 puy 5.0 40 3.0 2.0 1.0 5.0 b33 puy 4.0 30 20 1.0 0.5 1.0 15 w[rad sec1]

E'ig.3a: Added masses and damping coefficients of a rectangular cross-section for different water depths

107

0.5 1.0 15 05 1,0 15

a» pVL.a2 0.10 0.08 006 O 01. 002 -0.5 21. pV2 a -0.I. o 0.5 1.0 15 w Erad sec1l

2.0-©

=6,67 pV a

© °..

30 1,5 _a

o-1,0 2 0

o Bai & Yeung

+ Bai

0.5-Lebreton & Margnac 10 DIFRAC-K 0.5 1.0 1,5

ka

-05 b1.,, pwV4a2 -0.1. 0.05 b2,, pwV2a 0.01. 0.03 0.02 0.01 0.5 1.0 15 w E rad sec1]

Fig.4: Heave added mass and damping coefficient for a

rectangular cross-section

î

05 1.0 15 0.5 1.0 1.5

w Erad sec] w Erad sec

Fig.3b: Added masses and damping coefficients of a rectangular cross-section for different water depths

05 1,0

ka

L,0

b33

s50 1.5

H/2

deg.

1.0 0.5

o

1.5 M90 pg a3 1.0 0.5 Length = 120 n Beam = 40 n Draft =

5m

KG =

6.20m

d =

_121.5m

Radii of gyration = 15.2 m iy = 36.6 m = 39.1 n Experiment 109 Q H =

3m

E H =

6m

D H =

9m

Fig.5: Pitch transfer function _{Fig.6: Transfer function of the}

for a barge vertical bending moment

at midship

0.5 1.0 1.5 0.5 1.0 1.5

12 SR o

H/2

deg.

Lm

6 o O Si

60

_d

't f

0.6

1.2

w[rad sec1]

S30

H/2

1.2

0.6

O

12

H/2

deg.

Lm

6 O

d 200m

o

Radii of gyration

Fig.7: Motion transfer functions of a barge for different water depths

2a=L0m

0.6

1.2

w[rad secJ

= 14.40 In iy 17.64 n 21.34 in KG =

5.40m

3.0

SI o

H/2

1.5 o O OE6 1.2

w [rad sec1]

o

0.6

1.2

w[rad secJ

d=200m

0.6

1.2

w [rad sec1 I

(2) (2) Fk0

k0 =

₁ 2

,kzl,2

-pga (H/2)

P1.0 (2) fR

t

T d 0.5 J,

o

O = L,5

d=200m

d=30m

d=lOm

=

900

d=lOm

d=30m

d=200m

0.6

1.2

w[rad sec1]

(2) (2) F60 f

= ipga2(H/2)2

2

Fig.8: Drift forces on an barge for different water depths

0.6

1.2

w Erad sec1J

111

-0.5

F4

-

1.0

2a40m

F'

10 (7)

f.-

0

0 (2) O

d=lOm

d=30m

o

300 225 1 50 075 Fk

k=1,2,3

pg2aLH/2

Fk

fkR=

R ,

k1,2,3

pg2aLH/2

F_k1 ,

k12,3

pg2aLH/2

0kj

0kj

₌ 00 2 4 £ 8 10 12 14

k/a =.25 ALPHA = O DEG.

15 8 I J b o

k=1.,5,6

pg

2aL2H/2

Fk R

_k=L.5,6

pg

2aL2H/2

Fk1 fk1 =

k

L+,56

pg2aL2H!2

bkJ

pV7L

k

12,3

j =1,2,3

-

akJ

-

bk k=La,5,6

0kj

₂

'

bkJ= ₂

pVL

pVL7[

j =1.5,6

L 21

Dimensionless exciting forces (amplitudes, real part, imaginary part) added masses and damping coefficients in Figures 9, 10 and 11.

Reference point for moments on waterline

Fig.9a: Transfer functions of the exciting forces, added masses and damping coefficients for different water depths (1/a = 1, a/h = 4)

1 d/61 .50 2. d/h=2.00 3. n.=3.00 2 4. dlh4.00 5. 6. d1h6.00 7. d/12.0 1. d/k1.50 2. d1h2 00 4 . 4/h =4 .00 1 2

4dPiIr._

Un

.0 2 4 6 8 10 12 14 16

l/e 1.

k/a =25

ALPHA = ODEG.

ka

.00

75

16 04 20 . O 00 .60 .45 30 .15

Fig.9b: Transfer functions of the exciting forces, added masses and damping coefficients for different water depths (1/a = 1, a/h = 4)

1. 4/hl .50 2. 4/6=2.00 3 d/63.00 i 4, 5. 4 / 6 4 . OC 4/6=6.00 6. 4/6=8.00 7. 4 / h = 1 2 .

IJÍ

Ái

i 1. 2. 3. 4. 5. 6. 7. d/h=i.50 d/62.00 d/h=].00 4/6=4.00 4/h = 6.00 4/6=8.00 4 / h = I 2 0 2

I-iii'

1 . 4/6=1 .50 2. 4/6=2,00 3. 4/6=3,00 4. 4/6=4,00 5. d/h6.00 6 4 / 6 8 00 7. 4/6=12.0 li1d 3 L I, 4/61.50 2. d/62.00 3, d1h3,OO 4. 4/6=4.00 .

IIÍ.

_I..

5 I, d/61.50 2. 4/6=2.00 3. 4/6=3.00 4. d164.00 5. d/h5.00 b. d/h8,00 7. 4 /h = 1 2 . O 1

-1

A

A-_5'

'Î''

0.0 2 4 6 8 10 12 14 16

I/a 1. h/a =.25 ALPHA ODEG.

ka

0.0 4 6 a 10 12 14 0.0 2 4 6 8 10 1 .2 14 16

I/a =1.

h/a =25

a

_lIa

_1.

_{h/a r.25}

ka

0.0 4 8 1.0 12 14 16 0.0 2 4 6 10 12 14 16

I Ia 1. h/a .25

ka

l/a

1.

h/a =25

ka

.12

f5

08 80 60 4 20 150 333 10.0 4. 80 3 . 60 b33 2.40 20

a44

f

f 3P 40 .12 3 20 10 03 00 - .03 1 .20 BO .40 0.00 - .40 6

h/a r25

ALPHA = ODEG.

ka

b44

fi'

0h -.12 0.0

f3'

05 .06 .03 .0 00 - .10 - .20 - .30 8 10 12 14 16 ALPHA = ODEG.

ka

4 6 I/e 2. h/a =.25 8 10 12 14 16 ALPHA = ODEG. ka

Fig.lOa: Real and imaginary part of the exciting forces, added masses and damping coefficients for different water depths (1/a = 2, a/h = 4)

1 d/I1.50 2. dfh=2.00

-H

l'1.

74

1. 2. 7. dlh=1 .50 d/h=2.00 :;:. d/h12.O

JÍi"

V

A

ÌL!.d I

1 2 1, 2. 3. 4. 5 b. 7. d/h=i.50 dfh=2.00 d/h3.00 d/h4.00 d / h = 6 00 d/h=B.00 d/h12.O 2 1. d/h1.50 2. dfh=2.00 3. d/h3.00 4. d/k=4.00 1. d/h1.5O 2. dlh=2.00 3. d1h3.00 4. d/h=4.00 5. d/h=6.00 b d/h8 00 7 d/h=12 O

_i_t;Ii

0.0 2 .10 12 14 16 0.0 2 4 b 10 14 16

Il

=1.

h/a =25

ka

I/a 1.

h/e =25

ka

Fig.9c: Transfer functions of the exciting forces, added masses and damping coefficients for different water depths (1/a = 1, a/h = 4)

C 10 12 14 16 ka ALPHA = O DEG. Ile 2. h/a =.25 2 4 6 0 . O .2 4 I/a 2. h/a =.25 B 10 12 14 1h

05

f

f 2R 0.00 .02 0.00 -.02 - .04 06 Ob 04 .02 0.00 - .02 /a 2.

h/a =25

4 6 h/a .25 8 10 12 14 ALPHA =45 DEC. .04 O . 00

fi'

- .04 - .08 .8 10 1.2 14 16

.0.O

2 ka ALPHA = 45 DEC. /a = 2.

f

21 .0 .0 -.12 .18 1 6 0.0 ₂ a

_{I/a =2.}

4 6 h/a =.25 B lO 12 14 ALPHA =45 DEC.

Fig.lOb: Real and imaginary part of the exciting forces, added masses and damping coefficients for different water depths (1/a = 2, a/h = 4)

16 a 115

pull

:-r-s'-O

iiiii

A

Ïî

1. 2 3. 4 5. 6. 7. d/h=1 .50 d / b r 2.00 4(6=3.00 d / I. 4 00 d/h=6.00 4/6=8.00 4/6=12.0

--ji

567

1 1. 4/6=1,50 2. dIh2.00 5. 4/6=6,00 6. d(hr8.00 7 4/6=72 0 ==4$ II 1. 4/bn .50 2. d/h=2.00 3. d/h3.00 4. 4/6=4.00 5. d/6r6.00 6. d/hnS.00 7, 4/6=12,3

1L561_i!

lÈL1iÎ

1AIÍ

1, 4/bnl .502. 4/6=2.00 3. d/h=3.00 4. 4/6=4,00 5. 416=6,00 6. 4/6=8,00

'

i.

--

4/6=12,0

jIPUNI

-ÌÏ

Ìi

1, 2. 3. 4. 5. 6. 7. 4/6=1,50 4/6=2.00 d/h=3.00 d/h=4.00 4/6=6,00 4/6=8,00 d/h12,0 2 2 8 10 12 14 16 .0 8 10 12 14 16

I/a 2. h/a .25 ALPHA O DEC. a

Ile

2.

h/a =25

ALPHA = 0 DEC. ka

2 4

4 6 B 10 12 14 16

a

h/a .25 ALPHA 45 DEC. Ob

f

SP 03 20 15

f5'

.10 05

1.20-- .15 80 .40 0 00 - 2 018 f 4R 006 O . 000 - .006 .12 08 f 5R .04 0.00 -.04 0.0 2 4 6 I/a =2. h/a =25 I/a 2. h/a .25 8 10 12 14 16 ALPHA 45 DOG.

ka

-. 0.0 2 .01 .00

f

4' .015 .030 8 10 12 14 16

ka

ALPHA =45 DEG. I/a =2.

f5'

.16 .12 08 04 0.0 2 I/a =2. 6 h/a .25 4 6 h/a .25 8 10 1 2 1 4 1 6 ALPHA =45 DOG.

ka

8 10 12 14 16

ka

ALPHA =45 DEC.

Fig.lOc: Real and imaginary part of the exciting forces, added masses and damping coefficients for different water depths (1/a = 2, a/h = 4)

i

1 dIk1.50 2. 8/b2.00 3 d/f'3.00 4 d/64.00

ii

1. 4/b1 50 2 4/1=2.00 \'5.__ 3 4. 4/1=3.00d/h=4.00 5. d/h=6.00 6. d/h8,00 7. 4 / h 1 2 . O 1. i 2. 3. dfh=1.50 d/h=2.00 d/h=3.00 4. 4/1=4.00 5. 4/1=6.00 6. dII==8.00 7.

4111.

d/h12.0

-lì!

lii

1. d/h1.50 2. d/h2.00 3. d/h3.00 4. d/h4.00 5. d/h6.00 6. d/h=8.00 7. d/h12.0

w

1. 4/h1 .50 2. 4/h=2.00 3. d/h3.00 4 d/h4.00 5. d/h6.00 6. d/h8 00 7. d/612.O

pr

ilíl

1. 8fh1.50 2. d/h2.00 3. d/h3.00 4. d/h4.00 5. d/h6.00

A

iIlilI!i

4 8 10 12 14

lb

h/a .25 ALPHA 45 DOG. a

I/a 2. h/a =.25 ALPHA =45 DEG.

ka

I/a 2.

4 6 8 10 12 14 16 2 6 00

f3'

-.15 - .30

1.20 ea 40 0.00 -2

l/

=2. 4 6 hI0 =25 9 10 12 14

16.006

.25 O . 00 f 3t - .25 - .50 2

II. =2.

4 6

hie =25

8 10 12 14 ALPHA =90 DEG. a

Fig.lOd: Real and imaginary part of the exciting forces, added masses and damping coefficients for different water depths (1/a = 2, a/h = 4)

16 117 1. 2. 3. 4. 5. dih1 .50 d/h2.00 d/h3.00 d/h4.O0 d/h6.00 4t 6. d/h9.00 7. d/h12.0

A

iii

'Ii

1. d/h1.5O 2. d,2.00 3. d/h3.00 4. e1h4.00 5. dFh.6.00 6. d/kB.00 7. d / k = 1 2 0 1. 2. 3. 4. 5. 6. 7. d/h1 .00 d/h2.00 d/h=3.00 d/h=4.00 d/66.00 dfh8.00 d/k=12.0

'

-1. dfh=1 .50 3. d/h3.00 Al12. d/h=2.00 4. d/h4.00 5. dih=6.00 6. dlh=8.00

--}P5Ih»

/h = 12.0

iI_

1 . d/h1 .50 2. d/k2.00 3. d/h3.00 4/6=4.00 5. 4/6=6.00 6. 416=9.00 ÌLUU4. 7. d/h12.O 4

I1L

1. 4/h1.50 2. 4/6=2.00 3. 4/6=3.00 4. 4/6=4.00 5. 4/6=6.00 6. d/h8.00 7. d/612 0 -.0

lì

10 .10 05 0.00

f

2F 0.00 f 21 .10 - .05 - .20 000 2 4 6 .8 10 1.2 14 16 "0.0 2 .4 6 8 1.0 1 2 1 4 16

ka

I/e 2. hi0 ,25 ALPHA =90 DEC.

l/

=2. h/a .25 ALPHA 90 DEG. a

2 4 6

l/a 2. h/a .25 ALPHA =45 DEG. Ka

I/a =2.

h/a =25

ALPHA =45 DEC.

2 6 8 10 12 14 16 8 10 12 14 16

ka

ALPHA =90 DEG. 02 O 00 - .02 - 04 - 06 006 003

f6'

0.000 003

036 .024

f

4R .012 0.000 -.012 a22 4G .12 50 60 Jo

I/. =2.

h/a =.25 0.0 2

I/e =2.

2 4 6 h/a r .25 6 025 0.000

f4'

- 025 - 050 - .075 8 10 12 14 16

ka

ALPHA =90 0ES. b22 1.20 1.00 75 .50 .25 0.0 2

I/a =2.

Fig.lOe: Real and imaginary part of the exciting forces, added masses and damping coefficients for different water depths (1/a 2, a/h = 4)

14 16

ka

1. d/h1.5O 2 d/h=2.00 3. d/h3.00 4. d/Ir4.00 5. d/h6.00 1. d/61 .50 2. d/6r2.00 3. d/h=3.00 4 d/64.00 d/6r6 .00

in

d /6 12 .0

4

_

L 1. 2. 3. 4. 5. d/h=1.5O d/6r2.00 d/l=3.00 d/k=4.00 d/hrE.00

44

6 I 6 B 00 7. d/6r12.O

IriI

'3

1. d/F, 1.50 2. d/6r2.00 1 3. 4. d/k4 00 5. 6.d/6=B.00 4

I

______IÌi.

_____________

7.

-

---h

1. /h1.5Q 2. d/h=2.00 3. d/6r3.00 4. dIh=4.00 5. d/6r6.00 d/IB.00 7. //6=12.0 5 6

N6.

1. 2. 3 4. 5. 6. 7 d/61 .50 d/62.00 dIk3.00 /16=4.00 d/kb.00 /16=9.00 d '6 1 2 0 r,-.. 0.0 2 4 6 10 12 14 16 0.0 2 6 B 10 12 14 16

/a =2. h/a =25 a

l/a =2.

h/a .25 a

4 6 h/a .25 10 12 14 16 a B 10 12 4 6 B 10 12 14 16 a ALPI-4A = 90 DES.

I/a =2.

h/a .25 36 au .24 .40 30 b11 .20 .10

a 1 .00 - 75 50 25 b55 10.0 2 . 50 048 012 .40 30 20 .10

Fig.lOf: Real and imaginary part of the exciting forces, added masses and damping coefficients for different water depths (1/a = 2, a/h 4)

119 1. d/I,1.50 2. d/h2.00 3. d163.00 4. dIh4.00 5. d/h:6.00 6. 4/6=8.00 7. /.=l2.Q

ii

liii

1. 2. 3. 4 5 4/6:1.50 4/6:2.00 d/i3,00 d/64.00 8/66 00 .alll 6 4/6:8.00 7. d/F=12.

,Ji!U

nut

1. d/h:1.50 2. 4/6:2.00 1 3. 8/6=3.00 4. d/h=4.00 5. 4/6=6.00 6. 8/1:9.00 7. 8/6=12.0 2 76 5 i 2. 3. 4. 5. d/h:1 .50 8/1=2.00 4/6=3.00 d/I,4.00 4/6:6.00

Ai

b. 7. d/h8.00 4 / I, = 1 2. 0

ii

iup

'Pii6

4/6=1.50 2. d/62.00 3. d/h:3.00

i

4. 4/6:4.00 5. d/hb.00 b 7. 8 / 6:8 00 8/6:12.0

lidi

_ri..

I-1 1. 8/6=1.50 2. 8/6:2 00 3. 8/6:3 00 4. d/k=4.00 5. 6. 8/6=8.00

i

7. 8/6=12.0

ÎÌ__

5 0.0 4 8 10 2 14 16 0.0 2 6 B 10 14 16

I/a =2.

h/a =25

ka

i/a

2. H/a .25

ka

00 2 4 6 8 10 14 16 Ira 2, H/a .25

ka

10 4 6 h/a .25 12 0,0 2

i/a

2. 1.4 1.6

ka

0.0 2 Ia =2. 4 6 H/a =25 0.0 2 b

lia

2 h/a .25 B 10 12 14 16

ka

1 .2 8

Ib

ka

30 0 833 15.0 7 50 7 .50 b3 5.00 036 b44 024 12 844 08 04

.100 .025 .060 .015 1 .20 30

3_-___

7

-1, d/h1.50 2. dIh2.00 3. dlh=3.00 4. dfh4.00 5. d1h6.00 6. dfh8.00 7. d/h=12.0 1. 2. 3. 4. 5. d/I1 .50 d/h2.00 d/h=3.00 d/h=4.00 d/h6,00

A

6 / h = 8 00 7. d/h=12.0

l'li

2

ii

/1

-1 . d / h = 1. 50 2. 3. d/h3.00 4. d/h4.00 5. d/h6.00 6. d/h8 00 7. dIh12.0 1. 2, 3. 4, 5. d/h1.50 d/62.00 d/h=3.00 d/64.00 d/h6,00 i 6. dlh=8.00 7 d / h = 1 2 0

Iii.

1 1, 2. d/h1.50d/h2.00 0.0 2 4 6 8 10 12 14 16 0.0 4 6 8 10 12 14 16

I/. =2.

h/a =25

ka I/a 2. h/a .25

ka

Fig.lOg: Real and imaginary part of the excitingforces, added masses and damping coefficients for different water depths (1/a 2, a/h = 4)

0.0 4 6 8 1 0 1 2 1.4 1 6

I/a 3.

h/a =25

ALPHA = O DEC. ka

0.0 10 12 14 16 0.0 2 6 8 10 12 14 16

I/a =3.

h/a .25 a I/a 3. h/a .25 a

Fig.11a: Transfer function of F, added masses and damping coefficients for different water depths (1/a = 3, a/h = 4)

048 036 024 .012 .045 b66 .030 .075

fi

050 1 .60 1 20 822 .87 .40 50 b22 60

30 . O 22 5 a33 15.0 7 50 .0 02 0.0 b h/

=25

4 S 10 12 14 16

ka

121 1. d/hl.5O 2. dlh2O0 3. d/h3.0Q 4. 5. dih=6.00

III

6. 7. d/h=12.0d/h8.0Q

iRI

6 1. d/h=1 .50 2. d/k=2.00 3. d/h=J.00 4. d/h4.00

//

5. d/h=6 00 6. d/hB 00 2 7. dl h 1 24

/3

1. 5 7/ 1. d/k1.50 2. dlh=2.00 3. d/h3.00 4. dih=4.00 5. dlh6.00 6. dl h 8.00 7. d/h=12.0

76Ì

0.0 2 4 .6 8 1.0 12 14 16

I/

rJ h/

=25

ka

Fig.11b: Transfer function of Fx, added masses and damping coefficients for different water depths (1/a = 3, a/h

r

4)

0.0 2 4 6 10 12 14 16

I/e =3.

hie =.25

ka

16.0 12.0 b33 B . 00 4 . 00 o a44 .04

MB0 20 MB0

pga3-20

3.0 1.0 3.0 1.0

ka

Length = 120 m Beam = 40 m Draft = .

5m

KG =

5m

Radii of gyration 1.0 2.0 30

ka

Fig.12: Transfer function of the midship bending moment

d/a = 0.5 d/a = 1.0 d/a = 2.0 d/a .-. 1.0 20 3.0 ix iy iz 15.0 ci 42.0 ci 44.3 ci

15 Q0

pgo2-1-1.0 0.5 1.5 Q0

pga2-1-10 0.5 1.0 2.0 3.0

ka

Fig.13: Transfer function of the vertical shear force

1.0 2.0 3.0

ka

Fig.14: Transfer function of the torsional marnent

123 = 0.5 d/a d/a = 1.0 d/a = 2.0

d/a - co

1.0 2.0

ka

3.0

I

(2) o (2) o 2.0 1.0 2.0 1.0 Length = 120 m Beam = 40 m Draft

2m

KG =

2m

(2)

F2

_lo

f10 =

ipga(H/2)2

2 17) f-o Radii of gyration iX 15.0 in

42.0 in

=

44.3 in

1.0 0.5 d/a d/a d/a

d/a

-o cx = L5

Fig.15: Drift forces for different water depths (1/a = 3, a/h = 4)

d/a = 0.5 d/a = 1.0 d/a = 2.0 d/a -+ = 0.5 =

1.0

= 2.0 1.0 2.0

ko

3.0 10 20

ko

30

Fig.16: Drift forces for different water depths (1/a = 3, a/h 10) 2.0

1.0

1.0 20 30