A practical method of calculating vertical ship motions and wave loads in regular oblique waves

6 _{NOV. tgn}

ARCHIEF

Bibliotheek van Q...1ørfde I n ische DOCUMENTATIE ogesC 00,

:4'J8

1

Lab.

.v. Scheepsbouwkunde

Technische Hogeschool

e Summary

A practical method of calculating response oper.tors of verti-cal ship motions, shearing force and bending moment in _regular oblique waves is described in _{this note, along with results of} comparison between calculation and experiment. The calculation method is based on the linear _{strip theory, which is}

originally established for the case in longitudinal waves, and extensively applied to the case in oblique waves. Primarily, heave and pitch

of a ship in regular oblique waves are solved. Then, vertical

motions, shearing force and bending moment at a certain longitudi-nal location of the hull will be calculated by using the _solutions of have and pitch.

Contents

1 _Introduction

2 Modified Strip Theory

3 Heave and Pitch 4 _{Vertical Motions}

5 _{Vertical Shearing Force and Bending Moment} 6 _{Comparisons between Calculation}

and Experiment References

August 1968

* Prepared for the meeting of the Committee 2 on 'Wave Loads,

Hydrodyna,mjcsTt _{- ISSC, in Rome, August 1968.}

** Professor of _{Naval Architecture, Kyushu}

University.

J1

DATUM:

A Practical Method of Calculating Vertical Ship o'8-t _{Motions and Wave Loads in Regular Oblique Waves*}

1 Introduction

About a decade ago, a method of calculating heaving _and pitch-ing motions of a ship in regular longitudinal waves was proposed by Korvin_Kroukowskyl), _{which was based on the linear strip theory}

with application of Munk's method for a problem of airship fluid-dynamics. This method was immediately followed _{by Watanabe's}

modified one2) _{and widely developed for calculating vertical}

ship

motions and wave loads in longitudinal waves with the aid of

calculation methods of two dimensional added _{mass .and damping for} shiplike cross sections according to Grim, Tasai4 _and

Subsequently, a method of calculating _{response operators of} verti-cal motions, shearing force and bending moment in regular oblique waves was given by the author6'7'8), which was based on Watanabe's

theory (originally established for _{the case in longitudinal waves)} and extensively applied to the _{case in oblique waves. At present,}

the similar methods for calculating _{response operators in regular} oblique waves are adopted in japan9l41), Netherlands12) _and

Norway13). _{Since details of the author's calculation method are}

described in his published papers814), _{its outline is shown here} along with results of comparison between calculation and

experi-ment performed by other authors.

2 Modified Strip Theory

The application of Watanabe's method2) _{was attempted to the}

case in regular oblique waves, though the original theory was

es-tablished for the case in regular _{head or following waves. In the}

method descrjbed below, only the heaving and pitching motions _are

taken into consideration and the other motions; surge, sway, yaw, roll and drift are all ignored.

Consider the case when a ship goes forward with a constant

speed in regular oblique _{waves. As shown in Figs. 1 and 2, the}

coordinate system O-XYZ is fixed in space such that the XY-plane lies in the undisturbed water surface, and the coordinate _system o-xyz is fixed to the ship such that the origin is situated _{at the} midship. The ship goes straight on with a heading angle _{'/, to the} wave coming from the positive X-direction _{to the negative}

direc-t,ion. Then, the surface elevation _{of regular oblique wave} encoun-tered with the ship will be _{expressed as follows in the}

plane including the ship centre line:

h = hcos (k*x+wt)

₍₁₎

where

h: elevation of surface wave h0: amplitude of surface wave

= 1ccos, k = 27r/A, A: wave length, L': heading angle

We = cv + k*V : circular frequency of wave encounter

cv = _{: circular frequency of wave}

g: acceleration of gravity, _{V: velocity of ship, t: time}

When the ship is heaving and pitching in regular waves, the vertical oscillatory force per unit length acting _{on a section} distant x from the midship is given by follows _accordingto Watanabe's strip theory2):

p: density of sea water

half breadth of water line at x pN: sectional damping coefficient at x ps: sectional added mass at x

wr weight of hull and load per unit length at _x Z: vertical displacement of hull at x

Zre: relative vertical displacement between hull and effective wave at x

The values of sectional damping coefficient _{and added mass for} cross sections can be evaluated according to Tasai's

_method.

When the ship is heaving and pitching in _{regular oblique waves,}

Z and Zre will be given by: z = _{+ (x_xG)} Zre = _{+ (x_xG)# - he} (4) where dF dF1 dF dF dF

-=

+ 2 dx dx dx dx dx (2) where dF dF

_2pgYZ,

2 dx _dx

= -oNZ

' re dF3 d dF4 (3)

(psZ )

dt e dx

=---z

g

and it follows that:

=

_{+ (xxG)4, Z}

z=+ (xxG)4,

Z

As shown in Fig. 1, the surface elevation of regular oblique wave encountered with the ship can be described as:

h(x,y,t) = h0cos(kx cos(' - kysin\/'+ Wt)

at p(x,y) in a transverse section of the ship. And, the average

wave elevation in the transverse section of the ship will be:

: heaving displacement çb: pitching angle

xG: x-coordina.te of the centre of gravity of ship

he: effective wave elevation

1

ry

-

/

W

h(xyt) dy =

2y

_I-y

+ (x..x)q_

v4-

h

+ (xx)cp_ 2V- h

sin(kysin ii')

kysin

=Ch

_e

h cos(k*x+wt)

₀ e C = sin(kysin/i) /kysinfr ₍₆₎ e

The effect of decreasing orbital motion of wave for the wave induced force on the strip is introduced with:

exp(_kd*) = exp(-.ko-d) ₍₇₎ where

: sectional area coefficient at x

d: draught at x

CE: is the coefficient which represents the average wave

eleva-tion for the wave induced force on the strip and comes to a unity in longitudinal waves and in long waves. And exp(_kd*) represents the so-called "Smith effectt for the wave induced force. By intro-ducing Ce and exp(_kd*), the effective wave elevation for evalu-ating the wave induced force _{on the section may be given by:}

kd* kd*

h_e = C e_e h = C e_e

h cos(k*x+wt)

₀ (8)

e

By substituting the formulae (4) and (5) into _{the formula (3)} and taking into account of (c = _{- V), the vertical forces on the}

section are found as follows:

}

where

F*(x1) : oscillatory shearing force at

x1

_{(upward force on}

- the forward .side of section or downward force on

the after side of section is defined _{as positive)}

M*(x1) : oscillatory bending moment at

x1

_{(hogging moment}

is defined as positive)

Strictly speaking, the method described above _{is not exact for} the three dimensional problem. And it has weak points in

evaluat-The equations of heaving and pitching

integrating the force of the formula (2)

spect to the centre of gravity produced after end to the fore

to zero. Namely,

rFP

/ dxO,

dx

When the equations of heaving and pitching _{motions have been} solved, the shearing force and bending moment at a section distant

x1

from the midship can be calculated by follows:

rxl

rF-F.

F*(x )

₁ =

dx =

/ (11) j_AF dx J_x dx M*(x1)

[l

_dF A? dx

end of the ship and by putting them equal

F?

dF

(xxG)dx

= 0

(10)

(x-x1)dx

1FP

x 5

motions are obtained by

and the moment with re..

by the force from the

dx

(12)

dF

-2pg

₊ c..

_Ps..t+ (xXG)ee

ds +

+ (xxG)c

(x-xG)4h}

V92V4

-(xxc)

vq e} 'e} h -e (9) dx dF = dx dF 3

-=

dx dF4 = dx

.._._{+ (xxG)4

dx

ing hydrodynamic force and moment induced by the regular oblique

waves even though the strip theory could be admitted. At present, however, when the more elegant one such as a method of singularity has less applicability to the actual ship forms, the method de-scribed here will be adequate for the practical _purpose.

3 Heave and Pitch

Performing the integral operation of the left sides of equa-tions (10) and neglecting the secondary terms, _{we obtain the} equations of heaving and pitching motions in the f.orm of:

a

+ b.+ c

+ d4 + e + g _{= FoCOS(Wet+OC'p)} } (13) where A + + + + + G = M _cOS(wet PM 4o a = +

pfs

dx, W: weight of ship b

=pfNdx

c

2pgfydx

= p/s (x _xG) dx = 6 e A# = 2pg (x _{- XG)} dx - V b = I +

pfs

(x _-xG) dx, I: longitudinal moment of inertia of ship (14)

M cosp rm

Mc_h

+m+mt

MJ

J o

m+ mJ

COS k*x

fl=

_2pgfyCed*

dx sin k*xJ

f3

4):

= ocos cX, 4C = 40cosp4, WJDJN.Cee { sin kx dx cos k*x)

0ePf0e

_kd* <

ICOSk*X}cix

lsink*x m _kd* k*x) e

_{(X_Xc)dx}

m}

2pgfC

_(sink*x) 2 =

wp[NC e_*

Sin k*x xG) dx xn) s C0Sk*x COSk*x m3 =

_wwepf

SC e

Lflk*X}

e

mJ

+ wvPjr

kd* ( Sin k*x

sC e

e COSk*X}

The numerical integrations in (14) and (16) _{should be carried} out from the after end to the fore end of the ship.

The solutions of

equations

_{(13) are found in the form of:}

=

sSWet =

COS(wt+)

= 4)c05)et 5SiflWt =

40cos(w t +)

(17) where =

?0j

Q( = _sin.P# ) heaving amplitude

phase angle between the heaving motion _{and the wave} elevation at midship

pitching amplitude

phase angle between the pitching motion _{and the wave} elevation at midship

7

and

The vertical velocity and acceleration at can be easily ob-tained from (18) and (19).

The relative vertical motion between _{the hull and the} undis-turbed wave surface can be obtained from _{(1), (18) and (19) as}

follows:

z_r = z_rc

oswt - Z

_{e rs}

sincot = Z COS(Wt+E

zr1 where

Zro: amplitude of relative vertical motion at x1

zr: phase angle between the relative vertical motion at _x1 and the wave elevation at midship

Zrc = Zr COS _{5zr =}

c +

-

XG) _c

- hcos k*x1

21

Z_rs Z_rosine_{zr s} + (x -x )₁ - h sin k*x J

G S 0 1

And, the relative vertical velocity between the hull and the

surface wave will be given by:

v_r = v_rccosw-t - v sinw t = v

_{cos(w t+E}

₍₂₂₎

e rs e ro e vr"

where

vro: amplitude of relative vertical velocity at _x1

6vr phase angle between the relative vertical velocity at

x1 and the wave elevation at midship

and

Z:

0 amplitude of vertical motion at x1

phase angle between the vertical motion at x1 and the wave elevation at midship

Z

= Zcos

₌

+ (x1

_{- xG)}

Z

= Zsin

= + (x1 _.xG) P.

8

4 Vertical Motions

Various vertical motions of a. longitudinal location distant

x1 from the midship can be calculated by using the solutions of

heaving and pitching motions.

The vertical motion at x1 is derived from (17) as follows:

Z = ZCOS

_{Wet - Zs}

_10et

_ZoCO5()et+ (18)

v

=v cosE

=

::s

(X

_-

_xG)

S } - V + wh sin

= we _c + (x -x )_{1 G 'ci}.- Vçb_S

whcOsk*x1

The results of (18) (23) will be available for the purpose

of pursueing the _{eakeeping problems such as ship acceleration,}

deck wetness, propeller racing and slamming in rough seas.

5 _{Vertical Shearing Force and Bending Moment}

The oscillatory shearing force and bending moment at a section distant x1 from the midship can.be calculated according _{to (11)} and (12) by using the solutions of heaving and pitching _motions.

By performing the integral operations of the right side of (11)

and neglecting the secondary terms, the shearing force _{at x1 will} be obtained in the form of

F*(x1) = F*cosWt

- F*sinwt

_F*cos(wt+)

₍₂₄₎

where

amplitude of shearing force at x1

phase angle between the shearing force _{at x1 and the} wave elevation at midship

and, F and F are given by fàllows: F*) S C

pJC

+

2{

s}

Li

-

J

2{}.

C _S..

+Q1

+Q

+ h

fR1+

R2+ R3 (25)

R+ R

where

= -

_{{ -i- fw}

dc + pJs dx } + 2p fYW dx F2 =

OePfNdx + WVpS1

Q1 = -w

_{+fw(xxc)dx +}

pfs

_(xxG)dx}

+ 2p (x _{- xG)} dx + V2p s1 -

VpfN

dx

Q2_ePfNG)WeV{pfpsxl(xlxG)}

9

(23)

kd*

=

_2pf

y C e

cosk*x

LIflk*X}

we

R1 J 1

sin k*x

R2

-

cop/N

_Ce

e_*

{

Cos k*x

J R3

I

kd*

cos k*x

- WWePJ sCee

{sink*x}

+ciiVps

xle

C

e*{5*X1}

COS k*x1

B,' performing the similar integral

_{operations in the right side}

of (12) and neglecting the

_{secondary terms, the bending moment at}

x1will be obtained in the form of

= M*COS(,)t

_-

_{= M*cos(wt+S)}

.L. C

where

amplitude of bending moment at

x1

phase angle between the bending

_{moment at x1 and the}

wave elevation at midship

and, M

and

_{are given by follows:}

c

1Jc

I

s)

c}

+

q2}

L j M*

_'c

ps

C S

çr +r +r

+h

1 2 3 0

where

p1 =

q

=co{--fw

-

2pgfy

xG) (x-x1)dx

xG) (x-x1) clx

fw(x -xi) dx + p/s (x x) dx}

-

2pgfY

P2 = WePfN (x - x1) dx + W vpfs dx

)dx

+

pfs

(xxG) (x -xi) dx}

+

v2pfs

dx

+ vpfN

(x - x1) dx

1

(26)

(27)

(28)

'C

(x

(x

=

wePfN

r3 :.(

J=

OWf]5

=

2p9fYCe

= w *Jcosk*x Is Sjfl k*x _kd* sink*x

fNc

e e cosk*x _kd*

1cosk*x

ee

sin k*x

In the formulae (26) and (29),

be carried out from the after end x.,. And os

_Xl

in (26) denotes the

When calculating the shearing

ship, the formulae (26) and (29)

ing x1=O. ( (x

_kd*(_sk*x1

jPJsCee _I

dx

cosk*xJ

6 _{Comparisons between Calculation and Experiment}

A number of model experiments have been carried _{out fOr the} purpose of pursueing the response characteristics in regular head waves at experimental tanks in the world. And _{some of those}

re-suits were compared with calculated results based _{on the strip}

- . 9,14,15,16)

theory. Such investigations _{led us. to the general con-.}

clusion that the calculated response operators of _vertical

mo-tions, shearing force and bending moment in head waves were

suf-ficintly valid at least for the practical _purpose.

As to the case in oblique waves, however, we had not sufficient informations on the validity of the strip theory _{until the}

calcu-lations by joosenl2) _{and Nordenström13) were recently reported,}

where the calculated heave, pitch, relative bow motion and midship bending moment in oblique waves were compared with the results of experiments carried out at the Netherlands Ship Model Basin. In Japan, meanwhile, comparisons between calculation _{and experiment} for the vertical ship motions and midship _{bending moment in}

ob.-lique waves were tried by several authorsl719), _{where the}

cal-culation method described above is adopted. .

nunerical integrations should

to x1 or from the fore end to sectional added mass at x1.

force and bending moment at

12

Examples of the comparison between calculation and experiment performed by Japanese authors are shown below.

Fig. 3 shows results of calculaton and experiment for pitch and heave of a Series 60, 0.70 block coefficient ship model in regular

oblique waves carried out by Yamanouchi' and Ando17 at the

Sea-keeping Model Basin in the Ship Research Institute, Tokyo.

ionumal9) _{tried the calculation of midship bending moments on}

a T2-tanker model in reular oblique waves in order to compare

with the experiment carried out by Yamanouchi, Goda. and ogawa2O)' at the Ship Research Institute. The resuJ2ts are 'given in Fig. 4.

According to those calculations compared with the experiments

at the N.S.M.B. and the S.R.I. in Japan, 'the applicability of the strip theory in oblique waves seems tO 'be unexpectedly larger than the 'limitation 'of the theory. And it _{may be said that the strip} theory is valid for evaluating the response operators of _vertical

motions- and wave loads in oblique waves though further

investiga-tions are necessary for the applicability of _the,theory. ' '

Acknowledgement '

The author wishes to express his. deep thanks to Dr. Y. Yamano-uchi and his staff i,n the Ship Research Institute"and _toMr.,M.

Konuma for their sincere cooperations on,the occasion of this.

works. . "' ' ' -'

References

'1) _{- B. V. Korvin-Kroukowsky'and W. R. Jacobs: "Pitching' and}

Heav-.ing Motions of a.Ship in Regular Waves" TSNA,' _{Vol. 65,}

1957.

2) Y. Watanabe: "On the Theory of Heaving _{and Pitching Motions"}

Technology Report of the Faculty of Engineering, _Kyu.shu Uni-versity, Vol. 31, No. 1, 1958.

3)-P.. Grim: "A Method for. _{a More Precise Computation of Heaving}

and Pitching Motions in Smooth Water and in Waves" '3rd

Sympo-sium of Naval Hydrodynamics, Scheveningen, _1960.

4) _{F. Tasai: "On the.Daniping' Force and AddedMass of Ships}

Heav-ing and PitchHeav-ing" Reports of Research Institute for Applied Mechanics, Kynhu University, Vol. 7, No. _{26,1959 and Vol.}

8,:No.:,31,, 1960.'" '' ,

Coefficient for Cylinders Oscillating in a Free Surface" Uni-versity of California, Institute of Engineering Research,

:Series.No. 82, 1960.

J. Fukuda,, J. Shibata and H. Toyota: 'tMidship Bending Moments

Acting on a Destroyer in Irregular Seas" Journal of the

Soci-ety of Naval Architects of Japan, Vol. '114, 1963.

J. Fukuda and J. Shibata: "The Effects of Ship Length, Speed and Course on Midship Bending Moment, Slamming and Bow Sub-mergence in Rough Seas" The Memoirs of the Faculty of

Engi-neering, Kyushu University, Vol. 25, No. 2, 1966.

J Fukuda: "Computer Program Results for' Response Operators

of Wave Bending Moment in Regular Oblique Waves" The Memoirs of the Faculty of Engineering, KyushuUniversity, Vol., 26,

No. 2, 1966.

J Fukuda: "Theoretical Determination of Design Wave. Bending 'Moments" Japan Shipbuilding & Marine Engineering, Vol. 2, No.

3, 1967. . ', '

J. Fukuda: "Trends of Extreme Wave Bending Moment According to Long-Term Predictions" Journal of the Society of Naval

Architects of Japan, Vol. 123, 1968.

11). J. Fukuda: "Predicting Long-Term Trends of Deck Wetness of

Ships in Ocean Waves" To Be Read at the Autumn Meeting of ,the

Society of Naval Architects of Japan and Published at the End

of1968.' ' . . ,

12) W. P. A. Joosen, R. Wahab and J. .J. WoOrtman: "Vertical

Mo-tions and Bending Moments in Regular Waves. A Comparison

be-tween Calculation and Experimentt' ISP, Vol. 15, No. 161,

1968. '

N Nordenstrbin and B. Pederson: "Calculations of Wave Induced Motions and Loads. Progress RepOrt No. 6, Comparisons with 'Results from Model Experiments and F4l Scale Measurements"

Det Norske Veritas Research Department, Report No. 68-12-S, 1968.

J. Fu.kuda: "On the Midship Bending Moment of a Ship in

Regu-lar Waves" Journal of the Society of Naval Architects of Japan, Vol. 110, 1961 and Vol. 111,1962.

J. Gerritsma and W. Beukelman: "Analysis of the Modified Strip Theory for the Calculation of Ship Motions and Wave

16) M. Lötveit and K. Haslum:

Measured Wave. Bending Moments, Shearing

Motion for a T2 Tanker Model in Regular

Veritas Research Department, Report No.

17) Y. Yamanouchi and S. 'Ando: "Experiments

18) Y.

Forces and Pitching Waves" Det Norske

64-34-S, 1964.

on a Series 60, CE

0.70 Ship ode1 in Oblique Regular Waves" Proceedings. of 11th

ITTO, Tokyo,. 1966.

Yainanouchi, H. 0i1 .Y. Takaishi, H. Kihara, .T. Yoshino and M. J:izuka: "On the Ship Motions

_and

_{ccelerations of a}

Nude-ar Ship in Waves" Journal of'Che Society of Naval Architects

of Japan, Vol. 123, 1968.

:. Konuma: "Comparisons between Calculation and Experiment

for Vertical Motions aid Bending Moments in Regular Oblique Waves" Technical RepOrt of Nagasaki Shipyard,

Mitsubishi.

Heavy Indutries Ltd., 1966 (Unpublished).

Y. Yamanouchi, K. Goda. and A _{Ogawa: "Bending and Torsional} Moments and. Motions of a T2-SE-A1 Tanker Model in Oblique

Re1ar Waves" Note Presented to 2nd I,SSC, Delft, 1964

(Un-published.). . . .

14

Fig. 1 Coordinate System in a Sea. Surface W-.z'_e

A=d10 cas(1ec-rw.t)

PP' section

c,

tz/t.twv)

k

Fig. 2 Coordinate System in a Ship Centre Vertical Plane

-A

z

OX 5CtiO?

AVU Hood Scos c F..0 5 .0.I .02 5 -Hood Soc -.-z Bow Seos

,/

.\\ ; \\\ I 0i0 075 00 25 150 \ 0 W C F. s 0i .02

j

-Soco 075 00 25 L5Q 0., oorn 5sc3 Jr7 QcrtcrInQ Sc02 ,. isa F. 5 .01 --0.2 Foliowlnç Seoo

,aa is. CAg.a.ATsa.

,..o_.01

.02 5

0.50 075 .0 '25 150

Fig. 3 Calculations and Experiments for Heave and Pitch of a 0.70 Block Coefficient Ship Model in Regular Oblique

- 17)

Waves by Yamanoucn and ido ₎

Heaving Amplitude,

a Wave Amplitude 0a : Pitching Amplitude, Ic = 2?t/

0.32

= 0.37

.4tP!

rLU

L'i

o

_.

_

-:

_-

_!

-\J

I \/ L_.

-0

c0

0

U

C..

0.2

Fig. 4. Calculations and Experiments for Midship Bending Momert on a T2-Tanker Model in Bow Waves from 45° Direction (by Konuma19) C, = M*/ogL2Bh

'0

.-,', r,,

C

-./v i_._ - i .:, ;:i

0

0

0

0

0

.9

UL ON

0

_{MODEL TEST}

Fig. 4b Calculations and Experiments fdr Midship Bending Moment on a T2-Tanker Modl n Eow Waves from 45°.Direction

(by Konuma19) :

0M =

M/gL2Eh

.A/L= 1.38

0

0

0

0

Fig. 4c Calculations and Experiments for Midship Rending Moment on a T2-Tanker Model in Row Waves from 450 Direction

(by Konumal9)) : CM =

M/pgL2Bh0

X/L= 1.19

I..c

CALCULATION

TEST

0

_:MODEL

0.02

CM I

0.01

0

0.02

CM

001

0.I

_Fr

0.2

Fig. 1 Coordinate System in a Sea Surface

OX' section

h=.Aocos(tx-rw4t)

PP' section

(ooAi a/uu)

p0

z

I'

Fig. 2 _{Coordinate System in a Ship Centre Vertical Plane}

PITCHINI

HLAVINI

PITCHINI

Fig. 3 Calculations and Experiments f or Heave and Pitch of a

O70 Block Coefficient Ship Model in Regular Oblique Waves (by Yamanouchi and Ando17)

Za : Heaving Amplitude, _a Wave Amplitude

Pitching Amplitude., k = 27th L0- Os-

I-II

I I . 1

-Bia,,i Sics 050 075 00 L25 - L50 1 -1 Hsod S... I ---' .01.

// \

.0!.

1.0- _{\ *} o5 N U

,/

.\\

-050 075 100 125 50 LG I I - I..m Sic. OuortSrw.g S.c. I --

I-

-Following S... - UP. @.&*T. - 050 - 075 '00 'U ISO I S -__1 -Hood Sea I -I I I ,..0_.01 .e5 BowS... IO_

--- -

.

-I5T

5O 0'!

-.

0.0?

0

0.02

0.01

0

0.I

_Fr

0.2

Fig. 4a Calculations and Experiments for Midship Bending Moment on a C2-Tanker Mbdel in Bow Waves from 45° Direction

(by Konumal9)) : CM =

M/pgL2Bh0

CM

0.01

AlL = 0.37

:CALCULAT ION

0

:MODEL TEST

0

X/L =050

0

0

0

0

0

0

0

0.1

.rFr

0.2

Fig. 4b Calculations and Experiments for Midship Bending Moment on a T2-Tanker Model in Bow Waves from 45° Direction

(by Konuma19) : C =

M/pgL2Bh0

AlL =0.75

0

O

0

0

0

)

CALCULATION

:MODEL TEST

0

0.02

CM

0.01

0

0.02

CM

0.0I.

0

A/L =0.99

0

0

0

0

0

0

(j)

0

0.02

X/L=l.19

:CALCULATION

:MODEL. TEST

0

0

01

Fr

0.2

Fig. 4c Calculations and Experiments for Midship Bending Moment on a T2-Tanker Model in Bow Waves from 450 Direction

(by Konumal9)) : CM =

M/pgL2Bh0

0.02

CM

0.01

CM

00I

0

A/L= 1.38

0

0