The influence of water depth on the midship bending moments of a ship moving in longitudinal regular head waves

DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG-SWEDEN

THE INFLUENCE OF WATER DEPTH ON THE

MIDSHIP BENDING MOMENTS OF A SHIP

MOVING IN LONGITUDINAL REGULAR

HEAD WAVES

by

CHEUNG H. KIM

DIVISION OF SHIP HYDROMECHANICS REPORT NO. 45

C ONTENTS

Page

Abstract i

Nomenclature . . 2

Intr. duct1.11

Defijitie,n..f Ship Motions and Waves 5

The Coupied Equations and

Coeffieient

6

The Midship Bending

Moments

9

Dìmensi.nless Representation lo

Calcu.latin and Discussiin 11

Ackn.wiedgements 12

:Reerenoes o.... 13

The heaving and pitching motions and the midship bending

moments of a T-2 Tanker model moving in longitudinal regular

head waves of shaflow water are calculated by Watanabe's

strip theory [1

J, [2], [3], [4], [5].

The resiits are represented ïn

-2-NOMNC LkTTJRE

a.,b,c,d,e,g coefficients of hèave equation A,B4O,D,E,G co'efficients rØf pitch équation

Aw waterplane area

B(x) beam of a section

midship bending moment coefficient

exciting fOrce amp1itu.e

g gravity constant

center of avity ( C.G..)

h water depth

wave amplitude

half-beam draft ratio'

longitudinal rad, of gyration in % of L moment of waterplane area

moment of inertia of the ship abo'ut y-axis L length between perpendiculars

rn0 midship bendin omen.t at time t

ma amplitude of midship bending moment Ma exciting moment amplitude

N

sectional heave.damping coefficient

t time

T draft

mean dÉ.ft of a section

V

ship velocity

V displacement volume

W

suffix designating Wave

x,y,z bod7 coordinates

E E

ri

iviw C-mW ç X V V o p () w e

phase difference between heave and wave.

ti t! il _{pitch and wave.}

't t! t! _{pitch and heave.}

t, it t, _{exciting forbe and}

wave.

'I ti t, _{exciting moment and} wave.

it t! Il _{midship bending}

moment and wave

heave at. time t

heave amplitude

wave elevaiioñ at time t wave length

wave number _{( w2/} )

shallOw water wave number

water density

pitch at time t pitch amplitude circular frequen.cy

-4-THE INFLUENCE OP WATER DgPTH ON -4-THE MIDSHIP BENDING

MOME S OF A SHIP MOVING IN LONGITUPINAL REGULAR HEAD WAV

BY

C. H. Kfl

INTRODUCTION

By applTing Watanabe' s strip method [i]

,

[ 2]

,

[3]

,

[4]

,

[5]

, the

heaving and pitching motions as weil as the

midship bending moments öf a T-2 Tanker model moving in

regui..ar head waves of shallow water are calculated and the

effects of water depth Tare discussed

It is ievealed by thè calculation that the midship bend

bments are increased, while the motions heave and pitch are

h

D'INITION OF SHIP MOTIONS AND

/AV3

The coordinate systems here ùtilized _{are space- and}

body-coordinate system O-XYZ and G0-xyz respectively. X-axis lies

ön the undisturbed water surface and Z-axis points vèrticiay upward. x-axis is longitudinal passing through the center of gravity G0 of the ship, while y- and z-axis point port and upward, respectively. The coordinate system G0-xyz coincides with the system O-XYZ at the initial rest condition. «e fo1lo the òonvention

of right-handed

_{coordinate system.}

z

z

Z/ /'// ///7////////// // // //

/ /

Assuming only heaving and pitching motions of a ship

at the speed. V in .a longitudinally oncoming wave system,

wé describe the surface wave as follows

= i:i COS

_{(vx + Wet)}

, where

e.

-6..

viave amplitud e

y shallow water wave number, i.e.

0

2

( = y trnh y h)

g o o

We circular frequency. of encounter i.e.

(W+v

V

- o

The heaving

and

pitching motions

ef

the ship corresponding to the wave defined above

are

then expressed by

-

aCS

_et

₊ 1

(2) = aC (Wet + ethW)

J

respectively, where

Ca 4a

are heave and pitch

amplitudes

and c, e

phase

differences between heave

and

wave

and pitch

and

wave, respectively.

THE COUPLKD

EQUATIONS A1D COFICIS

The coupled equations of heave

and

pitch óf a s3iip moving in longitudinal

regular

waves [i] , [2] are written in the

form

a + bC CC

-

4

-

- g(

_{'a COS}

_(Wet

The' coefficients on the left-hand sides of the above equations are

b= pV+

f

m" dx L

b= fNdx

L 0 2pg

f y&

L x dx -F L

¡Ni dx

_{- vf}

m' dx

L

L

g= 2p

_J

yx dx -

_Vf

N

dx

L

A= I

_{+ f}

m"x2 dx

L

B ¡Ni2 dx

L

0= 2pg f

_{yx' dx - VE}

L

f

rn"x dx L

fNx

dx+Vf m"dx

L L

G=2pgfyxdx

LW

cosh y (h

_-

T) ICoS

y x

2pghfy

°

L

_{coshv h}

lain uxi

cosh y (h _-

T) (cos

y x

-

wh (w + y V)

fin

_dx .1

_coshyh

_LSVX

o o cosh y (h _-

_T)

Isin

y

X')

wli fN

° ° . dx L

cosh vh

_Lc0s

vxJ

E _{H cosh y (h} - T) reos. _{y xl}

-2pgy)x

. dx

Slfl

W _{cosh v0h} sin v0xJ

-

"

cosh y

(.h

T)

wh f (.N-V

)x

L

_coshvh

o Isin y X o. i ces v X o. g 'avity constant displacernen-i, volume

yw half-breadth of a section on the calm

water-line

longitudinal moment of inertia of the ship's mass about G0-y-axis

e,

rn sectional added mass of unit thic1ess for heavee

N sectional heave damping coefficient of unit thickness

The exciting forces and moments on the right-hand sides of the equations (3) are represented in the form

,Wiere h

water depth

meazi draft of a section

phase differences between exciting force

and wave and exc it

ing

moment and wave,

respectively

Sectional values o added mass and

_dornping

coefficient rn" and

N for heave are obtained from [6] . In the case of deep water

these values are

obtained from

[la]

If h. 00 then y and

Coshy (h-T)

T

o

O

are replaced byy änd

e ,

respectively.

cosh v0h

HL MIDSHIP BD ING MOMI'TTS

The preceeding discussions

are on the cloulations of the

heaving

and

piibching motions of a ship moving in regular

head

waves of shallow water. By making use of the motions calculated

above we obtain the midship bending moments

in the following

form:

m0= ma cos ( Wet + ) ...(4)

,

where ma

and are the midship bending moment amplitude

and the pho.se

difference between wave and bending moment

respectively. Assuming that C.G. lies at the midship section the sine and cosine components of the amplitude

are written as follows:

J'macos

CCOS

_c;;1

tma1

CrJ

aS±fl E.

fcsin

+

a

ewJ

[-2 Pg We (

f N xdx +V

Yr xdx

w(J m"xdx +

m" dx)] w

-

xdx g

aS1fl

2pgj Yx2dX.iNxdx

-e(I);i

-w

_{C i'm"}

x2cx-j-L

+ f

1\Txdx J - cosh v0( h-+ E

f (-wm"

_{+ 2pr1)}

X

S

a

i' coshvv ( h-T) wE

f Ç

N

-

V-)

X O.

.coshvh

h/T

V/JgL

= m/pgE12..

-lo--where is a wave length.

dx

whcro the integral is taken either between -L/2 and. O or

htwccn +L/2 and O and designates mass per unit lenght along

tfie

ship lenght.

DThIENB IQ11IES S REPRESEMIAT ION

In representing the calculated results, the

followirg

non-dimensional forms are used:

depth parameter.

wave length to ship length

ratio

Proude Number

heave amplitude ratio

pii ch amplitude ratiQ

midship bending moment ceÍ'ficient

fcos dx LSth. v0x (sin V X j O VOX:

Pr the nierical calculatiofls we adopt a model of T-2

Tanker having the fellowing particulars.

Length between perpendiculars ( L ) 3.066 m

Beam ( )

0415 m

DrÍt

(T )

0.183m

Displacement volie

( V ) 0.1725 m3

iÖckeoffjcièE.t

( CB)

0.741

Radius of gyration ( % of L) 0.23

,wii.ere B(x) beam o± a sektion

p (x) fullness coefficient of a section

H(x) half-beam dràft ratio

w/g nias s in kg p r i/i O (see Fig i ₎

Station B(x) 13(x) 11(x)

wg

1 0.168 0.442

0.9

6.15 0.351 0.749 0.959 12.62 5 0.406 0.911 1.115 20.01 7 0.415 0.960

1.L4.

24.65 9

0.415

0.980 1.134 11

0.415

_{0.980 1.134 24.71} 13

0.415

O980

1.134 24.74 15

0.397

04961 1.085 21.79 17

0.311

0.871

_{0.850 13.27} 0.109 0.837 0.298 5.00

-12-The calculations are cärried out for the following speeds.,

waves and depths.

F= 0.0,

01, 02

;/I

0.5,

06, 07, 082

0.9,

1,0,

1.1,

1.3v

15,

1.7e

2.0.

h/T=

co

10,0,

4.0,

2.5,

and

1.5

In the calculation the following assuìptions are made:

t, Although tria and parallel sinkage are

roduoed they are

not eonsidered.,

r

2. C.G. lies at midship section.

Tl'.e Heave and Pitch Amplitudes together with the phase

differences with respect to the waves are illustrated in Fige

2-7. In general the motions are rema"kably damped a

the depth

decreases. This tendency is more siificant as the Proude

Number increases

The Mïdhip Bendin Momens

together with the phase

difference with respect to the waves are illustrated in Pige

8-10. The bendin.g moments are generally increased as the depth

decreases. This tendency is quite opposite to the above

mentionS-ed motions, Pròbably it is Paitly causmentionS-ed by the decrease of

inertial bending moments due to the damped motions. The dc;uble

peaks are nearing each other as the depth decreases. This is

probably caused by the delayed position of the peaks of heaving

motions.

A0IUOWLD ENTS

The authour expresses sincerely his thanks to Prf. Palkemo

Head of the Division, for his constant support

He wishes to

thank Mr. Bemiet fr his kind advice on the wave bending moments

[1 ]Watanabe ,Y. " On the Theory of Pitch and Heave of a Sip Technology Reports of the Kyushi University, Vol. 31, No. 1,

1958

[2]Gerritsma,J. & Beukelmaii,W.." Comparison of Calculated and

Measured Heaving and Pitching Motions of a Series 60, CB ₌

0.7 Ship Model in Regular Longitudinal Waves.

Laboration Voor Sheepsbouwkunde Technische Hogesehol Deift

report., No. 139, 1966

[3'ukuda,.I.: "On the Midship sending Moments of a Ship in Re-gular Waves".

Journal of Zosen Kiokai, Vol.110 Dec.1961

[4 ]Pukuda, I.: "Computer Program Results for Response Operators

of Wave Bending Moment in Regular Obiigue .aves."

Memoirs of the Fakulty of ngineering Kyushi LJ1?liversity, Vol. )OVI, No. 2,1966.

[5] Kim,C.H. " The Influence of ater Depth on the Heaving and

Pitching Motions of a Ship Moving in LongïtudinalRegiflar

HeadWaves."

Division of Ship Hydromochanics Report No. 44. Chalmers University of Technonlogy, June, 1968.

[6jKlrn,C.H.:" Hydrodynaniic Forces and Moments f or Heaving,

Swaying and Rolling Cylinders on fater of Finite Depth."

Division of Ship Hydromechanics Report No. 43. Chalmers University of Technology, April, 1968.

[7] Lótweit ,M. , Murer,C., Vedeler,B. ,and Christensen,H.:

"Wave Loads on a T-2 Tanker Modl. Th. Influence of Variatin in Weight Distribution With Constant Mass Moment of Inertia

on Benaing Moments in Regular Waves." European Shipbulding, Vol.10, 1961.

-14-[8]Murdey,D.C." Ön the Double Peaks in Wave Bending Momn.t

ilesponse Curves»'

Advancepaper of R.I.N.A. 1969

[9 ]Joosen,W.P.A. and 1anab ,R.:" Vertical Motions and Bending

Moments in Regular Wives. - A óomparison between

calcula-tion and experiment."

I.ß.. Vol. 15, Jan 1968

[1O]Ivarsson,A. and Thomson,O. Jämförelse mellan Jiodalifbrsök och Berknade Värden f ör artygs Uppträdande i Reelbuiidna Vâgor."

Chalmers Telmiska Höskola, Institutionen f br Skeppsbygg-nadstebiik, Sept. 1965

[11]Grim,O. und Kirsch,M. "R-4 Programm zur Berechnung der Tauch- und Stampfschwingungen nach der treifen-Liethode."

Institut fur Schiffbau, Hamburg Jan. 1966.

[12]GrimO. " nine Methode fLu eine genauere Berechnung der

Tauch- und Stampfbewegungen in glattem Vasser und in W3llen."

HSVA-Dericht Nr. 1217, Juni, 1960.

[13]ickson,A,F. " Underkeel Clearance."

The Journal of the Instituteof Navigation, Vol. 20,

CHALME RS

TEKNSKA HOGSKQLA

Mass Distribution of

T-2 Tanker Model

0TH - SH

Report 45

p

o

c,'J

r

p4 e p4

CHALME RS

TEKNISKA HÓGSKOLA

140

120

100-..

80

60..

40.

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0

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4Q.

1, 0_

= 10.0.°.

0.6

0,8

Heave Amplitude Ratio at

= 0,0

1.0

1.2

2.5

10.0,

0.

1.4

1.6

Fi. 2

0TH - SH

Reort 45

1.8

2.0

CHALMERS

TEKNISKA HOGSKOLA

80

60

40

20..

'0

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-40

-60

0.6

Heávë

Ap1itudè

Ratio at

= 0.1

.5

2.5

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ÒTH- 8H

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CHALME RS

TEKNISKA HOGSKOLA

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80

60

40

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80

-loo

-120 1.2

-140

Heave Am1itude Ratio at

0.20

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CTE - SH

CHALMERS

TEKNISKA HOGSKOLA

80.. £4,1q

60

40

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o

20..

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60

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100..

Pitch Amplitude Ratio at

o.o

CTH - SH

Report 45

CHALME RS

TEKNISKA HOGSKOLA

140...

4'

120

100..,

80_

60....

40_

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-40

-60

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-100

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CHALMERS

TEKNISKA HOGSKOLA

0.6

Pitch Amplitude Ratio at

= 0.2

0.8

0TH - 3H

Report 45

$

Pig. 8

CTH - .SH

C HALME RS

TEKNISKA HOGSKOIA

Midehip Bending Moment at

= 0.0

0.6

0.8

1.0

1.2

4.0.10.0..

1.5

2.5

1.2

1.0

0.8

0.6

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100

80

60

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-.20

-40

-60

-.80

100

CHALME RS

TEKNISKA HOGSKOLA

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0.03

_0m

0102

0.00

1.8

1.6

1.4

1.2

h

_{= 1,5}

VA

10.0

2.. 5

100

80

60

40

20

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20

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-00

Midship Bending Moment at

= 0.1

CTH- SH

Report 45

0.6

0.8

10

1.2

18

2.0

Midhip Bending Moment at

= 02

Pig0 lo

CTH -5H

Report 45

CHALME RS

TEKNISKA HOGSKOLA

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