On the bending moments of a ship in regular waves

0

ARCHIEF

Iab

v.

Schee.psbouwkunde

Technische Hogeschool

Deift

THE FACULTY OF ENGINEERING, KYUSHU UNIVERSITY FUKUOKA, JAPAN

ON THE BENDING MOMENTS OF A SHIP IN REGULAR WAVES

ABSTRACT )

BY

JUJ-ICHI FUKUDA

On the Bending Moments of a Ship in Regular Waves (Abstract)

Jun-ichi Fukuda

1 IntrQductiOn

The theoretical analysis on the bending moments of a ship may be conducted by using the theory of three dimensional hydrodynaamlcs

taking into tile disturbance of water surface, or by means of so-called

strip method based on the Blender body theory of ship motions. In the

author's papers the method introduced by

tatanabe3when

he

de-iived the equations of motions on heaving and pitching was applied to calculate the bending momentB of a ship theoretically, and the strip method was also adopted here because of the simplicity of calculation. The numerical calculations were performed for T2-SE-Ai tanker, and the

theoretical results were compared with the model tests carried out by

(4)

Taniguohi . In the former paper , the effects of ship speed,

weight distribution, etc. on the midship bending moments were

con-sidered, in the latter (2) , the longitudinal distributions of the

bending moments -were treSted.

2 Theory

Consider the case when a ship goes forwards with a constant speed

V among regular head seas with heaving () and pitching () motiona-.

The co-ordinate system O-X,T,Z, fixed to the space and the co-ordinate

system Oo-x.,y,z, fixed to the ship are chosen as showxi in Fig.l. Oo

locates at the midship and the x-000rdinate of the centre of gravity

be expressed as

where

ho * amplitude of wave elevation

k -

27r/A

,

A :

wave length

* circular frequency of encounter

t * time

According to Watanabe1, the force acting on a

_u.ni.t

length of the ship

is expressed as

where

in above

p :

density of water

j

acceleration of gravity

yw* half width of the water plane

N damping coefficient of. the section

4=AL.c.d(4X+w.t)

--

(I)

+

+

+

d

tx

x

a

Z=x4'

cr

?-

Vs

(2)

0.

* additional mass of the section

w weight of ship per unit length

and

=

,

(4

z

+ w t)

where d, is the mean draft of the section

-44*

The term £ corresponds to so-called Smith's. effects.

The equations of motions of ship on heaving and pitching yield

In these equations the terms including

are arranged to the left aide of the equations, and the terms of

Re

and 11to the right. _{Then, the equations yield in the form of}

+

b+c+d4++,4 = F

(4)

A

=M

The equations (4) are the equations of motions derived by

Watanabe'5 andthe sameexpression as Korvin-Kroukovaky's _equations

If the terms of ,'% in F and M in the equations (4) were ignored, the

equations (4) exactly coincide with those of Korvin-Kroukovsky.

F and M in equations (4) can be expressed in the _{form of}

(3)

The solutions of the equations (4) _{can be obtained in the form of}

= cc&3w.toi.nwt

6cCWt-4IñtW*t

0cod(w4t+O()

0cd(w4tft)

(')

PP

fdr=o

and

(x-Z)dx= 0

The shearing force _z, and the bending moment at

the

po-sition of Z, are given by

fX

Jcir

AP _Pp and

m

Xii"

AP

rq)ix

(x-Z.)dz

where the shearing force acting upwards on the forward side of the section is positive and the hogging moment is positive in this paper. Substituting the solutions (6) of the equations of motions (4) into the above equations, one obtained

"IX, = PC

(Xi) Cöd Wa

t

-

'r,) Mm (h

_{EJs,(r)Ced(w.t+)'x)}

(7)

m,= WZCX,

C4d (Vat - m,(r,)6(A'Z

w.t

Z0(l,)COd(Wgt +

3 Example of Calculation

The exciting forces and, momenta, the ship motions (heave and pitoh),

the mid8hip bending momenta and the longitudinal distributions _of

bending moments were calculated for P2-SE-Al tanker by using the method

discussed in the previous section, and the theoretical results were

(4)

compared with the model tests carried out by Tanigchi . The

flu-..

merical calculations were performed for each condition of weight

dis-tribution shown in table 1. The weight distribution of (A) condition

corresponds to that of the actual ship in full load. _Taniguchi's

ex-periments were performed for condition of (A), (B) and (C), and not for

of section were obtained from Tasai's chart (6)

The main results derived from theoretical calculations are shown

in Fig. 2,3 and 4.

In Fig. 2 the exciting forces and moments are shown, and also the

heaving and pitching motions. in those figures,

F0

=

Fo/p.LB*,

A

=

=

_{0/ Jt0,}

=

where L ship length and B4 ship breadth

and (M',148M") denote the so-called Froude-Kriloff's

ex-citing forces and moments with Smith's effects, and ( F",'") and

(M",p') denote those without Smith's effects.

(,cr )

and

(',ft

) denote the statically calculated displacements due to F and M.

In Fig. 3 the midship bending moments for the conditions of weight

distribution of (A), (B), (C), (A') and (B') are shown. In these

figures,

c0=

o/p.L*

( Co',t5")

denote the statically calculated.values considering the dynamical pressure of regular Waves, and (

Ci', 5")

_{or ( C,", 6'")} denote the statically calculated values considering the. pressure of

regular waves baäed on Froude-Kriloff's hypothesis with Smith's _effects

or without.

In Fig. 4, illustrates the longitudinal distributions of bending

moments in waves having length of

A/L _I

. In the figures,

4 Discussion of Results

The result8 of theoretical calculations are considerabily well

confirmed by the model tests

()

qualitatively and approximately

coin-oide with them even quantatively, therefore the following conclusions may be given by theoretical results:

The length of wave in which the maximum midship bending moment occurs is nearly equal to the ship length.

The midship bending moment reaches the maximum value near the

synchronous speed of heaving, except in the waves having shorter length than the ship length, and decreases rapidly as the ship speed increases until it reaches the minimum value, and then it increases again with

the ship speed.

So far as the effect of weight, distribution on the midship bending moments is concerned, the hogging condition of weight diatribution

(in which the centres of gravity of fore and aft body are more distant

from the midship and the radius of ration is larger then in the

eaggjng condition of weight distribution) gives the smaller bending momenta than in. the sagging condition when the ship speed is less than that of heaving synchronism, and the change of bending momenta due to ship speed also become, slow in the case of hogging condition of weight

distribution.

The mazthum value of the midship bending moments does not exceed the statically oaloulatea value with SmitW'a effects.

The.longitudirial position where the maximum bending moment occurs is situated near the midship, although itmoea forwhrda with the in-crease of ship speed affected by pitching motions, and this maximum value of bending moment is not too much larger than that of the

0

References

J.Fukuda:"On the Midship Bending Moments of a Ship in Regular Waves" JSNA of Japan No.110 (1961)

J.Pukuda:"On the Bending Moments of a Ship in Regular Waves-Longitudinal Distributions of the Bending Moments" JSNA of Japan No.111 (1962)

Y.Watanabe:"On the Theory of Heaving and Pitching Motions of a Ship'! Technology Report of the Faculty of Engineering, Kyushu University Vol.31 No.1 (1958)

K.Taniguchi and J.Shibata:"Model Experiments on the Wave Loads of P2-SE-Al Tanker in Regular Waves" Mitsubishi Exp. Tank Report No.357 (1961)

B.V.Korvin-Kxoukovsky and W.R.Jacobs:"Pitching and Heaving Motions of a Ship in Regular Waves" TSNAME Vol.65 (1957)

F.Taaai:."Damping Forces and Added Mass of Ship's Heaving and Pitching (Continued)" Reports of Reaerch Institute for Applied Mechanics, Kyushu University Völ.VIII No. 31 (1960)

Loading conditions Radius of ration in % of L Weight in _% Afterbody Porebody C.G. frog

it

in _% of L

Z,z,A.,, F

+,M

+ x)

x,

V

Table 1

Main

particulars of model in loading conditiona

(A) (B). (C)

22.2

49.6

50 ₄

Notea _{Pull load and even keel for all conditions.}

z

(A') (B')

2316

49.6

50.4 8 Afterb ody -18.3 -19.3 _{-17.3 -18.3 -19.3} Porebody _18.8 19.8 17.8 _{18.8 19.8}

Fig. 2 a

Fig.2 C

Fig.3 a

Fig. 2 b

Fig.2 d

HEAVE iNO PITCH

..222L

_.I(.231. I ig. 3 b 9 EZCIT NQFCCE AM) MOMENT - - 1.00

a---&i\. J_A;.

-.

--- --- - .T?

PHASE AAfLE A. -o Al OJ

ECI rING FQQCE

Ah MOMENT -O.7S

PN.45E ANGLE

6

-0

f#).Z

0.3

EXCIT 1GFORCE

AND aIOMENT 1.I.S0

£

18 O 0 PHASE ANGLE 0,-0 O,'.-.02 QJ

EIC# TINO FORCE

A NONENT l2S T

::

0 -lao. PHASE ANGLE .1- --- _o; 0 OfF,-.-.aL o.

'I.

Q.J 0 -T

lEAVE AND PITCH

-

- .

PHASE ANGLE

I!.0J

0.ro

HEAVE AND PITCH -.. 125_A.

222L 2J8L,'

-.-.-.-'¼ PHASE ANGLE

-.

H HEAVEANOPITCk = 222L PHASE ANGLE Fik. 3 C

F.2

Fig. 3

Fig.5 a Fig.5 C Fig.6 a Fig.6 C Fg.5 b Fig.5 d Fig.6 b Fig.6 d

/0

00I-MOMENT /00

T1,L_

/

IA) --_:L e c.., c.. -rfl PHASE ANQLE C 9j 003 GENUINe MOMENT 075 e '0 PHASE ANGLE 0 0!0...-02 Oj 0.01 o0 BENDING MOMENT 1.30 00l -o C.. c._.,

c..r

'0 PHASE ANGLE 0

0/Nal

J 003

:r

0 BENDING MOMENT /.25 .c?.r PHASE ANGLE 0 OIN.-02 03 OEPiØIHQ atir - 1.00 '0. PHASE ANGLE 0 O,#..oZ 00 00 8ENO/N0.HJfT -?0 0 -PHASEANGLE

4'

0 0/ N. 02 03 003 BEP4O/NGMOMENT -lS0 V0 S PHASE ANGLE 0

0I..-02

Dl 0 03 OENO/Ne MLWENr .-- I ZS S 0 PHASE ANGLE ,,(A/ (A) 0 0.3

- L0O. - 0 C. - C.0..A.t.. 00 0.3 0.01 0 AP 0.02 0.0 I a AP C. 0 PP -,o '0. 0 -,0'

F.4

OP - I 00. F,.. - 0.25 C. -': C.I.bC... 0 Q 0.02 ./L. 1.00, Fr. 0.15

- C.I,.1.t..

00 OP C. Fig. 3 b Fig. 3d PP PP

'I

C. 002

f0

,0 0.01 0 0 qo 0 ./L-I.0o. F...-0.20 - C.h.ht... 00 C.