### 'JCH!EF

1 1 1### Bending

### oment

### of Ships in Regular Waves

### Junichi. Fukuda

### Kanaxne Taniguchi

### Yoshihiro Watanabe

### August, 1961

### lab. v.

### Schepsb.kte

### Technische

### FkgesccoI

### Ddt

Bending Moment of Ships in Regular Waves

Junichi Fukuda*,

Kaname Taniguchi and

Yoshihiro Watanabe

Synopsis

In this paper the theoretical analysis on the bending moment of ships in regular head seas was treated by means of the slender body

theory and numerical calculation, was performed on the T2-SE-Al tanker.

The effects of wave length, ship's speed and weight distributions on

the bending moment of ships were also studied.

1. introduction

In recent years the studies on the 1y-drodynainic forces acting upon

the ships in waves have been conducted with the progress in the theory of ship' $ motion, hence the problems on strength of ships among the waves have been closed up as one of the essential subjects in ship's design.

Among those problems the bending moment of the ship will be treated

### in

this paper when the ship goes straight with the constant. velocity in regular head seas accompanied by heaving and pitching motions of the ship. This bending moment of ship in regular waves may be divided into

two parts; one is the bending moment of ship in the condition of so-called statical equilibrium when the ship rests in the still water and

the other is the one c.aiis by waves

### when

the ship goes forward inregular waves with heaving and pitching motions. The former can easily

### be

determined hydrostatically if the geometical shape of the ship and the weight distributions are known, therefore it will be excluded in### *

_{Assistant Professor, Kyushu University}

** _{Assistant' Manager of Laboratory, Chief of Experiments). Tank,}

Mitsubishi Shipbuilding & Engineering Co.,Ltd.

Professor Emeritus, Kyushu University

### I

### this paper and only the latter will be considered.

### Namely, the latter

### consists of the following factors if we consider the problem linearly

### exclud.ing the bending moment caused by surging, slanmiing and other

### exceptional forces:

### Bending moment due to the hydrodynainic pressure of regular waves.

### Bending moment due to the pressure change in fluid, taking into

### account the surface disturbance of the sea, caused by ship's

### travelling, heaving and pitching motions.

### Bending moment caused by inertia forces due to ship' s motion.

### These bending moments vary with time - just like oscillation - about

### a certain magnitude of bending moment different from the one obtainea

### by calculation for the condition of statical equilibrium.

### This difference in the mean value of the bending moment frøñi the statical

### one can be considered to coincide nearly with the change in bending

### moment of ship when she goes forward in the still water with the freedom

### of trim and sinkage, - it is confirmed by tests that this relationship

### above mentioned holds almost good.

### -This difference is also excluded in this paper since it can be determined

### for each speed of ship if the ship's form is given.

### Therefore only the variations of bending moment of ships in reference

### to: the time will be treated in this paper among the bending. mome rits of

### ships in regular head seas, and such bending moment as mentioned above

### is simply. called "bending moment of ship" in the follaWing of this paper.

### The theoretical analysis on the bending moment of ships may be

### conducted by using the theory of three .- dimensions]. ldrodynamios

### taking into consideration the disturbance of water surface, or by means

### of so-called strip method, based on the slender body theory of ship's

type of ship' a. form although: it is not practical to apply thi,a method

to the actual form of ship ecauae of the complication of calculation.

An example of the latter case was studied by jacobe(2) tr means of the

Korvin-Kroukvskyt a theorey(3) on heaving and pitching motions, however

### IcKts.

.anthors feel bs anaiysis somewhat' unsatisfactory.

### Therefore,

in hie paper the method introduced by 'Watanabe when he derived the equations of motion on heaving and pitching was applied to calculate the bending moment- of ships theoretically and the strip method was also adopted 'here because of the simplicity### of

calculation.. The effçct' of . ship'Sspeed,' weight ditribution, etc. on

### the

bending moment of ships' was studied theoretical]y and explained clearly. In this paper, Only the bending moment at.miship was considered, however the longitudinaldistribttioñ of, the bending moment, was nOt treated here, but, authors

hope to have an opportunity to present the paper on this subject some

other dy. .

2. ' Ship's MotiOn . .

ConsIder the case vhen the ship goes forward with a constant speed

among regular 'head seas with

### heaving

and pitching motions. As shown in Fig.i, the co-ordinate system O-Z fixed to the space is employedsuch that th XY-plane

### coincides

with the still water surface and theZ.axis indicates the upward direction perpendicular to the still water

urface. The co-ordirte system Go z fixed. to the ship is chosen

such that the origin locates at the center t

### gravity Go of

the ship, the x-a±is points out### ahead

the ship and the i-axis .UpwárdB. The shipis assumed to gO. straight with a constant speed V in the direction of positive x.. The surface elevation h of regular head seas' can be

expres sed as. .

### -where

dF.ç### dx

### dx

### dx

### dF1

### \

### -

### =. ,-'

2 \_{::-'-'}

### -N.(Z-.h)

(2.2.1)### and

### z

is the displacement of the ship's section### with

respect to the### space 'do-ordinate

system,### j'

### the density

of 'water, y, the half width### of

the### water surface, 9

the### additional massUf

the section### and N

bhe. damping coefficient' of the section. The### displacement Z is

essed in terms of the,

### upward

displacement at the. point### 'x. (t)

with respect to the coordinate

### system

fixed to the ship.### such as

z. =

### *

### The: ieiatLve

uiward velocity### and

acceleratIon of the ship to thewater Induced by ship's motion under a constant speed V,is

### expressed by

Z = 2V

### respectively. Also the following relation exists;

### dt

### dx

/ 1,144

### hence the surface elevation of the wave encountered the

### ship is

### expreSsed

### in

### the form of the, equation (2.1) such as

h.=

### hoIat

### +W6t).

(2.1)### According to Watanabe

(4)### the force acting on a unit

length### of.

### the ship at the

paint### X of the space, co-ordinate systefl or at the:

### poiiitx(t) of the, co-ordinate system fixed to the ship at an

### arbitrary tims' .t is expressed as

Assuming that the effective elevation of the wave he at. the point x (t) of the ship is equal to the elevation of sub surface at the depth.,.

Z = - dm from 'the water surface of the mean 'draft of the section,. then

the effective elevation of the wave yields

h

### hC'k

cos (### + We t)

or considering the sub surface it' the depth Z - dm*:_{from the water}

surface of the nan draft along the total length of the ship, h'yields

### he =

ho e&* cos (kx + 4.1e t)Consequently the equation (2.2.1) is expressed as a function of ,

### $,. $,

; h0,. h, h. .The equation of motion of ship on heaving and pitching yields

### L±

### Jdx

where W, and I is the weight and the moment of inertia or the ship respectively.

In these equations the terms Including , _{; $,} and _{. are}

rearranged to the left side Of the' equations, leaving the ternt of

Se.'and h to the right. The equations yield in the form Of

(2.3)'

The equations (2.3) are the equations of motion derived by

watanat,e and the same expression as Korvin-Kroukovsky's 'equation(3)'.

### 1dF

### J-

dx### I).

If the terms of

the equations (2.3) exactly coincide with those of Korvin-Kroukovky'a.

### The

coefficients a 'g and A can be determined if the Ship's form, weight distribution and a speed are Iiown and### the

so-called### exciting

### forces. in the right band of the equations (2.3) are also

### obtained jf the

### hip' s form, waves and ship' a speed are given, resulting

in the form of (see Appendix)

### FFcco8 1etFSBinLc)et5Focoa(QJt+

)### (2.4.1)

M

### Mc cos C44t - Ms sin Wt Mo C08 (u)at +

_{M)}

### where

### Fc

_{COS 4?)}

_{=}

_{(fi +}

+
### ) ho

### Fe

### - Fo (gj

b}### + f2t + f3

I Mc CoB(3,4 Ms )### Mo (j

### The first, second and third term in the right hand side of the equations

### (2.4.2) corresponds to each term of 2 fhe, N

### and

_{(}

### ke) in the

### equations (2.2.2). respectively and especially the first term corresponds

### to the exciti.ng

### force obtained from the so-called Froude -Krioff' s

### assumption. (see

Appendix)### The solution of the equatIons (2.3) can be obtained in the

form of### SWetAWet

### 0os(Q)et+C

### 1* =

### S(Jet

### 405(&5tf/84)

### where

### ,cosC(- (RP + SQ)/(R2 + S2)

### -

### sin e(- (RQ

### SP)/(R2 + 2)

a### cos i= (aF'+SQ')/(a2 + S)

= 6_{#:I}

_{(RQt..SpI)/(a2 + S2)}

### p

### Q) =

### Pt

QI### ,m1 +m2 +m3

### ) ho

### m1' +m21

+### m3'

### (F\

### (M\

t### F)(e

### f

### t1S'

F.### u)E

### (F5)

### (pj

Fe..### w:A_)(F)±we

### (2.4.2)

### (2.5.1)

### (2.5.7.)

R (se.,### -&e

### *

1,144I

. ) Bénding.Mbment

When the Ship goes straight among regular head seas under. a con-i

'stant spied with heaving .ázid pitching motions, the forces acting on a

unit length of the cross section of the ship. at the point x are given

by t1ioaein the equation (2.2.1) together with the inertia force

### wu:

### (

is), namely### --Z8+d,

(3.1'W is the weight per unit length of the ship.

In this case.

and ZS is equal to Z in the equation (2.2.2), however s and s are related only to the ship herself and do not include the term of ship's

speed V different from Z and in the equation (2.2.2) which is the velocity, and the acceleration of the aurrounding water respectively

induced by the ship's motion, that is,

### -Zs+x

'The bending moment Mx1 of the ship t the position of x is given by

where the hogging moment is positive in this paper.

Substituting the solution (2.5) of the equation of motion (2.3) into

the equation (3.2), one obtains .

where

### QVL(esI

### -?3)(g)

### Wet)()

(### +%'+r3t

### -3.31Z)

I I I I I I I I I I I I I I I### *

1.144 ) I I I=bi cos wt - ibis sin _{et} 'COS (Wet (3.31)

/

### !(x- x1)dx

### r.

I### '!(-

### x1)dx

### (3.2)

### ()

### (

### 3.3.3)

then,

### if

both of the ship's form and the weight distributions are rmiietry about the midship, the first and the second, term of theequation (3.3.3) corresponds to the bending moment caused by the distributed forces symmetrically and asymmetrically along the ship' respectively. In general case it is different from the above mentioned,

### however it holds nly true.

Therefore the equation (3.1) may beexpressed as

dFm

### dFm(S)

.(A)

### --) +()

dFm (3) dPm (A)

where (

### -)

and ( - ) is the distributed force symmetrical].yand asymmetrically with respect tà the midship respectively.

Translating the origin of the x-coordinate to the midship and

The bending moment tt ='..

### j

) . . C!

### =

### j

### (fr) (L)d

### +

I L 1,144 'is given by### CA)

1-(A)### (&F..\

### using

.the notation ofd.Fm (S) for this ni (3) new

### 1ç

coordinate, system, ' dFm dFm### (--)

### (A)

dFm dFm (A) =### 2t(r)I(r).i

dFm , dFm### (j)_;

IThe first term of the right hand side of the equation (3.3.2) depends

the heaving motion, the econd, term

### on

the pitching and the-third 'term on tb wave motion.. (see Appendix). Rearrang'in theequation (3.3.2):.'### ±

### +

### I

### ±

### and

### therefore

### =

### ix

### -g-:

The bending moment '117.. at the midship is given by substituting

t 0 as

### there fore

### *

1,144### =.

### 94t)

### =0,

I### are

not zero for a general case, then### :;zx0=

### I

### (A)

### -(3.4)

### (3.5)

### (iL

_{cA)}G.4II

### F

### '

I### f.E!)

### (4,)4

### L +

### )e

### + J

'4%### (5)

### (A)

-tHence it is obvious that the bending moment at midship can be deter-L5)

mined by only the digtributed force symmetrically along

### the ship.

If both of the ship's form and the weight distributions are symmetry about the midship the bending moment at midship is given by onlythe first term of the equation (3.3.3) and the second term of the equation beComes zero.

In general it is supposed that the second term of the equation (3.3.3) is sifl as conared to the first term of the equation for an

CA)

### ()

### actual ship. Although 'bt0 is equal to zero, both of 'M

### and

(3.4). Therefore the longitudinal distribution of

### the bending

môinpnt### along

the ship is### not

symmetry about the midship (even though for the### case of ahip. aymmetzy about the midship).

',i 1.144

I

4. Statical Calculation of Bending Moment

The conventional method of calculation for the longitudiiial strength

of ship is based on the as8umption that the ship is ailöat on the waves

under the. condition of statical equilibrium with the pressure due to

waves, taking into con'ideration the wave pressure based on the

Fxoude-Kri].off's assumption, that is, equivalent to the term of(2f Yw he) In the first equation of the equations (2.2.2). However it. should be. borne

in mind that the pressure due to waves acting on an actual ship also
consists of the terms of Ne and _{t (? }e) in the second and the third}
equtionof the equations (2.2.2) besides the above méz1ioned term of
(2, Yw he).

In: this section the bending moment of the ship will be deterntne.d by.. assuming that the hip rests on the waves under the condition of statical

equilibrium with the pressure due to waves considering the three terms

.

of he, he and he. The, calculation of bending moment under the àondition

that the ship is statically in equilibrium with the. presaure due to

waves.,. iioring the dynamical effect of ship's heaving and pitching motions,

but taking into account the dynamical effect of wave motion, is called

calculation of bending moment in this paper.

(1) Displacement of Ship

-Under the .asèption that the ship is afloat in statical equilibium

-with the exciting forces, the terms of , , and

### 0

in the euqations-Of 'motion (2.3') can be ignored such as

### +

### =F'

### '4

### = N

.'### (44).

The displacement of ship is given as the solution of the equation (4.]. The solutions ' and

### 4"

are given byp

### .1

### 1

### =

c.o ,S We 8### s'

### &e

### '(L.2.1)

where

### c'\_c0sd\

.### (Mc

### 5t)iii 0(l)=%8

e4I### I

### i.

### 1.fFC

s' JIosinfl$i### I - 1..

Fsand the exciting forces are &.ven by the equations (2.4). (2) Bending Moment

The bending moment obtained from the statical calculation

### i8

ivéñby. using the equation àf displacement (4.2) :as

### ')7t.

### eo$&et.-1

### (4..1)

where### LI.'

### (S'

((10 1..### l'

### this

_{'5}

### )=

.1,144### .4

### )

### -

### c(:)},c._ cC)

I Mc### /

### - C)J/(gG_ cC)

### (4,2.2.)

### .+.-

### :()+

(3) . Conventional. Method of Calculatioh for Lontudin'al Strgth

In the previous sections the statical calculation was coth1uctedby considering the hydrodynamic effect of wave pressure, however the similar

forn to the equations (4.2) and (4.3) can be obtained for the displacnent and the bending moment respectively by considering only the teim of wue

pressure based on the Froude-Kriloff ! s assuntion (see AppencUc.) a

conventional method of calculation for the longitudinal strength of the ship corresponds to the particular case when the wave crest; .dth the

wave length ratio = 1.00 locates at the midship.

Generally the so-called Smith's correction is not consideredfOrthis;caae..

### *

1,144### 1

### 5.

### Example of Calculation

### The exciting. forces, ship's motion and the bending moment were

### cal-culated for T2-SS-A]. tanker by using the method discussed in the previous

### sections and the theoretical results were compared with the model tests

### carried out by Taniguchi6.

_{The main particulars for the actual ship}

### and the mode). of T2-SE-A]. tanker used for the calculation aje shown in

### table 1, and the loading condition (or weight distribution) of the mode].

### used is ahoi in Table 2.

### The weight distribution for the condition (A) corresponds nearly to the

### full load condition of the actual ship and the model rests

### on the still

### water in sagging condition slightly.

### The weight distribution for the

### conditions (B) and (C) are different from that of the condition (A) having

### the differen positions of the center of gravity in the fore and the aft

### half of the model from those of (A), however the radii of

_{rration for the}

### conditions (B) and (C) are the same as that of (A).

### The model shows severe

### hogging condition in still water under the condition (B), while it s)xws

### considerable sagging conditions under the condition (C).

### The conditions

### (A') and (B') have a larger size of radius of

_{ration thart that of (A)}

### and the weight distributions for the conditions (A') and (B')

### were adjusted

### so that the positions of the center of gravity in the fore and the aft

### half of the model for (A') and (B') coincide with those of (A) and (B)

### respecti'vely.

### Nunmrical calculations were performed for each

### condition of (A), .(B), (C),

### (A') and (B') by using the values of wave length ratio /L = 0.75, 1.00,

### 1.25 and l.5C).

### The added mass of section and the dampirg coefficient of

### section were obtained from tables and figures by Tasai

### '

### , however

### *

1,14### I

### The theoretical results were compared with the model tests by Taniguchi

### under the condition of wave height-length ratio Hw/L = 1/50.

### The mode].

### tests were conducted under the condition

### of self-propulsion and only the

### ship's motion and the bending moment were measured,

### but the exciting

### forces were not measured.

### (1)

### &citing Forces

### The results of calculation for the

### equation (2.4) on the exciting

### forces are shown in Fig. 2, where the enplitude is given by using the

### non-dimensial expression such as

### = Fo/

### hoLB*

### iio=

### Mo/ghoL2B

### In Fig. 2 double prime (") indicates the

### Froude-Kriloff

t_{exciting}

### force with Smith's correction and

### triple prime (") without Smith's

### correction.

### Both of Yo and

### o vary with the phip 'a speed,

### however the

### variation is not too much up to the

### design speed (about Fr = 0.19) except

### the extreme case of the test speed. Moreover, the exciting force due to

### heave Fo is much smaller than the

### Froude-Kriloff'S exciting force

### and

### is still smaller than

### FO" except the

### case of

### /L

### = 0.75.

### The phase angles

### of the exciting force

### C(F, CF" andF" are approxiiaately zero degree for

### the case when the wave-length ratio

### /L is greater than or equal to 1.00,

### but they reach about 180 degrees,

### that is, an inverse phase angle at the

### wave-length ratio /L = 0.75.

### This means that the wave-iength.ratio

### /L

### for which the exciting force becomes zero lies between 0.75 and 1.00.

### On the other hand, the exciting

### moment

### o due to pitch is much smaller

### than Mo" and also smaller than Mo"

### for all cases of the wave-length

### ratio tested, and their phase

### angles/3M,

### M" aridM" are nearly equal.

### to 90 degrees.

### Therefore it is supposed that the wave-length ratio

14

(2). Ship's Motion

### and Statical Displacement

Fig. 3 shows the results of calculation for the

### equation of ship's

nation (2.5) and the equation of statical displacement (4.2). The

np1-itude is given by using the following non-dimensional expreason;

= */ho, o = o/kho

and

### o' =/ho,

o' = O'/khoThe statical displacenisut due to the Froude-Krioff'8 exciting force with

Smith's correction is shown

### by

the notation of double prime### (n)

and### with.

no Smith's correction by triple prime (1t) as indicated in Table 4,,

The comparison beten calculation and teat results on the characteristic

period T of heave and TØ oi pitch is given in Table 3.

On the abscissa of Fig. 3 is shown the ship's speed

### at which jend

### A.

becomes unity, where### WTe

and

T

The theoretical result of T coincides fairly with

### the test result,

however-for the case of T

### the calculation gives slightly

### na11er value than

-that of the test result.

Frmi this fact it is supposed that the problem should be

### n14ed

as a-three dinensional problem so far as the

### iovgitm4(nn1

dietribution of the-added mass of section is concerned, because the error in the c*lculation of thç added moinnt of inertia becea considerable if this effect was

-ignored, though the error in the calculation of total added u rema1s

nj1 1 This three diiiiensional effect of the distributed added mass of

section along the ship

### may also introduce the error in the

### calculaticsi

of the

### befld1ng monnt.

### The se result as above may hold true in the

sees to be acculate enough to apply to the design of ship quantitatirely

as well as qualitatively.,

### 6.

Approximate Method of Calculation of Midship Bending MomentThe bending moment at. midship acting on .a ship in regular waves can be calculated by means of the strip method as mentioned in the previous

section. -

-If it i's possible to explain the feature of the problem approximately by considering only the important factors in the problem, one can easily find the nature of the bending moment of ship among waves.

This approximate method of calculation is performed for such _{a purpose as}

above.'

(1) Statical Calculation.

Assuming that the effect of ship's speed on the sending moment is

### *

1,1441

also shown in Fig. 5 for reference and the results of calculation on the'

conditions (A') ad (B') are compared with that of the condition (A) in

Fig. 6. _{Fig. 7 shows the comparison 'between the theory and the test on}
the conditions (A), (B) and (C).

The theoretical results are in good agreement with the test results so far as the tendency of the'chage in bending rnment due to ship's speed is

concerned, but numerically there exists a little differencebetween them,.

especially the difference seems to become large under the ship's speed at

which the anplitude of pitch and heave becomes large.. The reason why the

difference becomes large is due to the existence of the three-dimensional

effects on the added, mass and the -damping force _{as stated in the previous}
section as well as the effect of non-linearity.

However, if one considers the posible aceuracy expected in the test,

### Fnot

too much, it may be ignored in this approximation. One of the forces.acting upon the unit length of ship at the point , in front of the midship

is due to the wave pressure given by the following expression;

where the higher order of the force is ignored.in this expression.

In this equation.the term eIn expresses the Smith's effect and

) means the effect of the acceleration of waves in the

### 2cj.

perpendicular direction.

The former depends on the depth of the ship and the latter on the breadth

of the ship. (The term due to the velocity of waves in the perpendicular direction is ignored in this expression.) The conventional method of calculation on the longitudinal strength of ship corresponds to the case when these factors are unity.

Consider the mean value of the above expression along the ship,. or

.- Cl

kdrn

2 h0e

- d* (where d* is the draft) is used instead of d . Then the

C1w / .

in

-..__/ 2.

-term in the parenthesis is exnressed as - instead of (O

### '/2JiYw

### 42...

by using the following approximation;

where

Therefore the aiiplitude h of the effective wave is given by eo

### heo ='Ci

. C2### .h

### 0

### 'V

### ---)

rcos (k### +Wet

### y

(6.1)and Ow is the wat.er.plane coefficient and Cb is the block coefficient.

I I I I I_ I

-1,144

F .

### C2=

1$

then C1 C1

'I,144

Therefore using the following notation,

### = C1'

### 'fleo/á' =

C1 Cand

### C1C2 =C1

CEr _{F} 1.

' The midship bending' moment can be obtained by the conventional

stand-àrd calculation with the equation (6.1), which corresponds to the case

### wit-dut

considering the effect of ship' s motion on the bending moment. Strictlyspeaking, this: bending moment is

### equivalent to

the bending moment )zc of the equation (3.3.1)### with

We = 0. For the case when We 0, bo from the### equation (3.3.1)

coincides with.'hb' from the: equation (4.3.1)### of the statical

### -calculation.'

.The midship betiding moment lfló*' obtained

### from the

approximate method in this section is compared with '?)to'" ,1o''### and9)leo., where 'Tto'''

is. obtained by### -'the standard calculation without the Smith's correction and its

### non-dimension-a). expression is Co' '3 ,

'to'' rth the. Smith' s correction and### non-dimension-al. expresuion. 'Co'-'

### nd

### corresponds

to the bending moment obtained from test result atFr= 0 with the non-dimensional expression Co. StrictlySPkifl&, 'e.o is slightly'larger thaniflo*, however they are nearly equal to each other.

### Table 5 shows the

comparison between C1 C2### and C, C

obtained### £rornbo''.','h1o'' and '7eo

corresponding to the condition (A) in the test of'F 2 - SE - A 1 tanker. Ph'e value C1 gives good appro'rirnation for all values of

_{X/L}

_{.}

_{The product C1 C2 becomes large as}

_{k/L.becomes large because}

### the ratio 'To/ño*

### is greater than

unity as shown in the following### calculat-ion in spite of

the### assumption that 'b1o' is nearly equal to ')fleo.

### 'From' th.&s fact. 'C

C2 seems to give considerably good approximation.1

(2) Dynamica]. Effect due to Ship's Motion.

It is well known that the ship is subjected: to a considerable sagging

monent at midship under a certain speed of ship due to the waves produced by the advance of. ship herself. This sagging moment becomes maximum when the wave length A is neaily equal to the ship's length L.

The Froude number Fr.c at this speed of ship is given by

Fr.c

### = 0.4

### JTi/L

and the values obtained from the test result on T2 - SE - Al model are given by

Fr.c 0.37 and 0.85

The test result on other types of ship's form gives the same values as above mentioned approximately.

There is no need to consider the bending moment caused by the waves produced by ship's motion itself when the ship goes forward with the speed appoximate-

-ly below the above mentioned value.

The midship bending inoient can be obtained by integrating the product oi' distance and the mean value of the forces acting on the ship's section

at the distance ± from the midship along the half length of. the ship. These forces include the terms of pitching and heaving motions. There is a

combination of pitch and heave when the degree of asymmetry in the ship's form about the midship is large or the ship's speed is too high, however the probability of the combined motion reduces considerably for the case of less

degree of asymmetry about the midship or the case of lower speed such as the

case of T2 - SE - Al tanker. Therefore the combination of pitch and heave ignored in this calculation.

Assuming that the heaving motion is given b

### =

c cos Wet - s### sin &t

where

### =

_{:)(}

### 2&eB0k((sc

### e.i)eFs ) + ( &

### -and

### w is the

### mean

of the distance from the midship to each of thecenter of the fore and the aft

### half

of the water plane, and. .'g is themean of the distance

### from

the### midship

to each of the center of gravity (including the added### mass)

of the fore and the### aft half

of the ship.### Using the non-dimensional expression

### Co the midship bending moment

### is given

by### ht0

C C### U

### (1

### A

### )2}+

### (

_{A')1}

### H)

### (iAt) +

### + (GK

### CoS

### 4

I### 20

rr

-where and is th iaping coerricient of pitch and heave

respectively

### and

is related to R in the following equations.### (OeR .;--fr

### 2(1')

1,144 or### WeR

### CwB,2(0

.1..### ( 1+

### D )

### 16

Cb### (BMe)d*

### 20

_{c1}

Cw L2
or _{}

### u-c-There1ore &e R yields

### WeR

### CwBL2

### (+°

2

### The terms

### and

### ô $ are given by Tasai.'s approximation

### formula

### (8)

### And the term H is given by

### =

### )

### 'L05F

0Fig. 8

### shows

the-values of K### and

calculated by assuming that### the

### shape

of the water plane### is

expressed as the### m-th

order of parabola.### For the case of T2 - SE - Al

### tanker, it is reasonably assumed that

### 0,32

### and

### '2-$

### =

### O.2

### In the model test conducted by Taniguchi,

Ca)### /

c)### is given by

### =

### 0.926

for the case of

### = 0.823

hiieth' values, of g are given by 0.175L, 0.185L and O.195L for the

condition (C), (A).and (B) respectively considering the ship's weight only, then the ratio g/ w yields 0.13, 0.62 and 0.910 respectively. Th

ratio g/ 'w is nearly equal to the above even when the added mass is

taken into consideration,- therefore, in this section the values of

### g/ w

0.7', 0.8, 09 and 1.0 are used for the calculation. For the values of

### =o.86 (

H 0, K = -0.125), = 1.00 ( H = 0.155, K = -### 0.075.)

and & = 1.22 ( H = 0.38, K 0) are used for the calculation. The result of calculation is showii in Fig.9 with the abscissa ofindicating the corresponding values of the Froude number Fr in the figure. Comparing the calculation with the test result,. both are very close to' each other qualitatively and quantitatively for the case when the ratio

### g/'

takes the value of 0.8 to 0.9. However the calculation gives the'smaller value than the testresult for the large value of the Froude number, This seems to be because the effect of ship's speed is neglected ignoring also the effect of speed on the value of . Furthermore, the reason

why the value at = 1.22 and

### A= 0

is especially too high is simply because the estimation on### 'hto*

is too much. As seen in Fig.9 the shape of,the curve varies considerably as the ratio g/

### ,w

varies even though the value of is constant. And the bendingmoment becomes large when the ratio gJ'gw is small, corresponding to the sagging load condition.It should be noticed that the value of g/ w may be different

### even if the

radiue of rration is### equal to

each other as seen### from the

### conditions (A), (B)

and (C) in the test.1,144

L

3

7. Discussion on the Midship Bending Moment

From the result of calculation it is found that the maximum hogging (or sagging) bending moment at midship is caused when the wave crest (or

### wave'passes through

near the midship, being independent of the ship's speed for any.cases of the wave-length ratio. This fact is nearly confirmed by the test results. The amplitude Co Of the bending. moment at midship varies with the change in the wave length, ship's speed, and the weight distribution. Generally it increases slowly with theship' s speed and reaches the maximum value near the synchronous speed for

heave except the case of the wave-length ratio X/L 0.75. After

reach-ing the maximum, Co seems to decrease rapidly as the ship' a speed increases until it reaches the minimum value and then it increases again with the

ship's speed. The average value of Co is nearly equal to the extent of Co' obtained from the statical calculation except the case of extremely

high speed, and Co' is considerably smaller than Cotm with no Smith's correction and is slightly smaller than Co" with Smith's correction. And.

all of Co., ', Co" and Go" take their maximum values when the wave-length ratio

### X/L

is nearly equal to 1.00.Therefore the midship bending moment in regular head seas can

reason-ably be obtained by considering Co' at the wave-length ratio '-/L= 1.00 as a basis and taking into account the dynamical effect of ship' a motion which depends on the weight distribution. This basic value Co' reaches utmost Co" (with Smith's correction) even though the variation of the bending moment due to ship's motion is taken into consideration.

In the next, we consider the tendency how the bending moment varies

with ship's speed when the distribution of shij's weight is changed.

First of all, only the parameter p1 in the equation (3.3.3) is varied when

ship are changed, keeping the radius of gryation constant. Let the 'change

in p. be :

### zp1 , then

the chatige in bending moment_{and}

A ### __

given b.y .

### fof pitch.

Suppose the case when the center of

### gravity in

the fore and the aft ha]! of the### hip is removed toward the end of the ship respectively, then

two cases are possible because

### Ap

is negative, namely;(i) For

_{90°>}

### OX ): -

_{90°},

### 0 and

A### < 0

For:: .

### ±90°

' :0 and

### =0

### For

_{), 90° or}

_{o <- 90° ,}

' ### ( 0 and

### 4 ')n

_{.}

### <

### )For

_{180° >0.>O}

, and

### 4fbis <0

For o' =

### 00

or oç= 180° , =, 0### and:

### 0

For O <

### < C)

_{,}

### <0

and . O)ç### 0

### Genera].].y'it is supposed, that & is

nearly- equal to zero and alo### bt0

isnearly equal to therefore the

### conditions.

for 'the case (i) can also be applied to the change 4d)fl0 In_{dbt0,}

assuming that rt0 mainly depends
- on the, change 4
, in

a...

-When the center 'of gravity in the fore and

### the aft half 'of the ship

### -

is removed toward the midship### espectve1y,

_{then the opposite relations}may hold ijice p, is positive.

### For

exaii1e, consider the case when-

_{X/L}

_{1.00' and compare the conditions (B) and (C) with the condition (A)}

- which Is considered as a basis, then both

### dho

(B) and C%io (C) are'

-nearly equal to

### zero

at the ship's### speed corresponding to the condition

that

### d

= '-90°.### If.

the ship' a### speed is

_{lower than the above, then}

A.QV?o

(B) < 0 and ))Lo (C) ) 0. Again if

### the ship! s speed is

_{higher than the}

1,144

These changes in bending moment

### are caused by heave and

are independentabove, then hi.o (B) > 0 and

### A6hto

(C) < 0 (see Figs. 3b and 5b).- For the case when /L is equal to or greater than 1.00, the ship's - speed at which = -90° is close to the synchronous speed for heave,

- where becomes 900 at the synchronous speed for heave when the wave

- length ratio?%/L = 0.75. Hence the hogging condition

### of

the weight- distribution generally gives a smafl bending moment in the range of lower

- speed than the synchronous one for heave and a large bending moment in the

range of higher speed than the synchronous one, if the weight distribution - is changed under the condition of the constant radius of gyration.

However, the tendency becomes opposite for the case of short wave-length ratio such as = 0.75 (see Fig.5).

Next, we consider the case when the radius of gyration is varied,

keeping the positions of the center of gravity in the fore and the aft

half of the ship constant. For this case the ship's motion is also dif-ferent from the previous one, so the effects of radius of gyration on the bending moment are complex. In other words, all of c, s,

### 4'

c ands in the equation (3.3.3) are varied and q,' is also changed. For

the case when the ship is symmetry about the midship, the second term of the equation including the term of q

' becomes zero, therefore the result only depends on the change in the ship's motion. Generally the ship is not symmetry about the midship, therefore not only the change in ship's motion but also the change in q,' affects the bending moment of ship, though the effect of q1' seems to be smaller than that of ship's motion. According to the result of calculation, a large radius of gyration general-ly gives the tendency opposite to the case when the center of gravity in the fore and the aft half of the ship moves toward the end of the ship. respectively, keeping the radius of gyration constant, and the bending

moment becois large in the range of lower speed than the synchronous one

for heave and it becomes small in the range of higher speed than the

- synchronous one, having a boundary around the speed slightly lower than - the synchrous one. On the contrary, the tendency becomes opposite for

- the case of short wave-length ratio such as = 0.75 (see Fig.6).

By the way, the sagging condition of the weight distribution in

- ships generally results a large radius of rration together with the

- movement of the center of gravity in the fore and

### the

aft half of the ship toward the end of the ship respectively. For such a case like this, the tendency in the above mentioned two cases is superimposed and it isfound that the change in bending moment due to ship's speed becomes slow

(see the condition (B') in Fig.6). This tendency of bending moment. can clearly be observed in Fukuda's model tests (9) and Akita's model tests(]0) on T2-SE-Al tanker.

It may be concluded from these discussions that the hogging condition

of weight distribution in ship is preferable so far as the bending moments of ship among the waves are concerned if one considers the effect of ship's motion on them. It is sure that the weight distribution of ship should be determined also by considering the bending moment of ship in still water,

however at least the sagging condition of weight distribution is

undesir-able and the weight distribution of ship must be slightly in the hogging - condition.

8. Conclusions

In this paper the outline of the theoretical analysis on the bending moment of ship is developed by means of the strip method and the numerical calculation is performed on T2-SE-A]. tanker as an example..

### The

resultof calculation is considerably well confirmed by the tests qualitatively

and approximately coincides with them even quantatively. Although the

1

type of ship' a form used here in the calculation is only one exale of

tanker and it may be too early to, give the conclusion for a genera), type

of ship's form, the foUowings are found as a general tendency of the

i.dsbip bending moment among the regular head seas.

(1) The thaxiimmi hogging (or sagging) bending moment is caused when the

wave crest (or wave

### ?ssed through

near the midship.- () The bending moment becomes maximum when a ship goes among the waves

### having

the wave length nearly equa### to the hi's length.

( ) The bending oment generally increases as ship's speed increases and

reaches the maximum near the synchronous speed for heave except the

case of shorter, wave length than the

### ship' s

length and the case ofextremely high speed.

So far as the effect of weight distribution on the bending moment is concerned, the hogging condition of weight distribution generally

gives a smaller bending moment than in the' sagging condition when the ship's speed is less than that of heaving synchronism, and the change in bertdlng moment due to ship's: speed also becomes slow

### in the case

of liogging condition.The maximum .value of the bending moment in regular head seas can be

obtained approximately from the statical calculation under the condi-tion of the wave length ratio ./L = 1.00,

ship' a motion is taken into consideration,

is nearly equal to the one obtained by the

calculation with the Smith' a correction except an extreme case of weight distribution.

and even if the effect of

the maximum bending moment conventional method of

Acknowledgment:

This study is one part of researches in "Wave Load Committee" of the Society of Naval Architects of Western Japan, whose chairman is

Yoshihiro Watanabe. The authors wish to express our sincere appreciation to the Committee members for their cooperation in our study and Dr. Tadao Kusuda who translates this paper into English.

References

T. Hanaoka: _{"On the Calculation of Ship's Motion among Waves and}
the Bending Moment acting on Ship." Journal of the Society of
Naval Architects of Japan. No.101

_{(1957)}

W. R. Jacobs: "The Analytical Calculation of Ship Bending Moment in Regular Waves." Journal of Ship Research. Vol.2

_{(1958)}

B. V. Korvin-Kroukovsky and W. R. Jacobs: "Pitching and Heaving Motions of a Ship in Regular Waves." TSNAME Vol.

_{65 (1957)}

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)}

### (5.)

.### Y Watanabe:

"On Bending Moment ac tirig upon a Ship among Waves."Bulletins of Research Institute for Applied Mechanics. Kyushu
University. _{No.15}

_{(1960)}

(6) K. Taniguchi and J. Shibata: "Model Bcperinients on the Wave Loads of T2-SE-Al Tanker in Regular Waves". Mitsubishi Exp. Tank Report

### No. 357 (1961)

to be presented to the I.S.S.C.(7) F. Tasai: "On the Damping Force and Added Mass of Ship's Heaving and Pitching" Reports of Research Institute for Applied Mechanics. Kyushu University. Vol. VII, No.

### 26 (1959)

1,144

2

F. Tasai: "Damping Force and Miss of Ship's Heaving and Pitching (Continued)" Reports of Research Institute for Applied Mechanics, Kjushu University, Vol. VIII, No.

### 31 (1960)

J. Fukuda "Model Test of T2-SE-Al on Midship Bending Moments in

Regular Waves" (1961) to be. presented to. the I.S.S.C.

Y. Akita and K. Goda: "Experimental Determination of Bending Mont.

for T2-SE-Al Tanker Model in Regular Waves"

### (1961)

to be presented, to the I.S.S.C.### (1)

Coefficients in### the Equations

of### Motion

In the equations of motion

### for heaving and pitching motions

a

### ,+

b + c +### d4'

+ e4' + g FA

### 4'+

### B+ C

### D+ E

Nthe coefficients are given as follows;

1,144

### a

=### f

### J,

### b

### Appendix

### c=

= e =### g

=### L,L

_{\}

### The result of calculation by

### the values of the added rtass

s';

of section N.

### SiiIlar1y the

### A +

B C_{=}

### VE

### D=

d### -[

### G=

### flJ4rX

Tasai (7),(8)

### can be used

to### obtain

### of section

### ?

### and the damping coefficient

### exciting forces can be calculated by

}

### (2.3)

### Tasai' s method as shown in the following.

### The ranges of integration

### in the above equations are taken

cver### the whole length of the ship.

### (2)

### Exciting Force

### The terms of the exciting force F and the exciting moment N

### in the right hand side of the equation of motion (2.3) are given

### by the equations (2.4) as

### Fc\

### (C..odO(F\

### f1'f1-s- f3\

### 11

### ,1L0

Fs### /

### \ --'F /

4-_{+ 3 j}

### )

### Mc \

### (

C.i.)### M '

### f

+ 'Th3### Ns)

\ e.o### where

### I?

fAAS1-fU:\### e.

### ;

The exciting forces are given as follows when

### only the terms of

### 4

'Yfl and (P in the equations (2.4.2) are used; F". = F"

### Wat -

### F"tg,t

F0"### t

1.144 I, Ti### -f3

J3### )

### I

### N

1.. L### (

### -f -k

L### (si&(\c;&

### ff3,

### fAqe,(

(Th1'and 'Th,' for this case, the exciting forces yield 1

### je

_{L}

### )

1

These exciting forces correspond to those obtained from the so-called Frôude-Kriloff's asswnption, and in this case the

### factor

### in the

terms### f, f,',

Ii arid 'V't,' corresponds to the Smith's correction.### Furthermore, substituting that

### em = 1

into the### equations of

### f,

I I I MU =

### where

### c&e

### _frl;'A

Fç" / C.OQOF### =

F0 Fstt. Mct'### =

Ms"Where '1,

### F=

"I### M=

U, Fe ) I,, FsThese exciting forces in the above equations correspond o the' case without the riith' s correction, and the term f *, f *, in and

### in1' *

corresponds to f,

1'' and m1' respectively with the condition

that = 1.

### (3)

Bending MomentThe calculation of the. bending moment can be conducted by substituting the solution of the equation of motion into the equations (3.3.3). The coefficients p, q and r in the equation (3.3.3) are easily obtained from the coefficients a to g and A to G in the equations of motion in the Appendices (1) and (2) together 'with the calculation of the exciting forces f and in. Namely,

### 1jr4J

### -

### Jd

### -=

= L### -

### F"'AL'

### F0"c&" (t

### -coWt M'Atc)et

I,, ,,,### / CøO

### = F0

### X,

### =

)### -

0 . Cb'### (')et

/### /(pt1

### (

)11'### *

### iere

1,144### L'L

### 4xJ

### =

### _.vJ'4

### (4)

### Statical Calculation

### f

### ,:./c.oL\

(1### /cO.?(

### ae.

### r(.

### =

1### _k(X)

_{-,i}

2
### where theintegration is carried out

### from A.?. to

### or from

### F.P

### to

The

### statical displaceunts

### are calculated by using the exciting

### forces (F" , N't) or (F ", N") in

### the

### Appendix

### (2)

### as

### =

_{'A. ; -}

i& ### tc2..

.### (4)

-### '

S### fc.o?e.i..

_{\}

### I

_{/}

### e.

### :t

### j",j

:### -(

### :

### (.&'o'

### '\

Mc frJ5.### Ii?z4(%i5

3 cL%)) x:### ç UA? (4t(

### cJiL

C.84_{.7()}

### -

### z: \ e.e fQy*)

### -

### i "

_{T}

_{(}

### t

### *'

### (3)

### or

### where

and### or

### where

t I 1,144 II,### a

### =

### CO -'e-t

### -

### (we.

### t

+ 1 'I,### -=

### øø &

### -

"'### L)g't E

### c.ai

### t

### .* (24 )

### "(

### C-,

### f

C9.( J=C.)### SLr2

### From these coefficients the bending moment obtained by the statical

### calculation is given by

### -

### co

### (t

### where

### =

Q### {Jxd_

### r

### 2fl

### Qj:A

1.. I### 4

### ('I,,

### tOe-' )

The range of the integration is from A.P. to or from F.?. to

### 2g.

Nomenclature

O-XYZ Co-ordinates fixed to the space

Go Co-ordinates fixed to the ship

### L =

:### Ship's

length B*_{Ship's bredth}d* : Ship's draft

V :

### Ship's

speed### A.

Wave length### -&

, : Elevation of' wave, amplitudeNumber of waves

u-...

### I-/

:### Speed of wave

pen-Acceleration of

### gravity

### =

### k ,j-

Frequency of wave= (yj v-) : Frequency of encounter of the ship and waves

### F, Fo, o&

_{F}

: ### Exciting force, its

amplitude and### phase

angle### of heaving

N, Mo, _{(3M} : Exciting moment, its amplitude and phase angle

of pitching

### ', C'

: Displacement, its amplitude and phase angleof heaving

4), o,

### 1S,

: Pitching angle, its amplitude and phase angleI)fl, ' : Bending moment, its amplitude and phase angle

I I I

1, 144

I-Table

### 3

### Main particulars of T2-SE-A1

hip (ode1

Scale ratio 1 1/30.66

### Length between parpendiculars, L

153.31 m 5.óOO m Breadth, B 20.73 a 0.676 mDraft, D. 11.96 m 0.298 m

Displh,cement 21,770 ton 745.3 kg.

(salt water (fresh water)

### Block coefficient

0.741### Watérplane area coefficient

0.823### Centr& of buoyancy from midship

0.004 L fore### Centre of floatation frOm midship

0.014 L aftTable 2

### Main particulars of model in loading conditions

Note: Full load and even, keel for all conditions.

311

Loading conditions (A) (B) (C) (A') (B')

### Radius of

ration in % of L 22.2### 23.6

### Weight in %

Af te rbo dy### 49.6

### 49.6

Forebody### C.G. from

in % of L 50.4### 50.4

### Afterbody

-18.3### -19.3

### -17.3

-18.3 -19.3 Forebody. 18.8 19.8 17.8 18.8 19.8### Loading condition

### Heaving

Calculation Experiment### Pitching

Calculation Experiment Table 3### Natural oscillating periods

(A), (B), (C)

1.35 sec. 1.36 sec.

1.19 sec. 1.26 sec.

Note:

### Experiments were made for (A), (B) and (C) condition.

### 1.35 sec.

### : (o()

_&I,### I

### 'r0

### ft)

### c: ( s")

### '' (

### CX;)

### c: (

### cr'

### Note:

### c(,

### fi, 8

### in degree.

### Table

### 4

c1 (9### 0.75

### Statical calculations

### 1.00

### 1.25

### 1.50

### 0.079

### (160.2)

### 0.121

### (

### 8.1)

### 0.299

(### 2.5)

### 0.430

(### 1.2)

### 0.106

### ( 78.7)

### 0.303

### ( o3.7)

### 0.457

### ( 85.6)

### 0.563

### ( 87.3)

### 0.0236 ( -5.2)

### 0.0239 ( -.o)

### 0.0200 ( -2.0)

### 0.0161 ( -0.9)

### 0.081

### (155.0)

### 0.193

### (

### 9.5)

### 0.406

### (

### 3,7)

### 0.554

### (

### 2,3)

### 0,174

_{( 83.5)}

_{0.418}

_{( 85.2)}

_{0.585}

_{( 86.1)}

_{0.693}

_{( 86.7)}

### 0.0360 ( -4,5)

### 0.0329 ( -3.9)

### 0.0260 ( -3.4)

### 0.0202 ( -2.5)

### 40

### 0.75

### Tab1e

5### 1.00

### 1.25

### 1.50

### c.

### 0.653

### 0.728

### 0.770

### 0.810

CI### 0.633

### 0.725

### 0.780

### 0.817

CI_{Oz}

### 0.384

### 0.531

### 0.674

### 0.800

_CI C2### 0.407

### 0.520

### 0.600

### 0.655

### + Z,z,,tj',F

### I-,M +v'

### + X,z,V

'I $ 1### 04

### oz

### C

### EX.C/ T/AIGFORCE

### AND:.

.### MOHENT

### A A"!

### .

L.'!L_?### _

### . .

### -,_

### _

### -/

### F.

### -PHASE AN6LE

### -7

### cf

### =0.15

### o

### 0.! &._- 0.2

### 0.3

### 42.

### 43

### F;.

### a

### b

### EXCITING FORCE

### AND

### MOMENT

### = 1.00

### I

### AA ')

### -

_{F0 M0}

### __ -

### __-- __--

### __- __

### - - - -

0### ill

### -14

### ro

### -

### -

### -

### -F.

### /80

### o

### -

### J8O

### PHASE AIs/LE

### 0

### 0.1 Fr._- 02

### 0.3

### 44

### EXCITING FORCE

### AND

### MOMENT

### I.25

### o.4

### -

_{- ___}

### __

### __

### -

### - ___- ___

### 02

### U

### M

### r0

### 1800

S### c%,ft1b,

### 0

### -I80,

### PHASE ANGLE

### F

### - - - - -

### -0

### 0,lfr, s-az

### 03

### t

### 02

### 1800

### 0

### -

### 18O

### EXCIT:INQ

### FORCE

### A ND

### MOMEN r

### =

### f_SO

### PHASE ANQLE

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