I FEB. 913

### ARCHIEF

### Lab.

### v. Scheepsbouw6c

### Technische Hogeschøol

With the Compliments of_{tDeUithors}

### Theoretical Calculations on the Motions, Hull Surface Pressures

### and Transverse Strength of a Ship in Waves

By

Jun-ichi FuKuDA, Ryu-ichi NAGAMOTO, Mamoru KONUMA and Minoru TAKAHASHI

### ';4t e7p1,

### L

c&&Lj### j\Jj1

Reprinted from the Memoirs of the Faculty

### of

EngineeringKyushu University, Vol. 32, No. 3

FUKIJOKA JAPAN

### 1973

Contribution to

5th International Ship Structures Congress, Hamburg, 1973 DOCUPIEN liotheek van de

### /10

### Onera eing e

uw undeSc e_Hogeschoot,

0.CUMENTATIE

### Theoretical Calculations on the MOtions, Hull Surface. Pressures

### and Transverse Strength of a Ship in Waves*

By Jun-ichi FUKUDA ** Ryu-ichi NAGAMOTO *** Mamoru KONUMA and Minoru TAKAHASHI (Received Oct. 5, 1972) Summary

The authors tried the theoretical analysis on the motions, hull surface pressures and transverse strength of a gigantic oil tanker in regular waves.

In the first place, the ship motions in regular waves from different directions were analysed by assuming the coupled equations of heaving and pitching motions and those of swaying, yawing and rolling motions based upon the modified strip

theory The non linear roll damping was introduced into the latter coupled equations of motion.

In the second place, the hydrodynamic pressures induced on the hull surface

were evaluated theoretically by using the solutions of ship motions and the dynamic.

pressures of cargo oil due to the ship motions were approximately estimated. Finally, the transverse strength calculations were performed for several cases of the ship in waves by using the obtained hull surface pressures and the estimated

cargo oil pressures.

The large hydrodynamic pressOes were found in beam and quartering waves,

which produced the large bending moments on the bilge corner and the deck

corner of weather side and also on the bottom corner and the top corner of leeward side longitudinal bulkhead in the transverse ring structure of midship section or

S.S. 7-- section. Contents 1 Introduction 2 Calculation Methods 3 Calculated Results 4 Conclusions Acknowledgement References

Appendix 1 Hydrodynamic Coefficients and Exciting Forces and Moments in

Equations of Motion Appendix 2 Non-Linear Roll Damping

* The original paper was published in Japanese in the Journal of the Society Of Naval Architects of

Japan, Vol. 129, June 1971. ** Professor of Naval Architecture.

*** Nagasaki -Shipyard, Mitsubishi Heavy Industries Co. Ltd. **** Head Office, Mitsubishi Heavy Industries Co. Ltd.

186 3. Fti; R. NAooTo, M. KowMA and "M TAxAsrn

1. Introduction

The ship motions in waves are closely related to not only the sea-keeping problems, but

also the structural sea loads Significant advances in the theory of heaving and pitching

motions have led to the statistical approach of sea loads such as the vertical wave shearing force and bending moment by the aid of the linear superposition technique and the ocean

wave statistics. A number of different methods of predicting' the long-term distributions, of

vertical wave shearing force and bending moment have been proposed . and employed during

the last ten years in order to determine the design values of such sea loads. Such methods

will be available for determining, the design values of lateral wave shearing force, bending

moment and torsiOnal moment by the aid of the theory of swaying, yawing and rolling

motions.

Recent experiences, meanwhile, have shown that the study on the local or transverse strength of large bulk carriers, ore carriers and oil tankers' is a matter of pressing necessity. Many computer programs are now available for the transverse strength analysis by 'means

of two-dimensional and three-dimensional methods, but little accurate information on the

local' or transverse sea loads as the input data to those programs is available. It is evident,

therefore, that further research into the local or transverse hydrodynamic loading is necessary. Several works 1-5) including theoretical and experimental ones on the hydrodynamic pressures

induced on the 'transverse sections of a ship in longitudinal waves and in beam waves have

been already reported, but the informatiOns from those works are not sufficient to learn

the character of local or transverse hydrodynamic loading.

### In this report

therefore, a method to analyse theoretically the motions, hull surface pressures and transverse strength of a ship in regular oblique waves is proposed for the purpose of investigating the character of local or transverse hydrodynamic loading, and the 'method of such analysis is applied to a gigantic oil tanker in regular waves from differentdirections.

Supposing that a ship goes forwar in regular waves with a constant speed and a. constant

average heading angle, the ship motions can be solved by assuming the coupled equations

of heaving and pitching motions and. those of swaying, yawing and rolling motions based

upon the modified strip theory67>... The former coupled motions are equated as linear motions and the latter as non4inear motions by introducing the non-linear roll damping.

When the solutions of ship motions 'are obtained, the hydrodynamic pressures induced on

the hull surface can be evaluated theoretically according to Tasai's thethod2'3 and the

dynamic pressuies of cargo oil due to the ship motions can be approximately estimated.

Since 'the dynamic loads acting on a transverse hull section can be evaluated in this way,

the transvre. strength calculation can be made for the ship in regular waves. The more

precise method such as the three-dimensional strength calculation will be available, but the

simple two-dimensional method is adopted here for the transverse strength analysis of a

gigantic oil tanker.

2 Calculation Methods

2. 1 Ship Motions

Consider the case when a ship goes forward in regular waves with a constant velocity V

and a constant average heading angle X. As shown, in Fig. 1, the coordinate system O-XYZ

is employed 'such that XY-plane coincides with the still water surface and Z-axis indicates

the downward direction perpendicular to X Y-plane. The other coordinate system O-X Y1Z1

is determined so that X1-axis coincides with the average course of the ship oscillating among

system o-xyz fixed to the ship is determined

such that the origin o locates at the midship on the centre line of water plane and x-axis points out ahead the longitudinal direction.

The ship goes forward with a average heading angle x against the wave direction. Then

the vertical displacement of the encountered

wave surface will be written as follows.

h = h0cos(kX - O)t)

h0cos(kxcosx - kysinx - Wet) (1)

where

h0: wave amplitude, k = co2/g= 2ir/A 2: wave length,

g: acceleration of gravity &: wave circular frequency

= CO kVcosz: circular frequen-cy of wave encounter V: ship velocity,

z: average heading angle As for the subsurface of deep sea waves, the vertical displacement can be written as

dFB,j dx dFBZ2 dx

### - 2pgy,L,{C(xxG)q},

### - pN{(xx6)ç!+Vcf},

+Z1,Z ,h Fig. 1. Coordinates. I- Y, + x, dF1,1 - 2pgy,,h dx### dF2

- pNzvge dx + Ylh(z) = hoe_kzcos(kxcosx - kysinx - wet) (2)

and the vertical orbital velocity and acceleration, as

v(z) = wh0e sin(kxcosx - kysinx - Wet) (3)

z) = - Co2 hoe_kzcos(kxcosx - kysinx - Cost) (4)

and the horizontal orbital velocity and acceleration in the transverse direction, as

v5(z) = Wh0sinZecos(kxcosZkysinXCo,t) (5)

i'(z) Co2 h0sinxesin(kxcosZ - kysinx - coat) (6)

When the ship goes forward oscillating in waves, the hydrodynamic forces induced on the

ship cross section can be expressed as f011ows by taking into account the influeTcés of heave, pitch, sway, yaw and roll based upon the momentum theory, where surge and drift are

ignored.

a) vertical force (downward: positive)

dF, dF8,1 _{+} dFB2 _{+} dFB, _{+ dFB4}
dx dx dx dx dx
+

### dF1

+### dF2

+### dF3

+### dF4

dx dx dx dx where (7)188 J. Futrim, R. NAGAOTO; M. KONUMA and M. TAKAnASHT

dFBSS

### -

-(x

### -

xG)+2V}, - PSzzedFBr4

_{{(xxG)ct±VØ},}

dF124 _{v}

dx dx dx dx

p: density of sea water

Yw half breadth of water plane

xG: x-coordinate of the centre of gravity.

pN: sectional damping coefficient for vertical motion psi: sectional added mass for vertical motion

he = C1C2h = C1C2h0cos(kxcosx - coat)

C1Ccoh0sin(kxcosx - cost) - C1C2c2h0cos(kxcosZ - wet)

C1 = sin(kysinx)/ky,sinx

C2 = exp( - kd), d. = (sectional area)/(2y11.)

b) moment about the centre of gravity due to vertical force (zx-direction: positive)

dF._{(x}

xG)

dx dx

C) horizontal force (starboard: positive)

dF dFBUU _{+}

### dF,2

_{+}dF83

_{+}dFB4 dx dx dx dx dx +

### dF1

+### dF2

+### dF3

+ dx dx dx dx where dFBU1### -

dFBU2 dx dx dFB3 dxdFBU4

_{- v}

d(psU) _{{ ±}

_{(x - xG)ci' - Vçb +}

_{(ZG - l)} - VpsU}

dx dx - -

### dx.

dF01

_{-}

_{2pgho{\ exp(}

_{-}

_{kz:)sin(kyssinX)dzz} sin(kxccisz}

_{- coat)}

dx 0

### dF2

_{Nv}dF1,

_{-}

### dF4

_{-}

_{v}

d(ps)
dx " dx ?

dx. dx

pN.: sectional damping coefficient for horizontal motion ps: sectional added mass for horizontal motion

l: lever of sectional damping force due to rolling motion with respect to o'

which is the projection of c on the transverse section

l: lever of sectional added mass inertia fOrce due to rolling motion with respect to o' which is the projection of o on the transverse section

d: draught of the section

y3: ycoordinate of the section contour z8: z-coordinate of the section contour

V,Je coh0sinXe'2cos(kxcosX - cost)

### - ot)

- psU{71 + (x x)çb - 2V± (ZG - l,)O}

(8>

-moment about the centre of gravity due to horizontal force (xy-direction: positive)

dM

_{-}

### dFv()

(10)

dx dx

-transverse rotating moment about the centre of gravity (yz-direction: positive)

dM2 dMBol _{+} dMBB2 _{+} dMBOS _{+} dMB84
dx dx dx dx dx
+ dM91 + dM + dM93 + dM04
dx dx dx dx
where
dMBl _{- w(zG' - zG)O - pgs'n'}
dx
dZVJ2

- pIsjz6 - l) { + (x - x)q - V/i + ZGO} + pNl(z0 - l)O

dIVI3 _{- ps(z9 - l,7)j + (x - xG) - 2V + zJ} + psl,,(z0}

w: sectional weight of the ship

za: z-coordinate of the centre of gravity of ship

z': z-coordinate of the centre of gravity of w

s': sectional area, me': sectional metacentric radius

4: lever of sectional added mass inertia force due to horizontal motion with

respect to o'. which is the projection of o on the transverse section (that is

equal to the lever of sectional added mass inertia force due to rolling motion with respect to o')

4 = pi/ps,l,,

p1: sectional added mass moment of inertia with respect to o' 11 = (mIjmv.)/ffl = exp( - kz3)sin(ky3sinx)z,dz 0 rvw my = exp( - kz)sin(kysinz)y4y3 Jo fH = 0exp( - kz8)sin(ky3sinx)dz3

The coupled equations of heaving and pitching motions and those of swaying, yawing

and rolling motions can be obtained by putting as f011ows,

### !=r Ldx

_{JWL dx}

### dMxd

g WL dx dMBa### d{ps(zGl)}

_{+(xx6) V+z6O)} v}

d{psV4(zQ-4)}
a
dx dx dx
dM91 ### dF1

_{}

-(ZG _{i),}dMwø2 dF52 (ZG

### -

,) dx dx### dM3

dFw8 (Z dx dM94 dx V### d{ps(z-4)}

4), dx dx dx dx Ve } (12)190 J. FJXUDA, R. NAotoTo, M. Kocu and M. TAKAHASIII

### W...

### c.dF,

_{dx}JWL dx

### _!=S

dMxydx g WL dx### ±

dx### widx

(13>where the integrations should be carried out from the after end to the fore end along the

water line length, and

Wig: mass of the ship

4ig: moment of inertia of the ship for pitching motion Iig: moment of inertia of the ship for yawing motion

I9ig: moment of inertia of the ship for rolling motion

a31 + a32 + a33i + a34? + a3 + a364!i + a37O + a356 + a390 = M9 J

A11, Al2, - - -, A21, A22, - - -, F and M.1, in Eqs. (14) can be determined by using the

added mass and damping coefficient according to Tasai's method8) and _{a11, a12, - - -, a21,}

a22, -' - -, a31, a32, - - -, F,,, M9 and M in Eqs. (15) by using those according to Tasai's

method or Tamura's one'°>. _{Those methods are based upon the linear two-dimensional}

theory. The roll damping coefficient a in Eqs. (15), however, should be determined by taking into consideratiOn the non-linear viscous damping in addition to the linear wave making resistance. Here the non-linear roll damping, is introduced by using "N-coefficient"

according to Watanabe-Inoue's method'12 and also the speed influence, which is assumed

according to the model experiments carried out by Takahashi13), is taken into consideratioa

as follows.

a'8 = 2aa37{1+0.8(1e10) (co,,/o,)2} ' (16)

where

2aL = (2iir)co,,{a1 + bj(co,,ico)Oo}

a),, = \/ a39/a37

O: rolling amplitude, Fr: Froude number

a1 and b1 are the coeffiëients in the formula of extinction curve for the free rolling in stilt

water as follows.

40 = a10.,,, + b10,,

On the other hand, AC will be approximately given by follows.

### 40=N0,,

For the cases of Cm = 100 and °m = 200, N10- and N20- can be estimated according to Watanabe-Inoue's method. Then a1 and b1 will be determined by the following equations.

N10- = (a1/10°) + b1 N20- = (a1/20°) + b1

Eqs. (12) and (13) can be rewritten in the following forms.

A11 + Al2 + A13c + A14 + A1 + Aj6ç = F

A21 + A22 + A23c ± A24 + A25ç + A26 = M,,

a112 + a12i + a13 + a14c1 + a15ç1' ± a16b + a17Ô + a186 + a190 a21j + a22 + a23,i + a24ç1 + a25ç1' + a26çb + a27 ± a28O + a290

### = F

### = M

'(14>

Thus the non-linear roll damping is replaced by the equivalent linear damping coefficient a8 and Eqs. (15) should be solved by the iteration method.

The calculating formulas of hydrodynamic coefficients and exciting forces and moments in Eqs. (14) and (15) according to the linear theory are described in Appendix 1, and those for N10 and N20 in Appendix 2.

By solving eqs. (14) and (15), heave, pitch, sway, yaw and roll will be obtained as follows.

C = C1 cos (Wet C)

### = 0cos(wte)

= '2o COS (Wet - (17)

= 41'0cos(W9te) 0 = 0 cos (wet

### e)

where

CO3 qSo, v, and 00: amplitudes of heave, pitch, sway, yaw and roll

### c, c, c, c

and C9: phase angle of heave, pitch, sway, yaw and roll2. 2 Hydrodynamic Pressures

When the solutions of ship motions are known in the forms of (17), the hydrodynamic pressures induced on the point (y8, z) of the section contour which is distant x from the

midship can be evaluated theoretically according to Tasai's method 2,3) in the form of

P= P0cos( Wet c) = PcCOSWet + PaSflWgt (18)

This pressure will be expressed as follows.

P=Pv+PH+PR+Pw (19)

where

P: pressure due to vertical motion

PH: pressure due to horizontal motion with respect to o' PR: pressure due to rolling motion with respect to o'

P; pressure due to regular wave

Pv = pgh0 Pcoscot + F3sjnWt} PH _pghO{PHCcOSU)Ct + PHSinWet} PR = pgh0 {ccoso>et + FRssinwet} Pw = pgh0 {Pcoscot + Fw8sinwt} P'-c 1 - 1c0sc 1 1 sincc

### -(1+P)

Pve i h0 lsinCc J I cosc

cosC.i

_{-}

sinC
-- (x -- XG)° (1 + P
- I coscJ
I sin c'l Icosc. 'I
(V/W)-°- 2P ### +P

### (coscJ

Isinc J### -

fcosc) I sine -D"J 1_ D" ( 'aS) ( '98 Pff3J Isincqj### cosC

(22>o _{coscJ} _{sinC}

-tsin(kxcosx - ky8sinx)J

Accordingly, P, P8, P0 and e can be obtained as follows.

= pgh0(P + Pff + Ps + Pvc)

P3 =pgh0(Pv, + P,s + P3 + Pwa)

P0V'P32±P32 = tan

### The calculation methods of P, PCI, P, P, P

and P,### are given in detail by

Tasai 2.3,8,9)

2. 3 Dynamic Pressures of Cargo Oil due to Ship MotiOns

When the solutions of ship motions are known in the forms of (17), the hydrodynamic

pressures of cargo oil in a tank due to ship motions can be estimated approximately.

Con-sider the case, for example, when the centre tank is filled up with cargo oil and both side tanks are empty Assuming that the motion of a particle of oil would be just similar to

that of ship body, namely, there would be no relative motion between a particle of oil and

the tank, the dynamic pressures due to ship motions would be approximately given by

follows in the transverse section shown in Fig. 2. a) pressure due to vertical motion

The dynamic pressure is in proportion to the vertical acceleration and to the depth of

tank. On the bottom of tank, the dynamic pressure is given by follows.

### -

Icoscl sineP

### H P

+

(X_XG)I-o _{sinCJ} _{I. - cosc}
I SflCç Icose

_{D")}Icoseo'l laS) sinc6J I

### _D"

-sine6 cosc8### Zg

oPR3 Icose0)

### -

1cose9 I Sine8J+Yw PaR PdR

h0

PR. sin C9 sjn c0J - cose

Pwc Icos (kxcosx - ky,sinx)

- exp( kz3)1

Pws sin (kxcosx - ky3sinX)

### - exp( - kz) (/o)2P

fcos(kxcosx - ky3sinz)tsin (kxcosx - kysinX) + exp( - kZ,) (&/co)P I

sin(kxcosz - ky3sinZ) L. - cos(kxcos - kysinZ)

I sin(kxcosX - ky,sinZY

+ exp( - kz) (o/o)2P 'sinZ

I. - cos(kxcosx ky3sinz)J

Icos(kxcosx - ky,sinxY

+ exp( - kz3) (o/o) P sinX

4p1.= _Pcd2':

### pC{C(XXG)

(29)where

Pc dnsity of cargo oil, a; depth of centre taik

Z: vertical acceleration

pressure due to horizontal motion

The dynamic pressure is in proportion to thè horizontal acceleration and to the horizontal

distance from the tank centre line. On the longitudinal bulkhead, the dynamic pressure is

given by follows.

4p2

### p(b/2)Y= p(b/2){7+(xx)'}

(30)where

.: breadth of centre tank, Y: horizontal acceleration

pressure due to rolling motion

For the practical purpose, it is sufficient to consider only the increase of pressure due to heel angle, because the influence of rolling acceleration amounts to only J0'20 percent of

the former 14) The pressure increase due to heel. is in proportion to the rolling angle and

to the horizontal distance from the longitudinal bulkhead of the opposite side to heel. On

the longitudinal bulkhead of the heeled side, the pressure increase is given by follows.

4p3=pgbjOI . (31)

2. 4 Transverse Strength Calculation

The hydrodynamic pressures induced on the contour of a transverse hull section and the dynamic pressures of cargo oil on the inside of centre tank can be evaluated by the methods

Cc) p=pgSIeI

A

IN LONGITUDINAL WAVES

IN OBLIQUE WAVES

A

Fig. 2. Fluctuations of Cargo Oil Pressure due to Fig. 3. Support Conditions for Transverse Strength

194 .... j. FUKUDA, R. NAGAMoro, M. KONUMA and M. TAKAsHI . - -

-described above. Hence, the dynamical, loads on a transverse section of-ship body can be determined at any time dunng an encountered period in regular waves Under such a load condition, the two dimensional transverse strength calculation can be performed by assuming

appropriate support conditions. The more precise method such as the three-dimensional

strength calculation will be applicable, but here the simple two-dimensional method is applied

under the support conditions as shown in Fig. 3.

The transverse load distribution on a section of 'ship body is symmetric in lOngitudinal

waves but not in oblique waves. Therefore, two kinds of support condition are supposed

for those cases as shown in Fig. 3. Namely,

in longitudinal waves:

-deck centre and bottom centre are fixed,

bilge corner and bottom of longitudinal- bulkhead are supported such that -the vertical

displacement is constrained . and free otherwise. '. -

-in Oblique waves:

deck centre is supported suëh that the horizontal displacement is. constrained and free etherwise,

- bottom centre is supported such that the vertical and horizontal, displacements are

- constrained and free otherwise,

each bilge corner and each bottom of longitudinal bulkhead are supported such that the vertical displacements are constrained and free otherwise.

Under such support conditions, transverse loads are determined by superposing the static water pressures and the hydrodynamic pressures on the hull surface and also by superposing the static cargo oil pressures and the, dynamic oil pressures on the inside of center tank.

The method of transverse strength calculatiOn is based on "Stress Analysis of Plane

Frame Structure by Digital Computer" by Fujino and Ohsaka

### '.

3 Calculated Results

According to the methods desthbed in the preceding chapter, the calculations on the motions, hydrodynamic pressures and transverse strength were carried out for a gigantic oil tanker in regular waves from different directions. Main particulars of .the ship are

shown in Table 1.

Breadth/Draught (B/d) 2. 7191

Bloêk Coefficient (Cb) 0. 8403

Centre of Gravity from Midship (x0) 0. 0326L

Centre of 'Gravity from Water Line (z0) 0. 3202 d

Metacentric Radius (GM) 0.4264 d

Longitudinal Gyradilis (r) 0. 2336L

Transverse Gyradius (r) 0.3469 B'

Table 1. Main Particulars' of a Gigantic Oil 'Tanker.

Length between Perpendiculars (L) 310. 0Gm

Breadth Moulded (B) 48.40m Depth MOulded (D) 23.6Gm Draught Moulded (d) 17.80m Displacement (W) Length/Breadth (L/B) 230, 048 t 6.4050

3. 1 Ship Motions

The ship motions were investigated for the following cases.

Heading Angle

x = 0, 45, 90, 135, 1800 (x =00: _{following waves)}

Ship Speed

Fr=0.10, 0.15 (Fr: Froudé number)

Wave Length

= 0.3iS (L: ship length, 2: wave length) Wave Height

H =10 m (H = 2h0: wave height)

In the coupled equations of heaving and pitching motions (14), the hydrodynamic coefficients and the wave exciting forces and moments were evaluated by using the sectional added

mass and damping according to Tasài8)

And, in the coupled equations of swaying, yawing and rolling motions (15), the

hydrody-namic coefficients except a38 and the wave exciting forces and moments were derived by

Tamura's method 10) where. the influences of ship speed were not introduced into the wave

exciting forces and moments. The roll damping a35, was estimated by Eq. (16) including

the non-linear viscous damping and the influence of ship speed. Hence, Eqs. (15) should be

solved by the iteration method.

0 300 IO I.O- 20 5-0.5- 0 '10 w0 e0

### HI

L = 300M AlL = X = 90° Fr. 0. 15 0 5 0 15 -..-WAVE HEIGHT 20Fig. 4. - Amplitudes of Non-Linear Motions as Functions of Wave Height.

Preliminary calculations on the swaying, yawing and rolling motions were made for a

similar oil tanker of 300 meter length in order to examine the effects of non-linear roll

damping. The calculated results are shown in Fig 4, where the effect of non-linear roll damping is obvious in roll but not in sway and yaw. Other calculations on the swaying,

yawing and rolling motions were carried out for a similar oil tanker of 310 meter length

for the purpose of investigating the influences of non-linear roll damping and of ship speed,

and the results are shown in Table 2. According to those results, the influence of non-linear

roll damping is significant when the rolling amplitude is large and the influence of ship speed is moderate.

shown in Figs.. 6-40. The large rolling amplitudes are found in beam and bow waves. 3. 2 Hydrodynamic Pressures

By using the solutions of ship motions, the hydrodynamic pressures induced on the contour of transverse hull section were calculated for the following cases.

Heading Angle x =0, 45, 90, 135, 1800 Ship Speed Fr=0.10, 0.15 Wave. Length = 0.50, 0.75, 1.00 Wave Height

### H=l0m

Transverse Hull Section

Midship section and S.S. 7.-b-. section (0.25L forwards midship)

The calculated results of pressure amplitude are shown in Figs. 11-49.

The amplitude of hydrodynamic pressure is large on the hull side, especially on the water

### line, and is small on the hull bottom.

The large hydrodynamic pressures are found in beam waves and, following to that case, in quartering .and bow waves on the weather sidewater line. In beam, quartering and bow waves, a little difference is found between the

pressure amplitudes on the midship section and 'those on S.S. 7 section. In head waves,

however, the pressure amplitudes on .S.S. 7 section are generally larger than those on the

midship section, and, in following waves, the pressure amplitudes on S.S. 7 section are less

than those on the midship sectjon.

For the cases when the larger hydrodynamic pressures are found, the fluctuations of pressure during an encountered, period in waves are shown in Figs. 20-27, where the pressures include

the still water pressure and the hydrodynamic pressure.

According to the calculated results shown in Figs. 20-.27, the water surface on the deck

side exceeds. the deck edge when the hydrodynamic pressure becomes maximum. For example, in Fig. 21, 3.5 meter water head on deck is found at t = 30°/cog in head waves, and, in Fig. 22,

13.0 meter water head, on the weather side deck is found at t = 60°/o, in beam waves.

ilL Fr . . Rolling Amplitude in Degree

'(a) (b) (c) 0.7 0.10 13.9 ''' ' 13.9 - 13.6 0. 15 . 13.8 13. 8 13..5 1.0 0. 10 58. 3 36. 8 30. 4 0. 15 . . 58. 2 36. 7 28. 4 1.5 0. 10. 15.6 .. 15.6 ' 15.4 .0. 15 15.4 . 15.6 15.4 (a) (b) (c)

Calculated by using linear roll damping without speed influence, Calculated by using non-linearroll damping without speed influence, Calculated by using non-linear roll damping with speed influence

in.regular beam waves. (Wave. Height=i/20) .

-196 j. Fuuri, R. :NAOAM0To, M. KOX1JMA and M. TAKAHASHI:

3. 3 Transverse Strength Calculation

The transverse strength cakulations were performed for the cases when the large

hydro-lynamic pressures were induced on the su.face of hull section.

-The distribution of transverse load was determined by taking the water pressure induced

on the hull 'section contour at the time when the pressure on the weather side water line took the maximum value and also the cargo oil pressure in the centre tank including the

static pressure and the dynamic pressure due to ship motions at the same instant. Both

side tanks were assumed to be empty and the centre tank to be filled up with cargo oil. The load on deck was assumed to be linearly distributed from the weather side to the

other side by taking the water head in excess of the freeboard.

By assuming such transverse loads, the two dimensional strength calculations were carned out for the hull section with ring structure under the support conditions as shown in Fig 3 -which were supposed for the cases in longitudinal waves and in oblique waves.

The calculated results of shearing force and bending moment in still water are shown in

Fig. 28. And the shearing forces and bending moments calculated in waves are shown in Figs. 29-35.

According to the results of transverse strength calculations shown in Figs. 29.35, the following facts are- found.

The large shearing forces and bending moments are found in beam waves vihere the iiydrodynamic pressures are large and the deck load due to the water in excess of the

freeboard amounts to significant level. Following to the case in beam waves, the large

shearing forces and bending moments are found in quartering, bow and head waves. In beam, quartering and bow waves, the larger bending moments are generally found on

the bilge corner and the deck corner of weather side and also on the bottom corner and

the top corner of leeward side longitudinal bulkhead.

In head waves, the larger bending moments are found on the bilge corner and the deck

.corner.

Such trends as described above are found from the transverse strength calculations applied

to S.S. 7- section and to the midship section. As to the sections of forward position such

.as S.S. 8, 8 and 9, the more large shearing forces and bending moments will be found in quartering, bow and head waves.

Several-examples of the transverse strength calculation are reported here and the shearing

forces and bending moments are shown in Figs. 29-35. Therefore, the stress calculations

cn be made. easily for the parallel parts of the ring structure, and can be performed for the corner parts by the aid of curved beam theory or finite element technique. But such

'.analysises are left for another occasion.

4 Conclusions

A method to analyse the motions, hull surface pressures and transverse strength of a ship

in regular waves has been proposed here -together with examples of calculation. This is

-the first stage of the study on transverse strength design. The final goal will be reached

by means of the statistical prediction of transverse wave loads in ocean waves. In this

-paper, though the problem

### is dealt with for the case in regular waves

the importantcharacters of transverse wave loads and strength of a gigantic oil tanker reveal themselves. Due to the rolling characteristic of a large oil, tanker, the. larger hydrodynamic pressures re induced on the hull surface in beam waves, and, following to -that case, in quartering -and bow waves, which produce the large bending moments on the bilge corner and the deck

198 J FUKUDA, R. NAGAM0T0, M. Ko-imx and M. TAKAHASIII

longitudinal bulkhead in the transverse ring structure. In head waves, the large

hydrody-namic pressures will be induced on the hull surface of forebody, especially on the transverse

section in front of S S 7 (0 25L forwards midship) and the large bending moments will be

produced on the bilge corner and the deck corner of the ring structure Accordingly the

careful! consideration should be given to the structural design for those parts in the ring

structure as well as to that. for the horizontal struts.

In this paper the ship motions hydrodynamic pressures on the hull surface and dynamic oil pressures acting on: the inside of tank are calculated theoretically based on the modified strip theory and the two dimensional hydrodynamics and also on the experimental results of

non-linear roll damping. The propOsed method will be adequate to the present needs for

analysing the transverse wave loads However the theory has some week points in insig

ni'ficafit detail and the calculated results are not yet sufficiently confirmed with the

experi-mental results in oblique waves Therefore the following studies including theoretical and

experimental ones should be urgently promoted in order to improve and utilize the theory for the practical purpose.

on the non-linear roll damping of a ship running in still water

on the wave exciting rolling moment induced on a ship running in waves

on the wave exciting pressure induced on the hull surface of a ship running in waves

on the dynamic pressure of oil cargo in tank due to the ship motions on the dynamic pressure of ore cargo in hold due to the ship motions

When the studies on such problems as described above have been accomplished, the design

values of transverse wave loads can be reasonably determined by the aid of the statistical

prediction technique. The systematic research program of such problems has been planned in 1970 by the sponsorship of "The Shipbuilding Research Association of Japan" arid the research works have been continued by the research committee "SR-131 ". The works re-ported in this paper was planned and performed previous to the scheme of "SR -131 ".

Acknowledgement

The authors wish to express their deepest thanks to Dr. K. Taniguchi and Dr. T. Okabe

who have given them the conprehensive support to this research plan. They are also

### thankfull to Dr H Fujii Mr M lizuka and Mr T Takahashi for their cooperations in

this work. An enormous amount of numerical calculation has been performed by the digital computers "IBM-7040" and "IMB-360" in Nagasaki Technical Institute, Mitsubishi HeavyIndustries Co. Ltd. The authors are thankful to the staff of the computer office.

References

D. Hoffman: "Distribution of Wave Caused Hydrodynamic Pressures and Forces on a Ship Hull in

Waves" Norwegian Ship Model Tank Publication 94, 1966.

F. Tasai: " An Approximate Calculation of Hydrodynamic Pressure on the Midship SeCtion Contour of a Ship Heaving and Pitching in Regular Head Waves" Reports of Research Institute for Applied

Mechanics, Kyushu University, Vol. 14, No: 48, 1966.

F. Tasai: "Pressure Fluctuation on the Ship Hull Oscillating in Beam Seas" (in Japanese) Journal

of the Society of Naval Architects of West Japan, No. 35, 1968.

K. Goda: "Transverse Wave Loads on a Ship in Waves" (in Japanese) Journal of the Society of

Naval Architects of Japan, Vol. 121, 1967 and Vol. 123, 1968.

K. Goda: "Hydrodyriamic Pressure on a Midship in Waves" (in Japanese) Journal of the Society of

Naval Architects of West Japan, No. 35, 1968.

Waves" Appendix to the Report of Committee 2, Wave Loads, Hydrodynamics, Proceedings of 4th

ISSC, Tokyo, 1970.

F. Tasai: "On the Swaying, Yawing and Rolling Motions of Ships in Oblique Waves" (in Japanese>

Journal of the Society of Naval Architects of West Japan, No. 32, 1966.

F. Tasai: "On the Damping Force, and Added Mass of Ships Heaving and Pitching" Reports of Re-search Institute for Applied Mechanics, Kyushu University, Vol. 7, No. 26, 1959 and Vol. 8, No. 31,

1960.

F. Tasai: "Hydrodynamic Force and Moment Produced by Swaying and Rolling Oscillation of Cylin-ders on the Free Surface" Reports of Research Institute for Applied Mechanics, Kyushu University,

Vol. 9, No. 35, 1961.

K. Tamura: "The Calculation of Hydrodynamic Forces and Moments Acting on the Two Dimensional Body" Journal of the Society of Naval Architects of West Japan, No. 26, 1963.

Y. Watanabe and S. Inoue: "On the Property of Rolling Resistance of Ship and Its Calculation"

Memoirs of the Faculty of Engineering, Kyushu University, Vol. 17, No. 3, 1958.

Y. Watanabe, S. Inoue and T. Murahashi: "The Modification of Rolling Resistance for Full Ships"'

(in Japanese) 'Journal of the Society of Naval Architects of West Japan, No. 27, 1964.

T. Takahashi: "Analysis of the Mechanism of Rolling Motion and Its Application" (in Japanese,

Unpublished) Report of Nagasaki Technical Institute, Mitsubishi H. I. Ltd., No. 2842, 1969.

Y. Watanabe: "On the Water Pressure in the Tank due to the Rolling of a Ship" Memoirs of the

Faculty of Engineering, Kyushu University, Vol. 16, No. 4, 1957.

T. Fujino and K. Ohsaka: "Stress Analysis of Plane Frame Structure by Digital Computer" (in

Aii=_!'_+5ps4x

Al2 =

A13=2pg5Ywdx

A14 = - 5ps(x- XG)dX

A15 = - pN(x - xG)dx ±V5psIx

A16 = -2pg5Yw(X - xG)dx + VAl2 A21 = A14

A22

### -

5pN(x- XG)dx -V5ps4x A23 = -2pg5 yw(x - xG)dx A24 = ± PS2(X - x6)2dx A25 =5pNz(x- x6)2dxA26 = 2pg5 yw(x - x6)2dx + VA22

F = Fc0cos(&t ) = FcosaV + F,sinat M., = M.0cos(cogt - i9) Mcoscogt + M,,3sino.gt

### F)

1J1C+fC2C+f(SC = h0### F, J

### lfii +fs ±fs

### M1

### m1±m+nz.,3

= J%4, J m1, + + flZ,13 fc1 1 1 cosk*x =2pgC1C2y dx fCis J 3 sink*x fc2 1 1 sink*x O)IC1C2pNg dx fc28 J 3 - cosk*x (14. 1) (14. 4) (14. 5)200 J. FtTKUDA, R. NAGAMOTO, M. Komi& and M. TAKAHASHI

Appendix 1

Hydrodynamic Coefficients and Exciting Forces and Moments in Equations of Motion

In the coupled equations of heaving and pitching motions (14), the hydrodynamic coef-ficients A11, Al2, - - -, A21, A22, - - -, and the exciting force and moment F, M are given by follows based upon the linear strip theory.

(14. 2)

}

### wf

ai2=5pNsdx a13=O a14 = 5Psu(x- x6)dx a15 = pN(x - xG)dx V PS a16= Va12 a17 =5Psu(zG- 1)dx a13 =5PN1(zG a19 = 0 a21 = a14 a22 = pN,(x - x9)dx + v5Psudx a23 =0 a24 = ±5ps(x- XG)dX 1 sink*x### oVlCjC2ps3

dx .1 _cosk*x where k* = kcoszIntegrations in Eqs. (14. 1), (12. 2), (12. 5) and (12. 6) should be made over the water line

length.

Secondly, in the coupled equations of swaying, yawing and rolling motions (15), the hydro-dynamic coefficients a11, a12, - - -, a21, a22, - - -, a31, a32, - -, and the wave exciting force and moments F, M, M9 are given as follows by the aid of the linear strip theory.

(15. 1) fCSc 1 1 0)C0elCiC2PSz Icosk*x) dx fcao J J f Isink*xi cosk''x rn,13 = - 2pgC1C2y,,, (x xc)dx J sin k*x m,1,20 rn4,23 flçt,c rn,1333 sink*x

### ='lCiC2pNz

### (xx)dx

J - cosk*x### I

cosk*x = coo IC1C2Ps3 (xxG)dx J sin k4,x (14. 6>202 J. FUKUDA, R. NAG&zioTo, M. KONUMA and M. TAKAHASHI a25 = pN(x -xe)2dx a26 = - Va a27 pSU(ZG -

### l)(x -

xG)dx a28 = SpN,(zG### -

### -

x5)dx a29=0 a31 = a17, a33=0, a32 = a18 a34 = a27a35 = 5pNy(zc

### -

- x)dx - Va17a36= Va18

a37 = + 5pidx+ 2z0a17 -ZG5PSdX

a38 =5pNu(zo- l)2dx a39 = VV mt

where pi = ps11lle, rn: metacentric radius

F = F0cos(thgt - = F,cCScOet + F,7sSjfl(Oet

= M000S(o)t - j9) = Mccosa)et + Msina)et

M9 = M60cos(cot - = MOCCOSUt + Messina)et

F, 1
= h0sinx
F,,3 J
1W _{1} + m23 + rn33 + rn,
= h0sinx
M, J 1m1, + m2, + m3, + rn4,
ii?10c 1 rnSlC + me2,, + m033 + m4,
= Jz0sinz
J%4, J m9, + m02,+m93, + m943,

### I

i3 1 ( sin k*xi >=pglS1 dx f,,13 j .) I -cosk*x; = C3pN1, dx :: } Isink*xi f,,sc sin k*x = a)2C3ps dx .3### -

coskx### f'1

### I

sinkKx = - a)k*VICSpSY f,,43) .3 - cosk''x dx (15. 6) (15. 4) (15. 5) (15. 2) (15. 3)m1 1 ç sin k*x
(xxG)dx
mç1g ) _{)} _cosk*x
m2 ) f (cOsk*x
=olC3pN

### (xx)dx

m2 J J (sin k*x wVlCsps5 dx m43 J sin k*x sink*x = - cok*V 1Cs_{(z. -}

_{l)dx}m4 J

### -

coskx 4where s1 = f_{exp( - kza)sin(kysinZ)dzg}

sinz jo

C3=exp(kd/2)

Integrations in Eqs. (15. 1)<15. 3) and in (15. 6)(15. 8) should be perforEned over the -water line length.

In actual calculations of this paper, a33 in Eqs. (15. 3) is not used but a of (16) is

uti-lized, and M, M3, M0 and M9 are evaluated substantially by neglecting f,, f,

### in,

m4 and rn943 in Eqs. (15. 5)--(15. 8).Appendix 2

Non-Linear Roll Damping

When estimating the roll damping coefficient a33 in the coupling equations of swaying,

yawing and rolling motions (15), 4 according to Eq. (16) is used instead of a38 in Eqs.

<15. 3), where the non-linear viscous damping is introduced into the equivalent linear damping. n6jc m013 rn2 m2 m3 ) m038 J I sink'x - ak*V C3ps

_{(xx6)dx}

J ### coskx

sink*x = pg (zGll)dx - cosk*x cosk*x = Co lC3pN (ZG - l)dx J sin k*x sin k*x = wC3ps5 (zG - l)dx J - cosk*x (15. 8) m3 fli3 1.### sinkx

2cspsv J### coskx

### I

coskx### (xx)dx

(15. 7)'204 J. Fuiuo R NAGA0T0, M. KONUMA and M TAKAa&ssi

In order to evaluate a from Eq. (16), the coefficients a1 and b1 in the following formula of extinction curve for the free rolling in still Water should be determined.

40 = a10,,, + b1O

On the other hand, 48 can be approximately gisen by folloWs.

A0=N0

Accordingly, a1 and b1 call be determined by using "N-coefficients" for the cases of O,, 10°

and 0,,= 200 and equating as follows.

N10 = (a1/10°) + b1

N20=(a1/20°)+b1

N10. and IVQ can be obtained by the following foimulas given by Watanabe and Inoue12)

where

### ,

### __

4 6 4 1 + ,n+1 2m+1 3m±1 4m+1.### m=C/(lC)

W: displacement.L: length between perpendiculars

B,: breadth moülded

d: draught moulded

m: transverse metacentric radius

= (d/2) - z, zG: centre of gravity below water line

T9: rolling period

Ab: area of .a bilge keel

a0: given in Fig. 5 as functions of aspct ratio of bilge keel and of blo'ck

coef-ficient

C8: block coefficient

C: Water plaiie llteã coefficient

Njo.l nio. Ld _{8} d2

### f(C)

(16. 1) WmTI 1Q(l+### 4l

### -64

d N20J 2O" = (0.03 + 0.78C1 d/L) + 1.5a0 = (0.02 + 1.lCb dIL) + } (16 2)' 20' CoVT### 200

150 100### 40

Cb = 0O2 0C3 .005 .01 .02 .03 .05 .10### £2 sped tto

206 j FUKUDA, R. NAGAMOTO, 'M. KONUMA and M TAKAHASHI 0 0 .5 HEAVE 0.5 -0 0 Fr: 0.0 0.5 .0

### T[7

Fig. 6. Heaving Amplitudes in Regular Waves from Different Directions.

5

.5

0.5-PITCH Fr. 0.15

Fig. 7. Pitching Amplitudes in Regular Waves from Different Directions.

0

I 5-0 1.5 1.00.5 -0 0 11.0 SWAY -5 0.5 .0 IS L/A

Fig. 8. Swaying Amplitudes in Regular Waves from Different Directions.

5-S-. Jr
0.5-1.5
0
0.5
-SWAY
0.5- _{-4}

### \

55-5-s### \//

### ,.-

-0 0.5## -:

10### N

N \\ S### N

_-.QS__### \.

Fig. 9. Yawing Amplitudes in Regular Waves from Different Directions.

15.

0

L=310M, Hw=IOM 0 L3I0M. HWI0M
0 Fr. = 0.10 5-0 Fr, 0.15
YAW YAW
0 _{L3IOM. Hw-IOM} 0 L3l0M. Nwl0M
-5-.
Fr. 0.10 S 5--.
Fr.0.15
0 0.5 1.0 .5
1.5 .5
0.5 1.0 IS

208 J. Fuxun, R. NAGA0TO, M. KoNtms

### and M. TAsm

0 0 ROLL 4- L=3I0t. HwIOM### 2

Fr.=0.I0 001.1 0 0.5 1.0 .5 /L/Fig. .10. Rolling Amplitudes in Regular Waves from Different Directions.

4-0 ROLL Fr. 0.15

### \

0 0 0.5 1.0 .5NEAD WAVES

0.5 "

MIDSHIP SECTION,

Fig. 11. Amplitudes of Hydrodynamic Ptessi.iie-iñ -Regular Head Waves' (y 18O), Midship Setioii.

HEAD WAVES S.S.7)2 SECTION x P0/pgh0 0.5 - .0 1.5 1.5

Fig. 12; Amplitudes of Hydrodynamic Pressure in Regular Head Waves (y = 1800), S.S. 7 Section.

AlL Fr 0.50 0:10 0.50 0.15 - ô.s 0.10 - -o.$ 0.15 -I.00 0.10

### x .00

0.15 Lr3IOM HwIOM A/L Fr 0.50 0.10 -A--. 0.50 0.15 Q.-75 Ô.I0 - ---_{0.-75}

_{o:is}

### -.- 1.00

0.10 .00 0.15 L310M. HwIOu210 ;J. FuxuDA,.R. NAGAMOTO, M. Kosut& and M. TAKAHASHI.

0

0.5

O.5-FOLLOWING WAVES

Fig. 13. Amplitudes of Hydrodynarnic Pressure in Reu1ar Following Waves ( = 0°), Midship Section.

FOLLOWING WAVES

### Li

### Li

MIDSHIP SECTION S.S.7,i SECTION P0/pgh0 0.5 1.0 1.5Fig. 14. Amplitudes of Hydrodyathic Préssuré in Regular Following Waves (, = 0°), S.S. 7 Section.

AlL Fr O.5O 0.10 -L- O.5Ô 0.15 6.75 0.10 -- 0.75 0.15 1.00 0.10

### -x- 1.00 OI5

LIOM. Hw=IOM A/L Fr 0.50 0.10### -A- 0.50

0.15 - 0.75 0.10 0.75 0.15 1.00 0.10### -- 1.60

0.15 L310M. Hw=IOMFig. 15. Amplitudes of Hydrodynamic Pressure in Regular Beam Waves (x=90°), Midship Section.

BOW WAVES

MIDSHIP SECTION

BEAM WAVES

MIDSHIP SECTION

Fig. 16. Amplitudes of Hydrodynamic Pressure in Regular Bow Waves (x = 135°), Midship Section.

)'/L Fr 0.50 0.10

### -- 0.50 0.15

0.75 0.10 0.75 0.15 1.00 0.10### -X

I.00 0.15 L3IOu. Hw=IOM A/L Fr 0.50 0.10 0.5O 0.15 0.75 0.10 0.75 0.15 100 0.10### x

1.00 0.15 L=310M. HwIOM?12 -. -

### ...

J. Fuxuo, .R. NAOAMOTO, M. .KONUMA and M TAKAnASHI 1.0 0.5 BOW WAVES 1.0 S.S7Y2 SECTION - Po/pgho 05 .1.0 2.0Fig. 17. Amplitudes of Hydrodynamic Pressure in Regular Bow Waves (y = 1350), S.S. 7 Section_

QUARTERING WAVES

MIDSHIP SECTION

-P0/pgh0

0.5 0 2:0 30

Fig. 18. Amplitudes of Hydrodynamic Pressure in Regular Quartering Waves ( =45), Midship Section..

AlL Fr 0.50 0.10

### -A-

0.50 0.15 0.75 0.10 0.75 0.15### _._

I,.00 O.IO 1.00 0.15 L=310M. HW=IOM )'/L Fr### --- 0.50

0.10### -A-

0.50 0.15 0.75 0.10 0.75 0.15 .000.10### x .00

0.15 L=310M. HwlOMQUARTERING WAVES S.S.72 SCTI0N - Po4gho 0.5 1.0

### 7

3.0Fig. 19. Amplitudes of Hydrodynamic Pressure in Regular Quartering Waves (x=450). S.S. 7 Section.

A/L Fr
0.50 0.10
A _{0.50} _{0.15}
0.75 0.10
0.75 0.15
.00 0.10
---x=- .00 0.15
LOIOM. Hw=IOM

1.0 HEAD WAVES MIDSHIP SECTION

### Oet

0 = 3O 2: 6O 3: = 9O 4: l2O 5.: =I5O =I8O =2IO 5: =24O 9: =27O IO. =3O0 II: 33OFig. 20. Pressure Distributions at Time Intervals of Tell2 during an Encountered Period in Regular

Head Waves (x=i8O°) 2/L=1.00, Fr=O.15, Midship Section.

L=3IOM. X/L= 1.00, Hw'lOM, Fr.=O.I5

S.S.7,4 SECTION 0 20T/M2

### -Q:f= 0

I: = 30 2: 6O 3. = 90' " =1.20' = sd' 18O° =210° =240° =270° = =3O II: =330°Fig. 21. Pressure Distributions at Time Intervals of T/l2 during an Encountered Period in Regular

Head Waves (x='8O°) 2/L=1.00, Fr=0.15, S.S. 7 Section.

214 3. FITKUDA, R. NGAMOTO; -M. Kbxu1A -and M. TAXAnAsHI

o Wet 0° I : = 30° 2. = 60° ', = 90° =1200 =1500 6:" =180" 7o =210" "=240° =270" =3Ocf

### ii:

=330° Q:W0t= 0° = 30° = 60° = 90° 4.." = 120° 5.' = 150° 180° =210° =240" " =270° IC: " =300°### II:

=330° MIDSHIP SECTION MIDSHIP SECTION BEAM WAVESL=3!OM, )s/L'O.75, HW=IOM, Fr0.IO

Fig. 22. Pressure Distributions at Time Intervals of T6/12 during an Encountered Period in Regular

Beam Waves (x=9O°). 21L=O.75, Fr=O.1O, Midship Section.

BEAM WAVES

20 30T/M2

L310M, A/LI.00, HwIOu, Fr.0.I0

Fig. 23. Pressure Distributions at Time Intervals of T/12 during an Encountered Period in Regular

216 3. FUKUDA, R. NAGAK0TO, M. KbNUMA and M TAKAHABHI
o wet =
I: = 30°
° 60°
= 90°
=120°
=150°
=180°
=210°
=240°
" =270°
0: =300°
0:wet= 0°
I 30°
° _{= 60°}
= 90°
4 : = 120°
5: " = 150°
6 ° = 180°
=210°
0=240°
=270°
tO: =300°
II =330°
MIDSHIP SECTION

Fig. 24. Pressure Distributions at Time Intervals of Te/l2 during an Encountered Period in Regular

Bow Waves (x='359), )./L=i.00, Fr=0.15, Midship Section.

BOW WAVES

S.S.7/a SECTION

- BOW WAVES

L 310M, )/L 1.00, HwlOu, Fr.= 0.15

L= 310M. )'/L° .00, HwIOM, Frs 0.15

Fig. 25. Pressure DistributiOns at Time Intervals of Te/12 during a Encountered Period in Regular

o w0f 0
30°
2 = 60°
= 90°
°=I20°
'. =1500
=1800
7 =210°
8.: " =240°
9:, _{=270°}
10: =300°
'°330°
Q:et= 00
= 30°

### 2.0=60°

3. 0 = 900 °' = 20° = 1500 °_{= 80°}=2.10° 0

_{=240°}

### .9: o270°

0,: ° 300°### II:

=330° MIDSI-IIP SECTION QUARTERING WAVESL3IOM. )/L=0.50, Hw=IOM, Fr.=0.I0

QUARTERING WAVES

L=310u. A/L0.50, NW=IOM. Fr=0.lO

Fig. 26. Pressure Distributions at Tihie Intervals of Te/12 durihg 'an Enoüntered Pei9od in Regular

Quartering Waves (y=45 ) 2/L=O50 Fr=010 Midship Section

Fig. 27 Pressure Distributions at Time Intervals of Te/12 during an EncoüRtered Period i Regular

218 - - J. FUKUDA, R. NAGAMOTO, M. KoNUMA and M TAKAHASHI

HEAD WAVES

L3IOM, >/L=l.00, H=IOu, Fr0.15

wet=30°

-S.S.7Y SECTION

SHEARING FORCE IN TON

-469.7

SHEARING FORCE- tN TON _{BENDING MOMENT IN T-M}

Fig 28 Shearing Forces and Bending Moments Calculated in Still Water Midship Section

,,_-.303.7 208. 61.4 ) 147-0 82.3\ 33 29.0 30.8 \\=3597 16.1 HE-AD - WAVES

L-3IOM, /L=I.00, HWIOM, FrO.I5

wet = 30 S.S.7,V2 .SECTION 21.4 32ffl 15B 9260 46.2

### (

BENDING MOMENT IN T-MFig. 29. Shearing Forces and Bending Moments Calculated in Regular Head Waves (x='8O°),

S.S. Th Section.

IN STILL WATER _{IN STILL WATER}

MIDSHIP SECTION _{MIDSHIP SECTION}

824

112.3

912 341 I527

-®-2 98,3 114.1 89.9 32.5 "- -'423 32.6 129.7Fig. 30. Shearing Forces Calculated in Regular Beam Waves (x=90°). Midship Section.

BEAM WAVES MIDSHIP SECTION

3IOu , =0.75, Hw= IOu, Fr. 0.10 ()et60° 328.61 48.4 386 92 94.5 07.8 I75 207.6 00.9 66.5 8 11.9 53.8 '46.0 BENDING MOMENT 164,5 394.6 IN r-M 414 j73. 4322 15.2 105.5 266.0 - 429.4 73.8 344.1 . 148.0 243. 26.4 552.8 225.1 427 .2 4L3

Fig. 31. Bending Moments Calculated in Regular Beam Waves (z=9O°), Midship Section.

BEAM WAVES MIDSHIP SECTION

L= 310M, X/L0.75, H91'IOu, Fr.=0.IO et = 600

220 ." . J. Fuxtrni, R. NAGAMOTO, M. K0NUMA and M. TAKAHASHI

### -02

-BOW WAVES,

L310M. X/L=I.00, Hw=IOM, Fr.=O.15

()et 30°

SHEARING FORCE

878

-111,8 177.

S.S.7,V2 SECTION

Fig. 32. Shearing Forces Calculated in Regular Bow Waves (y=135°), S.S. 7- Section.

BOW WAVES S.S.7i4 SECTION

L= 3IOM. '/L= 1.00, H= IOu, Fr=O.15 = 30°

BENDING MOMENT

### EE

I64 272 440.7 1

### \69.6

'2### I72.I

410.3 I485 11.4 95.2 IN TON 9 181.3 l.5 321.2_{8.1}SHEARING FORCE 49.5 IN T-M Is 105.6 BENDING MOMENT =i.22742 217.6 223.1 2 31 16 22

### /

254.2 597 418 E### .$

154.1 251.4 39.5 I 3i.2Fig. 34. Shearing Forces Calculated in Regular Quartering Waves (y=45°), S.S. 7 Section.

QUARTERING WAVES _{S.S.7Y2} SECTION

L= 3ION, )/L.=O.5O, H ION, Fr.= 0.10 = I 8O

Fig. 35. Bending Moments Calculated in Regular Quartering Waves (y=45°), S.S. 7 Section.

I89.

QUARTERING WAVES S.S.7>'2 SECTION

L=310M, /L=O.5O, Hw=IOM, Fr.=O.IO

1.)et=180°