An analysis of ship capsizing in quartering seas

SCheepshydromechJ

Archief

Mekeìweg 2, _{2628 CD Deift}

O18.7ßß73.

An Analysis of Ship Capsizing

in Quartering Seas

by Masami Hamamoto Member Yoon Soo Kim* Member

Akihiko Matsuda**, Member Hiroyuki Kotani*n, Member

(From J.S.N.A. Japan, VoL 171, Jane 1992 _{and VoL 172, Dec. 1992)}

1. INTRODUCTION

Stability against in extreme

seas is one of the most fundamental requirements for

the àfety of ship at sea.

At the same time it is

one of the most complicated phenomena to investi-gate on the basis of analytical approach because the phenomena are concerned with extreme rnqtion both of ship and waves. What is the dengerous situations leading up to capsizing of ship in extreme seas? For this problem, several investigators carried out model

eriments'3) and pointed out capsze modes such as,

?onant roiling mode in beam wind and waves, pure

loss of stability at wave crest amidship'2', low cycle

resonance due to parametric eìcitationh2),

broaching-to due broaching-to successive waves''15'1, period bifurcation

of roiling36), surfriding phenomena 29), etc. -It is a big problem to investigate what is the essen-tial indices or mechanism to bring a ship to capsize. As well äs known, the metacentric height GM,

right-ing arm GZ and its area are the traditional indices

which, as pointed out by Paufling3), vary with respect to the relative position of ship to waves. These indices are hydrostatic feature for stability. In order to rna.ke

Osaka University

"Graduate School of Osaka University

An Analysis o

Ship Capsizing

in. Quartering Seas

by Masami Hamamoto*, Member Yoon Soo Kim, Member

Akihiko Matsuda**, Member Hiroyuki Kotani*, Member

(From J.S.N.A. Japan., VoL 171, June 1992 and Vol. 172, Dec. 1992)

Summary

This paper is concerned with the righting arm and capsizing of a ship in extreme quartering waves. The effects of wave height, wave length and heading angles of ship to waves on the righting arm are analytically investigated for a container ship. A time domain numerical Simulation program for motions and capsizing has been used to investigate motions in a variety of wave configulations. Numerical simulations based on a mathematical model are carried out to find out thè critical situation leading up to capsizing of a container ship in extreme quartering waves. Various phisicai mechanism

that could be responsible for capsize are pointed out by numerical experiments.

9

an analytical approach to the phenomena mentioned

above, it is necessary to consider the dynamics of

mo-tion and capsizing for stability. The purpouse of this

paper is to discuss the dynamics of motion by using a

mathematical model to describe the ship motion and

capsizing.

- For a mathematical model to conduct nunwical

simulations, it is necessary to siimrnrize the outline

of ship motion, hydrostatic and hydrodynamic forces

on ship. The frequency of wave encounter is low in following to quartering seas when a. ship is running at high speed. As a result, the ship motions will be determined largely by the hydrostatic forces

includ-ing Froude-Krylov forces which may be computed for

the exact position of ship and waves, and the contri-bution of hydrodynamic forces will be relatively not

so much. This enables us to retreat from the

dis.-culty and necessity of determining the hydrodynaxnic forces with great accuracy taking into account the ge-ometrica.l variation of the immersed hull. during large amplitude motions.

According to the considerations mentioned above,

we tried to drive a mathematical model taking into

account. the linearized hydrodynamic forces and non-linear hydrostatic forces which may be computed for

the exact immersed hull. Using this mathematical model, we conducted numerical experiments to

ana-lyze the ship motions and capsizing in extreme quar-tering seas.

2. ANALYSIS OF HYDROSTATIC STABILITY.

As pointed out by Paulling3, for waves of length nearly equal to the ship length the righting arm GZ

is increased at a wave trough near amidships and de..

creased at a crest near amidships. The variation Of

GZ in waves will affect the roll motion of the ship

and play an important role in extreme motion and

capsizing.

According to the coordinate system, the GZ3?)

tak-ing into açcount the exact position öf ship and waves

is given by

W . GZ pg f(

COS - Z9(1)

sin )A(z)dz

pg sin J (zB()

COS

+

_Ya(s)sin ₎

L

B().

sin(k.sm x)

k 2 xa B(z) e k ---- sin x

xA(z) sin k(G + z cosx - ct)dz

where p is density of water, g acceleration of grav-ity, a phase speed. of wave, i time, B(z) ship breadth

at draft, A(z) submerged area, and are

gravity center of submerged area. The first term of

right side in Eq.(1) is stability of hydrostatic pressure

gradient pg and the second term is stability by lateral

force of heading angle of ship to wave.

In order to analyze the relationships between the

GZ and the effect of following items:

Effects of wave length to ship length ratio

Effects of wave height to length ratio H/ Effects of heading angle i of ship to wave

Dynamical stability for the relative

posi-tion of ship to wave

We used a container ship as shown in Fig.l. Fig.2

shows GZ curves of designed GM, J.G regulation at C=1 and 1MO A.562 at C=1'9

(1)

-50

I.=L15.Q() 8=9.o() d=6.4() O13.S.3(t)

Fig.l Principal dimension

GZ(m) 0.2 0.1 -0.3 GM0.44(m) No A542 a C1) GM0.91(m) ed GM

Fig.2 _{GZ curves in still water}

Effects of wavelength to ship length ratio À/L The righting arm GZ is remarkably reduced wheñ the crest of a wave is. abôüt amidships nd increased

when the trough of a. wave is about amidships. Such variations of GZ in a wave are related to the re1ative position of ship to waves and the wave length to ship

length ratio. Fig.3 shows the variation of GZ with respect to the relative position /.\ of ship to waves

and the wave length to ship length ratio AlL. In

this figure, ¿e/A is equal to O at the trough, 0.25 at

the up slope, 0.5 at the crest and 0.75 at the down

slope amidships. The variation is smaller for larger wave length at çoast ant wave height. The madmum

variation comes in the same length as ship.

Effects of wave height to length ratio H/A

Fig.4 indicates the variations of GZ with respect to

the wave height at constant wave length equal to the

ship length. The variation is proportional to the wave height.

o ₅₀

GMO.28(m)

atCD-GZ(m) 0.7. -p0.

-

L.z0(deg) -a=5.23(th) --- /A=00 - i A/L=2 AtL=5 -60 -0.7 GZ(m) GZ(m) A/L=5_

.__._.-.

LLtATER

ÂJL1"

-STlLiTATE! GZ(m) íAqi.. :vA:j h-L vrn - I A/L=1

_---

---0... 6O

-60 '0.._ 60

-GMO.9l(m) . GM0.91(m)

- z0(deg)

-

xXdeg)

R=5.23(m) - H=5.23(m)j

/À=0.5.../A=0.75

-0.7 -01 -60

0.

---iL L_ - GZ(m) AJL=5 A!L2 A/L=1_ $TU.L lATER -60 - -- .z=O(deg)._.. - --- H=5.23(m> - - /Â0.25 -0.7

Fig.3 Variation of stability curves due to the ratio of wave length to ship length

-60 0 .60 -60

0__60

GM=0.91().

-- -'

GM=0.91(m)

À/L=1.0- -

--Vz=0(deg)

z0(deg) -0./ GM=0.91(m)-GZ(m) -0.1 GZ(m)

ii'vn_

- -

STU.L RATEd

.60 -60 --- 60 0.91(m). ' GM0.91(m) :1.0-....-

. ---.-. A/tL0j

:deg) :..

zÇe;).

L=0.5_ -

-

J).=0.75 -0. i

--

---r-- a Pv -

-Fig.4 Variation of stability curves due to the ratio of wave height to length

G2(m)

-1

Effects of heading angle ' of ship to wave

Fig.5 iñdicates the variations of GZ with respect to the heading angle x. The effect of heading angles

on the GZ can be sigDificant and the GZ in beam sea condition is about the same as that still water. But the GZ is smaller for smaller heading angles and the

smallest one is that in a following sea.

Dynamical stability for the relative position of ship to waves

The righting arm GZ vary with the relative

posi-tion of ship to waves, heading angles, wave height and length. Accordingly the dynamical stability vary with

the parameters mentioned above. Figs.6 and 7 show the variation of dynamic stability with respect to the relative position of ship to waves. In this case, the dynamicaistability is the area of GZ at the vanishing

angles of stability for )1/L = 1.0 and )1/L = 1.5.

E = W] GZ()dç6

(2)

o

The dynamic stability takes the smallest one near

wave crest and the largest one near wave trough. And

then the dynamical stability is seen to be closely re-lated to the capsize of ship in extreme following and

quartering seas. f-isL RATER. -60 -30 O...60 GM:O.91m A/L=1.0 JA=O.0 GZ(m) 0.7 1tL TATE! H -9°--60 GM0.91m B/A=1/22 GZ(m) 01 -0 7 GZ(m)

i.

0.i.

STU.i!ÄTEE r 30 6O I el -60 60 GM=0.91m HhA=1/22 -60 w GM=0.91m /A=1/22 -0 7 -0 7 G2(m) 0. 07 -- .60 A/L=1.0 /A=0.25 TU.L !.(TER.. 0.____ 60

A/L10

¿dA=O.75j

Fig.5 Variation of stability curves due to heading angle of ship to waves

£(vave)/E(still) z0(deg) -3 M0.91(m),H=5.23(m) A/L=1 2 _A/L=2.

-u

A/L5

0 0.2 0.4 0.6

0.8

z60(deg) GMO .91 (m),H=5 .23(m)

A/L1

.AIL=2 0 0.2 0.4 0.6 0.8 _{0 0.2 0.4 0.6}

_0.8

Fig.6 Variation ofstability curves due to the relative posItion _{of ship to waves}

E(ave)iE(sti11) - z=0(deg) - .... 3 GMO.91(m),A=115(m) fi/A =1/22 2

_'H/.2..=1/4O

-x30(deg)

GMO.91(m') ,A115(m) li IA =1/40 - E/A=1/80 0 0.2 0..4 0.6 0.8 0 0.2 0.4 0.6 0.8 ¿.IA o

Fig.7 Variation, of stability curves due to the relative position of ship to waves

3. A MATHEMATICAL MODEL FOR.

NUMERI-CAL EXPERIMENTS .

We used the equations of motion with respect to a new coordinate system called Horizontal body axes'

which have rotation about z' axis and no rotation

about z' and y' axes but a ship is able to rotate about z' and y' axes as shown in Fig.8. Theuse. of

Horizon-tal body axes are reasonably simple and convenient

for representing manoeuvring motion with large roll

angles in horizontal plane and seùeeping motion with

large roll angles in vertical plane, because horizontal

and vertical forces. on, ship automatically include the effects of large heel and roll angles. According to 'this

coordinate system, the equations of motion are

de-rived in the following'forms.

vve

k

F

GJc

A/L5

0.2 0.4 0.6 0.8 /A

Fig.8 Coordinate systems

z60(deg)

GM=0 .91(m), Â115(rn)

- H/A=1/4.0 H/A=1/80

Translational Mot!on's And Forces

(m+m)U -(m+rn,,)V

+mz09 +iflz(G9

= T(1-t)--R+X((G,,9,b)

-(1-t$FNsinö

(m+m±YvV+rnza +(m+m)Uò

YTn,1z0

= Y((G,&9,) +Yo.F.(iW,iW)

-(1+aH)FNcoS8

(in + m )Ç +

_{- Z}

± Z Ö +Z89

-= ZK((!G,Ó,9.'b)±ZD

Rotational Motions And Moments

(Irz+Jrr) +K

-mzU - m,zI' - (YV - Y,,i4 )z

= K;K((G,,e,)+(1 +aH)hRFNcosö

(I., +

+

- M96

-+TflrZÙ + (I + frz.) 6

= M.((G,,9,b)+ M0,.(,w)

(I +

+ (iVi + mXGU)Ç/) +

+J\ÇV - (Irz + J)ö ¿

- razU

= +(1 + aN)IRFN cosö (4)

There m is the mass of ship, rn, and m: the

'äaded mass with respect to the z, y and z axes

direc-tions, I, I, and I

the moments of inertia about

z, y and : axes, J, J, and J

the added

mo-ments of inertia. XG and z the displacement of added

mass center with respect to the origin of the

coordi-nate system fiiced in the center. of gravity of the ship,

U, V and W the velocities along the z,',

y' and :'

'axes, and , Ô and ' the angular velocities about

the z', y' and :' axes, T the thrust of propeller, t

the thrust deduction, and R ship resistance, Z,

Z, Z, Z9, M, M9, M and M

the

hydro-dynamic coefficients for seakeeping niocion,

i,

Nv and

are the hydrodynarnic

deriva-tives for rnanoeuvring ¡notion, .IÇ the damping

co-efficient of rolling4t. t,, correction factor of tesis-aiìce due to steering, ci,, coefficient of interaction of rudder tiormal force oit side force, a coefficient

M-_Ca

M

=. m.z,

= 2(Irr+Jrz[1+O.S(1-e_'°")]

of interaction of rudder force on yaw moment, _'R horizontal distance between rudder and C.G. of the

ship,

FN normal force on rudder2, X'K,

_F.K

and Z.K Froude-Krylov forces', K.K, M.K and

N.K Froude-Krylov moment, YD.F , ZD.F and

MD.F , ND.F the diffraction forces and moments?). Hydrodynamic Çoefficients for Seakeeping

and MPpeuvring Motions

=

j iV(z) dz

Z

=

fm(z)x dz

L =

_jN:(x)zdx+(m)U

uf

N:(z)dx -= IL N.(z)z7 dz M9

= U

f N(z)z.dx

JE IL N.(x) dz + (m - mr)U = +

1.4C3.]

N

=

_{.pL2dU ()}

!PL2dU[()]

=

PL3dU[(O.54_)]

where N(z) is the wave damping coefficient of Lesis

form section4, Tn(z) the added mass of Lewis form

section, L, B and d the ship length breadth and draft, C8 the block coefficient of ship.

Rudder Force

FN = pA,,fUstha,,

(7)

where AR is the rudder area, UR and ciR effective inflow velocity and angle to rudder, fa normal force coefficient of rudder force, â rudder agIe.

Froude-Krvlov Forces and Moments

0. . ti') -pg cos

x f F(.c)A(.r)siÌi k(G + cos

((G, ø,

8, t)pg sin

X f F(z)A(x)sink(G + zcosx

- c)dz

ZK((G,,8,v)

-pgf A(z)dz

-Pif

F(z)A(z)cosk(eG +zcsX

c)dz

-pg

_f(YB)

°S - Z9) Sin th)A(x)dz

-pg sin x j (zB() COS + YB(> SIfl Ø)

x F(x)A(z) sin k( ± z coax

c)dz

pgf z.4(z)dx

+pg j zF(±)A(z) cos k(G + cos

x - ct)dz

8, w) pgsin

x f zF(z)A(z) sin k(G + z cos

- ct)dz (8)

where A(z) is the instantaneously immersed area of

ship section, ¿G the relative position of shipto waves,

x the heading angle of ship to wave, c the phase veloc-ity of waves. Y9() and the center of buoyancy, , and z, are the velocities of orbital motion with

respect to y and axes, (,, a sinusoidal wave at any time and at the position z, y, and z written as

-(y coso

- zsinp)sinx

- Ct]

(G + z9 + a cos + cos X

(9)

and F(z) coefficient of pressure gradient ofwaves given as

B(z).

sui(k sin

F(z) = ah

B(z)

_(d(}

(10) Diffractjn Forces and Moments

Y; (i,i) = g sin j m.(x)F(z) Sin

+xcosX-ct)dz

+csin j N(z)F(z) cos k(

+zcos - ct)dx

=-g f rn(x)F(x) cos k(0 + z cos x - ct)dz

+cf

N(z)F(z)sink(e +zcosX -cfldz

M, ((W,W)

=g f m:(z)F(z)z cos k(,1 + cos

- ct)dx L -c f N. (z)F(z)z sin k( + z cos x - c)dz

N,(r},

) = gsin f rn,(z)F(z)zsin.k(

+zcos

-ct)dz

+csin X f JV,(z)F(z)z cos

+zcosX-ct)dz

4. NUMERICAL EXPERIMENTS

In order to find out the critical metacentric height

and waves leading up to extreme motion and capsizing

of the container ship mentioned in the previous

sec-tion by solving the equasec-tions of coupled six degrees of freedom, numerical experiments were conducted. For

the computation, a standard numerical procedure i

employed to integrate the equations of motion leading

to a step-by--step appro.mation of ship motion.

Effects of Wave to Ship Length .atio

At first , numerical experiments in time domain

were conducted for the ship running at. the heading

angles of 0, 30 and 60 degrees. to waves. Fig.9 is the examples of the time histories of yawing, pitching and rolling in case of GM=0.8(m) and ship-wave velocity

ratio U/c=0.7. Fig.10 stands for the results of nu-merical experiment of the critical GM leading up to

capsize against wave-ship length ratio AlL.

It is an interesting pQint whether the critical GM

is larger or smaller, than the GM=0.92m designed to

satisfy the weather criteria A562. The ship is safe in the critical GM smailer than designed GM. The ship will be unsafe in the range of waves AIL

= i.25 to

1.5 having he heading angle equal to zero and in the

waves of AIL = 1.25 to 1.75 having the heading angles equal to 30 degrees.

Effects of Wave Height to Lengt.h .atip

Fig.11 indicates the result of numerical experiments

to find out the critical GM with respect to the wave height to length ratio H/A. The ship will be unsafe in the wave height larger than H/A = 1/20.

Effects of Heading Angles of Ship to Wae

Fig.12 shows the results of numerical experiments to find out the critical GM with respect to the heading

angles of ship t'o waves of AIL = 1.0 and 1.5. The. ship will he safe for any heading angles in the wave

P=O(deg) U/c0.7

-8 _{GM=O.8(rn) A,'Ll.O} H/A=1/20 10(de& _Rol1itg -0.2 S O(de)

_Pitching

90ó4e)

_Ro1iing Yawing timea)

?=O(4eg) U/c0.7

GM=O.7() )./L=LO H/A=1/20 fl7v 50 z<de) Yawing 10 50 30 10 Yawing 25 8 8(deg) _Pitching ,ioq de(a) tiae(g) =30(deg)

U/c=O.l

GMO.8(m) A/L=j.O H/A=1/20 10 (deg) _Rolling 100 50 50 time(a) =3O(deg)

U/c0.7

GM=O.7() )./L=1.O

H/)1/20

90 (deej 90 60 30 8 P=6O(deg) GM=0.8(m) H/A=1/20 30 (degJ _Rolling

Fig.9 Time histories of ship motion in following waves

f )/L=1

z(deg) Yawing 8(deè) _Pitching ?=60(deg)

Ù/c=0.7

GM=O.7(m) A,'L=l.O

H/=1/2O

°L'

Rolling

Fig.1O Critical metacentric height leading up to capsize versus AIL

15

0 0 0.5 LO

1.5 2.OA/L

0.0 0.5 1.0

1.5 2OA,'L

0.0 0.5 1.0 1.5 2.ß.?,,/L

0.1_{:z<de) Yawing}

u,-z:34)

O Not Capsized Capsized

U/c t7 _{l/L1.I U/c.7}

ONot Capsized apsized

GM(m)

L6 L4 1.2 LO 0.8 0.6 0.4 0.2 0.0 0.0

GEm)

1.6

0.0

o

0.02

GM(m)

L6 z(deg) A/L:IS

o

Not Capsized L2 apsi'zed

LO'

L » 0.8

-DegGM

0.6 critilZ

20

40

0.04 H/A

0.0 0.02

0.04 FI/A

z:3(de)

.M.

U/cL7 Not Capsized - _Cpsizei

Fig. 11 Critical metacentric height leading up to capsize versus

60 (deg)

0.0 0.02

0.04 H/A'

0.0 0.02

_{0.04 H/A}

60

(deg)

Fig.12 Critic3l metacentric height leading up to capsize versus heading angle of a ship to

waves

G

z-d)

)./L:!5 CLpsized xCapsized U(/c:.1

óÌjot

-Desiig.

IHA

r.'

'I Iesi.ged

II

Critical I

Q Not Casied Capsized

A/L:1 H/A:1/2O U)"c:U.7

-Desuged G.

O Not Capsized

x

Ca sized

A/L:1.S J/A:1/2O _tJ"c:&7

0

20

40

0.4 0.2 0.0 0.0 0.02

0.04 H/A

0.0 0.02

0.04 H/A

1.2

0.8

0.4

-0.1 -10 Yawing o tie(a

O(deg) U/c=Q.2

GM=O8()

_A'I=i.o

H/A=i/20

10[(dag) Rolling °1E Yawing o 8 8(deg)

_Pitching

100 60 Ue(g) &=O(deg) U/c=O.8 GM=O.8(m) _A/L=1.o H/XF1/20 10 -10 50 30 10 8

of = 1.0 but unsafe for the heading angles from zero to 50 degrees in the wave of \/L = 1.5.

Effects of Ship to Wave Velocity Ratio

Finally, numerical experiments in time domain were conducted for the ship running at the lower and higher speeds. Fig.13 are the examples of the time

histo-ries of yawing, pitching and rolliñg. The ship will be

safe in the case of U/c = 0.2 but unsafe in the case of U/c = 0.8 for the heading angles of zero and 60

degrees. Fig.14 indicates the results of numerical ex-periment to find out the critical GM with respect to

U/c. .8 _{e(de) Pitchin.g}

b30(deg)

U/c=O.2 GM=O.8(m)

_/L=LO

10 d'(deg) Rolltn.g 10 A 100

A.A A £

° & A A

1VV

_Vi

_V Yawing 50 10O tine(s)

Ue)

030(deg)

U/c=O.8 GMO.8(m)

_L't=Lo

H/)=jj2

301(de& Rolling 30 90x(de) YaWing 50,

.ioo

L 8 6(4eg): _Pitching

Fig.13 Time histories of ship motion in followig and quartering, waves of X/L=1

Uà)

V 6O(deg) U/c=O.2 GMO.8(m) A/L=1.O H/...=1/2o 10 é(deg) ROUID.g A

A A..

¡A

I'

-10 100 100

The capsize is seen to be related to pure loss of

stability when a crest moves amidships at the speed

nearly equal to the ship for a safficient length of time

to capsize. The wave length would be of about the

same length as the ship.

5. CONCLUSIONS

This paper provides an analytical approach to ship

motion and capsizing in extreme quattering seas. The

problem is divided into the effects of waves on the hydrostatics of sway motion. First, the hydrostatic

indices, GM, GZ and E are evaluated up to taking

50 x(de) _Yawing 30 _L U/c=O.8 '=6O(4eg) GMO.8(m) H/A=1/20 9Q(de) Rol1ng

GM(S)

1.6

1.2

GM(m)

1.6

1.2

0.8

0.4

0.0

J./:tt

/Lf/

ONot Capsized Capgjzed

2eg

0

0.2

0.4 O.6U/c

0

0.2

0.4

0.6U/c

₀

_0.2

_0.4

_0.6U/c

Fig.14 _{Critical metacentric height leading up to capsize versus U/c}

into account the effects of the relative position of ship to waves, the heading angles, wave height and length.

Next, a mathematical model is proposed on the bar

sis of Froude-Krylov forces taking into account the

instantaneous free surface at each time instant and

linearized hydxodynaxnic forces on ship hull. Finally, using the mathematical model, a lot of numerical

ex-perirnents are conducted in order to find out the

ex-treme wave and ship motion leading up to capsizing. The results of numerical experiments are presented and discussed for the critical metacentric height with respect to the several parameters.

Some of the main findings are the importance of the

critical GM leading up to capsize in extreme quarter-ing seas. The critical GM seams to be larger than the GM desigued to satisfy the criteria A562 in beam sea

and wind. It will be a considerable problem for the

safety of ship at sea how to determine a reasonable

z:3d

_{li4. H4:ii3}

_jot Capsized

OCapsized J

_

OCapgjze >Not Cajsized ed

0.2

0.4

0.6U/c

GM and how to operate a ship in extreme quartering

seas. The critical GM is related to the reduction of GZ

at wave crest amidships and the reduction is largely dependent on the flare configuration, free boad of a ship and the size of waves. So that it is an important

implication to consider the dangerous_{wave size on the} basis of the risk analysis.

REFERENCES

Watanabe, Y., On the Dynamic Properties of

the Transverse Instability of a Ship due to Pitch-ings, J. of Soc. of Naval Arch., Vol. 53, 1934.

Grim, O., Rollschwingungen, Stabilitat und Sich

heit im Seegang, Schistechnik, 1952.

Kerwin, J.E., Notes on Roll in Longitudinal Wave I.S.P., Vol.2 (16), 1955.

- . 1 .

O?jot Ca'sized xCapgjzed

...

...4.... ...

.I

1tiz

I i t

I i I I I I

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