Statistical methods for predicting the non-linear stress induced on the ship hull in ocean waves

27 OKT. 1982

ARCI-IIEF

Reprinted from the Memoirs of the Faculty of Engineering

Kshu University, Vol. 42, No. 1

FUKUOKA JAPAN 1982

tab.

_{v. Scheepsbouwkund}

Technische Hogesckool

Deift

With the Compliments of the Authors

Statistical Methods for Predicting the Non-Linear Stress

Induced on the Ship Hull in Ocean Waves

by

Memoirs of the Faculty of Engineering, Kyushu University, Vol. 42,. No: 1, March 1982

Statistical Methods for Predicting the Non-Linear Stress

Induced on the Ship Hull in Ocean, Waves*

by

Junichi FUKUDA** and Akiji SHINKAI***

(Received December 5, 1981)

Abstract

Statistical methods are proposed. for predicting the extreme. value of non-linear stress such as the von Mises' equivalent stress induced on the longitudinal member of the ship hull in ocean waves. The methods are

based upon an assumption that the wave normal stress and the wave

shearing stress, which are combined to the equivalent 'stress, cvould be regarded as absolutely dependent stochastic variables and upon another

assumption that those stresses would be regarded as absolutely

inde-pendent stochastic variables.

The application of the methods is attempted for a large oil tanker

in a short-term storm seaway, and the calculated results of the probability density for the statistical distribution of the maxima of equivalent stress on the longitudinal m'ember of the ship hull have been examined in com-parison with the histograms of the maxima obtained from the nume'ically simulated time histories of equivalent stress. Satisfactory agreement is

found between both results and the validity Of the methods is

demon-strated.

Then, the short- and long-term prediction works have been carried out for the large oil tanker in order to evaluate the extreme value of

equivalent stress induced on the longitudinal member of the ship hull in ocean waves. Finafly, the short- and, long-term correlations between the

wave normal stress and the wave shearing stress are investigated in

detail to examine the utility of the present methods.

1. IntrOduction

Recent progress in the statistical technique for predicting the ship responses in the

seaway has made possible to evaluate the extreme values of longitudinal stresses, including

the wave normal stress and the wave shearing stress, induced on the ship hull in ocean

waves3>. In above statistical analysis, the ship responses in a short-term seaway such aTs

the wave normal stress and the wave shearing stress are considered to be the linear,

Gaussian and stationary stochastic variables, However, on the other hand, no available

* _{This paper is rewritten in EnglIsh based upon two papers1'2> originally published in Japanese,} and read at 2nd International Congress of the I.M.A.E.M., TRIESTE, 1981.

** _{Professor, Department of Naval Architecture}

22 J. FUKUDA and A. SHINKAI

method has been found other than a few works by Lin4, Vinje-Skjordal5> and

Jensen-Pedersen6> for predicting statistically the non1inear ship response in- a short-term seaway such as the von Mises' equivalent stress, which denotes a yield criterion. The equivalent stress is considered to be the non-linear response: to which different responses are combined

together, including different linear dynamic (or stochastic) responses, namely the wave normal stress and the wave shearing stress, and different static (or constant) responses, namely the still water normal stress and the still water shearing stress.

In this paper"2>, the theoretical methods. are proposed for calculating the statistical distribution of the maxima of non linear ship response such as the von Mises equivalent

stress induced on the ship hull in the seaway, based upon an assumption that the wave

normal stress and the wave shearing stress would be absolutely statistically dependent and upon another assumption that those stresses would be absolutely statistically independent according to a conception similar to that in a method of Rice"> or of Lin4>. The present methods are applied to the short- and long-term predictions of the equivalent stress induced on the longitudinal member of a large oil tanker in, ocean waves.

2.

Probability Distribution of the Maxima of Non-Linear Response

The von Mises' equivalent stress induced on the longitudinal member of a ship hull in a short-term seaway can be written as follows:

Y= {(cr+o)+3( r-+ta)2}"2 (1)

where

Y: equivalent stress

ô'-, VT: total wave normal stres and total wave shearing stress

o', r0 : still water normal stress and still water shearing stress

The equivalent stress given by Eq. (1) is considered to be the non-linear response, to

which different responses are combined together, including different random responses

(namely the wave normal stress and the wave shearing stress) and different constant re-sponses (namely the still water normal stress and the still water shearing stress).

By squaring both sides of Eq. (1), the following equation is obtained.

Y2=aX?+bXl+cX,+dX2+e (2)

where X, and X2, which correspond respectively to Cr and rr, are considered to be the.

linear, Gaussian. stationary stochastic variables with zero means and narrow banded spectra,

and a, b c, d and e denote certain constants, namely a1, b=3, c=20, d=6r0, e=c

In this case, the probabijity distribution of the' maxima of Y2 can be calculated by a

method of Rice7>. It is, however. considerably difficult to obtain a general sQiution to this

problem except special cases, namely the first special case when X1 and X2 are assumed, to be absolutely statistically dependent and the second special case when. X, and X2 are assumed to be absolutely statistically independent. The solution to the first special case

is obtained by Lin4>, and the method to solve the problem in the second special case is proposed by the authors'-2>, similarly dependent on a conception in Rice's method"1. Methods

Statistical Methods for Predicting the Non-Linear Stress 23

2. 1 Probability Distribution of the Maxima of Non-Linear Response Pioduced by Two.

Absolutely Dependent Linear Respoiises

By assuming that X1 and X2 in Eq. (2) Would be absolütel' statistically dependent, the following equation is obtained.

S=AX+BX1

(3)

where

S= Y2e, A=a±b1.z2, B=c+dp

,.z=±R2/)?1

for p=±l

R1, Rx2: standard deviations of X1 and X2 p : correlation coefficient between X1 and X2

The solution to the problem in this special case is obtained by Lin41 Its outline will

be described for the case of A>0 and BO. Similar results

_{can be given for the other}

case of A >0 and B< 0.

The derivative of S with respect to time is obtained as follows:

2AX11+Ek1

(4)

where

S=dS/dt, X1=dX1/dt, t: time

According to the previous assumption, _{X1 and X1 are considered to be the linear,.} Gaussian, stationary and independent stochastic variables With zero meáñs. Therefore, the

joint probability density function of X1 and X1 is given by

Px1x1(xi,

.j)

_{=(2nRx1Rx1)'exp [ xU2R)1i/2R)1J}

(5)

where

Px1x1(xi, ) : joint pr6bability density function of X1 and )jI fOr (Xi=x1,

Rx1: standard deviation of J

By using the relationship between S aild X1 a shown in Fig. 1 and that between

.' and J, the joint probability distribution function of S and . can be obtained as follows:

Fs(s,)=f8ds fp5s(s, )d

jcZdfB(

_)di1

(6)

Fig. 1 _{Relation between equivalent stress} and wave normal stress

24 J. FuxuDA and A. SHrr.acAi

where

F(s, ): joint probability distribution function of

Sand

.

for (Ss,

Pss(S s) joint probability density function of S and S for (S=s S=s)

B2/4A : lower limit of S

B2/4As<

i9=/I2Ari+BI,

x1*B/2A

O3<<oo

By differentiating partially F5(s, ) with respect to s and ,, the joint probability den-sity function of S and can be obtained as follows:

Pss(S, )d2F5(s, )/dsa

=(4As+B2)1{Pxii(ai

_2Aa+BI

)+Pxi1(a2, _I2Aaz+BI

_)}

-

()

According to Rice's theoryl>, the expected number of maxima of S above a positive

level s per unit time can be obtained by

M+(s)=fPss(s, )d

: s>O V (8)

where

M(s): expected number of maxima which are larger than a positive level

s per

unit time V

By using (5) and (7)Vfl Eq. (8), M(s) is obtained as follows:

M+(s)=

(2rRxi)' R,v1{exp[ a/2Ri]+exp[ aU2Ri] }

s 0 (9)

And the expected number of negative maxima of S above negative level s per unit time can be obtained as follows:

M_(s)= expected number of negative maxima, which are larger than a negative level s, namely in the interval (s, 0)

=expected number of minima in the interval (B/A, ai) =expected number of maxima in the interval (ai, B/A)

=

Ji Px1A,(ai, i1)d fii p1,(B/A, i1)dij

(2 2rRxiY' R1 {exp[ aU2R1]exp[ B2/2A2Rxi]

V

B2/4As<0

V V

(10)

Then, the expected total number of maxima of S per unit time is obtained as follows:

MT= M+(0)±M(B2/4A)

=(22rRxiY' R1{1 +exp[ - B2/8A2R1] } (11) where

Mr: expected total number of maxima of S per unit time

M+(0) expected total - number of positive maxima of S per unit time

M_(B2/4A) :

expected total number of-negative maxima of S per unit time

By using (9), (10) and (11), the probability that the maxima of S exceed a given level s can be obtained as follows:

Statistical Methods for Predicting the Non-Linear Stress 25

q(Smax>s)=M+(s)/Mr

={exp[ a?/2R1,] +éxp [ - I/2R,]}/{1 ±exp[ B2/8A2RL] }

s0

(12)

q(Srnax> s) {M+(0)± M_(s)}/M

= {1 +exp[ a?/2RiJ}/{1 +exp[ B2/8A2R1]}

B2/4As<O

(13)

where

q(Sma>s) probability that the maxima of S exceed a given level s

Above results of (12) and (13) coincide with those obtained by another simplified

method which is suggested by the authors8.

Then, the probability density function of the maxima of S can be obtained as follows:

pSX(s)={aM+(s)/as}/MT

s0

(14)

Psmax(S)={aM_(S)/dS}/MT

: B2/4As<0

(15)

where

Psmax(S) : probability density function of the maxima of S for (S=s)

Substitution of (9), (10) and (11) into Eqs. (14) and (15) results in

Psxnax( s) = { a2exp [

a/2R,J a1exp [ a/2Rx21] }/

(4As+B2)"2Ri{1 +exp[B2/8A2RL]}

: s0

(16)

Psmax(S) aiexp[ aU2Ri]/(4As+ B2)"2R11+exp[ B2/8A2R,,]}

B2/4As<0

(17)

Furthermore, according 'to the similar procedure, the probability density function of the minima of' S can be obtained as follows:

Psrn!n( s) = a2exp [ aV2 RL]/(4As+ B2)"2 R,1 { 1 + exp [ B2/8A2 R,J

B2/4As<0

(18)

From a practical viewpoint, meanwhile, the negative maxima of S might be ignored so

far as the larger maxima are seriously considered, and, in that case, the expected total

number of the positive maxima would be regarded as the expected total number of thaxima. When such a realistic assumption would be employed, the probability that the maxima of

S exceed a positive level s would be approximately given by q(Sm,> s)=M+(s)/M+(0)

= {exp[ aU2Ri] +exp[ aU2R.,]}/{1+exp[ B2/2A2R,]}

s0

(19)

and the probability density function of the maxima of S would be _{approximately given by}

ps,,,(s)= aM+(s)/as }/M+(0)

{a2exp[ d/2R,] a,exp[ a/2Rj,]}/

26 J.FuKuDA and A. SHINKAI

2. 2 Probability Distribution of the Maxima of Non-Linear Response Produced by Two Absolutely Independent Linear Responses

By assuming that X1 and X2 would be absolutely statistically independent, Eq. (2) can

be written alternatively as follows:

where

S= Y2e+c2/4a+d2/4b= Y2

YI=[(X!+C/2a)

1

Y2.f(X2+d/2b)

J

a>O, b>O.

The derivative of S with respect tO time is obtained as follows:

where

S=dS/dt,

dY1/dt,

'2=dY2/dt

According to the previous assumption for X1 and X2 descrived in the beginning of

this section all of Y1 Y2 Y and Y2 are considered to be the linear Gaussian stationary

and independent stochastic variables. Therefore,, the joint probability density function of

Y1, 12, and Y2 is given by

Py1y''z(Yi, Y2, Yi' Y2)

[

(yYi)2

(y2Y2)2

i

1

- (2,r)2RyiRy2RyiRy2 exp 2R1

2R2

2R1

2R2

J (23)

where

Pyy1(yi,'

y,: J2) : joint probability density function of 11, 12, Y1 and Y2 for

( Y1=y, Y2y2, Y2=y2)

R11, R2, R1, R2 :

standard deviations of Y1, 12, and Y2

mean values of Y1 and 1'2

By using the relationship between S and (Y1, Y2) and that between and )

as shown respectively in Figs. 2 and. 3, the joint probability distribution function of S and can be obtained as follows:

Statistical Methodsfor Predicting the Non-Linear Stress

F5,(s, s)=

dsf_j(S,

_{) d}

ffdyidY2 fJ Py1 y2 _{Pz(Yi, Y2,}

i, 2)d1 d2

=1imjdyj f-

dy2fd1f

F

(yii

(yz_V)2 _'?

exp:

_2R1

_2R2

2R2 JUY2

+iTifa(s)

7(5) _{- 5(i)}

dYif

dyzfdif

_{I') \2D. D D. D.}

(.2Z'J £ty1itY2LtyiLIY2

r (yi_,P)2 (ys_P)2 Y?

exp

2R1

2R2

2R,

?Rh j

Y

where

-Domain E=[(Y1, Y2)IITY1'T, Vs--Y12 Y2VsYfl

Domain G=[(,

_'2)Hoo<

°°<

Y1 _{'1/IYiI+s72IYzIi}

v(s)=Vsy

Y*O

By differentiating partially F5(s, ) with respect :to and the joint probability den-sity function of S and ' can be obtained as follows:

Ps(S, )=ö2Fss(s,

)/dsa

(25)

In order to perform the operations of Eqs. (24) and (25), the polar coordinates (r, 6) may be- employed instead of the rectangular coordinates (Y1, Y2),name1 the following

transformations will be introduced

yi rcos 0

y2=rsinO

:Or'T,

(26)

J(r; 8)=a(yi, y2)/8(r, 0)=r

(27)

Then the joint probability density function of S and S _{is obtained as follows}

Ps(S, s) f2?r(2,z)_312(4R

_{R2) {s(Ricos28+ Rsin29)}

}_112

exp[(fcos0 Y)2/2R2(/sjn8V)2/2R2

- 2/8s(R2 cos2 0+ R2sin2 0)] dO -- - (28)

When the dynamic parts of 11 and Y in (21) vanish, S of (21) denotes a static level

s, Which isgiven b -.

-S*=Yls+}7

_{(=c2/4a+d2/4b=e)}

- - (29)

By using Rice's theory7, the expected number of maxima of S above a positive level

s which is larger than ? per unit time can be approximately evaluated by

M(s)'fpss(s, )d

ss*0

(30)

27

28 J. FuKubA and A. SHINKAI By substituting (28) into (30), M(s) is obtained as follows:

M(s)=

_/ (2ir)312(Ryi Ry2)_hI(R.icos28+ R2sin2O)"2

-'0

exp[(/cos 9._

)2/2R,i_(/sin

Y/2R'2] dO

(31)

From a practical vieWpoint, the maxima of S lower than a static level s* might be

ignored so far as the larger maxima are seriously considered, and the expected number of maxima above a static level s* would be regarded as the expected total number of maxima.

Then, the expected total number of maxima of S can be approximately given by

MT= M(s*) . (32)

By using (31) and (32), the probability that the maxima of S exceed a given level s,

which is larger than s, can be approximately given by

q(S,>s)=M(s)/MT

=

f2R_(Ricos2e+R2sin28)2

exp[ (Icos 8 )2/2Ri_(fSjn9_

)2/2R?,z] d8/

.fZ/(Rjcos2e+R2sin2e)u12

exp[(/?cos 8

y)2/2R2 (fsin 8 Y)2/2R2]d8

(33)

And the probability density function of the maxima of S can be. approximately given

by

Psnmx(S)

=-

{aM(s)/as }/M

r2,r

-= J

(Ricos28+Rhsin2O)h1{_1/2f+(/c0S8_ Y1)cos8/2R'1

+('Tsin8 Y2)sin8/2Rz}

exp[ WTcos 8

Y)2/2R

(fisin 8 Y)2/2R2]d8/

exp[(/?cos8-- Yi)2/2Ri_(VTsin8_ Y2)2/2R2]

dO

ss*0

(34)

Furthermore, according to the similar procedure, the probability density function of the minima of S can be approximately given by

Psmin(S)=IPsrnax(S)I

: s*s0

(35)

2. 3 Probability Distribution of the Non-Linear Equivalent Stress

Statistical Methods for Predicting the Non- Linear Stress 29 stress can be calculated by the methods descrived above for two special cases, namely,

"Case 1" when the wave normal stress and the wave shearing stress are considered to be absolutely statistically dependent, and "Case 2" when the wave normal stress and the wave shearing stress are considered to be absolutely statistically independent..

"Case 1"

When X1 (that is the total wave normal stress ) and .X2 (that is the total wave

shearing stress VT) in Eq. (2) are considered to be absolutely statistically dependent, the

exact solutions to the probability distributiOn of the maxima and the minima of Y (that is the non-linear equivalent stress) can be obtained, by using (12), (13), (16), (17) and (18), as follows:

q(Y>y)=q(Y>y2)=q(S>s)

=

_{{exp [ - a/2Rr] +exp [ a/2RT] }/}

{1 +exp[ (a+3z r0)2/2(l +3p2)2R-]

yy

or

_{q(Ym>y){1+exp[aU2Rr]}/}

{1+exp[(ô+31zv0)2/2(i+32)2Rr]}

1 _-) 1 \__O 12

y

*2

y{a2exP[a/2RT]alexp[aU2Rr]}/

R{(1 +3p2)(y2y2) +(ô+3pro)2}"2 {1 +exp[(ø'o+3, To)2/2(l +3,z2)2R] }

yy

or Prrnax(Y)

ya1exp[a?/2R?,-,-]/

R7{(1 +32)(y2y2)±(o+3p

ro)2}U2 (1 +exp[(ô'o+3i r0)2/2(i +3p2)2R.] }

yL.y<y*

pymjn(y)= ya2exp[a./2Rr]/

R{(1 +3p2)(y2_y*2)+(o±3pro)2}112 {1+exp[ _(ôo+3Lro)2/2(1 +3p2)2RT] YLY<Y where

q( Ymax>y) : probability that the maxima of equivalent stress Y exceeds a given

level y

Pmax() probability density function of the maxima of equivalent stress for ( Y,

=)

probability density function of the minima of equivalent stress for (Ym

=)

30 J FIJKUDA and A SHINKAI

$=y2_y*2, y(o2+3v2)"2

YL= {y2(øo+3 o)2/(l +32)}1i2

a1

[(co+'3pvo)

+(6±3p r)2}]/(1 ±3A2)

a2J

EL±RT/RCT for p±l

: standard deviations of total wave oPmal stress 6T and total wave sheaing stress.

r-p : correlation coefficient between o- and Vt

And, from a practical viewpoint, the approxitnate solutions to the probability distribu-tion of the maxima of equivalent stress can be obtained, by using (19) and (20), as follows:

q(

{1+exp[-2(ao±3,zro)2/(1±3tt2)2Rr]}

yy

(41)

P(y)

=y{ a2exp [ - aU2 R7] - a1exp [ - a/2Ri-]}/

D2 _{+3,2)(yz_y*2)+(co+3ra)2}h/2.} 1 +exp[ 2(a0+3p T)2/(l ±3p2)2Rr]}

yy

(42)

'Case 2"

When X1 (t.he total wave no?mal stress) ãfld X2 (the tot1 wave sharing stress) in Eq. (2) are coni'dered to be absolutely statistically independent, the approximate solutions to the probability distribution of the maxima of Y (the equivalent stress) can be obtained by

using (33), (34) and (35), as follows:

( V

'-

IV2 -_

-

C

q, m .-

y, -

max- y / - q S 21r

J y(Rcos2O+3R.sin2O)h/2

exp [ - (ycos

ao)?/2R' (ysin 0 r/6Rr] 8/

y*(R,cos2O+3R,sin20)hI2

exp[ (ycos0 c0)2/2Rr(ysin 0VT v0)2/6R-]d0

Prmax(Y) =2ypsmax(s)=2yPsmaxy2)

-f2,r

Jo

yy

(43)

=f 22y(Rrcos2e+3Rrsin2o)h/2{ 1/y+(ycos0 c0)cos O/RT ±(ysin 0VT vo)sin0/3Rr}

exp[ (ycos 0 c0)2/2R. (ysin

VT

r0)2/6]d0/

f2'y*(Rcos20+3R2jsin29)1/2

exp[(ycos 0 ao)2/2R,_(yssin 0VT r0)2/6R] dO

y* (44)

where

s=y2

R&, Rj.r: standard deviations of b(=d/dt) and ir(=dv/dt)

(Note)

When B in Eq. (3) is negative, a in Eqs. (13), (17), (37) and (39) should be replaced

by a2, and

2 in Eqs. (18) and (40) should be replaced by a1.

3.

Statistical Predictions of the Equivalent Stress

3. 1 Numerical Simulations of the Equivalent. Stress

In ordei to examine the validity of the theoretical methods described in, the preceding section, the numerical experiments by means of a computer have been made for. simulating

the time histories of the equivalent stress induced on the longitudinal member of a ship

hull in long-crested irregular waves:

In the first place, the time history of the long-crested irregular waves might be

determined from the wave spectrum, as follows: ..

h(t)=,

hoCOS(wejt+ej) i='l = COS( Weit± e) = V2[f(wjP'4w COS(weit+i) (46) 1=1 where

ho: amplitude of the i-th component wave

random phase angle of the i-th component wave circular frequency of the i-th component wave

We1 w w Vcos8/g: encountered circular frequency of the i-th component wave

4ww+1w, Awe=4w(1-2wjVcos8/g)

wave pectrum as function of w

Lf(wej]2: wave spectrum as function of we V: ship's velocity: g: gravity accélaration 8: ship's heading angle against the wave direction

The circular frequency of the i-tb component wave is actüall' determined as follows:

w1=0.05(i±:5)s/2zrg/L

: i=1-31, L: ship length

-and the r-andom phase angle e is chosen so as to be equally distributed from 0 to 2r.

B

the aid of the impro ed strip theor

the response functions of the total wave

normal stress and the total wave shearing stress can be obtained in the form of

[aTi( WI)] = [are 0/h10]cos( Wejt . (47)

[rTI( WI)] = Err10! h10] cos( Welt (48)

where

.32 J. FuiuDA and A. SHINKAI.

T,o, rro : amplitudes of total wave nopnial stfess and total wave shearing stress in

i.th component wave

EØTi, er: phase angles (lag) of total wave normal stress and total wave shearing

stress in i-th component wave

Then, the time histories of the total wave normal stress and the total wave shearing

stress can be generally given as follows.:

[Tio/ hi0] ./2[f(w)]24w cos(wt+ e, er)

(49)

= [rrio/hio] I2(f(w)]24w COS( Wegt± £j. LtTi) (50)

However, in the first special cise when the total wave ñOrthal streSs 'aiid the total

wave shearing tress would be. absolutely statistically dependent, r(t) should be given as follows. instead of (50).

'Iz±RVT/Ro.r

for p=±l

51)

And. inthe second special case when the total wave normal stress and the total wave shearing stress would be absolutely statistically independent (namely p=O). the different random phase angles should be given in (49) and (50).

The time history of the equivalent stress can be given, by using (49) aid (50) or

(51), as follows:

211/2

ro; j (52)

Numerically simulated calculations have been carried out for a large oil tanker, main particulars of which are shown in Table 1. according to.the following calculation program.

Table 1 Main particulars of a large oil tanker

Metacentric height (GM) . 0.3305d

Longitudinal gyradius (XL) 02494 L

Transverse gyradiüs (Kr) ' . '0.3231B

Heaving period (TN) 11.6sec

Pitching period (l'p) 10.9sec

'Rolling period (TX) 14.0 sec

(a) Irregular waves . . .

The long.crested itregular waves are assumed according to the formulation of ISSC

Sectrüm9>, as follows: '

[f( w)]2 =0.11 H2wi'( W/WT) 5exp [ 0.44(w/wY4] (53)

Length between perpendiculars (L) 310.000m

Breadth moulded (B) 48.710m

Depth moulded (D) 24.500m

Draught moulded (d) . ' 19.000m

Displaernent (W) 250,540t

Block coefficient (Ci) 0.852

Water plane coefficient (Cr0) 0.903

Midship coefficient (Cm) 0.995

Centre of gravity before midship (Xc) 0.0331L

where

w: circular frequency of a component wave

ur=2iriT, T: average wave period du to visual estimation H: average wave height due to visual estirñation

Combination of T=9.5 sec and H=7.4m is adopted, which corresponds to the average.

sea state of Beaufort 10 in the North Atlantic Ocean'°.

Ship's heading angle

...

8=135' : bow seas . .

Ship speed . .

Froude number: Fr= V/VL=0.15 Location of longitudinal member

Weather side bilge of S.S. 5

Still water normal stress: o'= -1.29Kg/mm2

Still water shearing stress: r0 -0.38 Kg/mm2 Weather side gunwale of S.S. 7

Still water normal stress: c, -2.68 Kg/mm2

Still water shearing stress.: rc= -0.69Kg/mm2

The sign conventions for the normal stresS and the shearing stress are shown in

Fig. 4.

180

Statistical Methods for Predicting the Non-Linear Stress 33

Fig. 4 Sign convention for normal steSs

and shearing stress

Fig. 5 shows the response functions of the total wave normal stress nd the total wave shearing stress, together with the wave spectrum, as functions of

fL7

(A: wave length)

as well as of cv. On the bilge of 5 5, the response amplitude of wave normal stress is

KG.MM2M1 '.5

uI

p 1.0- _.0.101 0.5' _.0051 . 0.5 0 -.

---:

-, 0

_._ . o _.OT .EVT 0.5

I.0-

20 0 0.2 0.4 0.6 - w .0 SEC l.0 b 180 w -180 - TO aT 0

I.07 2.0

o 0.2 0.4 0.6 - U .0 SEC' Fig. 5 Response functions of wave normal stress and wave shearing stress and

LS.S.C. spctrum for the sea state of T=9.5 sec and HoEr 74 m

KGMI2.MI

'.5

SEC. 0.15

34 J. FUKUDA and A. SHINKAI

much larger than that of wave shearing stress, however, on the bilge of S.S. 7, the response amplitude of wave shearing stress is almost comparable to that of wave normal stress.

Examples of the numerically simulated time histories are .shown, in Figs. 6 and 7,

including those of the long-crested irregular waves, the wave normal stress, the wave

shear-ingstress and the equivalent .strees, which are obtained. for special cases of p=±l and

p=O. Those time histories have been calculated on every 0.5 second during 2,000 seconds, and the parts of those during 600 seconds are illustrated in the Figures.

Fig. 8 shows the theoretically calculated results of the probability density' function of the maxima of equivalent stress, including the exact solutions obtained by (38) and (39)

LIH IJhIHLftL.L

IIIILIt t4 I Ij',1 lii i_i l'

IIjIIIiIjflhII I hIftlIi IIIclhI uIiiliIiIlliujIuI,iI;I:I:i hIhIu! rIv

iIi?iIirNIuitiIiifiIiI:II ih!i liii iW

II 'II! lT' ii I, HI I p LONG-CREST 5.9.9 81. t 135.0' FR 0. 15 LONG-CREST S.S.S 81 1= 135.0' FR= 0.15 .O4G-CREST 5. . 5 81 t- 35.0' FR.- 0. IS OFT. 0 T- 9.5 5 7.4 OFT. 10 1- 9.S IS 1 7.4 151 P.1

I,

I,LH

lId

II iIIIL liii ItII i I U$iI1, IIiL

IIIIIIlIIl II UiIIlliiflU$l ?IIIIblilIIIrIllIihIftlIlñ II hiIIIIIiIfli?IiuhIiI

L.IIJII

,LjIIIlI,I 1,11 I,l.J1P

SIRE 55 8F1. IS I. '..5-5 H' i.4 SI P.O V. N JFF55 5HEH9j.5 TTRLs5... NE505.5J.5E OV STRESS

Fig. 6 Time histories of equivalent stress, wave normal stress and wave shear-ing.stress on bilge of S.S. .5 section and of long-crested irregular waves

LONG-CRESTES S. S. 7 GUNWALE ._'o 1- 135.0 F'R= 0.15 IY!!!

,ilIi

I 15C

911111 lIYYjjI TI1 I ITLIrPIIIU9 I115 IIIl E1I!I WWJ I 1

t ;M1!

Ii

300 360 420 460 ,i:.ItI I FI , I ii 116 ' LONG-CRESTED BFI. ID .9. -I - S. S. 7 GUNWALE 1= 9. 5 ISECI ...- S= 135. 0 H= 7. 4 (III

MLTJLA6. A.AEOUIVALENT STRESS

oI

Statistical Methods for Predicting the Non-Linear Stress 35

BrI. 0

1=95 5CC)

A7.4 III)

60 20 80 240 330

IlIllIlERIII iTT'T1ILTI RlW1 IllImlEl III III PNtJIP lW I 'P11)111 liii

uiitss:ii; 9 INI1 'PH F Ii

I''

1 II' II I H .11 iIIi I 'I

flg. 7 Time histories of equivalent stress, 'ave normal stress id wave shear-ing stress on gsinwale 'of S. S. 7 section and of long-crested irregular tvaves . . S.S. S SILGE cro.;s; 6.,35 B FT. 10 1-9.5 SEC. fl.74 M EXACT AFPR0X

- -La. I

_---..0

p.' rITE ISECI 420 489 540 600 1.0 :.: 0.4 0.2 .2 S.S. 7 GUNWALE Fr-O.I5, .I35 FT.I0 1.9.5 SEC. H .7.4 M EXACT A0X. . - 0

25

4 5 6 7 9 9 0 I 2 5 4 5 6 1 B

Fig. 8 Probability density functions of the maxima of equivalent stress

KGfll1V2

.ILhII.d Ii

iIiiuLijL1,iUi ,JjjIrIFiII A ' 0 I N 60 LVV)IllII U -. _!Il'

'11111

-. I 80 240 GO 360 IIII)IIIILL!iflVX1lIIIIlIIIIFIIIIIIlIL1 IIl1 420 l!1LYTIL1llI)l . IjjFIllI1' II 15CC 460 540 600 IIIIIITWW,IL N1,TTT I . I Pill1! III ..1 t1lll1I ilL I .. I III Il . I !I!!lr

ii j I . III.IWAVF FOR NORTAL

IJI.I)ø,I!L iIiIlIliIIIilLJlllI II)I LIIYIIIIl 1 N ILl l,I.J,AIAIIIJ,L,l I I!I 5T-j55 l,.I .

I lIIIip1iI I'!lIII! STRESS LONG-CRESTED OFT. 0 P.O

5.5.7 GUNWALE 1= 9.5 I5ECI 1. 135. 0 H= 7. 4 (III NI 1 600 INS STAFSS . 0nL..1LOE5S .

36 J. FUKUDA and A. SHiiickI Ic'.hSA2 1.0 !!IIIHIIIIIIiIui;..__. I. 2 0.4 0.2 1.2 .0

:

0.4 P-I S.S.5 BILGE F,-0.I5, d-135 BFT. 10 1.9.5 SEC. H.7.4 M

NO. OF SAMPt.E 38? MAX.

- 386 MIS.

-

MAX. - VACT P.DW NIX. tOW? POP. W MW. 2 3 4 5

_{- V}

6 7 8 9 KG/MM2 P S.S.5 BILGE F,-O.15, d.135 SF1.10 1-95 SEC. K7.4 N

NO. OF SAMPLE 399 MAX.

- 396 NUN.

-; APP.P.0.F. W MAX. - EXACT POP. W MAX.

. tOACT POT W MIS.

2 3 4 5 6 7 B 9 KG/M7A2 P0 S.S.5 BILGE F,.O.15. -I35 SF1. 10 1.9.5 SEC. H-7.4 M

NO. OF SAMPLE 353 MAX.

- 3B4MIN. - AP050XPO.PWMAA. APPROX PD F. N IN. MAX 0.2 .iI(IIIIIIiiliIiIIih..._... 2 3 4 5

-

6 7 9 9

Fig. 9 Probability density functions minima of equivalent stress

1.0 0.9 0.4 0.2 I 2 3 4

y

5 6 7 9 G/X,*A2 0.2 SIN: P.' 2 3 4 5 6 S.S. 7 GUNWALE F,.O.I5. 6-I35 SF1. 10 1 -9.5 SEC. H.?.4 N

NO.OFSAMPLE 269 MAX.

267 MIN.

-

W MAO.

- EXACT PDPWNIX.

EXACT ROT WNIN.

7 8 P - - I S.S. 7 GUNWALE Fr.0.15, -I35 SF1.10 1-9,5 SEC. H.7.4 M

NO, OF SAMPLE 420 MAX.

419N1N.

WP POT W MAR.

-: EXACT POT. OF MAX. - EXACT P.07W SIN. P - 0 S.S. 7 GUNWALE Fr.O.I5, d-135 SF1.10 1.9.5 SEC. 11.7.4 M NC. OF SAXeLE 283 MAX. - 294 MIN. rlrn

-:

0 I 2 3 4 5 6 7 8

- y

KG/MM2

and histograms of the maxima and the

for the cases of p=±l, the approximate solutions by (42) for the cases of p=±l, as well as the approximate solutions by (44) for the case of p=O.

Fig. 9 shows the histograms of the maxima and the minima of equivalent stress obtained from the numerically simulated time histories in comparison with the theoretically calculated probability density functions, where the theoretical solutions to the minima of equivalent stress are obtained by the exact method of (40) for the cases of p= ±1, and also obtained by the approximate method of (45) for the case of p=O.

As found in Figs. 6 and 7, and also in Figs. 8 and 9, on the bilge of S.S. 5, the os-cillating amplitudes of wave normal stress are much larger than those of wave shearing

stress, and therefore the trends of the probability density functions of the maxima and the minima of equivalent stress are not so different in the cases of p= ±1 and O=O. However,

on the gunwale of S.S. 7, the oscillating amplitudes of wave shearing stress are almost

comparable to those of wave normal stress, and therefore the trends of the probability density functions of the maxima and the minima of equivalent stress are considerably

,cci- MM2

1.2 1.0

Statistical Methods for Predicting the Non-Linear Stress 37

different in the cases of p=±1. and o=O.

According to the results shown in, Figs. 8 and 9, it may be said that the theoretical

solutions obtained by the approximate method are practically almost equal to those. obtained by the exact method. And., as shown in Fig.. 9, the theoretical, results of the probability density functions of the maxima and the minima, of equivalent stress, including the exact solutions and the approximate solutions, agree well with the histograms of the maxima and the mitiima obtained from the numerically simulated time histories of equivalent stress.

Therefore the theoretical' methods, including the exact one and the approinate one, are considered to be sufficiently valid for the statistical prediction of the equivalent stress.

And, from a practical viewpoint, th'e approxiniate method may be employed for the purpose of evaluating, the extreme value of the equivalent, stress.

3. 2 Short-Term Predictions of the Equivalent Stress

It is assumed that a ship navigates in a shortterm seaway with a constant average

velocity and a constant average heading angle against the average wave direction.

The shbrt-term seaway can be represented by the hort-restéd irregular waves, as

follows''>

Lf(w, v)]2=(2/ir)[f(w)]2cos2 v : 2r/2

2r/2

=0 : elsewhere

where

[f(w, .,)]2 : directional àve spectrum

[1(w)]2: ISSC wave spectrum given by (53)

y: angle between a component wave direction and the average wave direction

Then, the variances of the total wave normal stress, the total wave shearing stress, and also the derivatives of those with respect to time can be calculated, based upon the

linear superposition theory'°, as follows 2T2

RT=(2/,r)

_fI2

J [A0-(w, 8y)]2[f(w)]2cos ydwdy.

(55)

zrIZ

Rr=(2/,r)

_{f_7r,2}f [ArT(W, 8 y)]2[f(w)]2cos2 vdwdv (56)

2/2

R _{(2/ar) f,2 f w[Aor(w, 8 y)]2{f(w)]2cos2 ydwdy} (57)

Jr/2

Rr(2/2r)'f f w[Az-(w, 8

)]2

[f(w)]2cos2 ydwdy (58)

where

[A-(w,

_{8y)][o'ro//zo] :}

response amplitude of total wave normal stress

[A(w,8-v)]['r0/h0] : response amplitude of total wave shearing' stress

6TO, V7-0 :amplitudes of total wave normal stress and total wave shearing stress

ho: amplitude of a component waye

We w w2 Vcos(8 'y)/g: encountered circular frequency of a component wave

8: ship's heading angle against the average wave direction ship's heading angle against a component wave direction

By introducing the' calculated results of (55)(58) into (41) and (43), the approximate solutions to the short-term probability that the maxima nf equivalent stress, exceed a given

38 J. Fuxtm and A.. SHfNKAI

level can be obtained for special cases; when the wave normal stress and the wave shearing stress are considered to be abolutely statistically dependent (p=±1) or absolutely statisti-cally independent (pO).

According to those methods, the short-term prediction works have been made for the large oil tanker (Table 1) in the short-crested irregular sea which corresponds to the average

sea state of Beaufort 10 in the North Atlantic Ocean°. Thus, the short-term makima of equivalent stress are evaluated on the longitudinal members of S.S. 3, 5 and 7.

Examples of the calculated results are illustrated in Fig. 10, which shows the short-term

DECK C.L.j

-. p-ti

0 'Gnhi2 -.-: P. 0 I 80°

"a'

It N

..LF DEPTH OF SIDE SHELLI

c--O.29KG/1412 I -O.97IcGdl --- p. o 180° -P--I Io -2.68I" T0 -O.69Kt/MM I 80°

-P- I

---p. 0

w-wr

90° 90° KEEL CL.

HALF DEPTH OF LONGI. 8HDJ

0- 2.46As

-P- I

T0.-O.?IIItfll2 -

---P 0

180°

90°

Fig. 10 Short-terhi equivalent stresses on the longitudinal members of S. S. 7 sec-tion predicted with the exceeding probability of q=10 in the average

sea state of BIt: 10 (T=9.5 sec. H=74 m)

GUNWALEI I8ILGE

0- 2.61 ,IG/lm

KG1t 0

maxima of equivalent strss oi the longitudinal, members of S.S. 7 predicted with the

ex-ceeding probability of 10 in that seaway. As found in Fig 10, the short-term maxima of

equivalent stress take the largest values in the case of p=l or p= 1 in accordance with the case when the still water iiormal stress and the still water shearing stress take the same signs or opposite signs, and take generally the smaller values in the case of.p=0.

The similar trends are found in the short-term prediction resultson the longitudinal members of S.S 3 and 5.

3. 3 Long-Term Predictions of the Equivalent Stres

It is assumed that a ship navigates in long-term seaways always with a constant average velocity and a constant average heading angle against the average wave direction. Then,

the cumulative probability that the maxima of equivalent stress exceeds a given level can

be obtained as follows'2>

Q8( Ymax>y)

JooJcNq8(

Ymax>y)P(H, T)dHdT ₍₅₉₎

where

Qe( Ymax>y) : _{long-term probability that the maxima of equivalent stress Y exceeds}

a given level y when a constant heading angle 6 is assumed

q( Ymax>y) : _{short-term probability that the maxima o equivalent stress Y exceeds}

a given level y when a constant heading angle 6 is assumed, which

is calculated by the approximate method of (41) or (43)

p(H, T) : long-term pfobability density of occurence for the sea condition of visual

average wave height H and the visual average wave period T

Furthermore, by assuming that all headings against the _{average wave direction are} equally probable in long-term seaways, the cumulative probability that the maxima of

equiva-lent stress exceeds a given level can be obtained as follows12>

Qaii( Ymax>y)=(1/22r)f2'Q( Yxnax>y)d6 (60)

where

Q,1( Ymax>y) : long-term probability that the maxima of _{equivalent stress Y exceeds}

a given level y when all headings are equally ptobable

20 8 110 0 20 2 110 0

Statistical Methods for Predicting the Non-Linear Stress _'39

r

10 2 1' ' 20 110 4 5 6 7 0 ₄ 5 6 7 8 0

Fig. 11 _{Longitudinal distributions of long-term equivalent stress predicted with} the exceeding probability of Q=lO-8 in the North Atlantic Ocean

310M TANKER Q-I0, Fr.0.I5 ALL. HEADINGS

-:2-il

DECK'CL S.D. 3 4

5 67 6

KEEL CL. 31DM TANKER

-.8- I

Q-106,F,-'O,ID ALL HEADINGS -' :

HALF DEPTH OF SIDE SHELL

5S 4 5 6 7 6

HALF DEPTH (IF LONGI. 2ND.

(

-

':'

-31DM TANKER O-iO. F, -0.15 ALL HEADINGS

-:P- I

----: GUNWALE D.S.S_4 5 6 7 Ô- -BILGE' -

--.--.---.----40 J. FUKUDA and A. SHINKAI

According to those methods, the long-term prediction works have been made for the

large oil tanker (Table 1) in the North Atlantic Ocean, by utilizing the long-term wave statistics13)

The prediction results, for special cases, when the wave normal stress and the wave shearing stress are considered to be 'always absolutely statistically depebdent (p

±1) or

always absolutely statistically independent (p=O), are summarized and illustrated in Fig.

11, which shows the long-term maxima of equivalent stress on the longitudinal members

along the hull length predicted with the exceeding probability of 10 in the North Atlantic Ocean, as well as the static equivalent stress in still water.

As found in Fig. 11, the longterm maxima of equivalent stress take the largest values

in the case of p=l'or p=-1, and take generally the smaller values in the case

of PO

The largest equivalent stress is found on the gunwale of midship, and the larger'equivalent stresses are found on the deck centre line and on the bilge of midship and also on the

half depth of longitudinal bulkheads of S.S. 4 and 6.

Table2 Still water normal stress, still water shearing stress, total Wave

normal stress, total wave shearing stress and equivalent stress

.5 r, _{(Q=108) (Q=10)} Y YA YA/Y Y=MAX.(Y) (A) DECK C.L. 1.55 0. 14.84 5.30 18.79 1.07 (B) GUNWALE 1.41 -0.39 17.26 3.17, 1.05 (C) HALF DEPTH OF SIDE SHELL 015,

- .

053 1082 3 '11.63' 11.75 1294 110 (D) BILGE -1.29 -0.38 15.54 2,65' (E) KEEL C.L. -1.37 0 11.75 4.05 14.88 1.07 (F) HALF DEPTH OF LONGI. BHD. 0.15 -0.81 4.48 5.04 10.30 10.40 11.14 1. 70 Y: Q=108 YA: Q=10-8 STRESS IN KG/MM2

The extreme values of equivalent stress on the longitudinal members of midship, which are illustrated in Fig. 11, are shOwn also in Table 2 as compared with the conventionally estimated extreme values. The conventional estimates are obtained by

YA(Q=108)=V{ T( Q 108)+ }2+3{ rT( Q 108)+ro}2 (61)

where

YA(Q=lO8) : equivalent stress estimated conventionally with the exceeding prob.

ability of 10.8

CT(Q=lO) : total Wave normal stress predicted theoretically with the exceeding probability of 108

DT( Q 10_8) : total wave shearing stress predicted theoretically with the exceeding

probability of 10_8

and, in Table 2, Y denotes the equivalent stress predicted theoretically with 'the exceeding

probability of 10_B for the case of p=l or p= -1.

compared with the theoretical results dependent on the assumption of p=l or

_{p= 1.}

3. 4 Short- and Long-Term Correlations between the Wave Normal Stress and the Wave

Shearing Stress

By assuming that a ship navigates in a shortterm seaway with a constant average

velocity and a constant average heading angle against the average wave direction, he short-term correlation coefficient between the total wave normal stressand the total wave shearing stress can be obtained as follows

where

(8) : long-term correlation coefficient between total wave normal stress and total wave shearing stress, in the case when a constant heading angle 8 is assumed KrVT(H, T, 8) : covariance of total wave normal stress and total wave shearing

stress in a short-term seaway of the visual average wave height

- H and the visual average wave period T, in the case When a

constant heading angle 8 is assumed

[R.-(H, T, 8)12, [RT(H, T, 8)12: variances of total wave normal stress and total

wave shearing stress in a short-term seaway of the visual average

wave height H and the visual average wave period

T, in the

case when a constant heading angle 8 is assumed

Furtherrnoie, by assuming that all headings against the average wave direction are

equally probable in long-term Seaways; the average long-term correlation coefficient can be obtained, in a practical meaning, as follows:

(all) _{.K01'r(a11)/[R0.(al1)][(all)] (68)}

p

Jr/2

KTrT(2/r)f f [A(w,

8 y)][Ar(w, 8 )][f(w)]2

Statistical Methods for Predicting the Non-Linear Stress 41

(62)

cos[e,i-(w, 8 v)- ErT(W, y)]cos2 ydwd-y (63)

where

Korrr: covariance of total wave normal strss and total Wave shearing stress in a short-term seaWay

OT(W, 8 _{7), £rT(W, 8y) : phase angles of total wave normal stress and tothl wave}

shearing stress

And, by assuming that a ship navigates in long-term seaways always with a constant average velocity and a constant average heading angle against the average wave direction,

the long-term correlation coefficient between the total Wave normal stress and the total wave

shearing stress can be determined, in a practical meaning, as follows.:

(8)= ₍₆₄₎

Kcrrrr(8)=j f Krrr(H, T, 8)p(H, T)dHdT

(65)

T, 8)]2p(H, T)dHdT (66)

where

(all) long-term correlation coefficient between total wave normal stress and total

wave shearing stress, in the case when all headings are equally probable

According to those methods descrived above, systematic calculations of the short- and

long-term correlation coefficients etween the total wave normal stress and the total wave shearing stress have been carried out for the large oil tanker (Table 1) in the short-term

seaways and in the North Atlantic Ocean The obtained results are summarized and shown in Figs. 12-14.

Fig. 12 shows the short-term correlation coefficients between the wave normal stress

and the wave shearing stress on the longitudinal members of s_s. 7, as functions of the

average wave period which are evaluated in the short crested irregular waves from different directions of the starboard side.

10 0.5 10 -0. 0. 10 0.5 10 -0.5 GUNWALE, 5.0. 7 TANKER 31GM Fr .0.13 WEATHER SIDEI S. 0' - 45 90 -.135 - ISO LEEWARD SIDEI 4 6 8 10 12T l6 1.0 0.5 0.0 1° l.0 .0 05 05 .0 8ILGE, S.D. 7 TANKER L3IOM Fr .0.15 10 10 WEATHER SIDEI 0.5 ID 05 0.5 1.0 M.?WPNWI..L9., EL? -4- 0' 45 '-90 -'35 80, LEEWARD SIDEI 4 6 8 IC 12-T I6.

Fig; 12 Short-term correlation coefficients between wave normal stress and wave shearing stress derived in short-crested irregular waves

2ECI( C. L., SO. 7 TANKER L.3IDM F,.0.13 '-:1. ' . 0' 45 180 -KEEL CL, S.S. 7 I

42 J. FuKUDA and A. SH1NKAI

Kyrrr(all) ± (1/2 ir)f 8) do (69) {r(all)12=(1/2,,r)

f2F[i(8)]2do

(70) [T(all)]2= (1/2 it) f2 0)12 do (71) IU,JT9WEW9EU., I.!.? TANKER L.310M Fr .0.15' :4- 0' - - 45 . - 90 - .135 - I!0 WEATHER SIDEI -TANKER :S. 0' L310M : .45 F, .0.13

-.-:.l3S

- 90 WEATHER SIDEj .0 , 80 0.5 4 6 0 -10 12-.T I6. 10 LEEWARD SIDEI 4 6 8 10 12-T 16.,, 1.0 10 -0.5 -1.0 1.0 0.5 0

Statistical Methods for Predicting the Non-Linear Stress 43

IIIU!iNP

iIi!!IiI i.

_LEEWARD '0(111 0. TANKER. L.310M'. Fr-OlD TANKER,L.3IOM,Fr'-O.lS

4LI

WEATHER SIDEJ 1.0 -ISO -90 0 1 90 ISO'

Fig. 13 _{Long-term correlation coefficients between' wave normal stress and wave} shearing stress derived in the North Atlantic Ocean

LEEWARD 510(1 0.5 0.5

Jo

-0.5 1.0 S.S. 3 TANKER. L.SIOM.Fr-O.15

Fig. 14 Longitudinal distributions of long-term correlation coefficient between

wave normal stress and wave shearing stress derived in the North Atlantic Ocean

Fig. 13 shows the long-term correlation coefficients between the wave normal Stress and

the wave shearing stress on the longitudinal members of S.S. 7, as functions of heading angle, and the averaged ones for all headings. _{And, Fig. 14 shows the longitudinal} dis-tributions of the averaged correlation coefficients.for all headings.

As shown in Figs. 12 and 13, the short- and long-term correlation _{coefficients between}

the wave normal stress and the wave shearing stress take different values in _{the region} of 1--i in accordance with different conditions. However, as shown in Fig. 14, the

aver-aged long-term correlation coefficients.for all headings are generally considered _{to be nearly}

zero. Those trends of. the long-term correlation coefficient between the _{waye normal stress}

and the wave shearing stress will lead to the utility of the predictioti method for evaluating the extreme values of equivalent stress based on the assumption that the wave normal stress, and the wave shearing stress would be absolutely statistically independent (p=O). And, the other prediction method', based on the assumption that the wave normal stress

TANKER L310M Fr O.I5 0PW8GIJRWALE PI4ALFTHWDDED4ELL

P.SG OHS.

(C:L.AJSISEEl. CL. --e--p09BILGE 6---. P'O ALL HEADS -0.5 0.5

LEEWARD S'DEI WEATHER SIDEI

-1.0

-ISO -90 0 90 leO' 1.0- ISO

P OH HALF SEPTh W SIDE SHELL

S.S. 7 P D HALF WPTH WLI. RHO.

S.S. 7

-0---- P ON DECK CL.

P ON KEEL CL _{8.0. 7} -0-- P ON GUNWALE0 P ON BILGE

P

AD

-90 0 90 1800

44 J. FUKUDA and A. SHIr4KAI

and the wave shearing stress would be absolutely dependent (p= ±1), may be utilized to estimate the upper linlit of the extreme values of equivaltht stress.

4. Conclusions

Statistical methods are proposed for predicting the extreme value of non linear stress such as the von Mises' equivalent stress induced on the longitudinal member of the ship hull in ocean waves. The methods are based upon an assumption that the .'ave normal

stress and the wave shearing stress, which are combined to the equivalent stress. would be absolutely statistically dependent (p= ±1) and upon another assumption that those

stresses would be absolutely statistically indepeildent (p=O).

The validity of the methods has been proved by the numerically simulated experiments

on the equivalent stress induced on the longitudinal member of a large oil tanker in a

short-term storm seaway.

According to the proposed methods, the short- and long.term prediction works have been carried out for the large oil tanker in order to evaluate the extreme value of equivalent stress induced on the longitudinal member of the ship hull in ocean waves. And, the

short-and long-term correlations between the wave normal stress and the wave shearing stress have been investigated in detail to examine the utility of the present methods.

Following condusions are obtained from those calculation results.

From a practical viewpoint, the approximate methods dependent on the assumptions of p=±l and p=O might be employed for evaluating the extreme values of equivalent stress.

The prediction method dependent on the assumption of p=O may be adequately utilized for evaluating the extreme values of equivalent stress.

The prediction method dependent on the assumption

of pl or p=1 may be

utilized to evaluate the upper limit of the extreme values of equivalent stress.

The conventional estimates of the extreme value of equivalent stress will be over-estimated by 5 10 percent as compared with the upper limit of the theoretical estimates.

.5) The largest equivalent stress will be found on the gunwale of midship and the

larger equivalent stresses will be found on the deck centre line and on the bilge of midship of a large oil tanker in the North Atlantic Ocean.

Acknowledgement

The authors wish to express their sincere gt'atitude to Mr. H. Tsukuda for his valuable cooperation in this research work. They would also like to thank Mr. Y. Nozu and Mr. H. Hamamoto for their assistance in the numerical calculations. Furthermore, they are deeply indebted to the staff of the Computer Centre of Kyushu University for their help in the

computer work.

References

Fükudä, J., -Shiñkai, A. and Tsukuda, H., "01-i the statistical distribution of the maxima

of non-linear stress induced on the ship hull in sea waves" (inJapanese), Journal of the

Society of Naval Architects of Japan, Vol. 148, 1980.

Fukuda, J., Shinkai, A. and Tsukuda, H., "StatIstical prediction of the non-linear equivalent stress induced on the ship hull in ocean waves" (in Japanese), Trans. of the West-Japan

Statistical Methods for Predicting the Non-Linear Stress 45

Society of Naval Architects, No. 61, 1981.

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