Predicting the longitudinal stresses induced on a large oil tanker in sea waves

PREDICTING THE LONGITUDINAL STRESSES INDUCED ON A LARGE OIL TANKER IN SEA WAVES

by

J. Fukuda and A. Shinkai**

Summary

A method of predicting the longitudinal stresses induced on a ship hull in sea waves is presented. In short- and long-term Seaways, the total nOrmal stress and the total shearing stress on a longitudinal member can be predicted by using the variances of component 'sea loads and the covariances between

those component loads. Exact application of this method is.made for a large oil tanker in the North Atlantic Ocean and an approximate long-term correla-tion method is examined in comparison with the so-called square root method.

*) ProfessOr, Department of Naval Architeôture, Kyushu:Univèrsity1 FuküOka,

Japan.

**) Leáturer, Department of Naval A±chitectüré, Kyüshu University, Fukuoka,

2

1. Introduction

During recent years, the theoretical method of determining the design valuésof low frequency loads induced on a ship hull in sea waves has been es-tablished by the aid of modified strip theory to calculate the response func-tions of loads on a ship hull in regular waves, linear supeposition theory to evaluate the ship responses in short-term irregular seas and long-term predic-tion technique to estimate the extreme values of ship responses by utilizing the available ocean wave statistics. The validity of this method has been con-firmed by a number of model experiments in seakeeping tanks and a number of full scale measurements on actual ships.

Systematic prediction works have been continued in Kyushu University in cooperation with Mitsubishi Nagasaki Shipyard in order to estimate the design values of sea loads associated with the longitudinal hull strength including axial force, vertical and horizOntal shearing forces, vertical and horizontal bending moments and torsional moment. Those results have been succesively

pub-lished [1 - 7].

Following those prediction works on sea loads, a method of predicting the longitudinal stresses on a ship hull in sea waves is proposed here. In short-and long-term sea waves, the total normal stress on a longitudinal member of the ship hull can be evaluated by using the variances of three component sea loads, that is, vertical bending moment, horizontal bending moment and axial,

force, and the covariances between those loads, and the total Shearing stress can be also evaluated by using the variances of three component sea loads, that is, vertical shearing force, horizontal shearing force and torsional mo-ment, and the, covariancés between those loads, where the warping stresses might be ignored. Exact application of this method is made for a large oil tanker and an approximate long-term correlation method is examined in compari-son with the so-called square root method.

3

2. Ship responses in regular waves

Consider the case when a ship advances in regular waves with a cOnstant average velocity and a constant average heading angle. The coordinate systems are employed as shown in Fig. 1. The coordinate system O-XYZ is so fixed in space that XY-plane coincides with the still water surface and OX indicates the direction of average ship course. The other coOrdinate system O-X1Y1Z1 is so determined that OX1 indicates the regular wave direction. nd the coordi-nate system o-xyz is fixed on the ship, where the origin o locates at midship on the centre line of water plane and ox points out ahead the longitudinal direction.

Thus the vertical displacement of the encountered wave surface is written as follows.

h = h0cos(kX1 - wt) _{= höcos(kxcosx} kysin _{- Wet)} (1) where

h : vertical displacement Of surface .wave h0: amplitude of surface wave

2

k = 2ir/A = w /g : wave number

A ;wave length

w wave circular frequency

= w - kVcosX circular frequency of Wave encounter

V : ship velocity

X heading angle

The equations of motions for a ship in regular waves

can

be obtained by the aid àf modified strip theory [6 8], in the form of

A]1+ A1

+ B16 F + 826+ = M. + + Cl7 + Cl8Ô + Cl90 C2 _{+ C22T1 + C23T1 +} C24 + C25j)+ C264, + C27 + C28Ô + c298

=

_M+ 3l C32ri + C33fl + C34 + C35;

+

C364 + C37 + c38ë + C390 = M0 + A13 = + + B]4$ + B15 + B23 _{+ B244} + + C13i1 + C14 +

B11+ B12F

2l + B22 (2) (4)

where

surge, : heave, : pitch, sway, 4i yaw, 8 roll

Hydrodynainic coefficients A11, Al2, A13, B11, B12, - - , B21, B22,

-, C11, C12, - - - , C21, C22, - - - , C31, C32, - - - and exciting forces

and moments F, FC F, Mn,, M8 in Eqs (2), (3) and (4) can be evaluated

by using the added mass and damping according to Tasai's method [9, 10] or Tarnura's one [11]. However the roll damping coefficient C38 in Eqs. (4) should bedetermined by taking into consideration the non-linear viscous damping in addition to the linear wave making damping (12]. Coefficients A13, C13, C19, C23, C29 and C33 vanish identically and Al2 can be ignored 'approximately. Details on the equations of motions are described in the publications (6 - 8].

Solutions of Eqs. (2)., (3) and (4) can be obtained in the form of

= cocOs (wt -C = -C0cos(wt CC) = 0cos (w t - c) = 0cos(wt -.

= 4000S(W

t -o = B0c-os (wet - C0 where

0' Co

T, 4c,

80 : amplitudes of surge, heave,

pitch, sway, yaw and roll

' ' '

c, c0

: phase angles of surge, heave, pitch, sway,

yaw and roll

The low frequency loads induced on

a section

of the ship hull in regular waves can be evaluated by using the solutions of motions (5), in the form of

= FAOcos (w t - CFA)

Fv

=

F0cos(wt - C.v)

Mv = M70cos _et

-FoCOs

(w t CFH) =

M000S(W t

-MT =

MTocos(wet

._ .CMT) where 4

5

FA, Fv, Mv, FH, MHI MT : axial force, vertical shearing force, vertical bending moment, horizontal shearing fôrcé, horizontal bening moment and torsional moment

FAO. F\o' Mvo' FHOS _o' MTO : amplitudes of axial force., vertical

shearing force, vertical bending moment, horizontal shearing force, horizontal bending moment and torsional moment

CFA CFV _FH' CMT : phase angles of axial force, vertical Shearing force, vertical bending moment, horizontal shearing force, horizontal bending moment and torsional moment

Details on the calculation method Of those loads are described in the publications (6, 7, 13,. 14].

The loads of (6) can be. written generally in the form of r. = r. cos(w t - c

i iO e ri

where

: component load

rj0: amplitude of component load phase angle of component load

When analyzing the longitudinal strength of the ship hull in regular waves, the component stress induced on a longitudinal member by the component load r. can be obtained as follows.

1.

= k..r. k..r. cos(w t - .) (8)

3.

11.

110

e ri

where

Si

normal stress or shearing stress induced by the load r.

k. :. normal stress or shearing, stress induced by a unit load of and k. on the hull structure of Olosed sèOtioñ can be calculated based upon the beam theory, shear flow theory and torsional theory of Saint-Vénant, where

the warping stress is ignored. .

in that case, the total normal stress or the total shearing stress due to all compOnent loads Oan e obtained as follows.

s = Es. = Ek.r. = Ek.r. cos(w t .) (9)

T i.

ii

ii0

e ri.

where

total normal stress or total shearing stress

and Formula (9) can be written as Eollows, when calculating the total normal stress and the total shearing stress.

o = Za. = Ek.r. = Ek.r. cos(w t - c

P i.

11

ii0

e ri.

r Z. = Ek.r. = Ek.r.. cos(w t - : i 4, 5, 6

T

ii

ii0

e ri

where

total normal stress, _TT totaishearing stress

= normal stress induced by the vertical bending moment 02 = a : normal stress induced by the horizontal bending moment

03

0FA : normal stress induced by the axial force

r4 = shearing stress induced by the vertical shearing force = : shearing stress induced by the horizontal shearing force

MT shearing stress induced by the torsinal moment k1 = : normal stress induced by a unit of vertical bending

moment

= : normal stress induced by a unit of horizontal bending moment

= aFA : normal stress induced by a unit of axial force

= : shearing stress induced by a unit of vertical shearing

force

k5 = : shearing stress induced by a unit of horizontal shearing

force

k6 = : shearing Stress induced by a unit of torsona1 moment

r1=Mi r2_M.1

r3=FA

r4=F

rS.=FHI r6=MT

Systematic calculations of ship responses have been carried out for a large oil tanker in regular waves of 10 meters height; Main

particulars

of the ship are shown in Table 1. Sign conventions of the wave induced loads are shown in Fig. 2. Examples of the calculated results of ship motions are shown in Fig. 3, and those of wave induced loads in Figs. 4 - 6, where the following notations are employed.

L : ship length, B ship breadth, H = 2h0 : wave height.

Fr.: Froude number, p density of seawater .

gravity acceleration . .

, 2, 3 (10)

Mo =

M0/PgL2Bh0,

Fvo

F0/ØgIh0

F'HO =. F0/PLBh0l MTO

M0/gLBho

There are shown., in Fig. 7, the normal stresses induced on the hull sec-tion of middle body by a. unit of vertical bending moment, horizontal bending moment and. axial force, and in Fig. 8, the shearing StresSes induced by a unit of vertical shearing force, horizontal shearing, forde and torsional moment, which are calculated by Nagamoto [15].

-8-3. Ship responses in short-term irregular seas

Consider the case when a ship navigates in short-term irregular seas with a constant average velocity and a constant average heading angle. The coordi-nate systems are employed as shown in Fig. 9.

The spectrum of short crested irregular sea waves can be given by th pro-posal of I..S.S.C. [16], as follows.

[f(w,y)]2 = (2/it) (f(w)]2cos2y : -rr/2 y ir/2

(12)

= 0 elsewhere

= (13)

where

w : circular frequency of a component wave

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

= 2i1/T, T : visual average wave period

H : visual average wave heIght (significant wave height)

Then the variances of each component sea load, each component stress and total stress canbe obtained as follows [17, 18].

2 fit/2[ 2 2 2

R. (,2/iT)/

/ [f(w)] [A (w,-y)] cos ydwdy (14)

J-ir/210

ri

kR

(15) S = EkR + 2Ep. .k.k.R..R. : I <

j

(16) T

ii

ijijij

p.. K. ./R.R. (17)

1)

--1]

1)

2 =

(/u)jj

[f(w)'] _{[AriWsd_YU[Arj(Wó_Y)]}

CO5EEri(WS_1) - crj(W]c0sYdwd1

(18) where

: variance of component sea load r.

S : variance of component stress s. due to the component sea load

S variance of total stress

[Ari(Wiô1)] = r10/hO response amplitude of componertt sea load average heading angle against the' average wave direction

-9--average heading angle against a component wave direction correlation coefficient between loads r and rj

K.. : covariance between loads r. and r

.3)

- 1.

j

Formula (16) can be written as f011ows when evaluating the variances of total normal stress and total shearing stress.

S2T = + 2Zp..k.k.R.R. : i = 1, 2, 3, j = 2, 3, i < j (19)

ST =

Ek?R + 2Ep..kk.R.R. i = 4, 5, 6, j = 5, 6, i < j (20)

where

S2T variance of total nOrmal stress

S variance of total shearing stress R1 = standard deviation R2 = standard deviation = standard deviation R standard deviation R5 standard deviation = standard deviation p12 : correlation coefficient p13 : correlation coefficient correlation coefficient p45 : correlation coefficient correlation coefficient p56 correlation coefficient

of vertical bending moment Mv of horizontal bending momentMH of axial force

of vertical shearing force of horizontal shearing force of torsional moment MT between Mv and between M and FA between MH and between Fv and between Fv and MT between FH and MT

Thus the short-term probabilities of exceeding for given levels of compo-nent sea load, compocompo-nent stress and total stress can be obtained as follows

[19].

q(r>r) = exp(_(rp2/2R]

q(s>s) = q(kr>kr) = q(r>r)

= exp(-(s)2/2S] = exp[-(s)2/2UkR + 2Ep..k.k.R.R.) where

q(r1>r) : short-term probability t1at

exceeds a given level r*i.

q(s

hnrt-frm nrrhhi1iFv tht

the component. sea load r.

th

mnnnt strss s

--

---.

---i exceeds a given level ste.

or

- 10

Short-term probability that. the total stress ST exceeds a given level

By putting

q(r.>r) = q(s.>s) = q(s>s) = q* (24)

the following formula is obtained for responses on a given probability of ex-ceeding q*.

(s(q = q*)]2 = Ek2Er. (q = q*)]2

+ 2Ep..k.kr(q

q*)r.(q = q*)

:i<J

(25)

S(q = q*) = [Ek{r(q =

q*)}2 +2Ep..k.k.r.(q = q*)r..(q. q*)]l1'2.

= EE{s.(q =

q*)} ₊ = q*)s(q =

i <i

(26)

where

s(

= q*) _{the level of total stress} a given probability of

exceeding q*

r. (q = q*) : the level of component sea load r. on a given

proba-bility o exceeding q* s1(q = q*)

: the level of component stress s on a given probability øf exceeding q*

sgn(kk)

: sign of kk (plus or minus)

Formula (26) is strictly valid for short-term stationary responses arid no simplifying asSumption is introduced. For responses with small correlation co-efficients an approximate estimate is obtained by putting = 0, as follows.

= [k[r(q

q*)}2]1'2 = (E{s.(q = q*)}2]l'2 (27)

which is so-called "square root method".

For example, in head and following irregular. seas, :the correlation co-efficient between vertical bending moment and horizontal bending moment is zero because of the symmetric relationship of those responses in oblique c7m-ponent waves. However, in general, all correlation coefficients between all component sea loads do not vanish together so that. all correlations have to be considered.

Systematic calculations of Ship responses have been made for the large oil tanker in short-term irregular seas and examples of the calculated results are shown in Figs. 10 - 12. Inthose calculations based upon the linear

super-/

position theory, the equivalent response functions are assumed for the loads including horizontal bending moment, horizontal shearing force and torsional moment by using the calculated results of those loads induced on the ship hull in regular waves of 10 meters height, though they are not strictly linear re-sponses because of the influences of non-linear roll damping. The other loads, that is, vertical bending moment, axial force and vertical shearing force can be regarded in principle as linear responses.

The standard deviations of component. sea loads on the midship section are illustrated in Fig. 10 as functions of ilL/A

e = gT2/21r). In Fig. 11, the

correlation coefficients between vertical bending moment, horizontal bending moment and axial force on the hull sections of S.S. 3, 5 and 7 are shown as functions of average wave period, and in Fig. 12, the correlation coefficients between vertical shearing force, horizontal shearing force and torsional mo-ment on those sections. The correlation coefficients p12 (between vertical bending moment and .horizontal bending moment) and p23 (between horizontal bend-ing moment and axial force) in Fig. 11 and the correlation coefficients p45

(between vertical shearing force and horizontal shearing force) and p46(be-tween vertical shearing force and torsional moment) in Fig. 12. are zero jn

head and following irregular seas and their signs have to be changed when the sign of heading angle is changed.

In Fig. 11, the strong (negative) correlation is found between vertical bending moment and axial force, rather weak one between vertical bending mo-ment and horizontal bending momo-ment, and the weakest one between horizontal bending moment and axial force. Also, in Fig. 12, the strong (positive) corre-lation is found between horizontal shearing force and torsional moment, rather weak one. between vertical shearing force and horizontal shearing force, and the weakest one between vertical shearing force and torsional moment. Conse-quently, when considering the stresses on deck, the normal stress produced by vertical bending moment will be slightly increased by that produced by axial force and the shearing stress produced by horizontal shearing force will be also slightly increased by that produced by torsional moment. On the other side, when considering the stresses on bottom, the normal stress due to verti-cal bending moment will be slightly decreased by that due to axial force and the shearing stress due to horizontal shearing force will be also slightly de-creased by that due to torsional moment.

0 JO

12

-4. Long-term prediction of. ship responses

By asstnning that a ship navigates in long-term seaways always with a con-stant average velocity and a concon-stant average heading angle against the aver-age wave direction, the cumulative probabilities of exceeding for given levels of each component sea load, each component stress and total stress can be

ob-tained' as follows [20]. Q6(r>r*)

//exp[_(r)2/2{Ri(H,T,6)}2]p(H,T)dHdT

(28)

Q6(s>s) = Q6(k.r.>kr) = Q6(r>r)

(29) Q6(sT>s) fOfexP[_(s)2I2{$T(HvT,6)}21P(HlT)dHdT

=11

)2/2{EkR2 +2EP..kk.R.R}]P(HT)dHdT

J (30) where

Q6(r.>r) : long-term probability that the component sea load r

ex-ceeds a given level r when a' constant heading angle 6 is assumed

Q6(s>s)

long-term probability that the component $treSS s

ex-céeds a given level s when a constant heading angle 6 is assumed

Q6(sT>s) : long-term probability that the total stress sT exceeds a

given level s when a constant heading angle 6 is assum-ed

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

con-dition of the visual average wave height H and the vjsual average wave period T

Then, by assuming that all heading angles against the average wave direc-tion are equally probable in long-term seaways, the cumulative probabiliteS of exceeding for given levels of those responses can be obtained as follows.

aii

1r) =

(l/211)f Q6 (r.>r).d6

(31)

aIli>V = call

ir>krp

QalifrP

(32)

where

long-term probability that. the component sea load exceeds a given level r when all headings are equally probable

alli>P

long-term probability that the component stress s

ex-ceeds a given level s when all headings are equally probable

long-term probability that the total stress _ST exceeds a given level s when all headings are equally proba-ble

Formulae (30) and (33) are strictly valid for long-term prediction of the total stress and no simplifying assumption is introduced.

For short-term stationary responses the exact estimate of. the total stress on a given probability of exceeding can be given by Formula (26) and

the approximate one by Formula (27) for responses with small correlation co-efficients. Similar estimates are proposed here as the approximate ones for long-term extreme values of the total stress by introducing long-term

correla-tion coefficients.

Long-term correlation coefficients between the component sea loads can be determined as follows for a constant heading and for, all headings which are equally probable.

where

long-term correlation coefficient between loads r and for a constant heading angle

- 13

'ij

₌₁₀₁₀

K..(H,T,5)p(H,T)dHdT

()]2 =ff[R.(HcT151].2PHIT)dHaT

arid

p..(all) = K..(all)/[R. (all)] (.(all)1

f2rrf1

K..(all) (l/2r)J

'I J

(H,P,S)p(H,T)dHdTdS

0)00

[i(all)]'2 '=

(.1/2)jff(R(HT)]2P(HT)dTd

(P33) (34) (35) (36)

14

-(all) long-term correlation coefficient between loads and

for all headings which are equally probable

Then the approximate estimate for long-term extreme válües of the total. stress may be given by introducing the long-term ôorrelation coefficients . for a constant heading or all headings, as follows.

sT(Q = Q*) = tEk{r1(Q = Q*)}2 + 2E

kkr(Q = Q*)r(Q

= (Z{s(Q = Q*)}2 + 2Esn(k.k.)Pjs.(Q

Q*)s(Q

Q*)]1"2

:i<.j

. (37) where

ST(Q = Q*) the level of total Stress on a given long-term probability of exceeding Q* for a constant heading or all headings

r. (Q = Q*) : the level of component sea. load ri on a given long-term

probability of exceeding. Q* for a constant heading or all headings

s(Q = Q*) the level of öomponent stress s. on a given long-term probability of exceeding Q* for a constant heading or

all headings

long-term correlation coefficient between loads r. and r. for

Pu

.à constant heading or all headings

For responses with small long-term correlation coefficients the simplifi-ed estimate for the total stress Isobtairisimplifi-ed by putting = 0, as follows.

sT(Q = Q*) = [Ek,{r.(Q = Q*)}2]l1'2 = [E{s(Q = (38) Formula (37) is not strictly applicable to the long-term responses be-cause they are no longer stationary but would be probably valid for practical purpose of estimating long-term extreme values of the total stress and shuld be better than the so-called square root method of Formula (38)

Systernati calculations of long-term ship responses have been carried out

for the. large oil tanker by using the long-term wave statistics of the North AtlfltiO.Qceafl (20, 21], and examples of the calculated results are shown in

Figs. 13 - 19.

The long-term prediction results of component sea loads along the hull length are shown in Fig. 13, where all headings are equally considered.

There are shown, in Fig. 14, the long-term Oorrelation coefficients be-tween the loads which produce normal stresses, that is vertical bending nment,

15

-horizontal, bending moment and axial force, on the hull sections of S.S. 3, 5 and 7, and in Fig. 15, the long-term correlation coefficients between the loads which produce shearing stresses, that is vertical shearing force, hori-zontal shearing force and torsional moment, on those sections.

There are shown, in Fig. 16, the long-term prediction results of total normal stress and component normal stresses on the hull sections of S.S.. 3, .5 and 7, and in Fig. 17, the long-term prediction results of total shearing stress and component shearing stresses On those sections, where all headings are equally considered.

Finally, in Figs. 18 and 19, the long-term prediction results of total normal stress and total shearing stress according to the approximate methods are examined as compared with the exactly predicted results. There are shown, in Fig. 18, the prediction results of tOtal normal stress on the hull sections of S.S. 3, 5 and 7 according to the exact method, approximate long-term corre-lation method and approximate square root method, and in Fig. 19, the predic-tion results of total shearing stress on those secpredic-tions according to these

three kinds of methods, where all headings are equally considered.

As shown in Fig. 14, the long-term correlation coefficients between ver-tical bending moment and horizontal bending moment and between horizontal bending moment and axial force are zero but the long-term correlation

coeffi-cient between vertical bending moment and axial force will be -0.8 'u -0.6 when all headings are equally considered, and as shown in Fig. 15, the long-term correlatiçn coefficients between vertical shearing force and horizontal shear-ing force and between vertical shearshear-ing force and torsional moment are zero but the long-term correlation coefficient between horizontal shearing force and torsional moment will be 0.5 ' 0.7 when all headings are equally

consider-ed.

As found in Fig. 16, the long-term, total normal stress will be large on deck corner and on bilge, and the component normal stresses due to vertical bending moment and horizontal bending moment are generally significant but the component normal stress due to axial force is rather 'small. And, as found in Fig. 17, the long-term total shearing stress will be large on half depth of longitudinal bulkhead, on deck centre line and on bottom centre line,.arid the component shearing stresses due to vertical shearing force and horizontal shearing force are generally significant, but the component shearing stress due to torsional moment is rather small. '

long16

-term componnt stresses will lead to the validity of the approximate estimate of Formula (37) that includes the long-term correlation coefficients, and this method should be better than the simplified square root method of FOrmu].a (38)

that excludes the long-term correlation coefficients between vertical bending moment and axial forde and between horizontal shearing force and torsional mo-ment. As found in Figs. 18 and 1., the long-term prediction results of total

normal stress and total shearing stress according to the approximate correla-tion method show 'satisfactory 'agreements with, the exactly pedicted results and are superior to the prediction results according to the approximate squre root methOd. Nevertheless, the approximate square root method will be adequate ly valid for practical purpose of estimating long-term extreme values of total normal stress and" total shearing stress and seems to be preferable because of its simplicity, though it gives slightly underestimated total normal stress on deck and slightly overestimated one on bottom and also gives rather underesti-mated total shearing stress on deck and rather overestimated one on bottom.

- 17 -.

5. Conclusions

A method of predicting the longitudinal stresses, including normal stress and shearing stress, induced on a ship hull in short- and long-term seaways is

determined by using the variances of component sea loads and the covariances between those component loads. In addition to this exact method, an approxi-mate estiapproxi-mate for long-term extreme values of total normal stress and total

shearing stress is proposed by introducing the long-term correlation coeffi-cients between the component loads, which should be superior to the so-called square root method employed so often because of its simplicity. Applications of the exact and approximate methods have been made for a large oil tanker in

the North Atlantic Ocean and the long-term predictions of longitudinal stress-es have been carried out on the hull sections of S.S. 3, 5 and 7. Following

conôlusions are obtained from the long-term predictioh results on the assump-tiôn that all headings are equally probable.

The component normal stresses due to vertical bending moment and hori-zontal bending moment are generally significant but the component normal

stress due to axial force is rather small.

The long-term correlations between vertical bending moment and hori-zontal bending moment and between horihori-zontal bending moment and axial force are zero but the long-term correlation between vertical, bending moment and axial force is considerable when all headings are equally considered.

The total normal stress will be large on deck corner and on bilge, and the normal stress on deck due to vertical bending moment will be slightly in-creased by that due to axial force but the normal stress on bottom due to ver-'

tical bending moment will be Slightly decreased by that due to axial force. The component shearing stresses due to ver-tical shearing force and horizontal shearing force are generally significant

but

the component shearing stress due to torsional moment is rather small.

The long-term correlations between vertical shearing force and hori-zontal shearing force and between vertical shearing force and torsional moment are zero but the long-term correlation between horizontal shearing force and torsional moment is considerable when all headings are equally considered.

The total shearing stress will be large on half depth of longitudinal bulkhead, on deck centre line and on bottom centre line, and the shearing stress on deck due to horizontal shearing force will be slightly increased by that due' to torsiOnal moment but the shearing stress on bottom due to horizon-'

18

-7) The approximate correlation method is satisfactorily valid for

esti-mating long-term extreme values of total normal Stress and total shearing Stress and should be better than the so-called square root method. Neverthe-less, the approximate square root method will be adequately valid for practi-cal purpose of estimating long-term extreme values of total normal, stress and total shearing stress, though it gives slightly underestimated total normal stress on deck and slightly overestimated one on bottom and also gives rather underestimated total Shearing stress on deck' and rather overestimated one on bottOm.

Acknowledgement

The authors' wish to express their sincere gratitude to Dr.

H. Fujii

of Mitsubishi Nagasaki Technical Institute and Dr R. Nagamoto of Mitsubishi Nagasaki Shipyard for their cooperation in this reseach work. They would also like to thank the Ministry of Education for providing the Grant in Aid for Scientific Research, Project Number 346l27.and to express their appreciation for helpful calculation work in Kyushu University carried out by Messrs. A.

References

Fukuda, .3., Nagamoto, R., Tsukamoto, 0. and Shinkai, A., "Estimating the design values of vertical shearing force induced on the ship.hull in waves", Journal of the Soôiety of Naval Architects of Japan, Vol. 136,

1974..

Shinkai, A., "Estimating the design values of vertical bending moment in-düced on the ship hull in waves", Journal of the Society of Naval

Archi-tects of Japan, Vol. 138, 1975.

Fukuda, .3., Nagamoto,

k.,

Tsukámto, 0. and Shinkai, A., "Estimating the design values of hOrizontal shearing force induced on the ship hull in waves", Journal of the Society of Naval Architects of Japan, Vol. 139,

1976.

Shirikai, A., "Estimating the design values of horizontal bending moment induced on the ship hull in waves", Journal of the Society of Naval Ar-chitects of Japan, Vol. 140, 1976.

Fukuda, .3., Tsukamoto, 0., Shinkai, A. and Kamiirisa, H., "Estimating the design values Of torsional moment induced on the ship hull in waves'!., Trans. of the West-Japan Society of Naval Architects, No. 53, 1977.. Fukuda, .3., Nagamoto, R. and Shinkai, A., "Estirnatimg the design values of axial force induced on, the ship hull in waves", Trans. of the West-Japan Society of Naval Architects, No. 54, 1977.

Fukuda, .3., Shinkai, A. and Iwainoto, S., "Estimating the design values of axial force induced on the ship hull in waves (Continued)", Trans. of the West-Japan Society of Naval Architects, No. 56, 1978..

Fukuda, .3., 'Nagamoto, R., Kontina, M.. and Takahashi, M., "Theoretical

cal-culations on the motions,

hull

surface pressures and transverse strength of a ship in waves", Journal of the Society of Naval ArchitectS of Japan, Vol. .129, 1971, and Memoirs.of the Faculty of Engineering, Kyushu tJniver-sity, Vol. 32, No.. 3, 1973.

.9. Tásai, F., "On the damping force 'and added mas of ships heaving and

pitching'!, Reports of Research Institute for Applied Mechanics, (yushu University Vol. 7, No. 26, 1959, and Vol. 8,. No. .31, 1960.

Tasai, F., "HydrodynarniC .force and moment produced by swaying and rolling "oscillation of cylinders on the free furface", 'Reports of esearch

Insti-tute for Applied Mechanics, Vol. 9, No. 35, 1961.'

Tainura, K., "The calculation of hydrodynamic forces and moments acting on - 19

20

-the two-dimensional body", Journal of -the Society of Naval Architects of. West Japan, No. 26, 1963.

Fujii, H. and Tàkahashi, T., "Study on lateral motions of a ship i waves", Mitsubishi Technical Bulletin, No. 87, 1973.

Fükuda, J., "A practical method of calculating vertical ship motions and wave loads in regular oblique waves", Report of coimnittee 2 on Wave Loads, Hydrodynamics, Proc. of 4th I.S.S.C., Tokyo, 1970.

Nagamoto, R., Konuma, M., lizuka, M. and Takahashi, T., '!Theo_

retical calculation'óf lateral shear force, lateral bending moment and torsional moment acting on the. ship hull among waves", JOurnal of the Society of Naval. Architects of Japan, Vol. 132, 1972.

Nagamoto, R., "Study.on the longitudinal strength of a large oil tanker in sea waves", Doctor Thesis, Kyushu University, B-277, 1977.

Warnsinck, W. H., "Report of Committee 1 on Environmental ConditiOns", Proc. of 2nd Delft, 1964.

St. Dénis, M. and Pierson Jr., W. J., "On the motions of ships in confus-ed seas", Trans. S.N.A.M.E., Vol. 61, 1953.

. Dalzell, J. F., "A note on the application of multiple input spectrum theory tO combined wave induced stress", International Shipbuilding PrO-gress, Vol.. 21, No. 236, 1974.

19! Rice, S. 0., "Mathematical analysis of random noise", The Bell Technical Journal, Vol. 24, 1945.

Walden, H., "Die Eigenschaften der Meerswellen im Nordatlantischen Ozeãn", Deutschér Wetterdienst,, Seewetteramt, Einzelveröffentlichungen Nr. 41, Hamburg, 1964.

Fukuda, J., "Long-term predictions of wave bending moment, Part II" Selected Papers from the Journal of the Society of Naval Architects of Japan, Vol. 5,. 1970.

+xI

+ZI,ZIC,h

Fig. 1

Coordinate systems in

Fig. 2

Sign conventions for

wave induced loads

Vatical ShSeriflg Foic. Vertical Bending Mamint

14o,izentei Fo,ci M,lgsetai Bsading Moment

y

Asal Force Torsional Moment

Table 1 Nin particulars of a large ol tanker

Length between perpendiculars IL) 310.000 e Breadth moulded (B) 48.710 a

pth meulded ID) 24.500 rn

Draught meulded Id) 19.000 a

Displacement 1W) 250,540 t

9lod coefficient (C') 0.852

Watar plane coefficient (Ce) 0.903 Midship coefficient (Ce) 0.995

Centre of gravity before midship 1'G 0.03311.

Centre of gravity below water line 0. 2879d

Itacentrtcheight (Q4) O.3305d

Longitudinal gyradius (uc) _0.24941.

Transverse gyredius 0.3231B.

Heaving period (TH) 11.6 sec Pitching period (Tv) _10.9 sec

.-:

I575

---:

135.0 -

----:

112.5

ic\4\ \\

_---:

_45.O

;s

\\.:

22.5

-;x.iBo.o

---;.

157.5

---:

'35.0

--f-:

90.0 67.5

--'-;

450W

---:

225

-:

0.0 PITCH 1.5. Fr0.I5 0.5-

\

-:

---:

1515 135.0

----:

112.5 90.0

----3

67.5

---:

450

---:

22.5

-.

0.0 ,l.-310M.H-IOM F, -0.15 '.5

2

.m 1.0 0.5 SWAY YAW J ' ROLL L 310U. HuSIOM

f\

---: x.157

Fr -0.15 I'.

---:

1350

---:

675.

U( 1

---:

450

i/h

I

---:

22.5

Jff1 \/V'.

./i

Lt..

-.-'

I-:-

/

-

l

\

'

.

--

%. '%--C

-- -:

. I,

si..---.-- '\

.,\..

*

\

---,...

\ \

_{bql,__...}

S%..\

0.5

1.5 '-'171

.-.--..

---S

-.-:.57,5

- - -

135.0 112.5 900 67.5

---:

450

Fig. 3

1mplitudes of ship motions in regular waves

L310M, pL1,sIOM X.157 .Fr-0.15 135.0' 112.5 90.0

----:

67.5

---:

450

---:

22.5 0.5 1.0 '.5

-t71.

05 1.0 1.5 t71 0.5 1.0 1.5 0.5 1.0 '.5,

'I

21.0 0 10.5 4.0 p3.0 0 2.0 1.0

20b0 -S 0 0.03

2

0.02-0.01

0.05-S.S.5. Fi-0.I5 _{V.W.8ENDING MOMENT}

-;

_157.5 135.0 112.5 90.0 67.5 45.0 22.5 0.0

---:

135.0

----:

112.5 90.0 67.5 45.0 22.5 1.5

S.S. 5. Fr. 0.I5 _{W. AXIAL. FORCE}

-. Z I80.0

---:

151.5 135.0 1125 90.0

----:

61.5 45.0

_/

22.5

_.(

0.0

_/

/

r

-'

I.... S

/'A

..

,

_\%.',,,

0.5 10

1.5

-jO.I0 0 -J a 0.05-0 0.05 0.003 0.001 0 S.S. 5. Fr.0.l5 X.Ie0.0s '57.5 135.0 112.5 90.0 67.5 45.0 22.5 0.0 0.5 3.3.5. Fr.0.15J

---: Za575'

---:

135.0

---:

112.5

---:

90.0

----:

61.5

---:

45.0

---S

22.5

//7<t'\

,//

.A

,4

1/

J9

.-'\

-'.5 '%.. 1.0 W. SHEARING FORCE TORSIONAL MOMENT .5 1.0 1.5

Fig.4 Amplitudes of loads on the midship section in regular waves. '.5

-t7I

ai :'

S.S.5, Fv.0.I5 _{H. W. SHEARING FORCE}

/

/

/

/

I,

\

---.Z.I575

135.0 112.5 90.0 67.5 45.0 22.5 '.5 0 05 1.0 0.03

---:X.l57.5

H. W. 8 ENDING MOMENT,.--. S.S.5. Fr. 0.I5 0.5 1.0 1.5 _'E71

0.03 S 0.02 S 0.0' 0 0-iso a

-0

OS

I.0-.t7X 2.0

0.15 _0.03 I 0.50 '.O.02 0.05 0.05 0$

L0-.7

2.0

I

3400 0.55 0.03 I 0.10

20

0.02 0.051 10.05 0

Tn

0.5

5.0 -.'t7

2.0 0.55 I 0.50 0.02 0.05 0.01 o

__2

0.03 0 .Fr.Q.15 z/L.0.0

_,a0

__a10 --:'io

_C

I..

-0.5

0-

2.0 0.55 0.50 0.05 360 iso I Ii 0

I

.M0 __1M1, _1F

(

(,

£,

Fig. 5 Response functions of vertical bending moment, horizontal bending moment and axial force on the mid-ship section. S. 360o _0.5

_l.0-.DX

_2.0 .M.o __.PA,0___11

-:

360' leo

I

eNI 1,1 lil I 360

-c

0 I U 0 0 0 0 0

0.10 0 360 110 0 (,v 05

LO.C7

2.0

ieo

010 . 0.0011 1 o 0 360 05 (NW 1.0 'C71 20 .F,-o.l5 X-135 ilL- 02 :'NO __:ÜTO 1PV PIS It 05

_{1.0-4:71 20}

0.003 0.002

1 .! 005

0003 0 0.10.l',-o.tS C.0 i/I..- 02 o.0b3 0.002

2!

o.00iT (U,

Fig. 6 Response functions of vertical shearing force, horizontal shearing force and torsional moment on the midship section.

21) .Fr.O.U5 X.110

x/1.-OZ 0003

2.0 x KG/I2 PER 1 TON-NOFV.B.N.

Yc

2.0 x 1O' KG/I PER

1 TON OF A.F.

Fig. 7 Normal stresses on the midship section due to a unit of vertical bend-ing moment, horizontal bendbend-ing moment and axial force.

5.0 x 1O KG'1?2 PER 1 TON OF V.S.F. 5.0 x 1O KG/Il2 PER 1 TON OF N.S.F. -... 2.0 icr5 KGrrv, > ITON-NOPT.L

Fig. 8 Shearing stresses on the mid-ship section due to a unit of vertical shearing force, horizontal shearing

4$

uhI

Fig. 9 Coordinate systems in

0.004

I

0.0O3-I

-o.00a'-0.00I 0 a 0.003-Jo.002 0.001-0

S.S.5. Fr-O.I5 _{W. 8ENOING MOMENT}

0.004----:

157.5 135.0

---:

112.5 90.0 S.S.5. _{AXIAl. FORCE}

/ ..

-S.-s4;;.-'

-:4-00

---a

22.5

---:

45.0

----:

67.5 .: 25

0015- -:5 .80.0'

1575 1350 112.5 90.0 67.5 45.0 22.5 0.0 0.005-0 0.I-I V. W. SHEARING FORCE 0.5 1.0 1.5

-It7, 25

o.i5( - -:4.iao.0' 157.5 135.0

--.:

112.9 90.0 67.9 45.0 22.5 00 H. W. SHEARING FORCE

Fig. 10 Standard deviations of loads on the midship section in short crested irregular waves.

0.5 1.0 1.5

--t71

25 W. TORSIONAL MOMENT

-:4-0.0'

---:

22.5

- - -:

45.0

- ---:

67.5 0.5 1.0 1.5

t7I

25

-:4-190.0'

0.0'

---:

---:

157.5

-:4.

22.5 135.0

---:

45.0

----.

112.5

----:

67.5

----:

90.0 0 25 0.5 I.0 1.5

-Icz;

0.015 S.S.5. F,.0.I5

_{kw.eENoI MOT}

-:6-180.0'

(1110-si S.S.5. Fr.0.15j

0.4 -:4.190.0'

_---:

157.5

---:

135.0 112.5 90.0 i 002 0.01

0 -0.5 -I.0 0 p's -1.0 l.0 p.' 0.5 -0.5 0.5 p.. -0.5 0.5 -0.5 s.s. 51 s.s. ij TANKER. : 6 - 0 L 310W

.--: .45

F,-0I5 ----: - 90 s.s;sl : s.s. S S.S. 7 -.. -vm.wi a 10 12T 16..'

12T 6*..

0 p.., 1 -0. -1.0 0.5 0 -0.5 -1.0 6 'S 10 I2T IS.'.

Fig. 11 Correlation coefficients be-tween vertical bending moment, horizon-tal bending moment and axial force on the hull sections in. the North Atlantic

Ocean. , BETWEEN Mv AND MV TANKER L.310M :6.

--:

.2

0'

.e5

- 90 135 S.S. 31 . : P, BETWEEN Mv AND F TANKER L.310M F, -0.15 :6. 0' : -.45 : -1 90 -*135 5.5. 3.1 . : - ISO ---..--' ,

-s.s.,I

s.s.v1

---2

.135 0 'Ii I p's BE rwEEN MV AND F,

0 Ld 0 0

a

0 0 s.s_ 5 I

-.--.-.-.-.-.-.-.---ID I2T IGuc

8 10 2-1 IGuc 0 a

Jo

a

10

Fig. 12 Correlation coefficients be-tween vertical shearing force, horizon-tal shearing force and torsional moment on the hull sections in the North At-lantic Ocean BETWEEN F AND FM TANKER - ---:4. 0 L.310U

----:

-45

Fr. .0.15

----: .

90

---1

.135 S.S.31 . leo ----..--Pa BETWEEN Fv AND M TANKER L310M FO.I5 :4

----:

-.---:

---:

::

- 0° -45 .90 -135 -180 S.S.31 -:-- ______

-0 5 __-.

-s.s. 5 .

-J_

,----

-. S S.Sr?I._ P,BEIWEEN FH AND Iii

TANKER Fr .0.15 0 :45 -f---: _-!90

---: -t135

leo

-

-.-.

.

5 _-==.=-==

. . -0___ ---. 5:

- -;

-0

..

4 8

2 T I6sc

0.006 0 0 1 0. 0 4 5 6 7 8 9 FR I 2 3 4 5 6 7 8 9 FR 0.004 0.003 -i 002 0.001 0 AP 0.004 0003 .J 0.002 0.001 0 °AP I 2 3 4 5 6 7

I

Fig.].3 Long-term prediction results of loads along . -the hull length in -the North Atlantic Ocean. .. .L .L

20. 20

V. W. lENDING MOMENT. ALL HEADINGS

TANKER

-.:Q.IO

L.3IOMF0.iS

- lOs

V. W SHEARING FORCE ALL HEADINGS

TANKER L.3IOM

_:Q.I01 Ft.015

I0

%

H. W. SHEARING F0R. ALL HEADINGS TANKER L.3IOM

:.io4 F,-015

H.W. BENDING MOMENT. ALL HEADINGS

TANKER

-O-I0

L.310MFr-0.IS

W. AXIAL FOR ALL. HEADINGS

TAMER L.310M

-:Q.I0' Ff0.l5

W. TORSIONAL MNT. ALL HEADINGS

TANKER L3IOM Fr.0.15

-:O.IO

AP 2 3 4 S 6 7 8 9 FR _{2 3 4 5 6 7 $} 9 FR FR 9 FR V 8 6 9 5 3 2 4

A1

10

1.0

10

0.5 -0.5 -1.0 -0. I.

5 90

TANKER L310M, Fr .0.15. Da SETWEEN U, ANO U.

--

A. F1 s.s.5 U .J -I 4 180° -- ISO -90 0 - 690 I80

Fig. 14 Long-term correlation coeffi-cients between vertical bending moment, horizontal bending moment and axial

force on the hull sections in the North Atlantic Ocean. p.' 0 -0.5 1.0

10

.0 1.0 -03 .0 TANKER L310M Fr- 0.15 TANKER L310M. Fr-0.15 TANKER L310M Fr-0.I5

-

: BETWEEN F0 AND F, F1 My $5.7

a--- .

,. F My - 1.0 -ISO -90

-- :

BETWEEN F0 AND F, - F1 My S.S.5 -°-- : My I 800 U) 0 4 (a

I

1 4

Fig. 15 Long-term correlation coeffi-cients between vertical shearing force, horizontal shearing force and torsional moment on the hull sections in

th

North Atlantic Ocean. -TANKER. L310N, Fr0.I5. P , BETWEEN U, AND U, N, F0 U, F0 - -0---SS.3 -L I I

dliii

I I I U C

-a.

ILL.

pr

TANKER. L3I0M..Fi0.I5. --A. P., BETWEEN N, AND U. U, F.

Mu.

.

-D597

-AiII! _ UI

ii]pjP1I!i

I.0 -180 -90 .-6 90 I 800 - ieo -90 S.S. 3 ______ D, BETWEEN F1 AND F, -

:4

.

F0-

_MT

o---:D

F, My 0 90 -90 90 I 80 1.0 0.9 Joi -0.5 -1.0

NORMAL .0 STRESS 0 u1.#I a 10 I s.s. 7 I 10 10 10 10 NORMAL '0 STRESS 0 10 10 0

-310W TANKER F, e 0.IS 0-10 ALl. HEADINGS

- :

310W TANKER Fr 0.15

.i0'

AU. HEADINGS

- :

Cu,

-.---

-..

-310U TANKER Fr .0.15 0- 10 ALL HEADINGS

-; C,

C', O,A IS.s.51 NORMAL SO STRESS I0 SO 0

---T--TiiTi

Fig. 16 Long-term prediction results of total normal stress and component normal stresses on the hull sections in the North Atlantic Ocean.

I0

IS $ SHEARING STRCSS l0

-31DM TANKER Fr .0.15 0-Ia.. ALL HEADINGS ---It" T5 SHEARING 5 STRESS

it

3IOM TANKER I F-0.I5 Q-10 ALL HEADINGS I

L

-tt

L

_-.-:

.0

IS

-I

Fig. 17 Long-term prediction results of total shearing stress and cononent shearing stresses on the hull sections in the North Atlantic Ocean.

[ 0 S SO

- - -

_...

_r

-3 10 1 SHEARING 5 STRESS L. '.-t__. 4 0 I, _{310M TANKER} Fr .0.15 0.10-s ALL HEADINGS

-T,y ft. 0 IS I s.s. 5 I0 1 5 SO

NORMAL '0 STRESS 0 MIMI IT SO 0

-\

310M TANKER F, -0.13 0-10_I ALL HEADINGS

:.

(AMI MflMI .:taz,lITI tATMI MIMI

lOOM MIT MIMI

0 0 SO IT 31GM TANKER Ft .0.15

0-I0

ALL HEADINGS VXT MIMI MIMI MIT MTMI S-S. 7 NORMAL so STRESS

Is,

ao

L!J

NORMAL '0 STRESS 10 IT 31CM TANKER Fr -0.15 Q.10 ALL HEADINGS

IIT

MIMI

---:'3TI

MITMI MIMI

,amI

MOM MIT SlIM

SO

Fig. 18 Long-term prediction results of total normal stress on the hull sections in the North Atlantic Ocean according to the exact and approximate methods.

Js.s.31 SHEARING STRESS S. S. ₁ SHEARIPIG S STRESS 0 m so 310M TANKER Fr -0.15 ALL HEADINGS DT $ )IOM TANKER

Fr0.I5

a 0-I ALL HEADINGS

-.D

_aT

.:TE

ATUS 1I I I0 so SHEARING STRESS m s 510M TANKER Fr .0.15

0-10'

ALL HEADINGS

:

T5

*TE

RIOT TIRI .1

71

Fig. 19 Long-term prediction results of total shearing stress on the hull sections in the North Atlantic Ocean according to the exact and approximate methods.

Titles of Table and Figures

Table 1 Main particulars of a large oil tanker.

Coordinate systems in regular waves. Sign conventions for wave induced loads. amplitudes of ship motions in regular waxes.

amplitudes of loads on the midship section in regular waves.. Response functions of vertical bending moment, horizontal bending moment and axial force on the midship section.

Response functions of vertical shearing force, horizontal shearing force and torsional moment on the midship section.

Normal streSSes on the midship section due to a unit of vertical bending moment, horizontal bending moment and axial force.

shearing stresses on the midship section due to a unit of vertical shearing force, horizontal shearing force and torsional moment. Coordinate systems in irregular waves.

Standard deviations of loads on the midship section in short crested irregular waves.

COrrelation coefficients between. vertical.bending moment, horizontal bending moment and axial force on the hull sections in short crested irregular waves.

Correlation coefficients between vertical shearing force, horizontal shearing force and torsional moment on the hull sections in short crested irregular waves.

Long-term prediction results of loads along the hull length in the North Atlantic Ocean.

Long-term correlation coefficients between vertical bending moment, horizontal bending moment and axial force on the hull sections in

the North Atlantic Ocean.

Long-term correlation coefficients between vertical shearing force, horizontal shearing force and torsional moment on the hull sections in the North Atlantic Ocean.

Long-term prediction results of total normal stress and component normal stresses on the hull sections in the North Atlantic Ocean. Long-term prediction results of total shearing stress and component shearing stresses on the hull sectipns in the North Atlantic Ocean.

Fig. 1 Fig.. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 Fig. 11 Fig. 12 Fig. 13 Fig. 14 Fig. 15 Fig. 16 Fig. 17

Fig, 18 tang-term prediction results of total normal stress on the hull sections in the North Atlantic Ocean according to the exact and approximate methods.

Fig. 19 tng-term prediction results of total shearing stress on the hull sections in the North Atlantic Ocean according to the exact and approximate methods.

@ Ta.b!e

I

Table 1. Main particulars of a large oil tanker

Length between perpendiculars (L) 310.000 rn

Breadth inoulded (B) 48.710 m

Depth moulded (b) 24.500 rn

Draught moulded (d) 19.000 in

Displacement (W) 250,540 t'

Block coefficient (Gb) 0.852

Water plane coefficient (C) 0.903

Midship coefficient (C) 0.995

Centre of gravity before midship (xG) 0. 0331L

Centre of gravity below water line (zG) 0. 2879d.

Mètacentric. height (GM) 0.33O5d

Longitudinal gyradius

L 0.2494L

Transverse gyradius 0. 3231B

Heaving period (TH) 11.6 sec

Pitching period (Tv) 10.9 sec

+xi

® (FeZ)

Vertical Shearing Force

Vertical Bending Moment

Horizontal Shearing Force

Horizontal Bending Moment

y

y

AXiOI Force

+ FA

1

I57.5

135.0

I 12.5

900

67.5

---:

45.0

_-

22.5

oo

O.5

O 5

I. 0

I .5

1.5

0.5

HEAVE.

Fr= 0.15.

0.5

I'

-'p

',."

'

l .

\ \

S I

I

\ \

\

:

0..Ô

157.5

135.0

II2.5

90.0

67.5

45.0

-22.5

\.

\\

_\.

\\\

.5 S

1.5

X= 180.0°

5. S.

'Lu

Y/lp.

0'O

osgi,

góL9

O°06

S S 0

go.

-.

U S -I I

-

:---00081'.X :

HOJ.id

9.0

'JI.

0

,QSI

.1

OS'

0.5.

SWAY

'

L =310 M, H.w = 1 OM

Fr =0.15

a - -

_{- - a. - - -- - _ -}

-0.5.

..,-.

---:

135.0.

1,12.5

-, -

_{- - :}

.

90.0

---:

67.5

--

:

45.0

225

- - a

_a,

-\

-s

\

- .

..

\

\.

,.'

N

N

-l.0

1.5

.

I.5

0

0

0

-c

1.0

105

-YAW IH

L=3IOM, H

IO.M,

Fr= 0:. 15

-S a a

S

-a -a

-a

X=157.5°

135.0

:

II25

9OO

67.5

45.0

:

22.5

0.5

I.Q

1.5

4.0

1.0

ROLL

L310M, Hw

Fr =0.15

-

: X.= 157.

5°

-

:

135.0.

112.5

90.0

67.5

45.0

22.5.

I

I,

-,

-I..

/

-O5

1.10

1.5

. 1

-c

0

0

2.0

0.03

0

m

-

0.02

0

>

0.01

0

S. S, 5, Fr = 0.151

X= 180.0°

__-

:

157.5

0

0.5

V W. BENDING MOMENT

ISO

1.5

11

0.03

0

-C

0I

0.O2

0

0.01

S.S. 5,: Fr= 0.15

- __

S S . S S S S

135.0

112.5

90.0

67.5

45.0

22.5

H. W. BENDING

M0MENT,,-I

/

0.5.

I.0

1.5

oO.IO

-J

0

0.05-0

S.S. 5,. Fr= O.I5

X=180.O°

----:

157.5

135.0

112.5

.

90.0

-:

67.5

..

45.0

22.5

o,.o

0.5

W. AXIAL FORCE

1.0

1.5

2

0.10

-j

a'

0

LL

O.O5

S.S.5, Fr=O.15

X=180.0°

__-

:

157.5

-:

135.0

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