4 DEC. 1979
ARCHIEc
DEPARTMENT OFLab.
v.
Scheepsbouwkunde
Technische Hogeschoøl
NAVAL ARCHITECTURE
Ift
FACULTY OF ENGINEERING, KYUSHU UNIVERSITY,
e
HAKOZAKI, HIGASHI-KU, FUKUOKA 812. JAPAN
ISSC-1979-PARIS
WRITTEN DISCUSSION
TO
REPORT OF COMMITTEE 11.2
NON-LINEAR STRUCTURAI RESPONSE
BY
JUN-ICHI FUKUDA
(JAPAN)
FINAL PIANUSCRIPT FOR PRINTING
To Dr. C. S. Smith Dr. J. Bäcklund Prof. P. G. Bergan Prof. Y. Fujita Prof. N. Jones Dr. M. Kmiecik Prof. P. Meijers Prof. P. T. Pedersen Prof. K. A. Reckling Mr. A, B. Stavovy
Dr. J. L. Armand (Secretary of ISSC) Prof. E. Steneroth (Session Chairman)
JUN-ICHI FUKUDA (Japan)
Recent progress in the prediction technique for ship responses in sea
waves
has
made possible to evaluate statistically the longitudinal hull stresses including the total wave normal stress and the total wave shearingstress produced by the low frequency sea loads, that is, vertical and
hori-zontal bending moments, axial force, vertical and horihori-zontal shearing forces
and torsional moment El] However, from the viewpoint of yield stress analy-sis, an adequate technique is urgently required for predicting the
non-linear stress such as the equivalent stress combined with still water normal stress, still water shearing stress, total wave normal stress and total wave
shearing stress.
A non-linear statistical method is proposed for the purpose of
predict-ing the von-Mises' equivalent stress induced on the longitudinal member of a
ship hull in sea waves, based upon the assumption that the total wave normal stress and the total wave shearing stress would be considered to be com-pletely dependent random variables, and application of this method is made
for a large oil tanker of 310 metres length in short- and long-term seaways
[21.
The von-Mises' equivalent stress for yield criterion can be written as
follows.
2 = {(a )2
+ 3(T
+ 'tT)}
(1)where
Z : equivalent stress
still water normal stress
0
-still water shearing stress 0
total wave normal stress total wave shearing stress
In short-term seaways, by assuming that the total wave normal stress and the total wave shearing stress would be considered to be completely de-pendent, stationary Gausian, narrow-banded, stochastic processes, the
non-linear equivalent stress of (1) can be written as follows.
Z(CT) = {A
+ BeT +
ce}1"2 (2)where
A 2 + 3t2
, B = 2c ± 6'jit , C = 1 + 31.12
=
±RT/RT fOr
p = ±1R : standard deviation of total wave normal stress
aT
RT :
standard deviation of total wave shearing stressp : correlation coefficient between total wave normal stress
and total wave shearing stress
Time histories of the equivalent stress and the total wave normal
stress and the relation between those stresses are illustrated in Fig. 1 for
the case of .B 0. In this case, the characteristics of maxime and minima of the equivalent stress can be obtained as follows.
N1 =
N:
N1,
n:
Then the number of
obtained as follows. Case 1
: Z*?Vi
N(Z> z*) = n nf
(crT/ T)exp[dl./2R Id = aT T0
-nI
--D [cJ/2R2T]daT total number of N2, N3 : number number of maxima n = n.exp[_2D2/RT] maxima of Zof maxima of Z for Type 1, 2, 3 or minima of
maxima of Z that exceeds a given level Z* can be
2 2 2
- n
I
(cyRaT)exp[_clT/2RT]da
-= n{exp[_/2RT] + exp[_/2RTJ}
(7)= n{exp[_D2/2RT] - exp [_2D2/RT] (6)
When aT is equal to -D (= - B/2C), Z takes always the minimum value
which is equal to
Ii ( =
-
//E ).
When takes the minimum value which is larger than -D and less.
than zero, Z takes always the. minimum value which is larger than
/1
and less than /KWhen takes the maximum value which is larger than zero, Z takes
always the maximum value which is larger than . This type of
maxima of Z is called "Type 1".
When takes the minimum value, which is less than -2D, Z takes
always the maximum value which is larger than v' . This type of
maxima of Z is called "Type 2".
When takes the minimum value which is less than -D and not less
than -2D, Z takes always the maximum value which is larger than
/1
and not larger than . This type of maxima of 2 is called "Type 3".
In short-term seaways, by supposing that the number of maxIma or minima
of would be "n", the total number of maxima of Z can be obtained as
fol-lows.
N = N1 + N2 + N3 = n{l + exp[_D2/2RTJ}
Case 2 : N(Z>Z*) =
f
(aT/T)expE/2T]T
2 2 2 nf
(a,/RT)exP[_a./2RT]dcT-= n{l + exp[_/2RT]}
(8) where / } = { B +/82_
4C(A - Z)}/2C
(9)By using (3), (7) and (8), the short-term probability that the equiva-lent stress exceeds a given level Z* can be obtained.as follos.
q(Z>Z) = N(Z> Z*)/N 2 2 2 2 exp[-,/2R ] + exp[- /2R ] oT. T Z* (10) 1 + exp[-D2/2R]
1 + exp[_/2RT]
1 + exp(_D2/2RT]Similar results can be obtained for the case of B < 0.
A series of short- and long-term prediction works have been carried Out in order to estimate the equivalent stress induced on the longitudinal mem-ber of a large oil tunker in the North Atlantic Ocean by using the proposed
method and the available wave statistics [3] Main results of the short-term
prediction are shown in Fig 2, and those of the long-term prediction in Fig
3. The still, water nOrmal stress and the still water shearing stress are shown in Table 1 Following notations are employed in the figures
H visually estimated average wave height (significant wave
height)
T : visually estimated average wave period
heading angle against average direction of irregular waies
( 0° : following waves)
q short-term probability that the equivalent stress exceeds a
given level
Q long-term probability that the equivalent stress exceeds a
given level
Considering the assumption in the present method that the total wave normal stress and the total wave shearing stress are compl'tely dependent,
this method would be probably valid for estimating the upper limit of the
equivalent stress Furthermore, another statistical method should be studied in order to evaluate the non-linear stress combined with the linear stresses which are completely independent or slightly dependent.
REFERENCES
(1] J. FuJcuda and A. Shinkai "Predicting the Longitudinal Stresses In-duced on a Large Oil Tanker in Sea Waves", International Shipbuilding
Progress, Vol. 25, No. 291, Nov., 1978.
[21 J. Fukuda, A. Shinkai and T. Tanaka : "Long-Term Prediction of the
Non-Linear Stresses Induced on a Ship Hull in Sea Waves"1 to be read at the Autumn Meeting of the Society of N. A. of Japan, Nov., 1979.
[3] H. Walden "Die Eigenschaften der Meerswellen im Nordatlantischen
Ozean" Deutscher Wetterdienst, Seewetteramt, Einzelveröffentlichungen,
Table 1 Still Water Normal Stress and Still
Water Shearing Stress (in kg/mm2.)
2 0' 0{24o+ 6pT0}/2 { + 3j2 } E-3(pc0- tO}2,{I+32) -aD TYPE I TYPE I
A
TYPE I TYPE 2I
A
1Ii1ii
1!
1!
01. TYPE IFig 1 Time Histories of Equivalent Stress and
TOtal Wave
Normal Stress and Relation between Those Stresses
--- S.S.7 - S.S.5 - S.S. -0 0o DECK C L -2 90 0 1 55 0 -1 87 0 GUNWALE -2.68 -0.69 1.41 -0.39 -1.7o 0.44 HALF DEPTH OF -0 29 -0 97 0 15 -0 53 -0 18 0 60 BILGE 2 46 -0 71 -1 29 -0 38 1 56 0 43 KEELC.L. 2.61 0 -1.37 0 -- 1.65 0 -HALF DEPTH OF -0 29 -1 48 0 15 -0 81 -0 18 0 92 Z (a..)
Z(o)
DECK CLI
IHALF DEPTH OF SIDE SHELL
I GUNWALE-Fig. 2 1800 180° I 80° P ± I
-:
-p p.,NO.
9iyJ/
fKEELT CL. IBILGE-I.PtI
I 80°HALF DEPTH OF [oNGI. BHD.
I 80° 80° .p. I -: PU I :p_I
c
0 104jIO
4410
Short-Term Prediction Results of Equivalent Stress
Induced on the Hull Section of S.S. 7 in Average
Sea State of BFT. 10 for the North Atlantic Ocean,
H = 7.4m, 9.5sec, q= i0
-'p 90°
Is.s. 1 EQUIVALENT io STRESS 20 -tO-20 20 10 20 310M TANKER Fr -0. 15 0-10-' ALL HEADINGS
-:
I P--I to 20 Is:s. 7 20 EQUIVALENT 10 STRESS tO 20 0 'S 10 20 S 310M TANKER Fr-0. o- 10-s ALL HEADINGS : I - I - 0 20I
EQUIVA STRESS - Ocean, Q = 10-8. 0 KS/MU5 0 103 Long-Term Prediction Results
of Equivalent Stress Induced
on the Hull Sections of S.S. 3, 5, 7 in the. North Atlantic
30M TANKER Fr-0. IS
