r
ARCHIEF
THE PREDICTION OF THE LONG TERM
DISTRIBUTION OF SKIP BENDING MOMENTS
FROM MODEL TESTS
G. J. GOODRICH, B.Sc. 28th March, 1966
SYNopsIs.The long-term predict/on of ship bending moments has so far depended upon the collection of statistical stress data from ships in service over a
relatively long period of time. Tue object of tile present work is to demonstrate the feasibility of using model-test data for such predictions.
The first necessity in any such work as this is to have a reasonable representation
of the sea The representation must be in a spectral form in order that the sea can be combined with model results to produce tile ship responses in irregular waves, and statistics ofthe frequency ofoccurrenceofvarious sea conditions must also be introduced since the final predictions must be on a statistical basis.
The Darbyshire type spectra have been found to be representative of typical sea
spectra and have been used in a modified form in this analysis. Model-test results have been combined wit/i these spectra to give ship predictions and these are
compared with ship results obtained by B.S.R.A. Introduction
NVEST1GAT1ONS intended to establish the long-term distribution of
wave-induced ship bending moments have so far concentrated upon the collection of statistical stress measurements from ships in service over a relatively long period of time. The present paper shows that given
a reasonable representation of the sea, it is possible to predict such a
distribution from model-test data.
The representation must be in a spectral form in order that the sea
can be combined with model results to produce the ship responses in irregular waves, and statistics of the frequency of occurrence of various
sea conditions must also be introduced since the final predictions must be on a statistical basis.
Modified Darbyshire type spectra have been found to be representative
of typical sea spectra and have been used together with model-test data to give ships predictions. These predictions are shown to compare quite favourably with the full-scale ship results obtained by B.S.R.A.
The Sea (1)One-dimnensional Spectrum
A study was made of a large number of wave spectra obtained from analysis
of records taken on British weather ships using a shipborne recorder, and the
analysis of the records was carried out by the New York University. The results
of the study have been condensed into Figs. I to 3, where the distribution of
significant wave height and peak frequency and average period of the spectra are shown for various wind speeds. At wind speeds of 15, 45, and 50 knots the sample number was less than twenty; for the other wind speeds the average number of samples was fifty. The mean values of significant wave height for each wind speed have been plotted in Fig. 4 and the results compared with the curves of Darby-shire2 and Roll3. The curve given by Roll is based on visual observations from weather ships.
236 GOODRICH: THE PREDICTION OF THE LONG-TERM DISTR!BUTIO
The range of possible significant height for any wind speed is large, but if the
spectra are grouped within a limited range of significant height (regardless of
the wind speed), the Darbyshire type of spectrum based upon the mean significant
height does approximate to the mean of the measured spectra. The Darbyshire spectrum has therefore been used as a basis for calculating the cumulative
density of ship bending moments.
The equation of the Darbyshire spectrum is
fH1 \ (f-f0)2 1 df = 239 exp
-Lo.00847 { (f-f0) + 0042 ] df, (1) andH2 =
H1 spectral ordinate, f = frequency (secl),= frequency (see-') of the peak value of spectrum = I IT1,
T1 = 194 Wh12 + 25 W4 10-',
H = 0008l W',
W = wind speed in knots,
H"3= 00l33 W2 = 165 H.
An alternative form of Equation (I), expressed in terms of wave amplitude, is
The spectra in the form of Equations (I) and (2) cannot be combined directly
with bending-moment response-amplitude operators which are expressed in terms of wave-length to ship-length ratios. The equation can, however, be transformed from energy density per unit frequency to energy density per unit of wave length, and in terms of wave amplitude the equation is
'2 H2 1 I dA /H1\ 2
r (A) =-
/g ,l-/'
- -
df. (3)4 2V2ir
df\HJ
Wave Height
Full-scale stress data have beer collected on a C-4 type of ship. As well as the stress measurements, visual observations were made of the wave height. It
has been found that the observed heights agree with the results of Roll3 shown
in Fig. 4. When the curve of Roll was compared with the mean curve obtained from the spectra analysis it was found that the difference was of the order of
40 per cent over a wide range of wind speeds. This leads to the conclusion that
the observed heights represent the root mean square wave heights, rather than the significant heights. The ordinates of the Darbyshire spectrum Equation (1)
are directly proportional to the wave height squared and therefore spectra have
been used which have root mean square heights equal to the observed heights for given wind speeds. These spectra are shown in Fig. 5.
Two-dimensional Spectra
Uni-directional seas are rare and in order to give a more realistic represen-tation of the sea it is necessary to spread the energy of the wave spectrum to
give a two-dimensional spectrum. It. has been assumed that the energy varies as cos2 .w, where w is the wave component direction in relation to the
predom-inant wave direction. The factor is expressed as
2
f () = - CO52
i-Lw. (4)Using this function the energy is spread over ± 90 degrees, but most of the
energy exists within ± 45 degrees of the predominant direction. The
two-dimensional energy spectrum for the 40-knot wind is shown in Fig. 6.
( )2
r (wfl
I J F1'=
-4 I-
2iriH12
\F{J
df. (2)I
OF SHIPBENDINO M0ENTS FROM MODEL TESTS - 237
Response-Amplitude Operators
No response-amplitude operators are available for the C-4 type ship. It has therefore been necessary to use the results of the Vossers, Swaan, and Rijken model experiments,4 carried out on Series 60 models in oblique seas,
inter-polating for a ship length of 496ft.,
CbO648, L/B70, L/H l750, and F02,
these dimensions corresponding approximately to the C-4 type ship.
The results of Vossers' experiments are given in terms of a non-dimensional
vertical wave bending moment amplitude ,
M
i.e.
2v_
(5)where M = vertical wave bending-moment amplitude,pg h BL2
h = wave amplitude,
B = ship beam,
L ship length.
The -t values have been converted to amplitude-response operator c,
i.e.
= (M/h)2 = (pg BL2 i.
For the C-4 type ship L = 496ft. L/B = 70 (approximately),
= 4 x 248 x 10" tons ft. units.
Bending Moments
The two-dimensional bending-moment spectrum is obtained from the product
{r( x)) 2 c, and a double integration of the spectrum over the significant range of wave length and wave angles ±90 degrees gives the cumulative density of
vertical bending moment R. The root mean square bending-moment coefficient rm is derived from the equation
rm VR lip gL3B Cwp, (6)
where C, = waterplane area coefficient.
For the C-4 type ship
rm = 536
X lO-° VR.Since it is unrealistic to assume that the ship will always be in the head sea
condition, some account has to be taken of the effect of variation of ship heading
relative to the predominant sea direction. A study was made of this effect and an approximate reduction factor of 85 per cent was found to be reasonable.
This factor, however, should be studied further.
The calculated root mean square bending-moment coefficients are shown in Fig. 7 and are compared with the results obtained from the C-4 type ship. As
an alternative to the presentation of the prediction in terms of bending-moment coefficient, the results can be expressed in terms of an effective wave-height to ship-length ratio. The effective wave height is defined as that height which when
used in a static calculation, results in a bending moment equal to that derived
from the dynamic condition in waves.
The static bending moment obtained from a conventional calculation in a
wave of ship length may be expressed as
H
static bending moment M2
= cpg - L3B C,,
(7)L
where H = wave height.
For the calculated dynamic root mean square bending moment He
= M = cpg - L3B Cwp,
(8) Lwhere He is an effective wave height. Then the bending moment coefficient
m =
Mpg L3B C,
H= C-.
238 GOODRICH: THE PREDICTION OF THE LONG-TERM DISTRIBUTION
Hence H m
(9)
L c
H,,
For the C-4 type ship
- = 0538 (m x l0) per cent.
L
He
The scale of - R.M.S. is shown in Fig. 7.
L
Long-term Distribution
In order to determine a long-term distribution of bending-moment coefficient
for North Atlantic routes, it is necessary to assume a frequency distribution of Beaufort numbers for this area. The distribution used is one given by Bennet et a!5 and is shown in Fig. 8. This curve reads that the probability of having Beaufort numbers up to and including Beaufort 5 is 81 per cent, up to and
including 7 is 96 per cent, and therefore Beaufort 6 to 7 is 15 per cent.
The curve of root mean square bending-moment coefficient is grouped for the following Beaufort numbers: 0 to 3,4 and 5, 6 and 7, 8 and 9, 10 and 11. If m is the bending-moment coefficient and m/rm = v, then the probability of
exceeding v is
P1
=e"
The cumulative distribution of m is therefore
Qi =
P,where P is the general weather probability distribution.
For example: let m = 2 x 10 (H/L = I P075)
The values of Q1 have been plotted as logarithms taking various constant values of m, using the curve of rm given in Fig. 7 to derive the ship prediction
in Fig. 9.
S/zip Results
A comparison between the model prediction of the long-term distribution of wave bending moment and the C-4 type ship is shown in Fig. 9. Use has also been made of results obtained by B.S.R.A. and published by Johnson and Larkin.° Ships No. 1 to 5, 10, 12, and 14 have been re-analysed in terms of HCIL and the number of stress reversals n per total running time N, and the
results are shown in Fig. 10. To compare directly with the prediction from model
results it is necessary to extrapolate the ship results to zero and to assume this zero value is the 100 per cent probability. By this means the ship results have been superimposed on the model results and are shown in Fig. 11.
Sea state x 10"rm (m/rm)2= v2 e-v2 Weather distribution, P1 Qi
0 - 3
O25 4Ø '-
.520-4 - 5
O71 795 0O00354 29O 001036 - 7
1I0
332
00364 150 054608 - 9
143 1960l410
35
0494010 - 11 160 156
02l00
05
01050=11553
OF SHIPBENDING MOMENTS FROM MODEL msrs 239
Conclusions
The agreement between the model prediction and the C-4type ship is good.
The model did not, of course, correspond to any of the ships used by B.S.R.A. The ships fall broadly into two classes: Nos. 4 and 12 are of full block coefficient
O75O76, the others are of finer block in the range O63O68. Ships 5, 12,
and 14 show a tendency to have higher stress values of low probability and it
is suggested that these high stresses are the result of slamming.
The model results are sufficiently encouraging to suggest that the methodis
worthwhile pursuing. In order to continue the work it would be necessary to
obtain ship data on the long-term distribution of ship bending moments for further comparison with model results. Detailed information of thesea would be required in the form of energy spectra, preferably two-dimensional, together
with bending-moment or stress spectra in order to obtain the ship
bending-moment amplitude-response operators for comparison with model results.
Following the model experiments, which would have to be carried out over a range of wave lengths and directions, a long-term prediction of ship bending moments could be made and compared with statistical results obtained from
the ship.
A number of assumptions made in the present analysis should be investi-gated. For example it has been assumed that the mean curve of root mean
square bending-moment coefficient is sufficiently accurate for such predictions. Iii fact, ship R.M.S. coefficients will probably be distributednormally about a
mean line and such a distribution should be included in the analysis. The
arbitrary reduction of 15 per cent to allow for directional effects should also be considered in more detail.
Acknowledgements
This paper is based largely on work carried out while the author was on
short-term detached duty from the Ship Division, National Physical Laboratory,
and working at the Webb Institute of Naval Architecture, New York, with Professor E. V. Lewis and with Professor R. Bennet who analysed the C-4 type ship results. The work was carried out under contract to the American Bureau of Shipping and is published with the approval of the Director of the National Physical Laboratory.
REFERENCES
PIERSoN, W. J., MosKowrrz, L., and MEHR, E., "Wave Spectra Estimated from Wave Records Obtained by O.W.S.". "Weather Reporter and Weather Explorer," New York University, November 1962.
DARBYSHIRE, J. "A Further Investigation of Wind Generated Waves,"
Deutsche Hydrographische Zeitschrift, 1 (1959), p. 1.
ROLL, H. H., "Height, Length and Steepness of Seawaves in the North
Atlantic, and Dimensions of Seawaves as Functions of Wind Force,"
Translated and presented as Technical and Research Bulletin No. 1-19.
S.N.A.ME., New York, December 1958.
VOSSERS, 0., SWAAN, W. A., and RUKEN, H., "Vertical and Lateral Bending Moments Measured on Series 60 Models," Ins. S/iipbuild. Progress, 8
(July 1961), p. 302.
BENNET, R., IvARsor4, A., and NORDENSTRÔM, N., "Results from Full Scale
Measurements and Prediction of Wave Bending Moments Acting on Ships". The Swedish Shipbuilding Research Foundation, Report No. 32. 1962, p. 29.
JOHNSON, A. J., and LARKIN, F., "Stresses in Ships in Service". Paper
240 GOODRICH: THE PREDICTION OF THE IONG-TERM DISTRIBUTION 04 DO 0 00 SI
DI5TrUSIITI4 OF SIGNIPIC8RT RAVI HEIGHT FOR 5251005 0100 SPEEDS
FROM NZU ANALYSIS CF SPECTRA
04 DY 05 IS 0.5011 DI 0 24 5
DISTRIBUTION OF AVERAGE P0200 FOR VARIOUS WIND SPEEDS
FROM NYU ANALYSIS OF SPECTRA
H 005 005 00 Dli C?. DISTRISLITIOTI OF SPECTRA P5GM SI Fig. 1 -00 50401' 25 550Y. I I Fig. 2 Fig. 3 SO 014015
PEAR FRCOUCNCIIS FOR 5251005 WIND SPEEDS N.Y.U. ANAI.YAS OF SPECTRA
40 x 30 0 tO 00 30 50 60
R WIND SPEED (oTS)
CURVES OF SIGNlFICAN1 WAVE HEIGHT VARIOUS SOURCES
Fig. 4
IAODIFIED DARBYSHIRE SPECTRp
FOR VARIOUS WINO SPEEDS
0
0 0.a 04 06 03 2 I IS
FAMILY OF MODIFIED DARBYSHIRE SEA SPECTRA USED Ill CALCULATIONS
Fig. 5
242 GOODRICH: THE PREDICTION OF THE LONG-TERM DISTRIBUTION
TWO DIMENSIONAL SPECTRUM FOR WIND SPEED 40 KNOTS
es rM/L.RWS EFEtCTIVE KEIGHr 2.0 06 lmXIO4 -06 1.0 -06 0.4
03
06 0
x 0OF SHIPBENDINO MOMENTS FROM MODEL TESTS 243
0,,
0
40 50
SPEED 1*4015
Fig. 7
X MODEL USINC MODIFtED DARBYSHIRE EPECRA 0 MEAN MEASURED (34 SHIP WOLVE.RINE S1TE"
K
I4ESE INTS b.E
L$4CERTM4 DUE TO SCARCITY
OF DATA
123
4 5 6 7 6 9 0 II 12BEAU FORT
OBSERVED AND PREDICTED WAVE BENDING MOMENTS IN DIFFERENT WINDS
CO 10- 60 r
0
0 tO 20 30
244 GOODRICH: THE PREDICrION OF THE LONG-TERM DISTRIBUTION
0 I 5 8 5 6 7 B 0 6 II IS
REAUFORT NUMBER.
FREOUEY DISTRIBUTION OF BEAUFORT NUMBERS ON GENERAL N ATLANTIC ROUTES (FROM REF 5)
006 004 002 00I 7 0 I
PREDICTtD AVESACE wu,n F3M MODEL TESTS FULL SCALE TESYS
0000I 000001 00001 0001 001 Di
PROBA8ILITY OF EXCEEDING
FIG. 9 LONG TERM DISTRIBUTION OF WAVE BENDING MOMENT FOR
TYPICAL NORTH ATLANTIC WEATHER
Fig.9
-b 4 -. 1 0 1 P 3
10 OMPARIS0N OF B.S.R.A. SHIPS DISTIDUrION OF H./L VALUES
Fig. 10 I00 4 IL 4 14 245 OF SHIPUENDINO MOMENTS FROM MODEL TESTS
6
5
4
3
246 GOODRICH: THE PREDICTION OF THE LONG-TERM DISTRIBUTION
OF SHIPBENDTh1O MOMENTS FROM MODEL TESTS
0000001 0.00001 0.0001 000I 0.01 0.1 pceAeIY cc EXCEPG
LONG TCPM DISTRISUTION OF Hu/ FOG VARIOUS LS.LA. SHIPS AS SHOWN
Fig. 11
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