FURTHER ANALYSIS OF FULL SCALE
MEASUREMENTS OF MIDSHIP
BENDING MOMENTS
by
NILS NORDENSTROM
Report from the Division of Ship Design
Chalmers University of Technology.
Supported by a grant from the Swedish
Technical Research Council.
Gothenburg May 1965
Page PART 1 REGRESSION ANALYSIS
I Introduction i
II, Distributions of the independent variables
11:1 Distribution of state of sea. 2
11:2 Distribution of wave direction 3 11:3 Distribution of loading condition 4 III Distribution of the dependent variables
111:1 Distribution of wave load 6
111:2 Distribution of ship speed 6
IV Wave load as a function of the other
variables
IV:1 Successive regression analysis 7 IV:2 Wave load as a function of state of sea 11
IV:3 Wave load, as a function of wave
direction 17
IV:4 Wave load as a function of loading
condition 18
IV:5 Wave load as a function of ship speed '18
V Ship speed as a function of thé .other
variables
V:1 Successive regression analysis 19
V:2 Ship speed as a function of state: of sea 19 V:3 Ship speed as a function of wave
direction 20
V:4 Ship speed as a function of loading
condition 2O
VI Summary of par-t 1 22
PART 2 ESTIMATION OP LONG TERM DISTRIBUTIONS OP WAVE. LOADS
VII Introduction 23
VIII The long term distribution of r is assumed
to follow the Weibull distribution
VIII.:1 Pitting of Weibull distribui ions
to the estimated r values 27
VIII:2Transfornjatjon of a long term
distribution of r values to a ong
term distribution of wave loads 30
VIII:3 A practical applicatión 36
Appendix I
Tables and figures.
Error of visual estimates of wave
CHAPTER I
INTRODUCTION
This first part of the report presents the results from a triai to get some further information out of the full scale measurements of midship bending moments earlier presented by Bennet /1/ and Nordenström /9/, /io/.
Naval architects (and many others) often present results from measurements (full scale and model) without any
statis-tical analysis. This is very unsatisfactory, because such
data. are always affected by random events. Therefore, results from measurements mus-b be statistically analysed in order to
investigate if it is advisable to rely upon found apparent relationships between variables, or if these relationships
are likely to be caused by chance.
The föllowing chapters present the results of a statis-tical analysis of the influence of the. state of sea, the
wave-to-heading angle and the loading condition upon the wave
loads and the ship speed.
Particulars of the ships in question are given in table
1.1. See also reference /1/.
CHAPTER II
DISTRIBUTIONS OP THE INDEPENDENT VARIABLES
11:1 Distribution of state of sea.
We have used the Beaufort scale to describe the state
Of sea during each record. The Thrces according to the
Beaufort scale were estimated by the ships' officers using a state of sea card with photographs showing the appearance
of the sea from force O to 12. Table 11:1 gives a
descrip-tion of the Beaufort scale. The officers are accustomed to estimates of Beaufort numbers and, therefore the Beaufort scàle is believed. to give the best description of the ea
state in this case when we have to make. visual observations.
A certain sea state is fully characterized 'by the energy
spectrum, which can be estimated from a wave record.
Un-fortunately it is rather complicated and expensive to get such a record onboard during ordinary service so we did not have any possibility to get wave records. There is, however, a certain relation between the Beaufort scale and the cor-responding spectra so information regarding the influence of
Beaufort on ship speed or on wave bending moment can be
transformed to show the influence of spectral properties.
Regarding the accuracy of these visual estimates of the sea state, we believe that e.g. force 5 sometimes is esti-mated as force 4 or force 6 or even less correct. However,
such a spread in the estimates does not quantitatively affect the estimated average relations between sea state and wave
load or ship speed if the estimates of force are in mean
correct. The regression analysis in chapters 4 and 5 shows
a very close relationship between sea state on one hand and
wave load and ship speed on the other. This gives reason to
believe that the state of sea is rather accurately estimated by using the eaufort scale.
The distributions of Beaufort forces for all records
analysed are shown in figure 11:1.. We see in the figure that force 2 to force 8 are very well covered by records. Porces
The numbe± of records in force O to 2 is not at all repre-sóntatjve for the number of tines these forces óccur. This
depends on the fact that records usually were ndt taken in smooth water, as values in smooth watér do hot affect those
long term dis-tribttions of wave loads we originally intended -to estimate. This lack of information regarding the lower
'forces does not seriously affect the regression analysis of this report.
11:2. Distribution of wave direction.
The wave direction was visually estimated by the snips'
officers on the bridge. The angle between the longitudinal axis of the ship and the direction of translation of the wave
system is given by 16 numbers. Each number denotes a sector of 22.5°. Head sea is denoted O and following sea is denoted
8. In this report we consider the condition of symetry by
transforming all directions on the port side to the
corre-sponding. direction of the starboard side. We thus describe the wave direction by the numbers O to 8.
The distributions of wave direction for all records
an.alysed are shown in figure 11:2. The figure shows a very remarkable. skewness of the distributions.. Head seas are
estimated to be much more common than following seas in spite of the fact that seven different ships on different trades
during., several years are expected to get a uniform
distri-bution of wave direction. Consequently, an error is probably introduced by the. thethod of estimating wave direction.
There are principally two ways to make visual
obser-vations of wave direction.
To look along the wave crests at right angle to the
-direction of translation of the waves.
To look in the direction where the waves seem to come
-D)
For otir ships
iii
question we get the following formulas.Method No. i gives a correct observation. Method No. 2, however, does not give a correct observatioñ, because the
ship speèd introduces an error that makes the apparent wave
direction different from the true wave direction.
It is shown in Appendu I that the dstribution öf the
apparent wave direction (from method 2) closely resembles our
estimated distributions if:the true wavedirectjon is
uni-formly!dist±ibuted on all wave directions. We thus consider
the true wave d eötion to be
uniformly
distribüted., 1±1Appendix I we give, the simple ilation between true
and
appare.t wave direction. This relation
can
b.e sed tocor-rect for the error introduced by the method of observation.
11:3.. istribution of loading condition.
The loading condition during, each measurement is simply defined by the nond±mensional number D.
DW
D=-
; OD1
DW =' doadeight at measurement DWM = maximum deadweight. We. introduce = displacement at measurment = maximum displacementand
get
.D is thus approximately a linear function o± draught and still water bending mbment.
The distributions of .D are shoi in fig. 11:3.1
and
11:3.2. The dry cargo ships (No. iand
2) are subjected tòmany differen-t loading condit-ions, but th oro carriers
(No. 3
and
4)and
the tankers (No.5-7)
have two very distinct.loading cònditions viz, full load and ballast.
1 0,305 +
0.695
D 2 0.374 Ö.626 D 3 0e 192 + 0.808 D 4 0.246 +0.754
D 5 0.218 ± 0.782 .D 6 0.208 + 0.792 D 7 0.218 + 0.782 DCHAPTER III
DISTRIBUTION OP THE DEPENDENT VARIkBLES.
111:1. Distribution of wave load.
The midship bending stresses during each record of about
a quarter of an hour is described by the parameter r of the
Rayleigh distrjbut±on. The value of r is a measu±e of the severity of the load situation. See chapter VII. Different
types of long term distributions of r values are handled in
some detail in Part 2 of this report. Here we are only
in-terested in the distribution of recorded r values in order to get ranges of r value where the regression analysis below is valid. Distributions of the recorded r values are shown in
figure 111:1.
111:2. Distribution of ship speed.
The ship speed was reported by the ship's officers from readings on he ship's ordinary log. The distributions of
speed shown in figures 111:2.1 and 111:2.2 are affected by the fact that records were seldom taken in Beaufort O-2 (see 11:1) but the distributions are believed to be typical for the different types of ships with their different trades. Note the great difference between the trial speed T and the mean speed m of greatest interest to the owner.
CHkPTER IV
WAVE LOAD AS FfJNCT ION 0F THE OTHER VARIABLES.
IV:1. Successive regression analysis.
The midship bending stresses during each record are
characterized by one value 0±' the parameter r of the Rayleigh
distribution (chapter VII). By means of regression analysis
it is possible. to determine values öf regression coefficients
a1 in functions describing the influence of certain
inde-pendent variables on deinde-pendent vs.riable r. We write the regression equation in the. follöwing way:
r = f1(B) +
f2(A) + f3(V) +
f4(D) ... (i)r = parameter of the Rayleigh distribution
f1(B) = fUnctioì of state of sea B (Beaufort, see II1)
12(A)
function of wave direction A (see 11:2)f3(v) = function of ship speed V (knots, see 111:2)
f4(D) = function ofloading condition,D (see 11:3).
We have, chosen f 1(B) as a polynomial of the 3rd. order
in B.
f1(B)
= + B + c3B2 + 4B is a constant.
2' are. regression coefficients.
Full scale measurements (see Bledsoe /2/) show that the function f2(A) may well be described by a polynoial of the
4th order iii A. .
f2(A)
= 2 ±
3A +
4A2 + 5A3 + 6A4On acóount of the symmetry about the ship's centerline we choose the coeffióients
2 is a constant
a59 a6 are regression coefficients.
We have chosen polynomials öf the 2nd order in V and .D to describe the functions f3 (V) and f4 (D). (Later we had to revise the function f4).
f3 ( )= +
(D)
= ± D +
39 4 are cönstante
a7 - are regression coefficients.
By inserting the functions - f in ecl. (i) we get the
following theorêtiöal regression eQuation.
dA Í'2(8) = We then get = + a (0) O head sea
r =1 + a2 B +
3 B2.+ a4 2_12) + a6 A2 (A2 128) + V + a8 V2 .D± a10 D2 where following sea.(A-12) ± a6A .2i 28)
(2)
An equation of this kind is easily solved in an
values of the variables. All the variables of eq. (2) are
random variables and those values of the variables used in
the solution are also affected by errors of measurement.
Therefore the solution öf eq. (2) does not give any
exact values of the regression coefficients but we get a set
of values a1 - a10 as estimates of the coxresponding
quanti-ties
- We thus get the following empirical regression
equation.
r a1 + a2B ± a3B2 + a4B3
+ a5A2 (Â-12) + a6A2 (A2-128)
+ a7V + a
+ a9D + a10
Moreover the functions f1 - f are only approximative
hypothetical descriptions of the corresponding factual
rela-tions.
For the above thent±oned reasons a solution of eq. (2)
does not tell anything about the influence of the variables
in question unless we investigate if the "apparent"
relation-st.ips are likely to be "caused" by chance or if they are probably caused by "true" relationships. Such an
investiga-tion has been carried out using the statistical analysis ordinarily used for problems of this kind (see /6/). The
s1atistical analysis gives ánswers to the following questions.
Is it probable that the variation of r is "caused" by
a1y
of the chosen independent variables?Which independent Variable has the greatest influence on r, and how important are the independent variables as compared
to each other?
That do we loose in accuràcy by neglecting one or more
Do the regiession functions fit the data well?
Should it be justified to make the regression functions more (less) complicated?
These five questions are discussed below.
Question No. i is answered by testing the hypothesis Hi
that the regressiòn coefficients equal zero.. Tests of the
hypothesis Hi are applied to the following four regression
equations. r = f1 (B) - 3 =
=0
r f2 (A) 1112: == o
r f3 (V) ; 1113: == o
r = f4 (D) 1114: 9 = Q'10= o
Table IV:1.1 shows that H11 and
13 in all cases st
be rejected at the 0.1 % siificance level. 1112 must in
all cases except one (ship No. 2) be rejected tt the 2.5 %
significance level and H14 must in all cases except two (ship
No. 6, 7) be rejected. at the 2.5 % significance level.
Consequently, we can no .oubt state that there are
func-tional relationships between r and the variables B, A, V and
D. Note that V is not independent of B, A or D arid that. the test does not tell anything about the relationship between r
and V when B, A and D are constant. However, B, A and D are
considered to be independent of each other.
Question No. 2 is answered by means of a successive
regression analysis, The independent variables are then
in-troduced successively, starting with one independent variable and ending with all four independent variables at the same time. Tables IV:1.2.1 - IV:1.2.7 show the result of this analysis.
It is quite obvious fronithe tables that the influence
of A and D is almost negligible as compared to the influence
The remarkably close relationship between r and V is caused by the fact that V ±s closely related to the. severity
of the state of sea (see chapter V). In three cases (ships
No. 3, 6 and 7) the wave load is more c]osely related to
the ship speed V than to the visually estimated state of sea
B. In all cases except two(ships No. 2 and. 4)
the wave load
r is more closely related to the wave direction A than to the
loading condition D. We can thus raflk the variables in. the
decreasing order B, V, A, D, the influence of A and D being
very small as compared to the influence of B and. V. Remember that V is not indepéndent öf B, A and D.
Question No. 3 is closely related to No. 2. Prom Table IV:1.2.1-7 it is obvious that the accuracy of the functions is seriously affected by neglecting B and V, but that either B or V can be neglected without a very remarkable loss of
accuracy. Consequently, V gives a good description of the state of sea. If both A and D are neglected the loss of accuracy is comparatively small.
Questions No. 4 and5 are. handled in the following
para-graphs of this chapter6
IV:2. Wave load as a function of state of sea.
- We have used the following four equations to estimate
the functional relationship between wave load r and state of
scaB.
r(B) = f1(B)
r(B, A) = f1(B) + f2(A)
r(B, D) f1(B) + f4(D)
r(B, A, D)= f1(B) + f2(A) + f4(D)
Ship speed V is not included here because V is not
in-dependent of B, A and D. If B, A and D are independent of
each other, the above four functional relations betwèen r
the functions differ from each other to a considerable extent
the data calls f or further investigation. Figures IV:2.1 -IV:2.6 show that the four functions in all cases are rather
similar so we consider that no unexpected relationships be-. tween B, A and D exist. However, according to tables 111:1.1 and. IV:1..2 the result frorn ship No. 7 is considerably inferior
to the results of the other ships so the result from ship
No. 7 gives reason to doubt. B can for instance be badly
estimated. It also seems as if ship No. 7 has sJ-owed. down (see figure V:2.4) at lower values of B than have the o'ther
large tankers o
Table IV:2.1 shows the result of tests regarding the form of the regression curves r f 1(B). Only for two ships
(No. 2 and 6) the data is significantly (only 5 % level)
dif-f ering dif-from the regression curve, so on the whole we consider
the chosen polynomial of the 3rd order to be satisfactory.
It must be kept in mind that a test of this kind does not prove that the chosen function does fit the data. We have
failed to show that the function does not f±t the data but
this result also applies to a great many other functions.
Reg.rding the. general trend of r as a function of B,
all functions except those of ships No. 3 and No. 5 have got
a minimum a-t Beaufort 2-4. This is probably explained, by the
fact that swell dominates the states of sea denoted Beaufort
O-1 and that the dominant wave period of swell is more close
to the natural pitching period of the ship than s.the shorter
waves in Beaufort 2-4 (see table IV:2.2 and figure V:2.9). The upper part of the curve has got a maximum in two cases
(ships No. i and No.
7).
This result must not be taken to indicate any maximum because there are very few measurementswithin the upper Beaufort groups.
In order to compare the functions of the different ships
we haVe computed moment factors r'.
rZ
r' =
r = r (B, A, D)
Z = section modulus at measüring point = density of sea water
B = moulded. breadth
L = length between perpendiculars
The quantity r' is the root mean square of the
nondirnen-sional midhip bending moment factor.
Figure IV:2.7 shows r' as a function of B. Figure
IV:2.8 shows r' as a function of L (cross curves from figure IV:2.2) for B = 6 and B = 10. It is interesting to note that
fr B
= 6, r' decreases when the ship length increases, butfor B = 10, r' increases when the ship length increases (as mentioned above we do not rely on the result from sh1p No. 7). These different trends of r' are explained by figure IV:2.9
showing the ship length L5, corresponding to synchronous
pitching as a function of B.
The following four ways have been used to estimate L5 as. a function of B. For simplicity in order to get a rough estimate, we assume that synchronism occurs in waves of length
equal to the ship length (,L5 = x) and that the most
uflfavour-able case occurs when the period T0 of the maximum ordinate of the wave spectrum corresponds to the mentioned wavelength.
1. According to Moskowitz and Pierson (see /14/) the period
T0 corresponding to the maximum ordinate of a fully developed
spectrum is
w
T0
- 2.66
T0 = wave period in seconds W = wind speed in lmots
Using the well imown f ornula X = 1.56 T2 we get L8 = X = 0.22 W2
According to Darbyshire /3/
= 1.94 y 1W
where y is a parameter depending on pitch (y = i correspönds
to fully developed sea). X = 1.56 T2 gives
s = 5.87 y2 W
traced in figure IV:2.9 (curves 2) with y = 1.0, 0.9, 0.8 and 0.7. In reality thê spectrum is seldom fully developed.
When y lies in the interval 0.8 < y < 0.9 we get the area shaded. in the figure.
According to Warnsinôk et.al. ./14/ the spectrum can be expressed in terms of the. visually estimated apparent quanti-ties wave period T and wave height H. Prom /14/ we. get
= 1.3 T
V
X = 1.56 T2 gives
L5 = X = 2.64
We insert the values of Tv given by Roll /11/ (see table IV:2.2) and get the curve traced in figure IV:2.9 (curve 3).
Using the function
L5 = X = 1.56
(X = 1.56 T2 is used by Roll /11/)
we get a fourth curve traced in figure IV:2.9 (curve 4).
The four above mentioned relationships between L5 andfl or W show a wide scatter but it is evident that a short ship of length about 100 meters will reach synchronism, i.e. large stresses as compared to the state of sea, already in Beaufort
J(1_A2)2
+ b
A2a and b are constants related to the type of ship.
T
A = = tuning factor (ship speed and heading can also
ö be taken into consideration).
= synchronous pitching period of the ship.
T0 = wave periOd correspond-jng to the maximum ordinate of the wave speótru.m.
This could probably be a simple but in principle useful method to estimate the response of a shIp using state of sea
statistics.
We have made a very simple trial to compare formula (i)
to the data from ship No. 2. This ship is chosen because it
has operated mainly on the North Atlantic (Sweden-Nerth
America) and. this area is well covered by the wave statistics of Roll /11/. Prom this reference we get the mean of. the
vieually estimated wave height arid the mean of the
visual-ly estimated wave period as function of B. See Table
IV:2.2..
Assuming that the energy of the wave system is proportional to H, that synchronism occurs when the wave length equals 0.9 L and that T0 = 1.3 T (see 3. above) we get
6. The longer ships of length 200-250 meters usually need
Beaufort 10 to get synchroisrn, Consequently, the remarkably different trends of r' as a function of L for B = 6 and B =
= 10 is to be expected.
In consequence with the general theory of oscillations it might be possible to estimate r2 (= double responso
speot-rum area) as a function of r (= area under wave spectrum) from the following type of formula.
a
r' a viz
w-
- ; A = 0.585r
V
\j(l-42)2
+ b A2 VFigure IV:2.10 shows a plot of r'/H as a function of A
for ship No. 2. For comparison we have traced the fori1a
above using a = 0.075
and
b = 0.0125. The two curves arerather similar. This fact supports the following proposal regarding estimation of long term distributions of wave loads
(or any other variable) by using the visually estimated
properties H.and T.
Estimate r'/H as a function of
T.
Thiscan
be obtained by means of model experiments, full scale measurements or theoretical calculations (strip-theory).Describe .a long term distribution of states of sea by
finding the long term distributions of H within intervals
of T. These distributions of H can obviously (see the data presented by Warnsinck et.al. /14/) be described by means of
the Weibufl distribution.
H. - H
P (H., H) =
-
exp ( (IY')
P (H.. H) is the probability that H is smaller than or
equal to H.
H0, H0 and y are parameters of the. distribution. These
parameters are functions of
T.
The long term distribution of r' is obtained by summing. the distributions obtained (from 1. and 2.) within each
interval of T.
4. The final long terrn'distribution of the variate in.
question is obtained from the long term distribution of r'
IV:3. Wave load, as a function of wave direction.
In analogy with IV:2 we have used the following four equations to estimate the functional relationship between
wave, load r and wave direction A.
r (A)
=f2 (A)
r (A, B) = f2 (A) + (B)
r (A, D) = f2 (A) + f4 (D)
r (A, B, D) = f2 (A) i- f1 (B) + f (D)
Figures IV:3.1 - IV:3.4 show the result. There is a
rather wide. scatter of the mean values of the groups around
the regression equations, and the four regression equations of one ship differ from each other to a óonsiderable extent
in many cases. This depends on the fact that the state of sea has a much larger influence on the wave loads than has the wave, direction. Therefore the variation about the regres-sion curves is expected to be large and tests (see table IV:3) regarding the foin of the regression curves show that the
var±ation is not larger than is to be expected.
Figures IV:3..5 and IV:3.6 show comparisons between the different ships. The curves are made non-dimensional by
dividing the values of the different headings by the head sea
values. These non-dimensional curves show quite different
trends. Figure IV:3.7 shows the non-dimensional mean values and. following sea values as a function of ship length. There is a remarkably strong influence of ship length. As the
number of different ships is small, one has to be cautious about any conclusions, but the result is taken to indicate that the large tankers get comparatively smaller wave loads
in following seas than do the other ships.
The non-dimensional mean values are of particular in-terest because such values make it possible to estimate the wave loads in head seas only, and take the wave direction into account by applying this type of non-dimensional méan
factors are summarized in chapter V
111:4. Wave load as a function of loading condition.
Wave load as a function of loading condition is shown. in figures IV:4.1 and IV:4.2. On account of the fact that tìe loading conditions generally are concentrated around two distinct conditions, the second order polynomials which wore
originally chosen to describe the wave load as a function of loading condition turñed out to be unsatisfactory. The result
presented is therefore a linear regression analysis taking no other independent variables than loading condition into
account. In one case only (sh1pio. 1) the spread in loading
conditions motivated, a second order pöliomial. In this
particular case sea state and heading is included in the ana-lysis.
It is quite obvious (see figures IV:4.1 and 111:4.2) that
the wave loads increase with increasing values of the loading
condition variable D. The influence of D on the wave loads
is, however, rather small. The non-dimensional correction
factors (mean wave load divided by wave load of maximum
dead-weight) are found in chapter VI.
IV:5. Wave load as a function of ship speed.
It Was shown above (IV:1) that there is a close
relatioii-ship between wave load and relatioii-ship speed. These two variables
are, however, both dependent upon the independent variable
state of sea. It is therefore not possible to estimate the
influence, of the independent variable ship speed from this
CHAPTER V
SHIP SPE AS A PIJNCT
ION OP THE OTIR
IJARXABLES.V:1.
Suöcessive regression analysis.In analogy with chapter IV we have made a regression analysis concerning the influenc of stato of sêa B, wave directIon A. and loading condition D upon ship speed V. The following equation has been used.
V = f (B) +
f2 (A)
+f4 (D)
an f4 are the same functions as those denoted in the same way in chapter IV.
Tables IV:1.2.i IV:1.2.7 and V:1.1 show the expected result that V is strongly dependent of B
and
that thein-fluence of A and D is almost negligible as compared to that
of B. It is also documented that it is very unlikely that the obtained functions are caused by chance so the obtained
functions can 1n most cases be considered to describe TItet!
relätiönships.
V:2, Ship speed as a function of state of sea.
The following four equations are used to estimate the functional relationship between ship speed V and state of sea B.
V (B)
=f1
(B)v(B,A)
=f1(B)+f2(A)
V (B, D)
= f1 (B) +f4 (D)
V (B, A, D) =
f1 ()
+ f (A) f4 (D)For all ships except ship No. i there is a remarkably good fit between the regression equat-ions and the data (see figu±es V:2.1 - V:2.4). Thê unexpected result for ship No. 1 depends on the
unlucky
còincidence that, all 8 valuêsin. Beaufort 9-10 were taken in following seas. Consequently the mean speed in Beaufort 9-10 should be lower than is shown
in Íigure V:2.1.
No comparison between the different ships is made. This
is due to the fact that a ship slows down according to the judgement of the officers on the bridge so a comparison be-tween ships may be nothing but a comparisön bebe-tween officers.
Tb.is fact does not seriously affect the comparison between wave loads of different ships because ship speed has cempa.ra tively little influence on wave load. See /2/.
Ship speed as a function of wave direction.
The following equations have been used to estimate ship speed as a functiöi of wav direction.
V (A) =f2 (A)
V (A, B) = f2 (A) + f1 (B)
V (A, D) = f2 (A) + f4 (D)
V (A, B, D) = f2 (Aj) + f1 (B) + f4 (D)
The result is shown in figures V:3.1 and V:3.2. 1h this
particular case there is a strong reason not to rely on the
functions where B is included in the analysis. This is due
to the fact that i. certain increase of B which causes a speed
reduction n head. seas may cause a speed increase in following seas, so that the introduction of B in the analysis tends to
equalize the difference between head sea and following sea. This can be observed in figures V:3.1 and V:3.2.
The non-dimensional correction faàtor (mean speed divided
by head sea speed) is found in chapter VI.
Ship speed as a function of lÔading condition.
been used instead of the 2nd order polynomials originally ôhosen. For ship No. 1 only, the 2nd order polynomial is
retained. Figures V,:4.1 and V:4.2 show the expected result t1.at most ships get lower speed at full load than at ballast.
here is, höwever, one remarkable exception ship No. 2 -which shows the opposite trend. This rnay be explained by
the fact that a fast cargo liner of this type, operatng on the.North Atlantic, is seriously subjected to slaming,
especially in the ballast condition.
line of short dashes showing a4 inöreasing trend ship No. 7 refers to a result ïicluding 17
measure-zero speed during a machinery breakdown.
non-dimensional òorrectiön factor (mean speöd dided
at full load) is found in chapter VI. The
also for ments at
The by speed
CHAPTER VI
STJ1ARY 0F PART 1.
Data from a tota], number of 1577 measurements of midship
bending stresses in 7 ships of lengths about 100-240 meters,
have been used to estimate the average influence of the
in-dependent variables state of sea (Beaufort), wave direction
(heading angle) and loading condition, (deadweight) upon the
dependent variables longiudiz.a1 midship bending moment and
àhip speed. Functional relationships between the independent and dependent variables were obtained by means of regression
analysis.
Statistical tests show that the obtained relationships between the independent and the dependent variables probably
are not caused by òhan.ce but by "true" dependence.
When estimating wave load and ship speed by means of
rnodel experiments in waves or theoretical calculations it is
cOnvenient to use only head sea and full lOad and then obtain the average result in fl wave directions and loading
con-d.itions b applying correction factors. The correction
Part 2.
ESTIMATION OP LONG TERM DISTRIBUTIONS OP WAVE LOADS.
CHAPTER VII
NTRODtTCTION
The problem of estimating long term distributions of wave loads from long term distributions of the parameter r of the Rayleigh distribttion (defined below) has reòontly been handled by Bennet et al. /i/ and by Nordenström
/9,
10/.The principal idea of these references is that the life-time probability of exceeding a cèrtain wave load level can be obtained as a sum of probabilities from a great number of
short term distributions. Many full scale measurements have verified that each short term distribution of wave loads can be well characterized by the Rayleigh distribution which we
write
P [x
x1]
= i - eo
P is the probability that a variable x. (stress, möment etc.) takes on values smaller than or equal to a certain
value x1, and r is -the single parameter of the distribution (the notations r2 = R = Eis also found in the literature).
Introducing g [r] as the distribution function of r we get the total.pröbability Q [x > x1] that x éxceeds x1 during
the life-tithe of the ship by integrating (slimming) over all valués of r.
Q
[x
> X1] =
$(i
-
P [xx1])
g [r] dr (2)Prom (i) and (2) we get
(i)
In /1/ the long term distribution of r vlues was divided into groups defined by Beaufort numbers and the r values with-in each group were assumed to be normally distributed. Eq. (3)
then takes on the form
(r_F)2
+ 2
2s.
[x]=
(4)where subscript i denotes Beaufort group no.
i.
F and s denote mean and standard deviation of r.The total probability Q was obtained as a weighted sum
of the probabilities within groups.
Q
[X]
= E w. Q.i_i i
[x] (5)is the probability for the ship to meet the weather
defined by Beaufort group no. i.
In /9, 10/ it was shown that the method outlined above.
lads to
a doubtful estimate of the long terth wave löad,distribution because Q [x] is almost solely dependent on the.
weather group that containC the ost severe weather. This. group contains a number of measurements that is very small as, compared to the total number of measurements and therefore, the estimate will in reality be based on only a small part of. the ¿.vailable piece of information. It was also shown that. the accuracy of the estimate can be increased to a considerable.
extent by uniting all groups to one single long term distri-.
bution of r values. Therefore, the problem was to find a.
function that well describes the entire long term distributions of r values.
Jasper /7/ considered the long term distribution of r
values to be log-normal. In that case Q is obtained from
Q [x] = ____ j' .- exp
f.
(log r/)2
1211y
O LQ [x] =
/ 2ii
?
(X\2
'r'
(log (rc)/))2
2 2log is the mean of log (r +
o).
y is the standarddeviation of log (r +
e).
(7)
is the mean and is the standard deviation of log r.
In
/9/
the log-normal distribution was fitted to a totalnumber of 1577 r values of 7 ships. The fit was in all casés
unatisfactòry. It was also shown that the resulting long term distribution obtained from eq. (6) leads to wave loads that are 2-3 times larger than the result of eq. (4) and (5).
Therefore it cannot be em.hasized too stron:l that the
lo-normal distribution (which again is mentioned in a report to
ISSC 1964 /14/) must not be used in this connection.
In order to find another function fitting the data,
normal distributions were fitted to the entire distributions
of r values /io/. The resulting long term distribution of
r values is then obtained from eq. (4) applied to the single
long term distribution of r values. It was shown that the normal distribution was not satisfactory ano. that the
differ-ence. between the enpirical and theoretical distributions leads to an underestimation of the wave loads. The resulting long term distributions of wave loads were, however, rather similar to the results of the method using Beaufort groups.
For the above mentioned reasons it was concluded that the long term distribution of r values should be described by a function in between the normal and the log-normal ones. Such a function is easily obtained by adding a constant o to
the r values and fit the log-normal distribution to the
variable r + e. For e = O we get the log-normal distribution
and for o we approach the normal distribution. Q was obtained /9/ from the following equation.
In /10/ "log (r + e) - normal" functions were fitted to the r values of the 7 ships entioned above. For c = 2 k/
the fit was excellent. The integral (7) was evaluated numerically in an electronic computer and the result Was
presented /9/ in dimensionless graphs. By mans of these
graphs, eq. (7) was applied to the measurements of the 7 ships. The resulting long terfl distribution of wave loads gave i-n all cases roughly 10 % larger wave loads than did the
normal distribution of r values mentioned above. The log (r + c) - distribution was considered to give the best reult so fàr but in order to get a still better estimate in a
CHAPTER VIII.
THE LONG TER1E DISTRIBUTION OF. r I
ASSUE
PbtÏÒW TIIWEIBULL, DISTRIBUTION.,
VIII:1. Pitting of Weibull distributions to the estimated
r values.
The log (r + o) - normal distribution described above has two qualit±es that can be considered as a drawback: the
distribution is not valid for small aluos of r and there is.
no method to estimate c which is both simple and completely.
satisfactory. These two problems are in fact of secondary importance because the lower r values are negligible
and
anyc value above 2 kg/mm2 gives practically the same result as c = 2. Ìt is anyhow felt -that there is need for a better
way to describe the entire distribution Of r values.. In
order -to find such a siple function the Weibull distribution
has been fitted to our data. This was originally suggested
by Bennet.
We Write the Weibuil distribution in the following way:
.P [r r1] = 1 - e
P is the probability that r takes on values smaller than
or equal to r1.
m
and
a are parameters of the distribution. From (i) we get(- in (i - P)) = m (in r -in a) (2)
Where. 'in stands for the natural. logarithm. Introducing the reduced variable
y
= ln.(-
in (i P))(3)
we get
y = m (lñ r in a)
Consequently the Weibuil distribution is represented by
a straight line in the diagram shown in figures VIII:1.1 -VIII:1.7. This type of probability paper was originally proposed by Gumbel.. The diagram can be used to test whether
the Weibuil.distributjon fits the data or not and also to estimate the parameters. The parameter m is determined fom
the slope of the line and the parameter a corresponds to y = O. See eq. (4).
As plotting position we have used the conventional method of grouping according to classes of r values and we have estimated the cumulative frequency H of the boundaries
of the class intervals from
H0
[rd] =
(s)
where n. is the number of r values equal to or smaller than
class limit r and n is the total number oÎ r values. Gumbel
/4/ has analysed several plötting positions and hàs proposed a better one but we use (5) because it is more easy to
com-bine with the. weighting procedure described below.
The weighting procedure is introduced in order to correct
for unexpected weather conditions during the period of measure-ment and make the results of the.different ships comparable. The weighting is made by dividing the r values of each ship into groups defined by the Beaufort numbers O-12 (mthcimum
i groups). The r values within each group is then divided into r value classes. Let i denote the Beaufort groups and j denote the classes of r values. We tien get the following numbers
ni. of r values.
Beaufort
V. n
w.
i
Efl..
w1 is a weighting factor.
w = 1
gives H0 in eq. (5).y1
is the probability for the ship to meet Beaufort i.Est±ates of V1 are obtained from long term weather statis-tics. Prom ref. /1/ e et the following values of y used
in this report. Beaufort y Beaufort y o
11.2
75.30
110.3
82.65
214.0
91.05
31.5
100.36
416.0
11O.i
513.5
120.03
.69.0
Please note that these values are obtained by visual
estimation. The. a'e not öomparable to the forces in /11/.
.
n
e
S.S..,...
flj,The weighted curnu:Lativo f.requcncy H,, [rJ corresponding the class limit r. is estimated as
nu w1
n= E
E n1. i'i J(6)
f111 S .These values öòrrespond. to a normal distribution öf B (Beaufor±) with mean mB 3.4 and standard deviation s = 2.37.
Figures VIII:1.1 - 7 give the'unweighted and weighted
distributions of r values from ships no. 1-7! It is evident from the figures that the Weibull destribution is very well. suited. to describe the long term distribution, of r value. The plotted oiygons scatter at random around the straight
lines and no systematic deviation can be observed.
Por refined methods to fit Weibuil distributions to
observed data see references /4., 5, 12, 15, 16/.
VIII:2. Transformation of a long term distribtion of r
values, to a long term distribution of' wave loads.
We start with eq.VII;(3)
Q [x]
o
and insert
g [r] = _
rm1
e(r)m
according to the Weibull distribution. We then get
Q [x] e e - a dr (i)
In
order to get a result of widér applicability we ana-lyse the functionQ [x] = e g [r] dr VII: (3)
(X\l
'r" (r\m 'a' dr (2)i.e. a Weibu.11 distribution with parameters (r, 1) and the
parameter r distributed according to a Weibull distribution
with parameters (a, m).
By substituting 1 m V (r)m we get Q [A, B] = - [y + A v_B) dv
Prom figures 11111:1.1 - VIII:1.7 we lmow that m usually
lies iii the interval 1.2 < m< 2.1. Por i 2 we get
0.9 <B < 1.7. Therefore the oase *hnB = 1 which can 'bé handled anaiytiöaily is of special interest. B i gives
Q [A,
fe -
[y + A/y] dv (5)o
comparison with the Laplace transform
h
Ii)
= e
shows that s = i and h2/4 = A gives
Q [A, 1] = 2
IA
K1 (211)
(6)
[s t +
dt
(3)
This result was also obtained by Sjöström /13/ for the
special case i = m = 2.
The ±'unction K1 is the modified Bessel function of the
second kind and of order one (K is a second solution of the.
differential equation
¿d2
rdx dx
The function K1 is tabulated. See e.g. /17/. The föllowing asymptotic sèries
can
be used for, arge values ofz.
i
TPf-\
/ lT \Zj =
+ n2) y 0) 1.3 + I'TI.
123.5
!(8 z)2 12.32..57 - .1.) 3!(8 z)3Fröm (6)
and
(7) we thus obtainQ [A, 1'] e
2/
(i
+ 3/16I - ...) (8)and
for large values of Aìn Q in +
mA -
21Ä (9)which shows
that
in Q as a function of JA approaches astraight line for large values of A.
Por B i the. evaluation
of
(4) has been carried throughnumerically in an electronió computer. We.
have
öhosen B = 0.5 (0.1) 2.0 and increasing values of A = (2 n)2,(n=' , 1, 2, ,...) for Q The result is presented
table VIII:2.1. We have also traced JA as a function of
10log Q for constant values of B (figures VIII:2.1 - 2) and
/A
as a function of B for constant values of Q (f±gureVIII:2.3).
iX
is chosen as ordinate because i 2 (Rayleigh)then directly gives x/a
= ¡L
Equation (8) has beenused.
asan extra check of the calculations.
B = O gives
Q [A] = $
[y
A]o
which simply corresponds to one single value of r. The func-tion (io) is included in figure VIII:2.l.
The appearè.nòe of the curves in figures VIII:2.i and
VÏII:2.2 gives reason to expect that it is possible to
describe also the long term distribution of x by the Weibull dis-tributiön. Similar results were öbtained in /1,
9,
10/and
the results presented by Johnson /8/and many
others support this statement. Pigure VIII:2.4 shows lnIA
as a function of y (see. eq. VIII:i (4)) for constant values of B. The special case B = 0 gives a straight line corresponding to a Weibull distribution, but increasing values of B giveincreasing deviations from the Weibull distribution. However, for the case B = 2, causing larger deviation than we have
to expect in practice, the deviation from the Weibull distri-bution is for practióal purposes very small. See figure
VIII:2.5 showing the difference between the obtained
distri-bution
and
a Weibull distribution, It is thus possible toobtain a useful approximative relation between the parameters of a iesulting Weibull distribution
and
the initial Weibulldistributions. e (A)k
0f
e -[y +
A v_B] dvThe parameters b and k have been determined in the following way from the result of the numerical integration.
The parameters b and k are: both functions of B. As the obtained distribution of A is not an exact Weibull
estimate these parameters. We have used the twó pi'obability
levels Q8 = 1O8 and Q4 = 1O4 because the region in between
is of greatest interest to naval architects. We have used the following formulas which are easily obtained from the
Weibull distribution function.
in Q4 A -ln .tt4 .Lt4 j_n Q 8 0.693146 A8 A8
1-
1 C- in Q8)k 2079442kThe values A8 and A4, corresponding to Q8 and Q, were
obtained by linear interpolation in in A and in (- in Q) from
table VIII:2.1. The result is given in table VIII:2.2 and
figure VIII:2.6. For increasing values of B, the parameters
b and k approach zero.
When integrating eq. (4) we utilize certain ranges of
the initiai distributions of x and r. The reliabie range of the resulting long term distribution of x is dependent on the
reliable ranges and the parameters of the initial
distribu-tions.. n other words we have to extrapolaté the initial distributiOns in order to reach small probabiliiies of
ex-ceedance in the resulting distribution. The extrapolation of the initial distributions must not be driven too far and therefore we want to ]mow the "upper reliable limit" of the resulting distribution as function of the corresponding limits
of the initial distributions. In order to estimate this
fund-tion we start with eq. (4) and determine the "most important
value" of y as a function of A and B. y "most importait value" of y e mean the value of y giving the largest
con-tribution to the probability Q. This value of y corresponds io th,e maximum of the function
f = exp (-i.
df/dv = O gives
where
1
* P []
is the return period of î and From (3) we get
(r)mlflN
(r)m
P[]=1e
aProm (13) and (14) we get
1
in (A B)B+i
or
-
(In N)1)
The correspondthg return period
- i
- P[J
of the short term distribution is obtained from (Weibufl distr. )
giving -, A = (A B_B) or (16) (17)
lnN
1 +BJ
From (15) and (17) we get
1nN
1nN
(18)The functions (15) and (17) are traced in figares
VIII:2.1 - 2. These functions show how much it is advisable to extrapolate the resulting long term distribution os
func-tion of the "upper reliable limits' of the initial
distri-butions. Inversely the functions show what we demand from the initial distributions in order to make a prediction
regarding a certain number of cycles of the resulting
distri-bution. If for exarnple B = 1.65 and we rely upon the initial
distributions up to lO3 and = it is possible to
predict the largest value out of çycles of the resulting
di stribu-tion.
VIII:3. A practical application.
The result obtained from the assumption that r is distri-buted. according to the Weibull distribution will now be corn-pared to the results of previous assumptions /10/. We have
made a prediòtion regarding the long term distribution of
longitudinal midship stre,sse.s for the seven ships in question.
ob-tamed by visuafly fitting straight lines to the distributionC
plotted in figures VIII:1.1 - VIII:1.7. Figure VIII:3 shows
the new result traced in figure 2 /iO/. The. compared methods
are:
distribution of r
method i normai within Beaufort groups
method 2 normal
method 3 log (r + e) normal
new method Weibull,
It is evident from figure VIII:3 that the new method gives a result that is very similar to methOd 3. this is to
be expected since the Weibull distribution and th log (r + e)
normal distributiöii both fit very well to the measured data. However, for -the reasons mentioiied above (VII and VIII:l) the
new method is ±o be preferred and t1e Weibtill distibution is recommended for the description of the long term distribution
CHAPTER I!
SITh/UVIARY OP PART 2.
A short review is given of some recently proposed methods to predict the long term statistical distributions of wave
induced bending rnoments on ships.
The Weibull distribution has been fitted to long term distributions of the parameter r (= /) of the short term
Rayleigh distribution of midship bending moments. Pull scalo
data from 7 ships ranging from loo to 240 meters in length were analysed and, the Weibuîi distribution was found to fit the data well.
The resulting long term distribution of a variable having a Weibuil short term distribution, and the characteristic
parameter of this short term distribution distributed accord-ing to another Weibull distribution, has been obtained by
numerical integration. The result is presented in tables and graphs.
The resulting distribution was also fòund to be very similar to a Weibull distributiòn. A siple but rather accu-rate approximative, transformation between the initial Weibull
distributions and the resulting Weibull distribution is given.
Long term distributions predicted by the new method, are
shown, to be similar to long term distributions obtained by some other methods. The new method is shown to have advan-tages over those other methods. The new method is therefore recommended.
References
Berinet, R., Iarsson, A. and Nordenstöm, N.:
"Re'suits from Full Scale Measurements and Predictions
of Wave Bending Moments Acting on Ships", Swedish Shipbuilding Research Foundation, Report
32, 1962.
Bledsoe, M.D., Bussemaker', O. and Cummins, W.E.: "Seakeeping Trials ön Three Dutch Destroyers", S.N.A.M.E. Vol. 68, 1060.
Darbyshire, J.: "A Further Investigation of Wind
Generated Waves", Deutsche Hydrographisehe Zeitschrift
12 (1959) 1,
p. 1-15.
Gumbel, E.J.: "Probability Tables for the Analysis of Extreme-Valúe Data", Applied Mathematical Series,
Nat. Bureau Standards, Vol. 22, July 6, 1953.
Gumbel, E.J.: "Parameters in the Distribution of
Fatigue Life", Journal of the Engineering Mechanics Division, ASCE, Vol. 89, No. 1M5, Proc. Paper 3667,
October,
1963, pp. 45-63.
6 Hald, A.: StatisticaiTheory with Engineering
Applications, Johan Wiley and Sons, Inc., New York, 1960.
Birmingham, J.T., Brooks, R.L. and Jasper, N.H.:
"Statistical Presentation of Motions and Hull Bonding
Moments of Destroyers", T1V Report 1198,. September
1960.
Johnson9 A.G. and Larkin, E.: "Stresses in Ships in Service", Tran.s.R.I.N.A.,
1963.
Nordenström, N.: "On Estimation of Long Term Distributions of Wtve Induced Midship Bending Momênts in Ships", Report from the Division of
Ship Design, Chalmers University of Technology,
Gothenburg 1963.
1O,a
Nördénström, N.:
"Statistics and Wave Loads",
Re.ort from the Division, of Shi. Desi:
,Chalmers
University of, Teólmology, Gothenburg 1963.
Roll, H.Uf:
"Height Length and Steepness of Sea
Waves in the NOrth Atlantic", S.N.A.M.E. Technical
and Research Bulletin No. 1-19, 1958.
Sahran, A.E
,and Greenberg, B G.;
Contributions
to ord.e.r statistics, John Wiley and Sons,
nc.,
New York, 1962, pp 406
431.
Sjöström, S.:
"On Random Load Analysis", Trans.
Qf the Ro al Institute of Technolo
,No. 181,
Stockholm 1961.
Wärnsinck, W.H., et.al.:
Report of Counnittee 1,
"Bivironmental Conditionst1, to ISSC, Delft, 1964.
Weibull, W.:
Fatigue Testing and Anal.rsis of
Results, Pergamon Press, New York, 1961.
"The Weibull Distribution Punction for Fatigue
Lif e", Material Research and Standards, Vol. 2,
No. 5, May 1962.
17.' Mathem. Tables. Vol. 10:
Bessel Functions.
Part 2:
Punötions of Positive Integer Order.
Appendix I
ERROR, OP VISUAL ESTIMATES OP WAVE DIRECTION.
Then. estimating the wave direction it is common tö look
in the direction from where the waves seems to come straight
down to the observer. The error introduced by this method
is handled below.
We i-xtroduce the followinìg notations:
V =ship speed
W = wave speed
a = tne wave direction
apparent wave directiOn. f (a) = istr1butioi function of a g () = distribution function of .
1>
0H-
Vi-ì4EVrkca*
A 'VVi-Simple trigonometrical calculatiOns give
sin a
tg=
(i)COS a
=
-
!
sin eQsW S1flOE
The distribution function g () of is obtained from
i
.. dof
/
(uniform, O ir) we get
g ()
= u (i - co
(3)
-.
'rom equations (i) and (3) we get the distiibutïòn
func-tion g () shown in figure A 1.1. The following elation
holds for all values of if
1.
g ()
+ g (U 1)
For some special values of we get
g (0) = g (ir) = u O gives
g () =
= V i gives V + w 1-rf ()
i) (2)g ()
For fg () =
(o <Calculated statical
Ship data
Ship No. 1 2 3 4 5 6 7
Type of Dry
Dy
Ore Ore Tanker Tanker Tan4tervessel cargo cargo Carrier Carrier
¡Tanker /Tanker Deadweight tons 4600 8600
14100
21700
34200
48500 68500 Length bet-ween pp. m 97.8 138.1140.2
170.7 198.1 214.9 238.4 Breadth moulded rn 14.5 19.2 19.5 22.7 26.8 31.1 35.4 Depth moul-d.ed m 9.22 11.7 12.9 13.5 14.3 15.2 17.1 Draught Ïn69
7.8 8.9 9.4 10.7 11.5 13.1 Machinery 3000 9000 6200 8300 2x9150 16500 22000 BHP II IHP II SI SHP Speed lmots 14 17 15 14.5 16.8 16.8 16 Biockoeff. 0.68 0.66077
0.79 0.77 0.80 0.80 Wateline area coeff. 0.81. 0.79 0.82 0.84 0.86 0.88 0.87 Prism. cOeff. 0.69 0.68 0.78 0.80 0.81 0.81 0.81 Section mo-dulus Meas. point Calcu-lated n326Ó
5.67 6.32 11.6 18.7 25.0 43.9 stress vari-ations in sine L/20-wave kg/mm2 Hogging4.1
6.3 7.0 8.6 9.719.7
9.2 Sagging 4.9 8.4 8.6 10.5 11.5 12.7 11.0CRITERION' LAID DOWN BY THE WORLD IVJETEOROLOGICAL ORGANIZATION.
PÓR' WIIW
SPEED KT.
7 - lo
SEA CRITERION
O < i Sea like a mirror.
i i - 3 Ripples with the appearance of scales
are formed, but without foam crests.
2 4 - 6 Small wavelets, still short but more
pronounced - crests have glassy appearance
and do not break.
Large wavelets. Crests begin to break. Poam of glassy appearance. Perhaps scattered white horses.
Small waves9 becoming longer; fairly
frequent white horses.
Moderate waves, taking a more pronounced
long form; many white horses are formed. (Chance of some spray).
Large waves begin to form; the white foam
crests are more extensive everywhere. (Pröbably some spray).
Sea heaps up and. white foam from breaking
waves begins to be blown in streaks along
the direction of the wind.
8 34 - 40 Moderately high waves of greater length;
edges of crests begin to break into the
spindrift. The foam is blown in well-marked streaks along the direction of
the wind.
9 41 - 47 High waves. Dense streaks of foam along
the direction of the wind. Crests of
waves begin to topple, tumble and roll over. Spray may affect visibility1
10 48 - 55 Very high waves with long overhanging
crests. The resulting foam, in great atches, is blown in dense white streaks along the direction of the wind. On the
whole, the surface of the sea takes a
white appearance. The tumbling of the
sea becomes heavy and shocklike.
Visibility affected.
4 11 - 16
5 17 - 21
6 22 - 27
rD'
SPEED KN. SEA CRITERION
11 56 - 63 Exceptionally high waves. (Small and
ediumsized ships might be for a time lost to view behind the waves). The sea
is completely covered with long white atdhes of foam lying along the direction
Of the wind.
Eve
here the edges of thewave crests are blown into froth. Visibility affected.
12 64 - 71 The air is filled w±th foam and spray.
Sea completely white wit1. driving spray; visibility very serlously affected,
The Beau:fort Scale actually extends to Force 17 (up to 118
kt.) but Force 12 is the highest which can be identifêd
y
= variancef = degrees of freedom
P=v2/v1
significant at the 0.1 % level
* *
f, It lÌ 1 * II It II 5 % t, P99 strongly significant Fvalue
1 - P pröbabiiity to reject the hypothesis if it i traeShip Independent Residual Between
no. variable vi f. V2 f2
99.9
B0.034 167 1.5903
46.6
5.7
>99.9
* * *
1 A0.056 168 0.505 2
9.02
7.2
>99.9
* * *
V 0.049 168 1.070 221.9
7.2
>99.9
DÖ.059
1680.230 2
3.90
7.2
>97.5
* B0.058 189 3.670 3
63.3
5.7
>99.9
* * *
2 A0.113 190 0.165 2
1.46
7,2
<90.0
0.079 190 3.435 2
43.5
7.2
>99.9
** *
D0.108 190 0.675 2
6.25
7.2
>99.5 **
B0.104 316 8.290 3
79.8
5.6
>99.9
* * .*
3 A0.177 317
1705 2
3.99
7.1
>97.5
* V0.093 317 14.11 2
1527.1
>99.9
* * *
D0.177 317 0.680 2
3.85
7.1
>97.5
* B0.080 424 20.34 3
254
5.5
>999
A0.21.2 425 2.345 2
11.1
7.0
>99.9
V0.108 425 24.37 2
2257.0
>999
* 4E *0.209 425 2,925 2
14.0
7.0
>99.9
B0.147
77 3.250 3
2.1
6.Ó
>99.9
A0.241
78 1.150 2
4.77
7.5
>97.5
* V0.152
78 4.610 2
30.4
7.5
>99.9
D0.241
78 1.145 2
4.75
7.5
>97.5
*Ship
Independent
Residual Between
no.
variable
y1 y2 2 B0.257 117 3.140 3
12.2
5.8
>99.9 * * *
A0.306 1.18 1.695 2
5.54
7.4
>99.0 **
6 V0.141 118 11.43 2
81.1
7.4
>99..9
* .*, *
D0.32.4 118 0.595 2
1.84
7.4
<90.0
B0.13.3 .259 1.090 3
8.19
5.6
>99.9
* * *
7 A0.131 260 1.740 .2
13.3
7.1
>99.,9 * * V0.132 260 1.685 2
12.9
7.1
>9909* * *
D0.143 260 0.160 2
1.12
7.1
<70.0
Ship no. i
SSD = sum of squares o.f residual deviations about the empirical regression equation.
f = number of degrees of freedom.
s = residual staidard deviation, 2 = SSD/f. i) second order polynomial.
Depend Iidependent SSD s no.
10.44
170.?478
B5.68
167.1845
1 7 A9e43
168.2370
3 13 V8.30
168.2222
2 129.98
168.2437
4 15BA
5.49
165
.1824
2 6 B V5.62
165.1845
3 8 B D4.97
165.1736
1 3AV
6.60
166.1994
4 10 A D.9.32
166.2370
6 14 V8.19
16
.2222
5 11 .BA V
5.19
163.1785
3 5BA
D4.92
163.1737
2 4 BVD
4.90
163.1734
1 2AVD
-6.51
164
.1992
4 9BAVD
4.72
161.1711
1365
1701.47
B254
1671.23
Bi)
268
1681.26
A 333 1681.41
3 7310
1681.36
2 6 .B A198
1651.10
1 2 B 235 1651.19
2 A. D 284 166 1.31 3 .5BA
D 178 163 1.05 . .1SSD = sum ol' squares of residual deviations about the empirical regression equation.
f = numberof degrees of freedom,
s = residual standard deviation. 2 = SSD/f.
second order polynomial Ship no. 2
Depend Independent SSD f s no.
21.87 192 .3375 B 10.88 189 .2400 1 8 A 21.54 190 .3367 4 15 V 15.0b 190 .2810 2 12 D 20.52 190 .3286 3 1 B A 10.24 187 .2340 3 7 B V 9.49 187 .2252 1 4 B D 10.04 187 .2317 2 6
AV
14.77 188 .2803 5 11 A D 19.68 188.325
6 13 V D 11.60 188 .2484 4 10 BA V
9.17 185 .2227 2 3 B A D 9.57 185 .2275 3 5 B V D 8.77 185 .2177 1 2 A V D 11.37 186 .2473 4 9 B A V D 8.48 183 .2153 1 V 2052 192 3.27 1471 189 2.79 1 4 A 1934 190 3.19 2 6 1996 190 3.24 3 7 B A 1312 187 2.65 2 3 B D 1093 187 2.42 1 2. A D 1913 188 3.1 3 5 B A D 1040 185 2.37 1 B 1489 190 2.80SUCCESSIVE REGRSSI0NMTALYSIS,
Ship no. 3
SSD = sum of squares of empirical regressi f = number of degrees s = residual standardsecond. order polynomIal
residual deviations about the on equation.
of freedom.
deviation. 2 SSD/f.
Depend Independent SSD f s no.
r
57.57
319
.4248
B32.73
316
.3218
2 9 A56.16
317
.4209
3 14 V29.36
317.3043
i 856.21
317
.4211
4 15B A
32.57
314
.3221
5 12 B V24.11
314
.2771
2 6 B D32.55
314
.3220
4 10A V
24.10
315
.2766
1 5 A D55.08
315.4182
6 i 3V D
26.01
315
.2874
3 7 B A V21.73
312
.2639
1 2 A D2.36
312
.3220
4 11 B V D22.78
312..2702
3 4 AV D
22.51
313.2682
2 3 B AV D
20.99
310.2602
i V2627
3192.87
B 1 294316
2.02
i 4 A 2241 3172.66
2 52588
3172.86
3 7BA
992314
1.78
i 2 B D 1185314
1.94
2 3 A D2229
3152.66
3 6BA
D949
312
1.74
i B 1359317
2.07
second order polynomial
SSD= sum of squares of residual deviations abouut the
empirical regression equatioi.
= nti.mbe' Of degrees oí freedom.
residual standard deviation. . 2
= ssD/f. Ship no. 4
Depend Independent SSD f s no..
94.73. 427 .4710 B
