Proceedings of the Eleventh (2001) InternationalOffshore and Polar Engineering Conference
Stavanger, Norway, June 17-22, 2001
Copyright © 2001 by The InternationalSociety of Offshore and Polar Engineers
ISBN 1-880653-51-6(Set); ISBN 1-880653-54-O(VoLI11); ISSN 1098-6189(SeO
Joint Distribution for Wind and Waves in the Northern North Sea
Kenneth Johannessen, Trond Stokka Meling and Sverre Hayer
Statoil
Stavanger, Norway
ABSTRACT
For design purposes, it has been common to estimate the
100-year response by exposing the structure to the simultaneous
action of 100-year wind, 100-year wave and 10-year current.
Present design codes, see e.g. NORSOK N-003, recommend a
less conservative approach by stating that the aimed response
extremes can be predicted accounting for the actual correlation
between the environmental processes. This requires a joint
probabilistic model for the weather parameters of interest for the
problem under consideration. In this paper a joint probabilistic
model of mean wind speed, significant wave height and spectral
peak period will be presented. Such a model will be needed if a
long-term response prediction of motions and anchor lines for a
floater is carried out.
PREDICTION OF LONG TERM EXTREMES
For design purposes, the 100-year response has often been
estimated by combining 100-year wind, 100-year wave and
10-year current. A more consistent approach is to estimate the
long-term responses by a full long term analysis, i.e.:
F x ( x ) = f IIFxlwH°.or, (x[w,h,t)-fWH.oV~ (w,h,t).dhdwdt (1)
Fxrvm,,or° (x [ w, h, t)
is the distribution function of the 3-hour
extreme value given the weather parameters and
fwH,,0r, (w,h,t)
is the joint probability density function of the
weather characteristics of interest for the problem under
consideration. The choice of a 3-hour duration of the short-term
condition is of course somewhat arbitrary. A consistent estimate
for the 100-year extreme value, Xjoo, is now obtained by solving:
Simultaneous wind and wave measurements covering the years
1973 - 1999 from the Northern North Sea are used as a
database. The wind speed is chosen as the primary parameter
since the wind is assumed to have the strongest influence on the
loads on the mooring lines of a semi-submersible structure. The
significant wave height is assumed to have second most
influence and the spectral peak period is assumed to have least
influence on the loads.
1
Fx(Xloo) = 1 - - N10o
(2)
where N~oois the number of 3-hour conditions in 100 years.
The challenge related to Eq. (1) will often be to establish the
short-term distribution of X in the case of complicated response
problems, e.g. the horizontal motions of a cantenery moored
semi submersible. A possible approach is to assume that the
conditional distribution of X can be modelled by a Gumbel
distribution, i.e.:
The joint model is used to establish a contour surface, giving
combinations of the weather parameters for which the
exceedance corresponds to a return period of 100 years. The
paper is closed by briefly indicating the application of the joint
model to the mooring line loads.
KEY W O R D S
(3)
Waves, wind, contour plots, line tension
19
parameters, e.g. H,,,o and Tp, a fractile in the order of 85-90% is
recommended when the 100-year value is to be estimated in
Haver et al. (1998). If the problems involve 3 slowly varying
weather characteristics, a fractile in the order of 65% is
recommended as 100-year values are to be predicted (Meling et
al., 2000). In the end of the paper this will be illustrated.
For a given combination of weather characteristics, wi, hi, tk, the
distribution parameters can be estimated by fitting the model to
a simulated sample of 3-hour maxima. The simulated response
is obtained by exposing the floater to the simultaneous action of
a wind spectrum characterised by wj and a wave spectrum
characterised by hj and tk. A 3-hour time history of the response
quantity under consideration is simulated and a realisation of the
3-hour maximum is identified. For a case where the Gumbel
model is expected to be the "correct" asymptotic model, a
reasonable number of 3-hour simulations could be 20-30.
Provided that point estimates for the distribution parameters for
a sufficient number of combinations o f weather characteristics
are carried out, response surfaces can be fitted to the point
estimates, i.e. #(w, h, t) and [3(w, h, t) are estimated by
continuous functions and Eq. (1) is convenient solved
numerically.
It is seen from this brief discussion that in order to predict
consistent estimates for response extremes corresponding to a
prescribed return period, one will in principle need a joint
probabilistic model o f the slowly varying environmental
parameters of importance for the problem under consideration.
Characteristic parameters
The weather is described by the following 3 parameters:
Although one mainly need accurate distribution parameters in a
rather limited subspace o f the total weather space, it is quite
obvious that a rather large number of 3-hour simulations will be
necessary. For a complicated response problem, a 3-hour
simulation may well take 3 hour or more in computational time.
For such a problem a full long term analysis will be impractical
and it would be more convenient to estimate proper long-term
extremes by exposing the structure to a short-term sea state.
This can be done utilising the principle o f contour surfaces, see
e.g. Meling et al. (2000). Based on the joint model,
Significant wave height, Hmo
•
Spectral peak period, Tp
L',,'~,,,~'i:,,(w, 17: t) = j;~.(w) •f,.,-,,,o:,;,(hi w)..l:,:..~.; ,,,:.,(t ih, w) (4)
years. In practise this is done by transforming fwn,,oL (w, h, t)
into a non-physical space consisting o f independent standard
Gaussian variables, Uj (reflecting the marginal variability in W),
U2 (reflecting the conditional variability of H~,o given W) and Us
(reflecting the conditional variability of Tp given W and Hmo),
utilising e.g. a Rosenblatt transformation scheme, Madsen et al
(1986). In this space we know that all 100-year combinations
(u~, u2, us) will be located on a sphere with a radius, r, given by
1
~b(r) = 1
NI0o . With Nloo = 292000, this yields r = 4.5.
Transforming this sphere back to the physical parameter space
yields a "sphere" consisting of "100-year combinations" o f W,
-
1-hour mean wind, W
•
Simultaneous description of wind and waves
We seek a joint density distribution of the characteristic
parameters, W, H,,o and Tp. In this analysis the response will
probably be dominated by the variability of the wind and
therefore, W is chosen as the primary parameter. Based on this
the following joint density function seems reasonable:
fWH,,oT°(w,h,t) , one can estimate "sphere" where each point
at the surface corresponds to an exceedance probability of 100
-
•
-
H,o, and T~
If the short term distribution could have been modelled by a
delta function, i.e. no inherent variability in the 3-hour extreme
value, an accurate estimate for the 100-year response could have
been found as the most probable 3-hour maximum value in the
most unfavourable combination on the "sphere" surface. This
would be valid for all response problems, but of course the
unfavourable combination (i.e. the point on the surface) would
in principle be different from problem to problem. In practise
one can not neglect the inherent variability of the 3-hour
maximum. This can be accounted for approximately by selecting
a somewhat higher fractile of the 3-hour extreme value
distribution as the characteristic short-term value. For a
response problem characterised by two slowly varying
20
Marginal distribution for the wind, W
We will assume that the marginal distribution of the l-hour
mean wind speed at 10 m can be described by the 2-parameter
Weibull distribution:
r
~,[-
,.t
For each wind class, the Weibutl scale and shape parameters a
and 13 were estimated from regression analysis (the least squares
method), including all the wave classes where measurements
were available. To evaluate the goodness of the estimated
Weibull parameters, the corresponding cumulative distribution
was also plotted in Fig. 2. The corresponding density function is
plotted in Fig. 3.
(5)
where a and 13 is the shape and scale parameters, respectively.
3.0<=Wind<4.5 (m/s)
5
4
3
Based on measurements from the Northern North Sea in the
period 1973-99, and the method of moments, the values a =
1.708 and fl = 8.426 were determined for these parameters.
These parameters seem to give a good description of the wind
speed distribution and correspond to a 100 years extreme wind
of 39.0 m/s. The distribution based on the measurements is
plotted together with the fitted Weibull distribution in Fig. 1.
In(
/f
2
in( 0
1- -1
~ -2
-3
. '''/
-S
-1
I
I
1
2
4.5<=wind<6.0 (m/s)
4
3
In(
1
8:,-
in( - 1
0.999999
0.99999
0.9999
WeibuJl Model 1
0
In(Sign. wave height)
The data basis from the Northern North Sea consists of
composite measurements from the fields Brent, Troll, Statfjord,
Gullfaks and the weather ship Stevenson. For periods where
measured data are missing, model data from the Norwegian
hindcast archive (WINCH, gridpoint 1415) have been filled in,
thus a 20 year long continuous time-series has been used.
~ ] r ~
0 Measurements
- - Weibull
¢ A/?
f
..-
-5 --,¢.
-6
-1
0.99
- - Weibull
f
-3
-4
0.999
<> M e a s u r e m e n t s
<~ Y~"
1- -2
0
8
1
2
In(Sign. wave height)
"5
>,
0.9
.Q
0.8
0.7
f]e
0.6
16.5<=W<18.0 (m/s)
0.5
4
0.4
J
/
/,,-
./
2
0.3
A
0.2
,~-.2
c
0
z
3
4
s
!1!
<> Measurements
-- Weibull
..t
w
-¢ 4
/'0
0.1
-8
0
0.5
1
1.5
2
In(Sign. wave height)
0,05
2.S
W i n d s p e e d (m/s)
24.0<=Wind<25.5
Fig. 1. Cumulative frequency distribution of 1-hour mean
wind speedfor the Northern North Sea.
(m/s)
2
1
O
Conditional distribution of H,,o for given W
The 2-parameter Weibull distribution was suggested as the
conditional distribution of significant wave height for given
wind speed. Based on the measurements from the Northern
North Sea, the distribution of wave heights within wind groups
with a bin size of 1.5 m/s was found. Next, the adequacy of the
conditional 2-parameter Weibull distribution was considered by
plotting the cumulative distributions on Weibull scale. The
2-parameter Weibull distribution would be considered suitable
if these curves were approximately linear.
Measurements
Weibull
-3
1111
t f I I
~ I I I III
I lil
t II
1.2
1.4
1.6
1.8
2
I I I I
I I I I
I I I l l
2,2
2.4
I
I
I
(
I
2.6
In(Sign, wave height)
Fig. 2.
21
Cumulative distributions for significant wave height
within different classes of wind speed Based on
measurements and a fitted 2-parameter Weibull
distribution.
The graphs in Figs. 2 and 3 indicate that the adapted probability
distribution could be improved for the lower classes, of wind
speed. For higher classes, the 2-parameter Weibull distribution
seems to follow the measurements better.
3.0<=W<4.5
,~
o.,
•
, , , .,.,. .,., , ' '
'I',I
o.g
0
1
2
3
5
7
0
II/I
o
Fig. 3.
2
4
Wave helghl
3
i
~
2
/
3
4
s
Wave height
0
10
:;:,
Fig. 5.
3O
50
4O
The Weibull shape parameter, a, estimated from the
measurements versus a fitted estimate.
//
lO
2O
W,~ weed (rrVs)
F
g
Fit
2
o
i~
~
'0. M e a s u R r n e n ~
i\
'
0'15
..........
24,0<=W<25.5
o.,I J I ! !.U,[ [ t [
~,0< ~
--
X
Wave height
o.3~
o~
4
0.1
~
/.
5
,.,
16.S<=W<tS.0
.... III
sp
e
,i,u
~ 0.3
II
4
/<
6
/v
i J-~. k l
i i i
) i I Xk~L I I I
11.1
/
7
4.5<=W<6.0
~i i i i } i i J
0.4
.
1~ I ~ I ) I I
--Mea~lll~rnent$
. . . . . . . . . . . . .
11 II I
~eibllll
o,} bAI . . . . . .
8
The smooth parameters a and fl (Eqs. 6 and 7) were further
used to calculate the conditional mean and standard deviation
for significant wave height given wind speed as given by Eqs (8
and 9). The mean value and the standard deviation based on the
measurements and on the parameterisation are plotted in the
Figs. 6 and 7.
s
lO
Wave height
o
The probability density function for significant wave
heights within different wind classes. Based on
measurements (the "un-smooth " curve) and a fitted
2-parameter Weibull distribution (the smooth curve).
(s)
E(H,,,0) : [~-F(+ + 1)
Further, a and fl were plotted versus wind speed. Based on these
plots the following parameterisations were proposed for c~ and
and
STD(tt,,,0) =/3-[~(~ + 11)- r ~ ( + + 1)] °5
(9)
(6)
Ctfit = a l + a2 . IJV
and
/ ~ : b i + b2 - t~'b:'
From regression analysis (least square method) the parameters
a~, a:, b~, b:, b3 were estimated. All the points were included in
the analysis. The Figs. 4 and 5 show the fitted Weibull
parameters versus the Weibull parameters based on the
measurements.
o
15
/
~
0
10
20
/"
30
1>
-.'
40
Measurement=
50
Wind speed [m/s)
Fig. 4. The Weibull scale parameter, ,8,, estimated from the
measurements versus a fitted estimate.
22
.~a-a
•
From rneasuren
.,,~'
,~
o *** _**.,.x*"
Fig. 6.
10
,,
~ '°
~ 5
20
so
...t-
~: t5
(7)
10
l
20
30
Wind speed (m/s)
Expectation value for significant wave height versus
wind speed according to measurements and a smooth
Weibull representation.
pz), and CrVpare the mean value and standard deviation of T~, from
the measurements for each of the 600 wave-wind classes. Fig. 8
shows a selection of the ~,-distributions for random
combinations of wind and wave magnitudes.
2
,I("
•
1.5
""
,~
1
L~k. & : ~ " ~
-
Parameterised
•
From measurements
21 rr~os~emontl
33 me~sulement~
o~
a35
0.5
- W~.75.HsB.7 s
Lct~tmal
0
10
20
30
40
SO
Wind speed (m/s)
o,o~
o
Fig. 7. Standard deviation for significant wave height versus
wind speed according to measurements and a smooth
Weibull representation.
~J
5
I
&!X2
10
o.os
o
L_
15
20
I
o
971 rne~osurements
5
r
]
f" In(t)--'Uln(rP) "~ 2 1
/
p ~;.,
]
0
~0
IS
Uo~o~al
20
* Uog~o,mal
i °'
0,1
0
~
~0
15
20
P A R A M E T E R I Z A T I O N OF T H E M E A N V A L U E
IN T H E L O G - N O R M A L D I S T R I B U T I O N
To get an idea of how Tp vary with H,,o and W, the mean value
of Tp was plotted in a 3D diagram shown in Fig. 9. The figure
shows that for a constant value of H,,o, the period decreases with
increasing wind speed and for a constant Wthe period increases
with increasing wave height. To describe this behaviour the
following function was proposed:
--r,,(w, h) = -r ( h ) . [ ] + o . [ '.'-~(h)') >"]j
(10)
(14)
where T(h) is the conditional mean peak period for a given
value of the wave height. Correspondingly, w(tO is the
conditional mean wind speed for a given value of the wave
height. The term ill + 0 . \ ~(h~ y J
will then adjust the
expected peak period according to whether the actual wind
speed is above or below the expected wind speed for the
particular wave height. The coefficient's 0 and 7 will take into
account how the expected peak period will vary with wind speed
for different wave heights.
(11)
(12)
where
~;"
S
............
'
Fig. 8. A selection of the Tp-distributions for random
combinations of wind and wave magnitudes. The
dotted lines show the calculated log-normal
distribution while the other lines show the distribution
found directly from the measurements.
and
O-!n(7),) = In[o~), + 1]
!!!I
0.t
where ~o,,crp) and o),crp) are the expectation value and standard
deviation of ln(TA, respectively. The mean value and standard
deviation of ~, were calculated for each combination of H,,o and
W and used for the calculation of/-0,,crp~ and crz,crp), i.e. the mean
value and standard deviation in the log-normal distribution,
according to the relationships:
f
/21 O).) = 113.
I/V
°.,,',
In order to find a suitable distribution for the peak period, Tp,
for given wave height and wind, the distribution of Tp within
each wind-wave class was plotted. Due to the somewhat limited
number of data within each class, the plots were not particular
smooth. Still they seemed to indicate that a log-normal
distribution would be suitable for the distribution of Tp. The
log-normal distribution is given by:
fr,,(t)
A
15
137 m e c ~ u r e m e n t s
~3
C O N D I T I O N A L D I S T R I B U T I O N OF T~ F O R G I V E N H~o
AND W
F I N D I N G A S U I T A B L E D I S T R I B U T I O N F O R T~
The wind speed data used in this analysis were sorted into
classes with a bin size of 1.5 m/s in the range 0 - 36 m/s, that is
24 classes. Subsequently, the data for significant wave height in
each wind speed class were sorted in wave classes with a bin
size of 0.5 m in the range 0 - 12.5 m, that is 25 classes. This
lead to 600 combinations of wind speed and wave height and
with a limited number of data for each combination.
10
(13)
23
.... :....... :........
As for the period, a regression analysis was performed and the
result is shown in Fig. 1 1.
~i....!....
"i' " - . . . . .
35
30
. '"" "'''i
/
' ""
/
25
/
20
1.1.1
/
-- Measured
¢/--
-- Fit: F:(W)=l.764+3,426*h^0.78
/
10
:ii "
o.
3
o
~ "....:"..
'""
.....:..
5
12
/
/
0
0
10
5
15
20
Hs [m]
Wind
Fig. 9.
t,'.'J'-'s'
0
0
Hs [m]
Fig. 11. The conditional mean wind speed as a function o f
significant wave height according to measurements
from the Northern North Sea and a smooth
parameterisation.
The conditional mean peak period as a fimction o f
significant wave height and wind speed.
The mean peak period, ~'(h), was plotted as a function of wave
height and the following parameterisation was proposed:
To give reasonable values for the two coefficient's 0 and y, a
relation between wind speed and expected period had to be
found. Eq. 14 was rewritten as:
05)
7"(h) = ct + cz • h ¢:+
_
7~,,c,%,~>.-.?:~,(+)
_ 0.
(]7>
For given wave height, the normalised period
20
--
k( ,+-----+c+>
. ~ y 't +'
Tp(/O
The parameter's ci, c2, c3 were estimated using regression
analysis (least squares method). All points from the
measurements were included in the regression and the result is
shown in Fig. 10.
15
2.0<=Hm0<2.5
/
0.3
0.2 ~ , ~
/
if
,t
~
~ o
+-+.
..o+2
- " .a,.+
-I
2
4
6
8
Significant wave
10
12
14
.o.s
o
~
)
0,2
~
"7 ~
~ o
+++.,
+
.o,3
0
w-..i~(h i
4.0<=Hm0<4.5
0.3
10
:
was plotted as a function of the nonnalised wind \
,
. Fig.
12 shows a selection of these plots for different wave heights.
Fit:
E[Tp]=4.883+2.68*h^0.52 ¢
,,,
I +%,,x,...__]](:,
~(/+) ) 1
o+
I
"-_.
.is
~--j%
-i++
+1
++s
mo~limd wind
16
o
o.s
I
1.+
mofmal;sedwind
6.0<=Hm0<6.5
h e i g h t (m)
,, +-.~.
z -o.2
8.0<=Hm0<8.5
0.25
o.2
....
o+1 "-
Fig. 10. The conditional mean peak period, =i,: ~1~t0as
, a function
of significant wave height according to measurements
from the Northern North Sea and a smooth
parameterization.
-1
.o.s
•
1+ ....
+
-0.05
-1.s
o
Nor~li~d w;nd
Correspondingly, the mean wind speed was plotted as a function
of significant wave height and the following parameterisation
o.s
1
~o+6
-1+.+I++
I
.o.4
~.2
o
0.2
o.4
o.6
N o ~ l i ~ d wind
Fig. I2. The normalised expected peak period as a function o f
normalised wind speed. The data points are from
measurements while the solid lines are based on
regression analyses (least squares).
was proposed for ~:(h) "
~ ( h ) = dt + d2 " h c!s
+ ~
06)
24
-0.16
As can be seen from the plots, the trend is nearly linear
indicating that y is close to 1. The slope however, seem to vary
for the different wave groups. To investigate this, 0 from all the
wave groups were plotted as a function of the significant wave
height. This plot is shown in Fig. 13.
-0.17
-0,18
-0.19
-0.2
-0.05
-0.21
-0.22
-o.1 1
-0.23
-0.24
ie
~_ -0.15
i
e
-0,2
-0.25
o
]
1
0
iii
2
1
2
4
S
Significant
3
wave
6
7
height
8
9
10
(m)
Fig. 15. The slope, 0,, Eq. (17)between normalized period and
normalised wind speed as a function of the wave
height. Outlayers are excluded from the data.
3
4
5
6
7
8
10
9
Hm0 (m)
By considering the Figs. 13 and 15 no obvious expression for 0
did occur and thus based on best fit by eye the constant value 0
= - 0.19 was proposed for the slope. From the above analysis all
the parameters in Eq. (14) have been estimated. To evaluate the
goodness o f the proposed model, the periods were calculated for
all 600 wind/wave combinations. The difference between
measurements and estimate was investigated and the result can
be seen in Fig. 16. Table 1 gives the estimated peak period for a
number of combinations of wind speed and significant wave
height.
Fig. 13. The slope, O, Eq. (17) between normalized period and
normalised wind speed as a function of the wave height.
It is not straightforward to see any simple relationship between
the wave height and 0. Further, with intention on finding a
general trend for the slope, 0, as a function of significant wave
height, the wind classes with a very limited number of data were
excluded from the analysis. The peak period versus wind speed
when the outlayers were excluded from the data is shown in Fig.
14. Fig. 15 shows the slope, 0, versus significant wave height
when the outlayers are excluded.
• . . ' " " i ......... " ' " " " i " " " ! " . .
i
2.0<Hm0<=2,5
o.s
~
.........
4 1 ........
4.0<Hm0<=4.5
o,3
.
"''":'" • ,. i
:'"" ' . , .
°'2
o.
~°.
+os
~
.o.2
"~
N~~ri=e¢~ ~n~ Sl~=d
N ~ a I~'aed q'~nd~Peed
6.0<Hm0<=6.5
8.0<Hm0<=8.5
2.
h_0-4 :
4O
°"'"'"~'J"'""*""lttlttlI~ll~lllllli
:2,
" Illl]lllllllllill~-+41
~,/]llll
I IIt111111 IIII1
~.$ ~.4 -o3 .0.2 .0.1 0 0,1 0.2 0.3 0.4 0.5
Norm=lised~n~ speed
........
. '":
10
.0,2
Wind [m/s]
.0.1
0
0,1
Nonm~liscd~+~ndSl:~md
0,2
0.3
Fig. 14. The normalized expected peak period as a function of
normalised wind speed. Outlayers are excluded from
the data.
.." . . . . . . . :~'.1
....... ~'7~,.
i
.................
2°
=~~,'
"..
~
5
0
0
Hma I m l
Fig. 16. Difference in peak period between measurements and
calculations for all combinations of wind speed and
significant wave height for 0 = -0.19. A negative sign
means that the estimated peak period is lower than the
measured peak period.
25
PARAMETERISATION OF THE STANDARD
D E V I A T I O N IN T H E L O G - N O R M A L
DISTRIBUTION
Eq. 14 gives an estimate on the mean value recommended to be
used in the log-normal distribution for the period. As for the
standard deviation, Eq. (19) is proposed based on experiences
on the relationship between standard deviation and mean value
from the measurements:
Table 1. Estimated peak period for combinations of wind speed
and significant wave hei~ ht.
H~0 (m)
12
W
(m/s)
30
Estimated Tp
(s)
14.4
15
20
17.1
15
30
16.1
15
40
15.1
18
30
17.7
aTi~ = [ - 1 . 7 - 10 .3 + 0 . 2 5 9 . e × p ( - 0 . 1 1 3 .h)] .fz~
RECOMMENDATIONS
The joint distribution for wind and waves can be expressed by
inserting the parameters given in Table 2 in Eq. (4). Based on
these parameters, a contour body expressing different
combinations o f wind speed, significant wave height and peak
period was found for the 100-year level. This is shown in the
Figs. 17 - 19.
From Fig. 16 two conclusions can be drawn:
The estimates and measurements were consistent in the area
containing the majority o f the measurements.
For more rare wind speed and wave height combinations,
the estimated peak period differ from the measured peak
period. The difference for large wind speed and wave
heights is of the order up to 2 seconds.
Table 2. Parameters recommended for use in the joint wind and
wave distribution (Eq. 4)for Veslefrikk.
Wind distribution
To improve the estimate in the area o f interest, which is for
large waves and wind speed, two separate courses were tried:
1.
2.
Shape = c~ = 1.708
Scale = [3 = 8.426
An iteration was made where 0 was the parameter to be
optimised. The value giving the smallest RMS value, i.e.
the sum of the difference squared for all the 600 possible
combinations of H,,o and W. would be considered as the
best fit. In addition the mean value of the difference
between the measurements and the estimate should be as
small as possible.
By assuming the relationship between wind speed and peak
period not to be linear as originally was proposed, the
following equation was assumed to describe the relationship
better:
Conditional distribution,
Significant wave height
Shape: a = 2.0 ~ 0. t 35 • w
Scale: fi= 1.8 ~ 0. ll)0. w ~.:<"
Conditional distribution,
Peak period
Mean value:
u
r.,.a=ln
,-=-~:-i
Standard deviation '
08)
LS(h)
-
(l)(,~...~(;0))
(19)
k v,,{h) )
where
where 2 is a constant.
, , 7 ) , = ( 4 , 8 8 3 + 2 . 6 8 . h ~..... ).Ll ......0,19.i,
Eq. (18) is based on the idea that the outlayers in Fig. 12
indicate some curvature.
t.7(,4<~4:?e,;~,~7~
and
None of these two methods gave significant improvements to
the estimated wave periods, therefore it is recommended that
Eq. (17) is used with 0 = - 0.19. Further work in optimising the
parameter, 0 and possibly A, combined with omitting the low
wave classes could still lead an improvement in the model used
for calculations of the peak period.
o-7)~ = [ - l . 7 • 10 ....3 + 0.259. exp(-0.1 I3. h)] .,u :,)
w and h is the wind speed and significant wave height
26
Example of application of the joint model
The joint model is utilised for predicting extreme mooring line
loads for semi submersible with 12 mooring lines. The
application is presented into some detail in Meling et al. (2000),
and only some few results will be reviewed herein. Solving Eq.
(1), the following extremes are obtained for the line loads in the
intact case:
0
5
Significant waveheight He [nl]
1D
x~o0 = 3 5 2 3 k N
(20)
x 1o000 = 6_~.~4kb
(2 I)
Utilising methods from the field of structural reliability, one can
identify the most likely combination of environmental
characteristics as these values occur. The combinations are:
15
xt00 ( 7 8 % ) " w = 3 5 . 7 m / s , h~,0 = 14.1m, t ; = 14.3s (22)
Fig. 17. Contour surface o f the j o i n t distribution f o r wind and
waves - 100 year return period. 2D - Wind speed versus
significant wave height.
xt0000 (91%) :w = 42.1m/s, h,,0 = 16.9m, tp = 14.8s (23)
The percentages given in parenthesis, are the most likely
fractiles of the short-term extreme value occurring in
combinations with the given environmental conditions.
If instead of using a full long term analysis one will use the
contour surface principle, one will first of all have to identify
the most unfavourable parameter combination on the 100- and
10000-year surfaces for the problem under consideration. For
the mooring line problem, the following environmental
conditions are identified:
100-year"
0
5
lope ak period Tp Is]15
35.
,""
30.
'
/
(24)
~
lO000-year'w
Fig. 18. Contour surface o f the j o i n t distribution f o r wind and
waves - 100 year return p e r i o d 2D - Significant wave height
versus p e a k period
._.,......... 'i
"*
w = .36rms, hmo = 14.4m, tp = 13.3s
= 42.3m/s, hmo = 17.9m, tl~ = 14.8s
(25)
For these sea states we have estimated the median (50%) and
90% fractile of the 3-hour extreme value. We have also
estimated the fractile we have to select in order to equal Eqs.
(20) or (21), respectively. The results are:
........i " . .
......:., . .
1O0 - y e a r : .~.~481c~\;(~0,4), 3945kN (90%), 3523 (66%)
(26)
10000 -),'ear : 5820;~:N(50%), 6865kN (90%), 6334(76%) (27)
D>
The advantage by using the contour surface principle is that
good estimates for the n-year response may be achieved by use
of a short-term analysis. This saves time and resources at an
early stage in design. However, before final design a long-term
analysis must be performed. For more details on the response
example reference is made to Meling et al. (2000).
.'"
1
Signil@c~n
D
O
Fig. 19. Contour surface o f the j o i n t distribution f o r wind and
waves - 100 year return period.
27
Conclusions
The equation
--
~
r
(" " ~ ] w - ~ ( h7)
Tz'(w'h):z"h)'[l+O't.-~;i--JJ
(Eq. 14)
with 0 = -0.19 and y = 1.0 describes the peak period as a
function of significant wave height and wind speed.
Attempts to improve Eq. (14) have failed. For large classes of
wind speed and significant wave height the difference between
measured and estimated peak period is of the order up to 2
seconds.
The wind speed and significant wave height can be described by
a two-parameter Weibull distribution. The peak period can be
described by a log-normal distribution.
Simultaneous wind and waves can be described by the joint
density function:
/~,~o .,,~,( w, h, 1) = f ~ 4 w ) . J}.~,.o~,,.(hl w ) . j~',,i~i,.o'~.( t[ ;¢~,w) (Eq. 4).
An efficient method to predict a specified n-year response is to
apply the contour line/surface approach. However, this method
requires that the proper fractile of the extreme value distribution
of the response is known. When the short-term variability of
both wind and waves are included, a fractile around 66% is
found to be adequate in order to predict the 100-year response.
For the 10000-year cases, the proper fractile seem to be
approximately 76%. Further studies should be performed in
order to validate the proposed fractile levels. In addition,
different structures should be tested to check if the fractile levels
are case dependent.
References
Haver, S., Gran, T. M. and Sagli, G. (1998): Long-Term
Response Analysis of Fixed and Floating Structures,
Proceedings of Ocean Wave Kinematics, Dynamics and Loads
on Structures, ASCE, Houston, Texas, April 30 - May 1, 1998,
pp. 240-248.
Madsen, H.O., Krenk, S. and Lind, N.C. (1986): Methods of
Structural Safety, Prentice-Hall, Inc., Englewood Cliffs, New
Jersey, 1986.
Meling, T. S., Johannessen, K., Haver, S. and Larsen, K.
(2000): Mooring Analysis of a Semi-Submersible by use of
IFORM
and
Contour
Surfaces,
Proceedings
of
ETCE/OMAE2000 Joint Conference for the New Millennium,
no. OMAE2000/osu oft-4141, February 14-17, 2000, New
Orleans, LA, USA.
28