Actions on wind turbines - Concrete Structures for Wind Turbines - page 48

Gumbel distributions can also be used for other environmental actions, for example wind or
snow, and so employing such distributions is a great advantage for sea state actions,
primarily with respect to the combinations of actions that are to be applied and the
definition of partial safety and combination factors. Characteristic of this distribution is the
constancy of the standard deviation

s H,extr irrespective of the reference period considered.
Awave distribution diagram (e.g. Figure 2.27) is always valid for a defined period of
observation (reference period T 1 ). It is therefore necessary to convert this to the design
working life (reference period T N ) of the structure at the planning stage. The maximum
values in the reference period T N ¼ N T 1 can be obtained through exponentiation [25]:

¼ F extr ; 1 H ðÞ N

N

F extr ; N H s ; N ¼ exp exp a H s u 1 ½ ð ð ð Þ Þ Þ

¼ exp N exp a H s u 1 ½ ð ð ð Þ Þ Þ

¼ exp exp a H s u 1 ð ð ð Þ þ In ðÞ Þ Þ

¼ exp exp a H s ; N u N

This means that the Gumbel distributions develop through a displacement of ln(N)/a on the
H s axis. Accordingly, the modal value u 1 , or rather the mean value m H,1 , is also displaced
with respect to the initial distribution. On the other hand, the value of the standard deviation
s H,extr does not change. We therefore get

p 6

ln ð a ¼ u 1 þ ln ðÞ ln ðÞ

a

u N ¼ u 1 þ s H ; extr and m H ; N ¼ m H ; 1 þ

p

for the maximum values.

The statistical evaluation of the long-term wave distribution using the example of
Figure 2.27 with the help of a linear regression analysis leads to Figure 2.28.
The regression line has the following form:

y ¼ b þ m H s ¼ a H s u ð Þ¼ ln ln F extr H ðÞ ð ð Þ Þ

7963 m 1

a ¼ m ¼ 0 :

Fig. 2.28 Regression analysis based on a Gumbel distribution