Trans RINA, Vol 154, Part A2, Intl J Maritime Eng, Apr-Jun 2012
The calculations can be performed in two ways: Replacing the histogram built in each time window by theoretical distribution and calculating with it the POE of the maximal amplitude;
Dividing unity by the number of cycles in each time window.
The former method is time consuming but allows for other analyses to be performed beyond the purpose of this particular study. The latter method is simple and fast and is recommended for cases when only hull girder bending moments are analyzed. The effect of the two methods on the final results was analyzed and preference was given to the latter approach. Thus, a new statistical sample
is formed containing data for the maximal
amplitudes in each time window with their corresponding POE.
Step III: Dividing the statistical sample of maximal MW,h into several groups using as a criterion the POE of the recorded maximal MW,h,i.
It is not possible to calculate the Probability of
Occurrence (POO) of any given value with the theory of probabilities. Only the POO within two given boundaries can be calculated. Therefore, several
regions (with
corresponding two boundaries) of the random variables are created and the probability of getting into each of them is calculated. The POO of the random variable within each group “i” is formulated as:
10 i0.5
0.5 POE of max M h, i W, where i = 1, 2, 3, 4, 5, etc.
0 POE of max M 10 (13) For the sake of brevity, the POE of the max W, h, iM
0. 5 h 0 marked as: POE of max M h,i Pi W, (14) Thus, in each group “i”, the histogram of maximal MW,h,i
contains only maximal MW,h,i with identical POE/POO. The data in each group “i” is used to build a histogram having as abscissa the maximal MW,h,i represented as a
portion of the design MW,h. Then, the histogram in each group “i” is
computer program EasyFit criterion is utilized).
replaced by theoretical PDF using the (the Kolmogorov-Smirnov
Figure. 5 - Figure. 8 represent the results for maximal hogging wave bending moments from four “i” groups (the contribution of the other six groups is close to zero in this particular case). The time window’s duration is 60 minutes over one year full-scale measurements. The POE
A-92
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
histogram for max M(W,h,2) Beta PDF
The results refer only to max MW,h,2 ;
POO = 10-2 ; POE design
MW,h = 3.32x10-4
portion of design M(W, h) [-]
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Figure. 6 POE of the design MW when POO = 10-2 of the
Eq. (12) does not allow covering the range from zero to -0.50. For this case, the following equation is used: W, ,
is 10 i (12)
of the maximal MW,h is calculated as 1/N where N = number of cycles within one year. Figure. 9 represents
the results when the POE of the maximal MW,h is neglected which corresponds to the procedure following the EVT presently applied for calculating the POE of the
design MW,h.
One should note here that the probabilistic distributions of maximal MW,h in each group are not identical. The
POE of the design MW,h in each group, Pd,i , is calculated as:
PP Mmax d, i M h W, h,i design W,
It follows from Eq. (15) that the data window were considered as identical.
(15) in each time independent but not as
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
histogram for max M(W,h,3) Weibull (3P) PDF
The results refer only to max MW,h,3 ;
POO = 10-3 ; POE design MW,h = 0.0261
portion of design M(W, h) [-]
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Figure. 5 POE of the design MW when POO = 10-3 of the recorded max MW is used
©2012: The Royal Institution of Naval Architects
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