Practical Ship Hydrodynamics

Ship seakeeping 137
changing sea spectra. A typical example for T is the total operational time
of a ship. A quantity of interest is the average occurrence zL⊲rA⊳ of cases
when the reaction r⊲t⊳ exceeds the limit rA. The average can be thought of
as the average over many hypothetical realizations, e.g. many equivalently
operated sister ships.
First, one determines the occurrence z⊲rA; H1/3,Tp, 0⊳ of exceeding the
limit in a stationary seaway with characteristics H1/3, Tp, and 0 during total
time T. (See section 4.4.6 for linear ship reactions and section 4.4.7 for nonlinear
ship reactions.) The weighted average of the occurrences in various
seaways is formed. The weighing factor is the probability p⊲H1/3,Tp, 0⊳
that the ship encounters the specific seaway:
zL⊲rA⊳ D � � �
z⊲rA; H1/3,Tp, 0⊳p⊲H1/3,Tp, 0⊳
all H1/3
all Tp
all 0
Usually, for simplification it is assumed that the ship encounters seaways with
the same probability under n encounter angles 0:
zL⊲rA⊳ D 1
n
�
all H1/3
�
all Tp
n
�
z⊲rA; H1/3,Tp, 0i⊳p⊲H1/3,Tp⊳
iD1
The probability p⊲H1/3,Tp⊳ for encountering a specific seaway can be estimated
using data as given in Table 4.2. If the ship would operate exclusively in
the ocean area for Table 4.2, the table values (divided by 10 6 ) could be taken
directly. This is not the case in practice and requires corrections. A customary
correction then is to base the calculation only on 1/50 or 1/100 of the actual
operating time of the ship. This correction considers, e.g.:
ž The ship usually operates in areas with not quite so strong seaways as given
in Table 4.2.
ž The ship tries to avoid particularly strong seaways.
ž The ship reduces speed or changes course relative to the dominant wave
direction, if it cannot avoid a particularly strong seaway.
ž Some exceedence of rA is not important, e.g. for bending moments if they
occur in load conditions when the ship has only a small calm-water bending
moment.
The sum distribution of the amplitudes rA, i.e. the probability that an amplitude
r is less than a limit rA, follows from zL:
PL⊲rA⊳ D 1
zL⊲rA⊳
zL⊲0⊳
zL⊲0⊳ is the number of amplitudes during the considered period T. This distribution
is used for seakeeping loads in fatigue strength analyses of the ship
structure. It is often only slightly different from an exponential distribution,
i.e. it has approximately the sum distribution:
PL⊲rA⊳ D 1 e rA/r0
where r0 is a constant describing the load intensity. (In fatigue strength analyses,
often the logarithm of the exceedence probability log⊲1 PL⊳ is plotted