Page 281 - Practical Design Ships and Floating Structures

256

where &(ai) the phase transfer hction, and ai and yi are the wave amplitude and random phase for
component i out of I components. The time series of the accelerations are calculated by the same
principle. Insertion of these time series into Eqn. (4) gives F and N as functions of time.

2.3 Statistical Methods for Risk Calculation
The risk of cargo shifting is defined as the probability of at least one initial motion of a cargo unit
onboard, during the studied time interval. This is equal to the complement of the probability of no
cargo shift. The problem is similar to calculating the annual probability of capsize, as described by
McTaggart (1998). If the studied time interval T is a year or more, it is reasonable to assume that the
ship will encounter sea states according to the wave statistics. If the occurrence of sea states is
assumed to be independent, the vessel can be assumed to encounter a number of independent
conditions. For each condition i the probability of occurrence, denoted wi, can be determined from
wave statistics and knowledge of the ship operation. These probabilities of occurrence will have the
property Cwi = 1. Further a sea state only has a short duration d, about 3 to 4 hours. It is assumed that
the events of cargo shift during each condition lasting d hours are independent and identically
distributed random variables. If the probability of no cargo shift during d hours in a certain condition is
known as pi(d), the risk of cargo shifting can be written as
(i 1”’
P=l- Cwipi(d) .
In estimating the risk of cargo shifting under a certain condition, or rather the probability of no shift pi,
the time series of the left-hand side of Eqn. (3) is used. If the value of this expression exceeds zero the
cargo unit will start shifting. Therefore, the up-crossings of the time series through zero will be a
measure of the risk of cargo shifting. If these up-crossings occur seldom, i.e. if the limit (in this case
zero) is set high enough in relation to the mean value, the number of up-crossings occurring in disjoint
time intervals can be asymptotically regarded as independent random variables. Further requirements
for a Poisson process are the assumptions of stationarity and regularity (see CramCr & Leadbetter
(1 967)). These requirements are fulfilled for the simulated time series, and the events of up-crossings
asymptotically form a Poisson process The Poisson parameter h, Le. the intensity, is the number of up-
crossings per unit time. For a specific cargo unitj the probability of no cargo shift during d hours is
1.J . (d) = e-’jJd .
p. (7)
Under the assumption that the individual cargo units are independent the probability of no cargo shift
in a given condition will be the product of the probability of the individual units. However, due to the
large number of cargo units onboard and the long computational time for time-domain simulations a
sampling technique is used to reduce the number of calculations. Systematic sampling has been used
since units close to each other have a similar probability not to shift. If the total number of cargo units
is N and the sample size is n, a good estimate of the probability of no cargo shift in condition i is
pi (d) = fi p’”
.
j-I

3 CASE STUDIES AND RESULTS

In order to evaluate the methodology, case studies have been performed, where the risk of an initial
cargo shift during one year has been calculated for various ship and cargo parameters. A typical Ro-Ro
vessel (Lpp = 120 m and B = 20 m), in MIC between the Swedish and English east coasts, has been
used. In each case the loading condition and ship speed has been assumed constant. This limits the