Trans RINA, Vol 153, Part A4, Intl J Maritime Eng, Oct-Dec 2011
because enough statistical data is usually not available to develop the n-dimensional joint density function of the basic variables. Secondly, even when the joint density function is available, analytical or numerical integration is possible only
The global annual reliability index is obtained from the probability of failure as:
for a few simple cases. The
FORM/SORM methods provide a way of evaluating the reliability efficiently with reasonably good accuracy, which is adequate for practical applications as proposed by Hassofer and Lind [26], Rackwitz and Fiessler [27] and Ditlevsen [28].
Using a FORM/SORM technique and the S-N fatigue damage approach, the limit state equation for fatigue failure may be defined on the basis of Eqns 4, 5 and 9 for
gt x1
t , as:
C i X where i xpp x 23 57,
pp pp p ii
11 11 1 45
xp p x p8 1exp
32 7 4
tp9
p10
p (12)
p p p
6, p1 x are uncertainties on calculation and ip are
parameters defined in Table 1. For
the characteristics p 3iwi
hot-spot locations, i, 16 q
and p 6ici q 217.825, 11.427, the load are 12.511,
12.102, 13.956, 13.888, 9.796, 11.101 and 49.166, 192.470, 54.423,
42.061 MPa
respectively. A Normal distributed random variable
B was w
considered to take into account the uncertainty on fatigue stress estimation accounting for the wave-induced loading. As the stress calculation has several steps, each of which with its own uncertainty, the stochastic variable B w can be split into four components: B modelling
L,w
the uncertainty in the load calculation, B modelling the uncertainty on the
normal stress calculation, modelling the uncertainty of the hot B can be also defined. BH spot stress concentration factors and QB modelling the uncertainty
on the weld quality and on misalignment. In the same way the uncertainty c
The reliability calculations can also be performed using the total uncertainty on fatigue stress estimation
represented by the random variable B
BB i
and COV B ()
Ci i
1 with mean value
and coefficient of variation determined by: i
2 1 (13)
The stochastic model of the basic variables considered in this study is presented in Table 2.
Table 2 Stochastic model Variable Distribution Mean Value St. Dev.
B = 1 B
x L,w Bs
BH BQ
B = 2 x
w K x
B = 4x Log-Normal 1.0 B
B = 3 cor L,c
B = 5 c x
Normal 0.9 Normal 0.9
Log-Normal 1.0 Normal 0.85 Normal 1.00 Normal 1.00 Normal 1.00 Normal 0.85
Log-Normal 0.8 0.2
0.255 0.12 0.30 0.20
0.48 0.1
0.32 0.15 0.42
Figure 8 presents the results of the fatigue reliability assessment of the six hot spots of the very fast ferry operating during 25 years and accounting for the total damage including wave and car-breaking loadings. Figure 9 is the reliability beta index as a function of time only
accounting for wave induced load. The total
reliability index, composed by wave and car breaking load, presents severe condition as already was indicated by the results of fatigue damage assessment.
, t C 1()fP where 1 distribution function.
Table 1 Parameter descriptions Variable Units 1
p m p K
2
pT
4 d 1 m
5 22 1 2
21 2 m
p h deck 8 9
7
p d p C p10 t
exp km
m [-]
[m] 0.007 [m]
[year] 3.0 [year] 17.24 0.0023 28.4
[-] [-]
(14) is the standard normal probability
3.0 2.4E12 [Cycle] 4.42E8
©2011: The Royal Institution of Naval Architects
A-237
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