Trans RINA, Vol 153, Part A4, Intl J Maritime Eng, Oct-Dec 2011
A formal generalised series system reliability index is given as:
Pf s
. (20) 1 s
To evaluate the series system the second order Ditlevsen [29] bounds are used here:
s ij
1 mm i
ii j max
s ii j
max , ; ii
12 mm
2 ij 2 12 1
ii i, ; ,0
j
(21) (22)
The numbering of the failure elements influences the bounds. A good choice to arrange the failure elements is to account them to degreasing probability of failure.
1.2 1.4 1.6
0.2 0.4 0.6 0.8 1
0 0 5 10 15 Time, years
Figure 11 – Beta index, series system- second order bounds
Two series system are composed. The first one is related to the spot-weld structure including the hot-spots 1, 2 and 3 and the second one is for the hot-spots 4, 5 and 6 of the all-welded structure. The correlation coefficient of the probability of failure, due to the very close location of the hot-spots and similarity in the loading conditions, are assumed here as very high and equal to 0.99.
As can be seen from Figure 11, the lower bound of the all-weld structure demonstrates lower reliability in comparison to the spot-weld one. This is explained with the fact that the lower bound is related to a series system where the system components are 100 % correlated and the weakest element will dominate the system reliability. In the study here the hot-spot 4, which is a part of the all- weld structure, has the lowest reliability index.
The upper bound of the spot-weld structure is lower than the one for the all-weld structure.
©2011: The Royal Institution of Naval Architects
The work presented here analyzed the fatigue reliability of ship hull structural joints accounting for wave induced loads as well as vehicle operational loads. Different welding connections between trapezoid longitudinals and deck plate were examined. A set of case studies were analyzed taking into account global loading conditions, including two local finite element models performing spot-weld and all-weld analysis.
Analysis of the all-weld model revealed three areas of high stress concentration, one of them being the same as on the spot-weld model. The location of highest fatigue damage for the all-weld longitudinal and for the spot- weld longitudinal is the bending of the trapezoid.
As a result of the performed analysis the beta reliability indexes are defined based on the combination of low frequency wave induced loads and transient loads accounting for the corrosion deterioration.
The structure modelled as a series system composed by hot-spot elements was evaluated based on the second order bounds. The all-welded structure demonstrated a
A-239 20 25 30 7. CONCLUSIONS
1.2 1.4 1.6
0.2 0.4 0.6 0.8 1
0 0 5 10 15 Time, years Figure 12 – Beta index, series system
Eqns (21) and (22) presented a bound and it is convenient to calculate a specific value. Feng [30] developed a point estimate for the joint probability of failure as:
P P Pf , j f ,ij f ,i 1 arccos ij / (23)
Eqn (23) has high accuracy to be used in Eqn (21) and (22).
Upper (spot-weld) Lower (spot-weld) Upper (all-weld) Lower (all-weld)
The final result for the system reliability of spot-weld and all-weld structure is given in Figure 12. As may be observed the
spot-weld joint has a better reliably
performance during the service life of 25 years without accounting for the repair and maintenance actions.
20 25 30
F (spot-weld) F (all-weld)
Beta index
Beta index
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