Page 524 - Marine Structural Design

500 Part IV Structural Reliabilig

28.3 Reliability Updating Theory for Probability-Based Inspection Planning
28.3.1 General

Baysian models have been applied to reliability updating for probability-based inspection
planning. This Section shall present two major approaches that have been developed in the pat
30 years.
Updating Through Inspection Events to update the probability of events such as fatigue
failure directly, (Yang, 1976, Itagaki et al, 1983, Madsen, 1986, Moan, 1993 & 1997). A
simplified Bayesian method that only considers crack initiation, propagation and detection as
random variables and independent components in a series system was proposed by Yang
(1976) and Itagaki et a1 (1983).
Updating Through Variables to re-calculate failure probability using the updated probability
distributions for defect size etc. (Shinozuka and Deodatis, 1989). The change in reliability
index is caused by the changes in random variables. The distribution of a variable can be
updated based on inspection events. When the variables are updated, the failure probability
can be easily calculated using the updated variables. However, if several variables are updated
based on the same inspection event, the increased correlation between the updated variables
should be accounted for.
The approach for updating through inspection events will be further explained in the next sub-
section.

28.3.2 Inspection Planning for Fatigue Damage
Fatigue failure is defined as the fatigue crack growth reaches the critical size, e.g. wall
thickness of the pipe. Based on fracture mechanics, the criterion is written in terms of the
crack size at time t. By integrating Pans law, the limit state function can be written as, (See
Part IV Chapter 27 of this book, Madsen et al, 1986)

(28.4)

where, Y(a,X) is the finite geometrical correction factor, ES is the stress modeling error, EY is
randomized modification factor of geometry function, vo is the average zero-crossing rate of
stress cycles over the lifetime, r(.) is the Gamma function.
Basically, two most common inspection results are considered here, namely: no crack detected,
and crack detected and measured (and repaired), see Madsen et a1 (1986).
No Crack Detection
This means that no crack exists or the existing crack is too small to be detected. This
inspection event margin for the ith detail can be expressed as,

(28.5)
in which, a(ti) is the crack size predicted at inspection time ti, aD is the detectable crack size.