(()0 ()0)
()
(( ) 0)
PM h
PFI
PhXX
X
(12.4)
where X is a vector of random variables having the prior distribution fX()x.
This procedure can easily be extended to complex failure modes and to a set of inspection
results ( ). For further calculation, software packages such as STRUREL (1998),
PROBAN (1996) and VaP (1997) are available.
(^) hi(X) < 0
Finally it should be mentioned that individual random variables may also be updated by
inspections of events involving the outcomes of several random variables. This should
nevertheless be done with care. For instance it is important to realise that all the random
variables that are present in g(X) (and all the variables correlated to X are affected by the
inspection. For instance, if a crack length is measured in one weld of an offshore structure,
this affects the distributions of the load parameters, the stress concentration factors, the
residual stresses, and the parameters of the fatigue model. Moreover, all these parameters
become correlated, even if they were independent before inspection.
12.5 Decision Analysis in Structural Reassessment
In practical decision problems such as re-qualification of structures and inspection and
maintenance planning the number of alternative actions such as strengthening and
maintenance activities can be extremely large and a framework for the systematic analysis of
the corresponding consequences is therefore expedient. A framework suitable for this purpose,
which facilitates the utilisation of both subjective and frequentistic information, is the
Bayesian decision analysis, see e.g. Raiffa and Schlaifer (1961) and Benjamin and Cornell
(1971).
In the following a basic introduction to Bayesian decision analysis is given.
The Decision Tree
The analysis of decision problems is greatly enhanced by visualisation of decision / event
trees. Consider as an example the decision / event tree illustrated in Figure 12.5.
In Figure 12.5, drefers to a decision, :ᅧ refers to an uncertain state of nature and u is the
utility associated with the decision and the uncertain state of nature.
The example considers the steel bar subject to tension loading. The engineer is faced with the
problem that the yield strength of the steel bar is uncertain and that it is required to increase
the tensile loading of the steel bar by 10%. Assume that the engineer has two choices possible,
namely to do nothing or to exchange the steel bar with a new one with a cross sectional area
10% larger than the original one. The consequence of failure is 100000$ and the cost of
strengthening is 1000$. If the steel bar is not strengthened the probability of failure will be
higher than if the steel bar is strengthened.
The task is now to analyse such decision problems in a way making consistent use of all the
information available to the engineer. This includes prior frequentistic as well as subjective