The uncertainty model for the yield stress and the limit state equation comprise the
probabilistic model for the assessment of the safety of the steel bar.
The consequence for the present example is measured in terms of the annual probability of
failure Pf, the costs of collecting information about the yield stress and the costs of
strengthening the steel bar.
CI
CS
New information may be obtained by testing the ultimate yield stress of the steel bar.
Repeated tests of the same steel material will result in different results. This is partly due to
statistical uncertainty introduced by random fluctuations caused by e.g. the accuracy of the
testing device and the testing procedure itself. However, also inherent physical variations in
the yield stress of the steel will influence the results. Given a test result the a-priori
uncertainty model of the steel yield strength can be updated and an a-posteriori uncertainty
model of the yield strength can be established.
The first step in the reassessment is to establish whether the annual failure probability for the
steel bar is acceptable based on the available prior information. If not, it must be investigated
how a sufficient safety for the steel bar is achieved at the lowest costs. This type of analysis is
referred to as a prior decision analysis.
In practice one would plan and perform a number of tests and if on the basis of the n tests
results fˆˆˆ ˆyyy yn(ff f 12 , ,.., )T it can be shown that the failure probability satisfies the given
requirements no further action is needed. If on the other hand the results of the tests lead to the
opposite result, either more tests or a strengthening of the steel rod must be performed such
that the requirement to the annual failure probability PfT is satisfied. This type of analysis is
referred to as a posterior decision analysis, posterior because it is performed after the test
results are obtained.
Finally, it is of significant interest to be able to plan the number of tests such that the
requirement to the annual failure probability is fulfilled and at the same time the overall costs
including the costs of testing and costs of strengthening are minimised. In some cases it is
relevant to include the maximum acceptable annual probability of failure in the problem as a
decision variable and this is readily done if the costs of failure are included in the overall
costs.
The general idea behind this type of analysis is to perform posterior analysis for given test
plans even before the results on the tests have been obtained and to assume that the results of
the tests will follow the prior probabilistic model. This type of decision analysis, which is the
most advanced, is often referred to as a pre-posterior analysis.
The above example, which will be revisited in the subsequent sections, points to a number of
the most important issues when considering reliability based reassessment of structures. These
are:
Formulation of prior uncertainty models.
Formulation of limit state functions.
Establishing posterior probabilistic models.