Probabilistic procedures of reliability and lifetime assessment

The so far used deterministic calculation is based on the safety factor method (application of the coefficient 1.5) together with the method of extreme values (Design values, which enter into calculations, are determined as the extreme ones according to different criteria). The method guarantees certain structure safety, however there is lack of any safety margin. This is the main reason why the deterministically prescribed lifetime is not possible to exceed. Reduction of the safety coefficient 1.5 cannot be actually quantified in terms of safety change. Least of all, we are not able to specify extreme value changes, which, in itself, occur with very low probability and specification of this probability, is usually missing.

Generally respected view is that the behaviour of real structures is given by influence of many factors, which have random (stochastic) nature. Even if the lifetime is determined by any deterministic method there is a certain probability of failure, which is not expressed explicitly only. That is why the use of probabilistic methods seems to be an efficient tool in the lifetime and reliability assessment. These methods do not only describe the behaviour of structures more realistic but also provide a general safety margin– probability that the limit state is not reached. For formal reasons, the complement on one is used; i.e. probability (risk) of limit state reaching (for example crack initiation or instability, creep rupture).

Probability of crack initiation as a function of operational time

In itself, the quantification of the margin of safety (risk) is not the most essential aim, the main motivation is possibility of safety margin application. For example, if the lifetime is determined by a procedure with a reduced safety factor the probabilistic calculation is able to assess the influence of the safety factor change on the failure risk. However, the use of probabilistic methodology indicates more effective applications, namely lifetime extension above the deterministically determined limit.

The result of a probabilistic based calculation is the probability (risk) of a limit state reaching during operational time. If there is no failure after certain operational time (diagnostic testing is negative), it is possible to use this fact to recalculate the risk of limit state reaching in the next operational interval (a posteriori risk) or to determine the operational interval in which the failure risk is equivalent to the previous one. The results can be plotted in graph which shows dependence of failure risk on operation time.

Failure rate prediction as a function of operational time

The field where the deterministic methodology cannot be applied at all is the failure rate prediction in structures with allowable failures (for example pipe systems of heat exchanging surfaces). In a probabilistic calculation, it is possible to predict mean time between failures, probabilities and time tolerance bands of failure origin and to determine operational reliability indicators as a function of operating time.

Moreover, the probabilistic methods provide the possibility of a lifetime objectification – in connection with econometric procedures. The maximum of a function C(t) determines optimized lifetime t_opt.

C(t) = -Z + I(t) - O(t) - H.R(t)

Z is basic investment in plant construction, I(t) is income, O(t) are operating expenses for operational time t, H are expenses arising as a consequence of a failure and R(t) is probability of failure origin during operational time t.

In probabilistic lifetime extension, the need of diagnostic testing was mentioned. A material diagnostics is also useful as a part of testing. For example, the creep strength of steel 0.5Cr0.5Mo0.3V can be classified according to the current yield stress. After the current yield stress determination (and following creep strength classification), a mathematical submodel can be applied instead a general model which covers all strength categories. It results in a reduction of the creep-strength scatter of the used material model and in more precise lifetime and risk prediction.

Ekonometric optimization of failure probability Probability of crack initiation as a function of operational time (influence of revised prediction)