Fuzzy reliability
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In the case of uncertainty structural safety or reliability can be expressed by fuzzy failure probability, that is usually time-dependent.
In the general case that the structural reliability is influenced by nonlinear structural behavior and the resistence depends on the stress process the time-dependent fuzzy failure probability is defined as
(1)
Eq. (1) means, that the fuzzy failure probability has to be computed in the space of the fuzzy random basic variables. This space is constructed with the aid of one-dimensional fuzzy random variables obtained by discretization of all fuzzy random functions. The fuzzy joint probability density function
is defined as assessed set of trajectories dependent on the fuzzy bunch parameters 
. 
(2)
Furthermore, the space of the fuzzy random basic variables is subdivided into a fuzzy failure domain and a fuzzy survival domain by the fuzzy limite state surface 
.
. 
(3)
The fuzzy failure probability is then the solution of the fuzzy integral Eq. (1) by integrating of the fuzzy joint probability density function over the fuzzy failure domain. The computation of the fuzzy failure probability at different time points τi results a set of fuzzy variables. This fuzzy variables are functional values of a fuzzy process on the fundamental set Τ, i.e. the time-dependent fuzzy failure probability is expressed by a fuzzy function.
The numerical computation of Eq. (1) requires the bunch parameter representation and bases on the new methods fuzzy Monte-Carlo simulation and fuzzy adaptive importance sampling. Numerical procedures and example are shown in Sickert et al. (2005) and here.
In the special case of linear structural behavior the fuzzy failure probability can be computed with the aid of the fuzzy random load process
and the fuzzy random loadability process 
(4)
The fuzzy random load and the fuzzy random loadability depend on the time-dependent uncertain loading, geometrical, and material parameters, which are described by fuzzy random functions, random processes, and fuzzy functions. The parameter τ represents the time. The evaluation of eq. (4) requires time-discretization of and . At each time point τk, and are obtained as fuzzy random variables which can be described by fuzzy probability density functions. Using these fuzzy probability density functions the fuzzy failure probability can be computed. In Figure 1 a fuzzy random resistance process and a fuzzy random stress process are shown. The discontinuity of the fuzzy random resistance process at time point τSt results for example from rehabilitation or strengthening. As a consequence, the structural resistance increases.
and the fuzzy random loadability depend on the time-dependent uncertain loading, geometrical, and material parameters, which are described by fuzzy random functions, random processes, and fuzzy functions. The parameter τ represents the time. The evaluation of eq. (4) requires time-discretization of and . At each time point τk, and are obtained as fuzzy random variables which can be described by fuzzy probability density functions. Using these fuzzy probability density functions the fuzzy failure probability can be computed. In Figure 1 a fuzzy random resistance process and a fuzzy random stress process are shown. The discontinuity of the fuzzy random resistance process at time point τSt results for example from rehabilitation or strengthening. As a consequence, the structural resistance increases.
Figure 1. Fuzzy random stress-resistance-representation in time τ
References
- Möller, B, Graf, W, Sickert, J, and Beer, M (2006) Time-dependent reliability of textile strengthened RC structures under consideration of fuzzy randomness, Computers and Structures 8-9(84):585–603.
- Sickert, J, Graf, W, and Reuter, U (2005) Application of fuzzy randomness to time-dependent reliability, In: Safety and Reliability of Engineering Systems and Structures Proceedings of the 9th Int. Conference on Structural Safety and Reliability, edited by Augusti, G.; Schueller, G.I. and Ciampoli, M.. Millpress, Rotterdam, pages 1709–1716.
- Möller, B, Graf, W, and Beer, M (2003) Safety assessment of structures in view of fuzzy randomness, Computers and Structures 81:1567–1582.
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