Criticality problems probability variables

For a single-channel problem, only one term in the sum in Eq. (56) is to be considered. On the other hand, it is not unusual for multichannel chemical reactions to have a multiexponential form of reaction probability. Reduction to a single exponential form can depend on individual cases under study. For our cluster system, in which various isotropic properties dominate due to the identical particle composition (see Ref. 39), it is quite likely that there are very many critical points that are topologically equivalent or energetically similar to each other. In these cases, averaging should work. In the two familiar means over variables (X i = 1,2,..., M ... 

The structural reliability problem seeks the estimation of the probability that a structure exceeds a critical state defined by a state function indexed by a vector of so-called basic variables X, which obeys a joint density function fx X). Hence, the problem is written as follows ... 

Then, the problem consists in the determination of a set of design variables that minimizes an objective function for a specified reliability. The failure probability function Pp( y ) accounts for the uncertainties in the system parameters as well as the uncertainties in the excitation. The failure probability is given in terms of the probability that some stochastic dynamic responses exceed in magnitude within a specified time interval [0, T] certain critical threshold levels, that is... 

After the identification of the critical sections using deterministic FEM calculation, the probability of the containment failure for the critical structure areas are considered for the various level of the overpressure. Most problems concerning the reliability of building structures are defined today as a comparison of two stochastic values, loading effects E and the resistance R, depending on the variable material and geometric characteristics of the structural element (Haidar and Mahadevan, 2000 Kr ik, 2009, 2013 Bazant et al., 2007 Kala, 2011 Nov et al., 2003). The variability of those parameters is characterized by the corresponding functions of the probability density fg (r) and fE(e). 

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