General Form of Reliability Integrals

In probabilistic analysis, the following integral is commonly encountered  

For example, in reliability analysis, the quantity Q is considered as the probability that the structure with parameter vector 0 would fail, i.e., Q(0) = P(F 0, C). Then, the updated integral becomes the updated robust probability of failure for the structure, when it is subjected to some stochastic excitation [198]  

This updated robust probability of failure incorporates knowledge about 0 from C and from the updated information from the data. It is robust because the modeling uncertainties are taken into explicit account [198]. The updated structural reliability of the stmcture is P(5 P, C) = 1 - F(F D, C), where S denotes a safe status of the structure. 

Another example is the probability of damage. Damage may be quantified as the fractional reduction in the stiffness tfom that in the undamaged state of a substructure. Assume that the measurements of the structure are available for the undamaged and possibly damaged state, denoted by 27 and respectively. Then, the probability that damage in the /th substructure 

---