2. Modeling and Propagation of Uncertainties

The basic problem dealt with in this chapter consists in finding the distribution of a scalar quantity Y which is a function of a vector X of probabilistically qualified random quantities. The dependence of Y on X is not explicit, requiring algorithms for its determination. An approach that facilitates the solution of the problem was first adopted in problems of nuclear safety, and consists in expressing the probability of Y exceeding a given value (if Y has a monetary connotation), or a given structural Limit-State of a whole system, as a multiple integral of a Markov chain of conditional probability functions. This approach is described in this chapter with reference to buildings, but the approach can be equally applied to various industrial systems. A crucial step in the chain is the passage from the intensity of the action, in the present case the intensity of the ground motion, to the vector of the structural response variables on which the state of the system, and ultimately Y, depend. This passage involves consideration of the variability of the action, in the form of different samples of ground motion together with all the uncertainties inherent in the numerical determination of the response, which include those related to the selection of the structural model and the uncertainties of its parameters. A number of approximate techniques for dealing with this problem are presented, starting from the simple but inadequate FOSM method, to the approach based on the use of a Response Surface in the space of the structural variables, and concluding with the potentially more accurate Latin Hypercube Sampling technique, underlining however its practical limits due to the computational effort required for large number of uncertain quantities.