Model Uncertainty for Bilinear Hysteretic Systems

In structural reliability analysis at least three types of uncertainty must be considered, namely physical uncertainty, statistical uncertainty, and model uncertainty (see e.g. Thoft·Christensen & Baker [1)). The physical uncertainty is usually modelled by a number of basic variables. The statistical uncertainty -due to lack of information can e.g. be taken into account by describing the variables by predictive density functions, Veneziano [2). In general, model uncertainty is the uncertainty connected with mathematical modelling of the physical reality. When structural reliability analysis is related to the concept of a failure surface (or limit state surface) in the n-dimensional basic variable space then model uncertainty is at least due to the neglected variables, the modelling of the failure surface and the computational technique used. A more precise definition is given in section 2, where some different methods to treat model uncertainty are described. In section 3 a new method based on subjectively modelled conditional density functions is presented. It is shown that in some special cases this method is equivalent to existing more simple methods. In the analysis of dynamically loaded structures it is often assumed that the loading and the response can be modelled by stationary stochastic processes. Further, it is assumed that the structures can be modelled by non-linear systems showing hysteresis. This non-linear behaviour is essential to the design procedure from an economic and reliability point of view. In section 4 it is shown how the probability of failure of a simple bilinear oscillator can be estimated and in section 5 it is demonstrated by numerical examples how model uncertainty can be included in the calculations.