A compatible probabilistic framework for quantification of simultaneous aleatory and epistemic uncertainty of basic parameters of structures by synthesizing the change of measure and change of random variables

Uncertainty has been attached increasing importance in performance evaluation and reliability assessment of engineering structures. However, the logical framework for the quantification of simultaneous aleatory uncertainty and epistemic uncertainty of basic parameters of structures in a compatible probabilistic sense is still not readily available as yet, and the computational efforts are also usually prohibitively large. In the present paper, a compatible probabilistic framework is proposed for this purpose. Limited to the epistemic uncertainty that characterizing the uncertainty in aleatory uncertainty, i.e., the uncertainty in the shape or parameters of probability density function of the source random variables, it is found that the quantification and propagation of aleatory uncertainty is a problem of change of random variables, and the principle of preservation of probability holds. For dynamical systems the probability density evolution method (PDEM) can be adopted for this purpose. Whereas, the quantification of epistemic uncertainty is essentially a problem of change of probability measure, and thus the Radon-Nikodym theorem holds. Therefore, synthesizing the change of measure (COM) and the change of random variables (CRV) will provide a logically clear compatible framework for the quantification of simultaneous aleatory and epistemic uncertainties. The numerical algorithm by changing the assigned probabilities of representative points in the PDEM is then proposed. A nonlinear equation, the Riccati equation, is investigated to illustrate the proposed method. The result is verified by the exact analytical solution. Moreover, a 3-span 10-storey reinforced concrete (RC) frame structure modelled by the finite element method is studied. This exemplifies the quantification of simultaneous aleatory and epistemic uncertainties of basic parameters of real-world civil engineering structures. The examples demonstrate the effectiveness of the proposed method. Problems to be further studied are also outlined.