The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters

A new approach for evaluation of the extreme value distribution and dynamic reliability assessment of nonlinear structures with uncertain parameters is proposed. The approach is established based on the thoughts of the newly developed probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) and its evolution of the responses of nonlinear stochastic structures. In the proposed method, a virtual stochastic process associated to the extreme value of the studied stochastic process is firstly constructed in such a way that the extreme value equals the value of the virtual stochastic process at a certain “instant of time”. The probability density evolution method is then employed to evaluate the instantaneous PDF of the virtual stochastic process. This will yield the PDF of the extreme value as a natural byproduct. After that, dynamic reliability could be evaluated from the extreme value distribution, instead of the level-crossing process theory. A simple integration in terms of the extreme value distribution over the safe domain will give the dynamic reliability, requiring neither the joint PDF of the response and its velocity, nor the assumptions on properties of the level-crossing events. Numerical algorithm is outlined. Two examples, of which one is to capture the extreme value distribution of the random sampling, the other is to evaluate the extreme value distribution and dynamic reliability of a nonlinear stochastic structure, are studied. Some features of the instantaneous PDF and its evolution, the extreme value distribution and the dynamic reliability are discussed. The investigations indicate that the proposed approach is of versatility, accuracy and efficiency.