The purpose of the Reliability-Based Design Optimization (RBDO) is to find the best compromise between safety and cost. Therefore, several methods, such as the Hybrid Method (HM) and the Optimum Safety Factor (OSF) method, are developed to achieve this purpose. However, these methods have been applied only on static cases and some special dynamic ones. But, in real mechanical applications, structures are subject to random vibrations and these vibrations can cause a fatigue damage. So, in this paper, we propose an extension of these methods in the case of structures under random vibrations and then demonstrate their efficiency. Also, a Robust Hybrid Method (RHM) is then developed to overcome the difficulties of the classical one. A numerical application is then used to present the advantages of the modified hybrid method for treating problem of structures subject to random vibration considering fatigue damage.

Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables. It requires at each iteration the evaluation of some robust measures of the objective function and the constraints. In a previous work, the authors have proposed a method which efficiently generates a design of experiments with respect to the design variable uncertainties to compute the robust measures using the polynomial chaos expansion. This paper extends the proposed method to the case of the robust optimization. The generated design of experiments is used to build a surrogate model for the robust measures over a certain trust region. This leads to a trust region optimization method which only requires one evaluation of the design of experiments per iteration (single loop method). Unlike other single loop methods which are only based on a first order approximation of robust measure of the constraints and which does not handle a robust measure for the objective function, the proposed method can handle any approximation order and any choice for the robust measures. Some numerical experiments based on finite element functions are performed to show the efficiency of the method.