Control Variates Method to Estimate Stochastic Buckling Loads

Buckling is the most significant failure mode for thin-walled structures. In particular, geometric imperfections have a major influence on the buckling behavior. These spatially correlated imperfections are inherently random and can be modeled using random fields. Therefore, computationally expensive probabilistic buckling analyses have to be performed. For some structures, a linear pre-buckling behavior can be observed. In this case, the stability point can be calculated with a linear buckling analysis, which is widely used in engineering practice. However, the results of linear buckling analyses strongly differ from the correct buckling load in the case of a non-linear pre-buckling behavior. Then, a non-linear buckling analysis is required, which is computationally expensive for probabilistic safety assessments based on Monte Carlo simulations. This paper aims to estimate the second-order statistics of buckling loads for thin-walled structures exhibiting strongly non-linear pre-buckling behavior. The estimation leverages existing correlations between the outcomes of linear and non-linear buckling analyses. The proposed approach utilizes the framework of Control Variates, wherein the more expensive analysis (non-linear buckling analysis) is run a few times only, while the cheaper linear buckling analysis is run a considerable number of times. The proposed method is demonstrated on a variety of structures, including a folded plate with multiple types of stability points, a composite shell panel, and a cylinder with random geometric imperfections. In these numerical examples, stochastic buckling analysis using Control Variates is approximately 1.5 to 2.6 times faster than classical Monte Carlo simulation.