Directional sampling based reliability sensitivity analysis for independent normal variables problem is extended for the reliability sensitivity analysis involving correlated random variables. For the reliability and reliability sensitivity problem involving correlated random variables, independent normal transformation techniques, including Nataf transformation or Copula functions, are firstly employed before the implementation of directional sampling. And then the reliability and sensitivity are estimated by the directional sampling in the independent standard normal space. Employing the equivalently transformation techniques between correlated random variables and independent normal ones, the reliability sensitivity of failure probability with respect to the distribution parameters of correlated random variables can be estimated by the chain rule of derivative finally. After simple numerical example is used to demonstrate the validity and feasibility of the presented extended method, it is employed to analyze the reliability and reliability sensitivity for the aeroengine turbine blade with correlated random variables. From the results of examples, it is determined the important parameters with large influence on the reliability, and the presented method can significantly reduce the computational cost than that of classical Monte Carlo simulation. The Nataf transformation can give the equivalently transformation when the random variables are correlated normal distributed, and the Nataf transformation is a way to model the dependence structure of a random vector by a normal copula, parameterized by its correlation matrix.
To measure effects of the distribution parameters of input variables on the output response of engineering structures, the analysis methods are investigated to solve the sensitivity of the cumulative distribution function of the output response with respect to the input parameters. For a linear input–output response model with independently normal input variables, the properties of the cumulative distribution function sensitivity are analytically derived. For a complicated input–output response model, a novel subset simulation-based method is presented to solve the response cumulative distribution function sensitivity. By using the stratified subset Markov Chain simulation in the subset simulation-based method, the response cumulative distribution function sensitivity can be adaptively obtained at the threshold of each subset. The sensitivity with larger cumulative distribution function value can be treated as the byproduct of those with the smaller ones in the presented subset simulation-based method, thus greatly reducing the computational cost. Several engineering examples are analyzed by the subset simulation-based method to get the response cumulative distribution function sensitivities, and the comparisons in the example results show that the presented subset simulation-based method is more efficient than Monte Carlo simulation with acceptable precision.
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