Variance decomposition and global sensitivity for structural systems

Abstract

This paper applies the Sobol’ decomposition of a function of many random variables to a problem in structural mechanics, namely the collapse of a two story two bay frame under gravity load. Prior to introduction of this example application, the Sobol’ decomposition itself is reviewed and extended to cover the case in which the input random variables have Gaussian distribution. Then, an illustrative example is given for a polynomial function of 3 random variables. In the structural example, the Sobol’ decomposition is used to decompose the variance of the response, the collapse load, into contributions from the individual input variables. This decomposition reveals the relative importance of the individual member yield stresses in determining the collapse load of the frame. In applying the Sobol’ decomposition to this structural problem the following issues are addressed: Calculation of the components of the Sobol’ decomposition by Monte Carlo simulation; the effect of input distribution on the Sobol’ decomposition; convergence of estimates of the Sobol’ decomposition with sample size using various sampling schemes; the possibility of model reduction guided by the results of the Sobol’ decomposition.