A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.