Flexibility-based upper bounds on the response variability of simple beams

Spectral- and probability-distribution-free upper bounds on the response variability of both statically determinate and indeterminate beams are established in the present paper based on exact closed-form analytic expressions derived for the variance of the response displacement. A conjecture has to be made in the case of statically indeterminate beams in order to establish these bounds. The conjecture is supported through an argument postulating the existence of an integral form for the variance of the response displacement and through a brute-force optimization procedure providing numerical validation. Such bounds require knowledge of only the variance of the stochastic field modeling the inverse of the elastic modulus and are realizable in the sense that it is possible to fully determine the probabilistic characteristics of the stochastic field (modeling the inverse of the elastic modulus) that produces them. Furthermore, it is possible to fully determine also the corresponding stochastic field modeling the elastic modulus that produces these bounds. These spectral- and probability-distribution-free bounds can also be computed numerically using a so-called fast Monte Carlo simulation procedure that does not require a closed-form analytic expression for the response displacement, making this approach much more general. Numerical examples are provided involving a statically determinate and a statically indeterminate beam.