Review of the concept of variability response function and its application in stochastic systems

Accurate stochastic analysis of structures is often complicated by the need for detailed probabilistic information about the random spatial variation of the underlying material/geometric properties. For many relevant properties, such as the flexibility or the elastic modulus, it is often possible to determine only their mean and standard deviation from the available measurement data. On the other hand, the majority of available stochastic structural models require knowledge of both the marginal probability distribution function and power spectrum (correlation function) of the stochastic field describing the uncertain system properties. The concept of Variability Response Function (VRF) emerged more than 35 years ago as an alternative to such a full stochastic analysis. The VRF can accomplish the following at a minimal computational cost: (i) establish realizable upper bounds on the random response variability based only on the mean and variance of the system properties, (ii) compute the response variance for a given power spectrum, (iii) perform a complete sensitivity analysis of the response variance with respect to the form of the power spectrum modeling the uncertain material/geometric properties, (iv) provide valuable insight into how different wavenumbers/wavelengths/scales of fluctuation contribute toward the overall value of the response variance. Since its initial inception for the response displacement of one-dimensional linear elastic structures, the VRF concept has been expanded to address displacements, internal forces, eigenvalues, and homogenized (effective) properties of structures in multiple dimensions, with multiple stochastic material properties, exhibiting nonlinear elastic constitutive behavior, and having large stochastic variations in their properties. Given the long timespan and the large body of work on VRFs, this paper provides a much-needed overview of all these previous developments that should prove useful to researchers seeking to develop VRF methods further or apply the approaches to practical engineering problems.