Response variability of stochastic frame structures using evolutionary field theory

The integral form for the variance of the response of stochastic statically indeterminate structural systems involving the so-called variability response function (VRF) and the spectral density function of the stochastic field modelling the uncertain system properties is established for the first time in this paper using evolutionary spectra theory. The VRF is a function depending on deterministic parameters related to the geometry, boundary conditions, (mean) material properties and loading of the structural system. No approximations are involved in the derivation of the integral form. However, a conjecture has to be made that is validated using Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). Closed-form expressions can be derived in principle for the VRF of any statically determinate or indeterminate frame system using a flexibility-based formulation. Alternatively, a fast Monte Carlo simulation approach is provided to numerically evaluate the VRF. It is shown in closed-form and numerically that the VRF for statically indeterminate structures is a function of the standard deviation σff of the stochastic field modeling the inverse of the elastic modulus. Although the VRF depends on σff, it appears to be independent of the functional form of the spectral density function of the stochastic field modeling the uncertain system properties. For statically determinate structures, the VRF is independent of σff. The integral form can be used to compute the variance of the system response as well as its upper bound with minimal computational effort. It also provides an excellent insight into the mechanisms controlling the response variability. The upper bounds for the response variance are spectral- and probability-distribution-free requiring knowledge of only the variance of the inverse of the elastic modulus. The proposed bounds are realizable in the sense that it is possible to determine the probabilistic characteristics of the stochastic field that produces them. Several numerical examples are provided demonstrating the capabilities of the methodology.