Variability response functions for statically indeterminate structures

The response variability of statically indeterminate beam structures caused by uncertain material properties modeled by random fields is studied in this paper. By solving the governing equations of the statically indeterminate structure, the response bending moment along the length of the beam, M(x), is expressed as a function of its (random) zero-moment location. This approach, combined with a second-order Taylor series expansion of the random zero-moment location, leads to novel Variability Response Function-based integral expressions for the variance of the response bending moment, Var[M(x)]. Variability Response Functions have numerous desirable attributes including the capability to perform a full sensitivity analysis of the response variability with respect to the spectral characteristics of the random field modeling the uncertain material/system properties and establishing realizable upper bounds of the response variability. The proposed expressions are fundamentally different from all currently existing expressions that are based inherently on first-order Taylor series expansions, as they involve: (i) two distinct integrals in the expression for Var[M(x)] and, (ii) an expression for the mean value ɛ[M(x)] that depends on the stochastic field modeling the uncertain material properties. In general, the proposed second-order Taylor series-based formulation is expected to provide more accurate results. Extensive numerical examples are provided where the accuracy of the results obtained using the proposed formulation is validated using Monte Carlo simulations involving stochastic fields that follow truncated Gaussian and shifted lognormal probability distribution functions. These Monte Carlo simulation results indicate that the proposed Variability Response Functions are probability-distribution-independent.