Transient response of stochastic finite element systems using Dynamic Variability Response Functions

Abstract

In this study a methodology is presented for effective analysis of dynamic systems with stochastic material properties. The concept of dynamic mean and variability response functions, recently established for linear stochastic single degree of freedom oscillators, is extended to general finite element systems such as statically indeterminate beam/frame structures and plane stress problems, leading to closed form integral expressions for their dynamic mean and variability response. The integrand of these integral expressions involves the spectral density function of the uncertain material properties and the so called dynamic mean and variability response functions respectively, which are assumed to be deterministic, i.e. independent of the power spectrum as well as the marginal pdf of the uncertain parameters. A finite element method-based fast Monte Carlo simulation procedure is used for the accurate and efficient numerical evaluation of these functions. In order to demonstrate the validity of the proposed procedure, the results obtained using the aforementioned integral expressions are compared to brute-force Monte Carlo simulation. As a further validation of the assumption of independence of the variability response function to the stochastic parameters of the problem, the concept of the generalized variability response function was applied and compared to the steady state dynamic variability response function. The methodology is applied in a dynamically loaded statically indeterminate beam/frame structure and a plane stress problem. The dynamic mean and variability response functions, once established, can be used to perform sensitivity/parametric analyses with respect to various probabilistic characteristics involved in the problem (i.e., correlation distance, standard deviation) and to establish realizable upper bounds on the dynamic mean and variance of the response, at practically no additional computational cost.