An approximate stochastic dynamics approach for design spectrum based response analysis of nonlinear structural systems with fractional derivative elements

A novel approximate approach is developed for determining, in a computationally efficient manner, the peak response of nonlinear structural systems with fractional derivative elements subject to a given seismic design spectrum. Specifically, first, an excitation evolutionary power spectrum is derived that is compatible with the design spectrum in a stochastic sense. Next, relying on a combination of statistical linearization and stochastic averaging yields an equivalent linear system (ELS) with time-variant stiffness and damping elements. Further, the values of the ELS elements at the most critical time instant, i.e., the time instant associated with the highest degree of nonlinear/inelastic response behavior exhibited by the structural system, are used in conjunction with the design spectrum for determining approximately the nonlinear system peak response displacement. The herein developed approach can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the governing equations of motion. Furthermore, the approach exhibits the significant novelty of exploiting the localized time-dependent information provided by the derived time-variant ELS elements. Indeed, the values of the ELS stiffness and damping elements at the most critical time instant capture the system dynamics better than an alternative standard time-invariant statistical linearization treatment. This leads to enhanced accuracy when determining nonlinear system peak response estimates. An illustrative numerical example is considered for assessing the performance of the approximate approach. This pertains to a bilinear hysteretic structural system with fractional derivative elements subject to a Eurocode 8 elastic design spectrum. Comparisons with pertinent Monte Carlo simulation data are included as well, demonstrating a high degree of accuracy.