Stochastic Analysis of Fractional Structural Systems Subjected to Fractional Ground Motions

In this paper, the nonstationary stochastic response of fractional structural systems subjected to fractional excitations is examined. Fractional operators provide an effective mathematical construct to model energy dissipation and stiffness–damping coupling mechanisms that cannot be modeled by integer operators. In particular, the memory property of fractional operators makes them suitable for treating the non-Markovian features exhibited by systems with path-dependent responses, such as the damping and stiffness mechanisms in structural systems and soils under dynamic loading. In this paper, particular attention is devoted to fractional structural systems excited by a proposed stochastic ground motion model consisting of a nonstationary filtered fractional Kanai–Tajimi process, thus accounting for path-dependent energy dissipation in both the structure and the seismic waves’ propagating medium. The work attempts to elucidate the sensitivity and effects on probabilistic structural response predictions due to non-Markovian features in the excitation when both the system and the excitation are modeled using fractional operators. To analyze the fractional operators describing the structure and soil, the operators are discretized and approximated as the superposition of the solutions of a system of linear first-order ordinary differential equations, which, together with the system dynamics equations, are transformed into an equivalent first-order linear state-space model. The stochastic response evaluation for the system is conducted in closed form by solving the associated Lyapunov covariance equation. The analytical results from stochastic analyses are juxtaposed with pertinent Monte Carlo simulations to demonstrate the accuracy of the proposed method. Validation results using measured data from the 2023 Turkey–Syria earthquake demonstrated that the recorded ground motions exhibit fractional features with an energy rate of decay in the spectral density that is captured more accurately by the proposed fractional model than the traditional integer Kanai–Tajimi model.