Modal decomposition method for response spectrum based analysis of nonlinear and non-classically damped systems

A novel inelastic modal decomposition method for random vibration analysis in alignment with contemporary aseismic code provisions (e.g., Eurocode 8) considering non-classically damped and nonlinear multi degree-of-freedom (MDOF) systems is developed. Relying on statistical linearization and state-variable formulation the complex eigenvalue problem considering inelastic MDOF structural systems subject to a vector of stochastic seismic processes is addressed. The involved seismic processes are characterized by power spectra compatible in a stochastic sense with an assigned elastic response uniform hazard spectrum (UHS) of specified modal damping ratio. Equivalent modal properties (EMPs) of the linearized MDOF system, namely equivalent pseudo-undamped natural frequencies and equivalent modal damping ratios are provided. To this aim, each mode of vibration is assigned with a different stationary random process compatible with the excitation response spectrum adjusted to the corresponding equivalent modal damping ratio property. Next, an efficient iterative scheme is devised achieving convergence of the equivalent modal damping ratios and the damping premises of the excitation response spectrum corresponding to each mode of the system. Subsequently, the stochastically derived forced vibrational modal properties of the structure are utilized together with the appropriate mean response elastic UHS for determining peak nonlinear responses in modal coordinates. The modal participation factors are determined for the complex-valued mode shapes and generalized square-root-of-sums-squared (SRSS) is employed as the modal combination rule for determining the peak total responses of the system in physical space. The pertinency and applicability of the proposed framework is numerically illustrated using a three-storey bilinear hysteretic frame structure exposed to the Eurocode 8 elastic response spectrum. Nonlinear response time-history analysis (RHA) involving a large ensemble of stationary Eurocode 8 spectrum compatible accelerograms is conducted to assess the accuracy of the proposed methodology in a Monte Carlo-based context.