Closing the Duration Gap in RVT: The Energetic Duration

Rajesh Rupakhety, Victor Moises Hernández Aguirre 

Duration is a fundamental descriptor of earthquake ground motion, yet it remains ill-defined across engineering seismology, with numerous threshold-based measures adopted for specific applications. This ambiguity has allowed duration to function as a calibration parameter in analyses such as random vibration theory (RVT), where it is often adjusted to match observed response spectra. This paper introduces a new definition based on the participation ratio of the temporal energy distribution, yielding the energetic duration {\operatorname{D}\funcapply}^\ast. When applied consistently, {\operatorname{D}\funcapply}^\ast provides an objective measure of the quasi-stationary, peak-generating phase of ground motion and resolves the internal consistency problem of RVT. The framework is extended to a rotation-invariant formulation and to a duration spectrum {\operatorname{D}\funcapply}_R^\ast(T), providing a period-dependent duration consistent with oscillator response. The analysis further shows that conventional moment-based estimates of bandwidth and crossing rates are biased under finite-duration conditions, particularly when D/T is small. An empirical correction based on a sinusoidal baseline test, combined with time-domain estimation of zero-crossing rates, is introduced to mitigate these effects, improving the stability of RVT peak-factor calculations and reducing both bias and dispersion of predicted spectral ordinates. The proposed approach provides a physically grounded and analytically consistent definition of duration, clarifies its role within RVT, and supports its integration into ground-motion modelling and seismic hazard analysis.