Number of Equivalent Stress Cycles for Liquefaction Evaluations in Active Tectonic and Stable Continental Regimes

The number of equivalent cycles ($n_{eq}$) concept plays an important role in geotechnical earthquake engineering and underlies the accounting for ground-motion duration in simplified liquefaction evaluation procedures, whether explicitly or implicitly. In this regard, several $n_{eq}$ correlations have been proposed over the years that were developed using a similar variant of the Palmgren–Miner (P–M) fatigue theory. The correlations presented herein were developed using an alternative implementation of the P–M theory that better accounts for the nonlinear response of the soil and for multidirectional shaking. The proposed correlations are for shallow crustal earthquakes in both active tectonic and stable continental regimes. Additionally, two forms of the correlations are presented, one being expressed as a function of peak ground acceleration ($a_{\text{max}}$). This relation shows a strong negative correlation between $a_{\text{max}}$ and $n_{eq}$, implying that motions with high amplitudes have short durations and vice versa. This negative correlation is not accounted for in most previously proposed $n_{eq}$ correlations, which could result in the erroneous weighting of the unlikely scenarios of high amplitude–longer duration and low amplitude–short duration motions in liquefaction hazard studies.