Damping-Ductility Relationships for Flag-Shaped Hysteresis

Abstract

Damping-ductility relationships, as applied in direct displacement-based design (DDBD), are products of equivalent linearization techniques used to estimate the nonlinear response of structures. Currently, suitable relationships are lacking for an increasingly wide range of flag-shaped hysteresis that is possible in practice. This paper seeks to fill that gap. Two commonly used numerical procedures are shown to be equivalent in terms of the equivalent viscous damping ratio (EVDR) produced. From approximately 1 million damping-ductility datapoints, an expression for the EVDR is proposed for flag-shapes defined by four parameters: the fundamental period, loading-to-initial stiffness ratio, unloading-to-loading stiffness ratio, and ductility. Compared to Jacobsen’s curves, the EVDRs obtained are more sensitive to the first two parameters, and this is reflected in the proposed expression. As Jacobsen’s EVDR assumes resonant conditions, the period-dependency is ignored and potentially underestimates the EVDRs for short periods (and vice-versa). A DDBD example demonstrates the design of a flag-shaped hysteresis using the proposed expression. The results of this study offer a flexible damping-ductility formula for the displacement-based design of self-centering seismically resilient structures.