An equivalent damping ratio that accounts for ground-motion spectral shape for displacement-based design of flag-shaped hysteresis systems

Direct displacement-based design (DDBD) methodologies simplify the estimation of the seismic deformation demands of a nonlinear system by replacing it with an equivalent linear system that is characterized by an effective stiffness and equivalent damping ratio (EDR). The calculated EDR for flag-shaped hysteresis relationships vary greatly among existing DDBD methodologies, and these methodologies display a strong bias according to the shape of the ground-motion response spectrum. In the work reported here, the limitations of existing methods are demonstrated for three sets of ground-motions: a set of far-field ground-motions, a set of near-field ground-motions, and a set of physics-based simulated ground-motions that account for the effects of a deep sedimentary basin. The effect of spectral shape can be quantified by a new measure of spectral shape that is based on the spectral displacement ordinates between the system's initial and equivalent periods. To address the limitations of existing methodologies, a new equation for calculating EDR was developed based on numerous nonlinear history analyses for all three sets of ground-motions and for a wide range of systems with flag-shaped hysteresis. In comparison with existing equations, the proposed equation for calculating EDR improves the accuracy of the DDBD methodology by lowering the bias and dispersion of the ratio between the displacement calculated by nonlinear analysis and the displacement calculated with the equivalent linear system. The usage of the new EDR equation in DDBD requires iteration, but neither numerical modeling nor analysis is needed. Specific steps of the application are demonstrated for a system with a flag-shaped hysteresis.