Influence of Hysteretic Behavior on Equivalent Period and Damping of Structural Systems

Relations for the equivalent period $T_{\mathrm{eq}}$ and equivalent damping ratio ${\zeta }_{\mathrm{eq}}$ for the equivalent linear system approach for seismic design are proposed. These relations are based on optimal $T_{\mathrm{eq}}$ and ${\zeta }_{\mathrm{eq}}$ values derived from extensive time-history analyses of single-degree-of-freedom systems with six types of hysteretic behavior and a period range of 0.1 to 1.5 s. The proposed relations are more general (in their applicability to a wide range of hysteretic systems) and have stronger ties to physical behavior compared to existing empirical methods for evaluating $T_{\mathrm{eq}}$ and ${\zeta }_{\mathrm{eq}}\text{.}$ The proposed relations address differences in equivalent stiffness resulting from different hysteretic behavior. Simulations show that the optimal $T_{\mathrm{eq}}\text{,}$ associated with the stiffness of the partial load-displacement cycles of a hysteretic system, is the highest for the slip system (with a zero unloading/reloading stiffness) and the lowest for the bilinear elastic system (with an elastic unloading/reloading stiffness). The optimal ${\zeta }_{\mathrm{eq}}\text{,}$ associated with the energy dissipation capacity of a hysteretic system, is the highest for the elastoplastic system and the lowest for the bilinear elastic system.