On estimation of seismic damage from ductility and hysteretic energy demands in equivalent oscillators using linear response

Estimation of damage in a structure under anticipated seismic events is important for its performance-based design. This can be done in terms of the ductility and hysteretic energy demands for each of the anticipated plastic hinges under the anticipated ground motions. This study considers the possibility of quantifying the structural damage simply from the linear analysis based on the elastic design spectra of the ground motions and undamped mode shapes of the structure. A two-parameter model is first developed for the estimation of hysteretic energy demand in single-degree-of-freedom (SDOF) oscillators for five types of nonlinearities from the linear displacement peaks. The parameters of this model are estimated for various initial periods, nonlinearity types, and specific values of damping ratio, maximum possible ductility demand, and hysteretic parameters. Next, it is assumed that damage in each of the equivalent oscillators corresponding to different modes of vibration of the structure can be combined to quantify the structural damage. The hysteretic properties of these equivalent oscillators are estimated in the cases of 2-DOF and 3-DOF frames, and linear-peaks-based models for ductility demand and hysteretic energy demand are then used to estimate damage index for each of these oscillators. Finally, a combination rule is proposed to suitably combine these damage indices and thus estimate the extent of overall damage. A numerical study with the help of a suite of 100 ground motions illustrates how the proposed methodology estimates the damage levels of 2-DOF and 3-DOF example frames with strain-hardening bilinear and stiffness-degrading Riddell-Newmark type nonlinearities in the moment curvature relationships of their column sections.