Normalized energy-based methods to predict the seismic ductile response of SDOF structures

In this paper, normalization procedures for simple rectangular pulse and sine-wave ground excitations are proposed. Normalized hysteretic energy spectra are then developed for a simple SDOF system subjected to these simple excitations, and studied to determine how the seismic inelastic cyclic response is expressed in these spectra. The influence of damping on these spectra is also investigated. It is found that the selected energy normalization methods, one using maximum ground velocity square and structural mass as a normalization basis, the other using structural yield strength and displacement, both produce useful dimensionless energy values. Then, the applicability of these simple normalization methods is studied for systems subjected to real earthquakes. Prediction of hysteretic energy using the previously derived pulse spectra is attempted statistically by considering earthquakes as a sequence of equivalent rectangular pulses. It is found that the normalized predicted hysteretic energy can be easily obtained for actual earthquake excitations by: firstly, converting these earthquakes into equivalent pulses; secondly, summing the values read for each pulse from the normalized hysteretic energy spectra constructed for simple rectangular pulse or sine wave excitations; and finally, adjusting the total values by ratio spectra or equations statistically calibrated against a number of real earthquake records. This simple and rapid procedure allows direct and reliable prediction of hysteretic energies without the need to resort to complex and time-consuming step-by-step nonlinear inelastic time-history analyses.