Energy Considerations for Estimating Displacements of Oscillators with Different Hysteresis Shapes

This paper evaluates the number of cycles of deformation, Nc, of single degree of freedom (SDOF) structures with different hysteresis loop shapes subjected to earthquake loading because this influences both the peak displacement and the damage in a structure. The open-source software OpenSEES was modified to allow flagshaped hysteresis loops ranging from elastic bilinear (with no energy dissipation) to traditional bilinear. To address the current inconsistency in the literature the concept of oscillation resistance ratio (ORR) has been introduced before. A relationship to estimate (ORR) for oscillators with different displacement ductility and a flag-shaped hysteresis loop is developed under free vibration, assuming no damping and no strain hardening. This relationship, initially developed using simple mechanics considerations, is then compared with that found from a time history analysis. The relationship is then extended to consider the effects of tangent stiffness proportional damping. The number of oscillation cycles, Nc, is defined as the number of post-initial elastic displacement excursions obtained during shaking to the number in one full cycle to the same peak displacement. Relationships between Nc and ORR were developed for structures with lateral force reduction factor, R, period, T, and damping assuming no strain hardening using a suite of 9 earthquakes records. It was found that estimated ORR for the mechanics-based method and time history analysis considering free vibration were identical for all flag shape loop shapes. For a damping ratio of 5%, the ORR obtained increased by less than 12% compared to 0% damping for the flag-shaped parameters greater than 0.2. As a result, the ORR for a loop with no damping was used in the remainder of the paper. In general, as the ORR decreased to less than 0.3 (i.e. the hysteresis loop had less energy dissipation), Nc increased. It was found that Nc was insensitive to the lateral force reduction factor, but increased for lower damping when ORR was less than 0.3. Also, it increased for shorter period oscillators when ORR was less than 0.3, but the trends were less clear with greater ORR