Ductility-strength and strength-ductility relations for a constant yield displacement seismic design procedure

The modern engineering approach to design of structures exposed to rare but intense earthquakes allows for their inelastic response. Models and tools to rapidly but accurately assess the extent of the inelastic response of the structure and control its performance are, therefore, essential. We develop a closed-form $\mu -R^{\ast }-S_{d,y}$ relation between the ductility $\mu$ and the strength reduction factor R*, as well as its approximate inverse R*-$\mu$-S_d,y relation, both functions of the SDOF oscillator yield displacement S_d,y, not its vibration period T. The fundamental vibration period of the structure varies during the iterative design process focused on modifying its strength. However, the yield displacement of the structure is practically invariant with respect to the strength of the structure, as it depends primarily on its geometry and material properties. We use these relations to formulate a constant yield displacement seismic design procedure and exemplify it. Noting the structure of the developed relations, we use dimensional analysis to formulate a version of the ductility-strength and strength-ductility relations that are dimensionless and independent of the seismic hazard intensity. These novel, dimensionless master relations are the $\mu$-R*-H/B-$\kappa$ ductility-strength and the R*-$\mu$-H/B-$\kappa$ strength-ductility relations.