P-Δ Effect in Inelastic Seismic Demands

Structures are designed to dissipate input seismic energy by performing hysteresis cycles through repetitive inelastic excursions. P–Δ effect becomes prominent during inelastic seismic excursions and may considerably increase the ductility demand. Furthermore, ductility demand, prima facie, is regulated by lateral strength and hysteresis behaviour of structural material. In this context, this paper presents an attempt to capture inelastic seismic response incorporating P–Δ effect for single degree of freedom (SDOF) and multiple degrees of freedom (MDOF) systems with representative short period, medium period and long period structures. Four hysteresis behaviours, namely: (a) elasto-plastic, (b) only stiffness degrading, (c) only strength deteriorating and (d) both strength and stiffness degrading hysteresis behaviours are considered to cover all possible material characteristics. Rigorous nonlinear dynamic analysis is performed and the ratios of displacement ductility demands and overturning moments obtained incorporating P–Δ effect to the same obtained without P–Δ effect are computed for various response reduction factors (R). The results indicate that SDOF systems are relatively more vulnerable when compared to MDOF systems with the same fundamental periods. Further, long period systems are more vulnerable than short and medium period systems, irrespective of their degrees of freedom (DOF). The study further indicates that P–Δ effect may cause an instantaneous magnification of responses at some limiting values of R depending on material hysteresis behaviour. Such observation helps to fix the maximum limits of R to yield similar ductility demands intended by code-specified R values normally computed by ignoring P–Δ effect. The study as a whole aims to provide all round guidelines for modifying the codal provisions for performance-based design to incorporate the extra demand due to P–Δ effect. In the absence of consideration of the degrading behaviour so far, the applicability of the results in this area was restricted only to steel structures. The present study is an effort to overcome this limitation for enabling the design provisions to be extended for concrete structures as well.