P-Delta effects in displacement based assessment of R.C. hinged frames

In the last two decades, several performance based design approaches were developed and recently applied to the seismic assessment of existing structures. Among these procedures, increasing attention was placed on displacement based assessment methodology, which considers structural displacements, in terms of inter-story and roof drift, and material strain limits as the main seismic vulnerability indicators. This approach is based on the substitute structure theory and the structural response is evaluated by means of an equivalent single degree of freedom (SDOF) system, which accounts for the inelastic behaviour of the building using an effective stiffness and an Equivalent Viscous Damping (EVD). In this context, the appropriate definition and response evaluation of the equivalent SDOF system is fundamental, as it significantly affects the results. During an earthquake, the horizontal displacements achieved by the structure can significantly increase P-delta effects, especially in the case of laterally deformable buildings, as the R.C. hinged frames considered herein. The displacement based assessment procedure should take into account the contribution of second order effects, being P-delta effects influencing the SDOF system both in terms of shear-displacement relationship and in terms of EVD. In the literature, several EVD formulations are available, accounting for the non-linear force-displacement SDOF response associated to various hysteretic models: in the case of R.C. frame structures the Takeda model is commonly adopted with a post-yield stiffness ratio of 0.05 typically selected. Considering that P-delta effects reduce the post-yield stiffness of the force-displacement curve, the aforementioned formulations could underestimate both the equivalent viscous damping values and the effective stiffness at maximum displacement adopted in the assessment procedure. A parametric study is carried out to investigate the effective EVD accounting for P-Δ effects. The study is based on the evaluation of the dynamic response of a series of non-linear SDOF systems changing the post-yield stiffness ratio, while keeping constant the displacement ductility and the effective period. According to the parametric analyses results, new formulations are proposed in order to include P-delta effects in the common displacement based assessment procedure.