# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 1 Introduction  
Reinforced concrete (RC) framed structures, especially those conceived and designed in compliance with old technical standards and characterized by poor construction details, are notoriously vulnerable to seismic excitation, as demonstrated by the observed response during past earthquakes [1; 2]. The increasing need to carry out seismic vulnerability assessments on existing RC buildings over the last decades has motivated the development of a variety of numerical models and hysteretic rules that are accurate enough to describe the inherent material nonlinearity and, consequently, the seismic response of the RC structures [3; 4]. In the literature, two distinct strategies for modelling the nonlinear behavior of the RC elements have been proposed, namely the concentrated plasticity and the distributed plasticity approach [5; 6]. The first approach entails less computational effort but requires a preparatory calibration of the moment-curvature relationships of the plastic hinges (having an _a priori_ specified location in the RC structure) based on the cross-sectional characteristics and reinforcement details. On the other hand, in the distributed plasticity approach, the structural response of the RC elements is determined by integrating the stress-strain relationships of several fibers distributed over the crosssection that contribute to the calculation of the element stress resultants. Although this second approach is undoubtedly more accurate and general than the first formulation, it is unavoidably associated with higher computational cost because of the generally high number of internal fiber equations involved, depending on the discretization of the cross section. Furthermore, in most RC structures, one can reasonably anticipate the locations where the inelastic behavior is expected during a severe seismic excitation caused by the increasing lateral-load-induced stresses, such as the terminals of beam-column members. Based on this consideration, in recent years, a hybrid strategy between these two formulations has been developed and increasingly adopted, in which the element is assumed to be linear elastic for a major portion of its length, while the inelasticity remains concentrated over a specified hinge length at the element terminals--the so-called fiber-hinge model [7]. This approach, which inherently combines the accuracy of plasticity-based models and the computational advantages of concentrated plasticity formulations, is considered in this work to investigate the seismic response and recentering behavior of the RC frames within a broad parametric study. Modeling the nonlinear behavior of the RC structures under cyclic (seismic) loading is also highly affected by the hysteretic rules. Elasto-plastic rules, commonly adopted for steel members, are not suitable for reinforced concrete. Likely, the oldest and most well-known flexural hysteretic rule for RC members is that developed by [PERSON] (degrading stiffness model [8] and modified versions [9]), in which the stiffness was changed over repetitions of the cycles to simulate the lower energy dissipation capacity due to damage accumulation. Subsequently, other hysteretic models were proposed with different levels of versatility and accuracy, e.g., to include biaxial bending effects, variable column axial loads, and asymmetric sections [10]. In 1998, [PERSON] et al. [11] developed the so-called _pivot hysteresis model_ for the force-displacement or moment-rotation response of the RC elements, in which the effects of the cyclic axial load, strength degradation, and asymmetric cross section were automatically incorporated. This versatile model, governed by three parameters and able to mimic very complex hysteretic behaviors, was originally developed for RC circular bridge columns, and its main tuning parameters were linked to the axial load ratio and longitudinal steel ratio via design charts; subsequently, [PERSON] et al.