# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 2 Modeling Assumptions  
In this section, the modeling strategy adopted to perform the parametric study on the RC frames is illustrated. A fiber-based finite-length plastic hinge element is utilized for both the RC beams and the column elements. This element, pioneered by [PERSON] [13] and implemented in various software packages, such as OpenSees [14] and SeismoStruct [15], combines the accuracy of the distributed plasticity approaches with the computational advantages of the concentrated plasticity formulations. In particular, the present work is based on the so-called _fiber-hinge model_ (Fiber P-M2-M3) implemented in the structural analysis software SAP2000 [16]. This formulation describes the axial behavior of several axial fibers, each characterized by position, tributary area, and stress-strain curve, distributed along the cross section of the frame elements. The inelastic behavior is concentrated within a pre-specified length of the frame element, assumed here to be equal to the plastic hinge length \(L_{pl}\) for reinforced concrete members calculated according to the Italian Building Code [17; 18] expression:  
\[L_{pl}=0.1L_{V}+0.17h+0.24\frac{d_{bl}f_{y}}{\sqrt{f_{c}}} \tag{1}\]  
where \(L_{v}\) is the moment/shear ratio of the frame element, assumed to be approximately equal to \(h/2\) for columns (\(h\) being the height of the cross section) and \(L/3\) for beams (\(L\) being the element length), \(d_{bl}\) is the average diameter of the reinforcing bars, and \(f_{y}\) and \(f_{c}\) are the steel yield strength and the concrete compressive strength, respectively (both expressed in MPa). The axial force-deformation and biaxial moment-rotation relationships are given by the contribution of all the fibers, considering the above-specified hinge length. Axial force-biaxial bending interaction behavior, moment-rotation curve modifications, and plastic axial strain are automatically taken into account. With regard to the stress-strain relationships adopted for the various fibers, the non-linear elastic with plastic hardening stress-strain curve has been adopted for reinforcing the steel fibers, while the [PERSON] et al. [19] stress-strain model has been adopted for concrete, by distinguishing between the confined model for concrete core and the unconfined model for concrete cover. For each section of the frame elements, the confined concrete ultimate strain \(\varepsilon_{ccu}\) has been calculated according to the Italian Building Code [17; 18] and Model Code 1990 [20] expressions:  
\[\varepsilon_{ccu}=\varepsilon_{cu}+0.1\alpha_{h}\alpha_{v}\omega_{st} \tag{2}\]  
where \(\varepsilon_{cu}\) is the unconfined concrete ultimate strain, assumed to be equal to 3.5%, \(\alpha_{h}\) and \(\alpha_{v}\) represent the horizontal and vertical confinement efficiency factors, respectively, while \(\omega_{st}\) is the mechanical volumetric shear-reinforcement ratio. With regard to the hysteresis model governing the cyclic post-elastic response, the versatile pivot hysteresis model [11] has been adopted for both the beams and columns in order to incorporate the pinching effects ascribed to the weak transverse reinforcement and the poor construction details that might be observed in the existing RC structures. The pivot model is similar to the model proposed by [PERSON] et al. [8] for RC members but is equipped with additional input parameters that can be suitably calibrated to describe the possible pinching effects and degrading hysteretic loops. More specifically, in the pivot hysteresis model, the unloading and reloading tend to be directed toward specific points, called pivot points.