## 3 Case Study  
In this section, the implementation of a peak-oriented hysteresis function is presented and further used for modeling the nonlinear behavior of beams in a 3D reinforced concrete moment resisting frame structure. A dynamic nonlinear simulation is conducted, the structural response being obtained using numerical analysis algorithm developed in Matlab/Simulink which is validated in terms of displacements and bending moments using a commercial software. Although the plastic behavior of three-dimensional structures is very complex, the common seismic design procedures consider that beam end rotation is the only desired source of nonlinearity while shear and axial deformations are considered to be elastic. These assumptions are in close connection with the ability of structural members to dissipate high quantities of input seismic energy through sustained cyclic rotation. In this paper, in order to account for the nonlinear behavior of structural elements, the concentrated plasticity approach is used [27]. Considering that most of the time, for civil engineering structures, beams are incorporated in slabs, it can be accepted that axial force is negligible and plastic rotations occur only around the horizontal axis of the cross-section. Therefore, bending moment plastic hinges can be adopted for beams. In the case of columns, the plastic hinge mechanism is far more complex. The bending moment values are significant on both principal the direction of the cross-sections and depend on the axial force variation. Consequently, axial force-bending moments (P-m-m) interaction plastic hinges can be utilized. Moreover, the novel algorithm proposed by [10] is used in this paper to implement this type of plastic hinges in FAM dynamic nonlinear procedure. Moreover, laboratory tests conducted on the response of structural members [28] indicate that strength deteriorates as the number and amplitude of cycles increase, even if the monotonic displacement associated with the maximum strength has not been reached. Consequently, the cyclic model implemented herein, similar to the peak-oriented model in [29] that had been developed firstly by [PERSON] and [PERSON] and modified by [PERSON] and [PERSON], takes into account both strain hardening and softening behavior. The general bending moment-rotation hysteresis function are presented in Figures 2 and 3. Furthermore, as stated earlier, plastic rotation is considered to be the only source of nonlinearity, and therefore, a bending moment-plastic rotation hysteresis model is a better fit for the implementation in FAM equations. It is important to note that the cyclic model proposed here is only implemented for the beams of the model. Figure 4 together with Table 1 present in detail the five general rules upon which the cyclic behavior model is built. The rules describe elastic, plastic, and softening behavior together with loading, unloading, and reloading principles. The elastic and plastic behavior including loading and unloading are defined only by their corresponding fixed slope values. Softening behavior is also described by a fixed slope, but it occurs only after \(\theta_{s}^{\prime}\) is reached. Furthermore being a peak-oriented cycle model, the reloading branch is then computed using variable slopes aiming at the overall maximum (\(m-\theta^{\prime\prime}\)) point in the \(I\) and \(III\) quadrant in each corresponding case. Furthermore, it must be mentioned that when designing the numerical algorithm the most changing task in the implementation hysteresis models is the determination, for every plastic hinge, in each time step, of the proper \(k_{i}^{br}\). Being a history variable, it is dependent on the hysteresis path and therefore several key parameters must be memorized throughout the entire analysis in order for the appropriate contextual value to be assigned. Firstly, taking into account that the cyclic model is a peak-oriented type model, the overall maximum and minimum bending moments together with the corresponding plastic rotation values must be stored.