## 4 Conclusions  
The main scope of this paper is to introduce a straightforward numerical routine developed in Matlab/Simulink for nonlinear dynamic analysis of reinforced concrete (RC) frames using the insights of the force analogy method coupled with an explicit integration scheme. The accuracy of the proposed numerical algorithm is further validated through a comparison of the structural response with the results obtained by commercial software commonly used by structural engineers in structural design. Furthermore, an energetic evaluation of the structural response was conducted. The energy balance time history was calculated using a seismic input scaled with a scale factor of 1.4. In addition, the energy balance error was computed in each time step in order to evaluate the performance of the time integration scheme. Moreover, in order to showcase the capabilities of the algorithm, an incremental dynamic analysis was conducted on a structural model using Vrancea 1977 seismic inputs scaled with scale factors ranging from 0.6 to 1.4. The structural response data was processed and a damage evaluation procedure, using damage indices, was conducted. The results reveal damage distribution among different plastic hinges and different stories of the model, together with the time evolution of story drifts. The overall building hysteresis dissipated energy and damage index are also displayed. Compared to the well known classical approaches, the proposed routine has the following crucial advantages: Throughout the entire dynamic analysis, the initial stiffness matrix of the structure is used, and therefore no other numerical algorithm is necessary  
Figure 22: (**Up**): maximum story drift together with maximum damage indices evolution over IDA (with scale factors enclosed) for 1 st, 3 rd, 6 th and 9 th stories. (**Down**): RMSE after linear regression on above variables. The values above the bar plots, correspond to the minimum and maximum story’s damage index value in IDA. for the determination of stiffness matrix in each time step; the classical equation of motion described by a second-order differential system is reduced to a first-order differential system which is further transformed into an explicit time integration scheme that uses adaptive time steps; the technique used for implementation of hysteresis functions that describe the degrading behavior of concrete elements during cyclic loading is rather uncomplicated. To point out some of the important results, the nonlinear dynamic procedure proposed in this paper is the first to present a detailed methodology for the implementation of polygonal hysteresis models in the FAM 3D. Moreover, validation of the results using commercial software shows very good compliance of the structural response as illustrated in Figures 9-12. Furthermore, the performance of the time integration procedure is also evaluated using the energy balance error (Figure 13). The step error reaches a maximum value of 0.7%, which is situated under the limit accepted by civil engineering literature (2-5%). Another interesting result aroused form damage analysis is presented in Figure 16, where the percentage of dissipated hysteresis energy by plastic hinges of base columns and by the rest of the plastic hinges with respect to the total dissipated energy is illustrated. The total dissipated energy by the base columns varies linearly over IDA from 50% to 30%. Therefore, it can be concluded that the overall amount of energy dissipated through plastic deformation is mostly influenced by the cyclic behavior of the base columns. Consequently, as the seismic scale factor is raised the plastic energy amount dissipated by beams increases. To conclude, the authors' main contribution is the development of a force analogy-based algorithm that can be used to conduct, in a simple manner, nonlinear dynamic analysis for RC moment-resisting 3D frame structures. Investigation, R.I.M. and F.M. ; Conceptualization, V.C. and C.B. ; writing--review and editing, R.I.M. and F.M. ; Software, R.I.M., V.C. and F.M.