## 3 Case Study
### Results  
In this section the nonlinear dynamic structural model is simulated using the Vrancea 1977 seismic input. Energy response and structural damage quantification are presented and discussed. The energy balance is given in Figure 13, which illustrates a significant percentage of input energy being dissipated through inelastic incursions. These inelastic incursions have the significance of damage, distributed throughout the entire height of the structure. Another observation that ought to be made is related to the energy dissipation mode. The pulse-like nature of the seismic inputs leads to early dissipation of the induced energy through high amplitude deformations as illustrated in Figure 12. At the same time, it can be seen that significant values are also reached by the energy dissipated through viscous damping while low values are encountered by the kinetic and strain energy time variations. The second picture of Figure 13 presents the energy balance error (EBE) time history. Taking into account that each energy type is computed based on the displacement, velocity and acceleration vectors obtained from numerical integration in each time step, the EBE can be seen as a criterion for the evaluation of the global accuracy of the time integration algorithm. The EBE can be calculated using the following relation. \[EBE_{k}=\frac{|IE_{k}-DE_{k}-KE_{k}-SE_{k}-HE_{k}|}{|IE_{k}|} \tag{33}\]  
As stated by [32], a tolerance situated between 2% and 5% is reasonable for practical applications. Furthermore, the concept of damage index is used in this paper to give a numerical insight on the element story and overall damage of the structure. The Matlab/Simulink model introduced in the above sections has been used to make an incremental dynamic analysis (IDA). A series of scale factors were applied to the time history loading, spanning from 0.6 to 1.4, and a 0.1 incremental factor. Therefore, in Figures 14 and 15 is presented the development of damage indices for the plastic hinges highlighted in Figure 5, across the loading history, with respect to the scale factors used in IDA. Figure 13: Energy balance time history and energy balance error time history. The results shown in these figures highlight that damage index values recorded for columns are higher than those for beams. Figure 14: Development of damage index for base column plastic hinges along IDA scale factors: PH no. 1—X direction and PH no. 2—Y direction. Figure 15: Development of damage index for beams’ plastic hinges along IDA scale factors: PH no. 125—story,3 beam, X direction, and PH no. 651—story, 6 beam, and Y direction. This fact is also highlighted by the percentage of energy dissipated by the base columns, with respect to dissipated energy for the whole building, in contrast with the rest of the plastic hinges. In Figure 16 the hysteresis energy sum of all base columns plastic hinges together with the rest of the plastic hinges hysteresis energy sum is presented as a percentage from the total dissipated energy of the building (\(Eh_{b}\)). In all the analyses, a minimum of 30% from the total dissipated energy is due to the base columns' contribution. An obvious second observation is that, as the demand approaches the capacity of the columns, more and more beams are engaged in the energy dissipating process, implying that even though the plastic hinges of base columns are approaching peak-capacity, there are still some beams yet to be involved in this dissipating process. The damage index is shown for the base columns, even though higher than the conventionally understood value for collapse state of 1, should not be perceived as the limit state of the building. Damage occurring in base columns dominates also the global damage index value of the building, as displayed in Figure 17 and in correlation with Figure 18 where the evolution of 1 st story maximum drift at each time step and the story damage index (\(DI_{s}\)) is represented.