# Numerical Study of the Seismic Response of an Instrumented Building with Underground Stories
## 3 Numerical Model
### Numerical Models vs. Recorded Measurements Comparisons
#### 4.2.3 Analysis of Relative Errors  
The average relative error among most models showed differences greater than 10% with empirical data. Thus, no model was able to completely represent the shape of the record, as observed in Figure 7a,b, where there is a clear difference between the model results and the actual behavior. Despite the above, the models with the lowest error were those where the vertical ballast is considered at the foundation instead of the embedded foundation, as can be observed in Figure 8.  
Figure 7: (a) Recorded on-site acceleration measurements and model accelerations on 16 th story for model 12, excited by event 12, on sensor 14. (b) Recorded on-site acceleration measurements and model accelerations on 16 th story for model 40, excited by event 72 on sensor 14. The highest relative error for the maximum acceleration was given by the models that included cracking of the vertical elements (models 16 to 21), but there was no defined trend on the relative error for maximum accelerations among models. This parameter, across accelerometers and models, was relatively high, exceeding by 10% in almost every analysis. In Figure 9, the median relative errors on the maximum acceleration for all the models and sensors analyzed are depicted. In this case, it can be observed that the models that considered vertical springs at the base of the structure usually have greater differences with the measured data. From the models without vertical springs at the base, those with a better fit with the measured data are models 2, 4, 5, 12 and 13. All of those models consider the underground stories, and most of them consider the soil-structure interaction using horizontal springs. It should be observed that the relative error of model 1 (embedded at the base of the first story) is on the order of magnitude of the more complex models, but the accuracy is strongly dependent on the stiffness reduction factor (in this case, as the ground motions were not very strong, to not consider a reduction factor leads to better results). Figure 8: Median average relative errors for all the events analyzed. (**a**) Models without vertical springs at the base; (**b**) models with vertical springs at the base. Figure 9: Median relative errors on the maximum acceleration of the records. (**a**) Models without vertical springs at the base; (**b**) models with vertical springs at the base. The relative error of the number of crossover distortions was usually greater in models where vertical springs were included at the base level, except for model 41 (Figure 10). In this case, the behavior is similar to that of the maximum acceleration: the models embedded at the base level with horizontal springs are the ones with best-fit to the actual behavior of the structure. From the models without vertical springs at the base, those that included rocking or rolling vertical inputs (models 3, 6, 7, 10, 11, 14, and 15) presented larger percentages of relative error compared to those that consider the vertical input as the average of the records of accelerometers 3 and 4 (models 1, 2, 4, 5, 8, 9, 12, 13 and 40). This parameter had the same trend as the number of crossover distortions in the whole record and for the strongest signal. The relative error of the duration of the seismic event is greater for models where vertical elements were considered with a cracking factor less than 0.7. In general, the relative errors of the duration for the strongest signal are small. Among cases that do exceed 20%, the lowest relative errors are generally from models that do not consider rocking as an input (models 1, 2, 4, 5, 8, 9, 12, 13, and 40). Again, the models that consider vertical springs had a less accurate performance than models embedded at the base, as observed in Figure 11.  
The relative error of the dominant frequency of the recording is greater for models where vertical elements were considered with a cracking factor less than 0.7, except for the event 62.