# Numerical Study of the Seismic Response of an Instrumented Building with Underground Stories
## 5 Discussion and Conclusions  
The huge relative errors obtained for models considering vertical springs at the base of the foundation can be attributed to a poor calibration of the spring constant used. This parameter was obtained from the soil mechanics report and is primarily intended to design the slab using a pseudo-static approach, and therefore, the actual value may be very different. The complex process that is required to validate and accurately define this parameter, and the high sensitivity that the numerical analyses performed shown to this parameter, make it unsuitable for design purposes to model the structure using vertical springs, at least until further studies regarding the calibration of the spring constant makes easier to define it. Furthermore, although there are no explicit requirements for the consideration of SSI in the modeling of structures, some suggestions can be found in the PEER _Guidelines for Performance-Based Design_[44] and the ASCE/SEI 7-16 (Section 12.13) [1], mainly proposing the use of horizontal and vertical springs and dashpots, similar to the approach used in this study. However, neither of both codes provides explicit equations for calculating the springs and dashpots constants, whose values were shown to be very critical for the accuracy of the analyses. Larger relative errors were obtained for models with a cracking factor under 0.7, which suggests that the building behavior was slightly cracked for the low-intensity earthquakes studied. This confirms the recommended 0.7 stiffness reduction factor applied to building elements during design (Los Angeles Tall Buildings Structural Design Council, [45]) to verify deformations of a medium-rise building. For events with a maximum acceleration lower than 0.05 m/s\({}^{2}\), inter-story drifts were greater in the underground levels with horizontal ballast models, demonstrating that the soil-structure interaction should not be discarded even for small seismic events. Models without rocking best represented the structure response when embedded at the foundation level. This may be due to inputs from accelerometers located at level-3 in the foundation. The models that best-fit the on-site data were, based on relative error of crossover distortions, the relative error of duration for the strongest signal, and the relative error of the dominant frequency, the foundation-embedded model, with [PERSON] horizontal ballast; and the foundation-embedded model, with [PERSON] horizontal ballast, and a stiffness reduction factor of 0.7. The [PERSON] horizontal ballast model may model the soil using compression-traction springs. Indeed, the relative errors were shown to be similar for Kuesel with or without traction, as well as inter-story drifts and story shears, with no visible differences. This result is supported by [16], who reached a similar conclusion for a high-rise building. However, this conclusion is still to be tested for the response of the structure under strong motions. Notwithstanding the results presented above, it is necessary to obtain more data from instrumentation across a wider variety of structures to continue to evaluate better design and modeling practices. Similarly, further analyses, including nonlinear time-history and high-intensity events, are needed to best regulate building design. In sum, and for design purposes, the results allow for the recommendation to:  
1. Apply a stiffness reduction factor of 0.7 to all elements in deformation verification models for average-height buildings, including subterranean elements;
2. Embed the structure in the foundation level and not in the street level;
3. Whenever possible, include horizontal springs with [PERSON]'s model with traction for the analysis of the structure;
4. Select spring constants carefully, primarily if vertical springs are to be considered in the modeling of the structure. **Author Contributions:** Conceptualization, E.S. and G.L. ; methodology, E.S., G.L. and A.L. ; numerical modeling, E.S. ; validation, E.S., A.L. and M.G. ; formal analysis, E.S.