# On the Seismic Design of Structures with Tilting Located within a Seismic Region
## 4 Seismic Zones Analyzed
R\({}_{\text{BSCS}}\) (defined in Equation (4)) for zones A to G corresponding to a ductility demand of 2 (\(\mu\) = 2) and annual rate of exceedance \(\
u\) = 0.008 (corresponding to a return period, T\({}_{\text{r}}\) = 125 years). The horizontal axis represents the ratio of the fundamental vibration period of the structural systems to that of the dominant period of the soil corresponding to the seismic zone under consideration; while the vertical axis shows the parameter R\({}_{\text{BSCS}}\) which represents the additional strength required for asymmetric yielding systems to achieve a seismic performance and structural reliability, in terms of the ductility demand, equivalent to their symmetric yielding counterparts. Figure 7: Parameter R\({}_{\text{BSCS}}\) corresponding to (**a**) Zone A; (**b**) Zone B; (**c**) Zone C; (**d**) Zone D; (**e**) Zone E; (**f**) Zone F and (**g**) Zone G, for different values of \(\alpha\), and \(\mu\) = 2. Even though the results are presented for a single value of ductility demand and mean annual rate of exceedance, the main observations are valid for all the ductility values considered in this study. Besides, as reported by [22], the increment in the ductility demand of asymmetric yielding systems with respect to symmetric systems is sensibly independent of the selected mean annual rate of exceedance (return period). It is worth mentioning that in Zones A and B (firm ground) there is no a dominant period of the soil; however, it is considered in the same way as the other zones with the objective of standardizing the results presented in this study. It is observed in Figure 7 that the R\({}_{\text{BSCS}}\) parameter is smaller for firm ground (Zones A and B) than for intermediate and soft soils (Zones C, D, E, F, and G) in the zone where the effect of asymmetric yielding is more important (i.e., T\({}_{1}\)/T\({}_{\text{s}}\)\(\approx\) 1). This indicates that the detrimental effect of asymmetric yielding is, generally, higher on soft soil than on firm ground; however, this is not always true, particularly for T\({}_{1}\)/T\({}_{\text{s}}\) ratios away from unity. In contrast, it is observed that for Zones A and B the shapes of the graphs are almost constant for T\({}_{1}\)/T\({}_{\text{s}}\) greater than 1. However, for the other seismic zones, the strength requirement reaches its maximum where the vibration period of the system is close to the dominant period of the soil and decreases as the T\({}_{1}\)/T\({}_{\text{s}}\) ratio moves away from unity. In summary, Figure 7 demonstrates that the effect of asymmetric yielding is more detrimental for structural systems located on soft soils and whose vibration period is close to the dominant period of the soil where is located. Once the information was grouped into these seven categories, simplified mathematical expressions were fitted to the results using the least square method. It must be noticed that the shape of the graphs corresponding to firm ground is different from that associated with intermediate and soft soils (see Figure 7). The resulting expressions are in function of the parameters that influence the seismic performance of structures with asymmetric yielding such as: the level of asymmetric yielding (\(\alpha\)), the ductility demand (\(\mu\)), the fundamental vibration period of the structure (T\({}_{1}\)) and the dominant period of the soil (T\({}_{\text{s}}\)). The general forms of the mathematical expressions proposed here are an extension of those proposed by [PERSON] and [PERSON] [21], which are based on a statistical analysis of the response of SDOF systems, under one-directional ground motions, where the SDOF systems are idealized with asymmetric force-deformation relationship (Figure 5), and in accordance with a constant damage criterion using the Park and Ang damage index [47]. On the other hand, the expressions proposed in this paper are based on a reliability analysis, which estimates mean annual rates of exceedance of an EDP for a given return period.