# On the Seismic Design of Structures with Tilting Located within a Seismic Region
## 6 Ductility Transformation Factors between Simplified and MDOF Systems  
The above analyses regarding the estimation of the strength amplification factors for the design or reinforcement of structures with asymmetric yielding were performed considering simplified systems (SS); however, the ductility demand of asymmetric yielding MDOF structural systems may be different from that of its equivalent SS. Because of this reason, it is necessary to consider such difference by finding ductility transformation factors between the ductility demands of both structural systems (MDOF and SS).  
Figure 9: Comparison of the strength amplification factors obtained with Equations (6) and (7) with those recommended by MCBC-2004, corresponding to \(\mu\) = 2 and \(\alpha\) = 0.02. (**a**) Firm ground; (**b**) Intermediate and soft soil.  
There are some studies related to transformation factors of the dynamic response of conventional MDOF and SDOF structural systems. For example, [PERSON] et al. [48] proposed equations to obtain probabilistic response transformation factors for maximum ductility and inter-story drift using artificial neural networks; they concluded that an artificial neural network is a useful tool for reliability-based seismic design procedures of framed buildings. Similarly, [PERSON] et al. [49] estimated probabilistic and deterministic response transformation factors in order to evaluate the response in terms of maximum ductility and maximum interstory drifts of MDOF systems based on the response of equivalent SDOF systems. They analyzed five steel buildings designed in accordance with the MCBC-2004, and concluded that in both approaches, the structural demand of steel frames can be obtained with good accuracy by means of equivalent SDOF systems; however, the results documented in the above studies are not valid for asymmetric yielding reinforced concrete buildings. In the present study ductility transformation factors (DTF) for reinforced concrete structures exhibiting asymmetric yielding produced by tilting are calculated using Equation (9) as:  
\[\text{DTF}=\frac{\mu_{\text{SS}}(\
u)}{\mu_{\text{MDOF}}(\
u)} \tag{9}\]  
where \(\mu_{\text{MDOF}}(\
u)\) represents the maximum ductility demand estimated for buildings modeled as MDOF systems associated to an annual rate of exceedance value (\(\
u\)), and \(\mu_{\text{SS}}(\
u)\) is the maximum ductility demand calculated for an equivalent simplified system associated with the same value of \(\
u\). To estimate the expected ductility demand of the structural systems, associated with an annual rate of exceedance value, the corresponding ductility demand hazard curves (DDHC) need to be obtained using Equation (1). Figure 10 illustrates schematically the calculation of the DTF from the ratio of DDHC corresponding to MDOF and to SS systems, associated with a given \(\
u\) value.  
DTF were estimated for five reinforced concrete structures (8-, 9-, 10-, 11-, and 12-story buildings designed according to MCBC-2004) and their corresponding equivalent simplified structural systems, considering several values of \(\alpha\) (0.0-0.03). Regarding the number of buildings analyzed, it is worth mentioning that this study is based on the analysis of asymmetric yielding produced by differential settlements, which is more likely to occur in buildings built on soft soil and having shallow foundations, which are often used in medium-rise buildings. Conversely, taller buildings are usually supported on deep foundations, which reach the firm ground, and consequently, the tilting is not a common problem. To incorporate the influence of the annual rate of exceedance value in the results, DTF were calculated for \(\
u\) values between 0.01 and 0.001 (which correspond to return intervals between 100 and 1000 years).  
Figure 10: Schematic representation related to the estimation of the ductility transformation factors (DTF).