# On the Seismic Design of Structures with Tilting Located within a Seismic Region
## 4 Seismic Zones Analyzed
In addition, the structural ductility demand and the level of asymmetric yielding are explicitly considered. The mathematical expressions for the estimation of the strength amplification factors (AF) that were fitted to the R\({}_{\text{BSCS}}\) ratios (as those in Figure 7) are:  
\[\text{AF}=\frac{\text{a}\Big{(}\frac{\text{T}_{1}}{\text{T}_{s}}\Big{)}^{ \text{b}}}{\text{c }+\Big{(}\frac{\text{T}_{1}}{\text{T}_{s}}\Big{)}^{\text{b}}}+\text{d }\left(\text{firm ground}\right) \tag{6}\]  
\[\text{AF}=\frac{\text{a}\Big{(}\frac{\text{T}_{1}}{\text{T}_{s}}\Big{)}^{ \text{b}}}{\text{c }+\Big{|}\frac{\text{T}_{1}}{\text{T}_{s}}-1\Big{|}}+\text{d }\left(\text{intermediate and soft soils}\right) \tag{7}\]  
The values of a, b, c, and d are presented in Table 2 for the seven seismic zones in which the valley of Mexico was classified. Equations (6) and (7) were developed to be applied to structural systems with levels of asymmetric yielding up to 0.04 and ductility demands of 2, 3 and 4; they are not valid to estimate strength amplification factors for structures that exceed those values. However, most of the structures susceptible to present asymmetric yielding produced by tilting, fall into these ranges. It is noticed that a level of asymmetry larger than 0.03 has a very small probability to occur. Figure 8 compares the results obtained with Equations (6) and (7) with those shown in Figure 7, corresponding to R\({}_{\rm BSCS}\) associated with a ductility demand of 2. It can be seen in Figure 8 that Equations (6) and (7) fit appropriately to the R\({}_{\rm BSCS}\) data. It is important to mention that the values of the parameters presented in Table 2 do not correspond strictly to the least square method results, since their values could be more complex by introducing a larger number of parameters, and consequently, represent better the data obtained from the analyses; however, the authors consider that the proposed mathematical expressions (Equations (6) and (7)) can be used in a practical seismic design context, providing suffici \begin{table}
\begin{tabular}{c c c c c c} \hline
**Zone** & **Dominant Period (s)** & **a** & **b** & **c** & **d** \\ \hline A & T\({}_{\rm s}\leq 0.5\) & \((3.5\mu-1.5)\)\(\alpha\) & 13.4 & 0.1 & \(1.6\alpha+1\) \\ B & \(0.5<T_{\rm s}\leq 1.0\) & \((4.8\mu-3)\)\(\alpha\) & 8.8 & 0.1 & \(4.1\alpha+1\) \\ C & \(1.0<T_{\rm s}\leq 1.5\) & \((1.5\mu-1.4)\)\(\alpha\) & 0.7 & 0.08 & 1 \\ D & \(1.5<T_{\rm s}\leq 2.0\) & \((2\mu-1.6)\)\(\alpha\) & 0.5 & 0.1 & 1 \\ E & \(2.0<T_{\rm s}\leq 2.5\) & \((1.5\mu+0.8)\)\(\alpha\) & 0.9 & 0.12 & 1 \\ F & \(2.5<T_{\rm s}\leq 3.0\) & \((1.5\mu+1.1)\)\(\alpha\) & 0.7 & 0.13 & 1 \\ G & \(3.0<T_{\rm s}\leq 4.0\) & \((1.9\mu-0.05)\)\(\alpha\) & 0.1 & 0.12 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters involved in Equations (6) and (7), corresponding to seismic zones A to G.  
Figure 8: Comparison of the strength amplification factors (AF) obtained with Equations (6) and (7) with those presented in Figure 7, corresponding to a ductility demand \(\mu=2\). (**a**) Zone A; (**b**) Zone B; (**c**) Zone C; (**d**) Zone D; (**e**) Zone E; (**f**) Zone F; (**g**) Zone G.  
Comparison of the Proposed Mathematical Expressions with that Recommended in the Current Mexico City Building Code  
Figure 9a,b compare the strength amplification factors obtained according to the mathematical expressions proposed in this study (Equations (6) and (7), and Table 2) with those recommended by the MCBC-2004. The MCBC-2004 recommends amplifying the design base shear coefficient of structures that exhibit asymmetric yielding produced by tilting by the following factor:  
\[\text{AF}_{\text{MCBC-2004}}=1+\text{SQf} \tag{8}\]  
where Q is the seismic behavior coefficient, that is related to the ductility of the structure, and f is the level of asymmetry of the structure (which is equivalent to the parameter \(\alpha\) used in this study). The comparison is made for a ductility demand of 2 and considering a level of asymmetric yielding \(\alpha\) = 0.02.