# On the Seismic Design of Structures with Tilting Located within a Seismic Region
## 3 General Methodology
### Steps to Follow for the Proposed Methodology  
In order to find expressions that allow us to estimate the strength amplification factors such that asymmetric yielding structures present a seismic reliability similar to that of symmetric yielding structures, steps (1) to (8) are followed:  
1. Firstly, several nonlinear time history analyses are carried out for simplified structural systems with different characteristics of base shear coefficient (c), asymmetric level (\(\alpha\)), and vibration period (T\({}_{1}\)). This is performed with the aim of obtaining the EDP as a function of seismic intensity. The maximum ductility demand of the systems (\(\mu\)) is taken here as the EDP of interest. In order to calculate the maximum ductility demand of the systems, it is necessary to estimate first the ductility demand in both E-W and N-S directions (\(\mu_{\mathrm{x}}\), \(\mu_{\mathrm{y}}\)) considering the simultaneous action of both horizontal components of the seismic ground motions; \(\mu_{\mathrm{x}}\) and \(\mu_{\mathrm{y}}\) are calculated by taking the maximum horizontal displacement of the center of mass of the structural systems in each direction (d\({}_{\mathrm{X}}\), d\({}_{\mathrm{Y}}\)) divided by their yield displacement, d\({}_{\mathrm{Y}}\), which in turn is estimated by a nonlinear static analysis). Finally, the maximum ductility demand is defined in this study as the maximum of the ductility values estimated in each horizontal direction, as indicated in Equation (2). \[\mu=\max\left(\mu_{\mathrm{x}},\mu_{\mathrm{y}}\right)\] (2) Next, the median (D) and standard deviation (\(\sigma_{\mathrm{InD}}\)) of the ductility demand logarithms are calculated. 3. Fragility curves for several values of the maximum ductility demand are obtained using Equation (3): \[\mathrm{P}(\mathrm{EDP}>\mathrm{y}|\mathrm{IM}=\mathrm{im})=1-\Phi\left(\frac {\ln(\mathrm{y}/D)}{\sigma_{\mathrm{InD}}}\right)\] (3) where all the variables were defined before, and \(\Phi\) represents the Gaussian cumulative distribution function. 4. Ductility demand hazard curves (DDHC, Equation (1)) are obtained for symmetric yielding systems, and alternatively, for systems with different levels of asymmetric yielding. 5. Considering the ductility demand hazard curves corresponding to a wide variety of systems with different characteristics, ductility uniform exceedance rate spectra (\(\mu\)-UERS) are obtained for several mean annual rate of exceedance values. To explicitly display the increment in the expected ductility demand of asymmetric yielding systems with respect to symmetric systems, ratios of \(\mu\)-UERS corresponding to asymmetric yielding systems with respect to symmetric ones are calculated. 6. The next step is to obtain base shear coefficient spectra (BSCS) for symmetric as well as for asymmetric yielding systems employing a linear interpolation process. This procedure consists in selecting a value of the ductility demand and the associated values of T\({}_{1}\) and c, corresponding to a given \(\mu\)-UERS. The process is repeated several times in order to obtain a data set T\({}_{1}\) vs c for each value of the ductility demand considered. More details about this process can be found in [22]. Ratios between BSCS of systems with different levels of asymmetric yielding with respect to symmetric systems, R\({}_{\mathrm{BSCS}}\), are calculated with the objective of quantifying the additional lateral strength requirement of systems with asymmetric yielding to achieve a seismic performance equivalent to their symmetric counterparts. These ratios can be expressed as: \[\mathrm{R}_{\mathrm{BSCS}}=\frac{\mathrm{c}[\mathrm{BSCS}(\mathrm{T}_{1}, \mathrm{v},\mu,\alpha)]}{\mathrm{c}[\mathrm{BSCS}(\mathrm{T}_{1},\mathrm{v}, \mu,\alpha=0)]}\] (4)
7. A simplified mathematical expression is fitted to the ratios of the base shear coefficient spectra obtained in step 6.