# On the Seismic Design of Structures with Tilting Located within a Seismic Region
## 1 Introduction
They concluded that: (a) the ductility demand of systems with asymmetric yielding may be much higher than those corresponding to symmetric systems, particularly for structural systems with vibration periods close to the dominant period of the soil, (b) the effect of asymmetric yielding is more detrimental for systems with low lateral strength and with high ductility demand, and (c) the increment in the expected ductility demand of asymmetric yielding systems with respect to that of symmetric systems is independent of the chosen value of the mean annual rate of exceedance. Despite the valuable contributions of the studies discussed above, most of them were limited to the analysis of single degree of freedom (SDOF) systems subjected to unidirectional analyses, and the asymmetric yielding was considered by means of an idealized SDOF model. In addition, several important factors were not incorporated as the explicit consideration of the tilting angle, the influence of two orthogonal components of the ground motions, and the implicit levels of reliability in the development of the strength amplification factors were not considered. Some of these issues are addressed in the present paper. Hence, the specific objectives of this research are: (1) to apply a reliability-based methodology to estimate the additional strength requirement of structures with asymmetric yielding, located in sites with different soil conditions, (2) to develop simplified mathematical expressions for this purpose, which can be applied to existing structures that exhibit asymmetric yielding located in a seismic region, considering the influence of two orthogonal horizontal seismic ground motions, and (3) to estimate ductility transformation factors between the expected ductility demand of MDOF asymmetric yielding structural systems with respect to that of their equivalent simplified 3D systems, both associated to a given annual probability of exceedance. Buildings with Asymmetric Yielding Produced by Tilting in Mexico City  
Many buildings located in Mexico City downtown, which is a very populated area, suffer tilting problems. The main reason associated to structural tilting in this zone is that a considerable part of the valley of Mexico is located over an ancient lake, this fact indicates that Mexico City downtown is built on soft and very soft soils. Figure 2 shows the classification of the soil in the valley of Mexico according to the dominant period, Ts [3]. The Ts values were computed from spectral amplification functions corresponding to more than 100 free-field strong motion stations complemented with around 500 micro-treemor measurements [23]. The contour map of soils with equal dominant periods (iso-periods) is shown in Figure 2. It was constructed by means of a Bayesian spatial interpolation technique [23]. The firm ground (commonly known as Hill zone) presents relatively small dominant periods (around 0.5 s); however, the soft soil (commonly known as Lake Bed zone) have large vibration periods, up to 4 s.  
To illustrate the problem of buildings suffering tilting in the downtown of Mexico City, a small area (marked with a small red dot in Figure 2) was selected. The tilted buildings located in that area were classified as shown in Figure 3. The detection of these buildings was based on visual inspection. It was observed that the main cause of tilting in these buildings was that they are built on very soft soils, provoking differential settlements. However, there could also be cases in which tilting was due to the effect of previous intense earthquakes. It was found that exist over 50 buildings that present tilting problems in such a small area (approximately 1.5 km\({}^{2}\)). It indicates that there are hundreds of tilted buildings in the soft soil of the valley of Mexico. Most of the tilted buildings that were identified in this  
Figure 2: Curves of the dominant soil periods in the valley of Mexico. zone have 3 to 6 stories; however, there are some that have 8 or more stories.