# On the Seismic Design of Structures with Tilting Located within a Seismic Region
## 1 Introduction  
Structures built on intermediate or soft soils are those that commonly present structural problems. One of these complications is that buildings may suffer tilting due to differential settlements, leading to an uneven load-deformation behavior (hereinafter referred as asymmetric yielding behavior); this means that such structures exhibit different yield strength in opposite directions. Since symmetric (without asymmetric yielding) structures tend to oscillate around their un-deformed position, their plastic deformation demands tend to counteract each other. Otherwise, asymmetric yielding structures tend to accumulate the plastic deformation demands in the weaker direction (i.e., the direction of tilting) when they are excited by seismic loading. In other words, originally symmetric structures develop \"strong\" and \"weak\" directions due to tilting. There are several reasons why a structure exhibits different yield strength in opposite directions, for example: asymmetry in vertical loads, structures with sloping facades, the presence of mezzanines, tilting, etc. (see Figure 1a-c). This undesirable behavior may significantly affect the seismic performance of these kinds of structures, particularly, when they are subjected to long-duration seismic ground motions. As discussed later in this paper, the effect of asymmetric yielding significantly reduces the seismic performance of structures, and this, combined with other factors such as constructive deficiencies, poor design procedures, among others, may cause the failure of such structures when subjected to intense seismic ground motions. Therefore, it is necessary to take action to counteract the effect of asymmetric yielding. There are different types of solutions to improve the performance of buildings with asymmetric yielding. The best solution will depend on the particular problem under consideration. These solutions could be geotechnical, structural or a combination between them. The present study is focused exclusively on structural solutions. Using the approach proposed in this paper, buildings with asymmetric yielding must be designed for a higher lateral strength than symmetric ones, requiring the development of strength amplification factors (AF) of the design pseudo-acceleration spectrum in order to take into account the detrimental effect of the yielding asymmetry. On the other hand, future design codes must be oriented not only to guarantee a satisfactory seismic behavior, but also an adequate level of reliability, which will allow engineers to design for a certain pre-established risk associated with structures. Over recent years, the development of structural reliability methods has provided a more rational basis for the design of earthquake-resistant structures. Numerous reliability-based code calibrations have been formulated by several researchers [1; 2] and have also been included in guidelines like MCBC [3], National Building Code of Canada (NBCC) [4], Federal Emergency Management Agency (FEMA 445) [5], FEMA P-58 [6], among others. Hence, in the present study, a reliability-based methodology is proposed and applied to develop simplified mathematical expressions to estimate strength amplification factors for existing structures exhibiting asymmetric yielding produced by tilting. The structures are supposed to be distributed within a seismic region. It is important to emphasize that this methodology can be applied not only for existing buildings, but also for new structures with asymmetric yielding caused by other reasons, as illustrated in Figure 1.  
The mathematical expressions proposed in this paper are based on the analysis of simplified one-story one-bay three-dimensional (3D) structural systems subjected to orthogonal seismic ground motions. The methodology is illustrated by applying it to develop strength amplification factors for the valley of Mexico.