# Reliability Analysis of Response-Controlled Buildings Using Fragility Curves
## 1 Introduction
According to [8], it is assumed that the conditional probability of DM to IM satisfies the lognormal distribution. This assumption is based on large amounts of data statistics experience with existing research results on engineering structures, and the advantage of this assumption is that the logarithm of DM, (In(DM)), and the logarithm of IM, (In(IM)), are linear in the logarithmic coordinate. [PERSON] et al. [9] obtained the seismic fragility curves using lognormal distribution function to analyze 250 reinforced concrete structures in L'Aquila that were subjected to the 6 April 2009 L'Aquila earthquake and came up with a damage scenario that matched the real one. Fragility curves are obtained using analytical methods such as Incremental Dynamic Analysis (IDA). The IDA is widely used to capture the overall seismic performance of structures. Alternative procedures for IDA include procedures such as Multiple-Stripe Analysis (MSA, [10]) and Cloud Analysis (CA, [11]), where MSA involves performing a series of nonlinear dynamic analyses at specific intensity levels, whereas CA aims at deriving IDA-based fragility curves by choosing ground motion intensities strategically in order to minimize the amount of scaling. CA requires linear regression predictions based on the results of the structural analysis of the un-scaled records to identify the range of intensity values near demand to capacity ratios equal to unity prior to performing actual CA. These nonlinear dynamic analyses are used to characterize the relationship between DMs and IMs and to perform fragility assessments based on recorded ground motions. [PERSON] et al. [12] simplified and standardized IDA's general procedure, creating a strong foundation for its future implementations. IDA is a way of subjecting analytical models of structures to a suite of Ground Motion Records (GMRs), and each GMR is scaled to several intensity levels designed to force the structure from the elastic range to the nonlinear range. IDA can analyze a structure's seismic performance, from elasticity through plasticity to collapse. IDA includes the selection of appropriate IM and DM, selection of a suitable and adequate number of ground motion intensities, and appropriate scaling of GMRs for higher intensities to cover the entire range of structural responses by a series of scale factors [12; 13]. Then, it is possible to obtain the IDA curves that describe the relationship between the IM of the GMRs and the DM of the structure. The selection of appropriate IM and DM is the first important step in performing IDA. A sufficient and efficient IM should be selected based on depicting a good correlation with the DM of choice, possessing low dispersion, and predicting a relatively better structural response using a relatively low number of GMRs [8]. The peaks of the ground acceleration, velocity, and displacement signals (PGA, PGV, and PGD), as well as the spectral acceleration value Sa(T\({}_{1}\)) corresponding to the fundamental period of a structure, are now used as typical ground motion IMs. Considering that a structure's displacement response, Sa(T\({}_{1}\)), is the most efficient IM, since it allows records to be selected regardless of magnitude, distance, or duration, as well as predicting the response with less uncertainty [14]. It is known that the spectral acceleration, Sa(T\({}_{1}\)), has a good correlation with the seismic damage measures in first-mode-dominated structures [15]. However, for taller or asymmetric structures where higher modes become important, improved IM alternatives should be sought [14]. On the other hand, the maximum inter-story drift ratio, 0 max (hereafter referred to as story drift ratio) appears to be an effective damage measure index [14]. [PERSON] et al. [16] studied the predictive ability of nine spectral IMs for base-isolated structures subjected to near-fault earthquakes using three engineering demand parameters. [PERSON] et al.