# IMPA versus Cloud Analysis and IDA: Different Methods to Evaluate Structural Seismic Fragility
## 1 Introduction  
Performance-based earthquake engineering (PBEE) procedures allow the prediction and evaluation of the probabilistic seismic performance of bridges and buildings in terms of system-level decision variables, such as loss of use, repair cost and casualties. In the United States, the first generation of PBEE assessment and design procedures for buildings (SEAOC Vision 2000, FEMA 273, ATC-40 [1; 2; 3; 4]) took significant steps toward achieving performance-based earthquake engineering. Since then, the Pacific Earthquake Engineering Research Center (PEER) has been working on developing a more robust methodology that involves four stages: hazard analysis, structural analysis, damage analysis, and loss analysis [5]. In the third stage, damage analysis, fragility functions describe the conditional probability of component, element or system to be damaged for a given intensity measure. The first attempt to determine fragility curves can be dated back to 1975, when the Seismic Design Decision Analysis (SDDA) procedure was proposed in the US [6]. Further developments [7; 8] were initially applied in the field of the nuclear industry to define a probabilistic relationship between an intensity measure representing seismic input and a damage measure representing the seismic failure of a component of a nuclear power plant. From then on, several methods to estimate fragility (expert-based, experimental, analytical, hybrid, empirical) have been developed by researchers worldwide, relying on different assumptions and restrictions to overcome prevalent intrinsic uncertainties. However, dueto the extremely high subjectiveness, lack of data and other drawbacks typical of expert-based, empirical and experimental methods, the common practice has aroused its interest in analytical and hybrid methods during the last two decades. Among analytical approaches to derivate fragility curves (probabilistic seismic demand model, elastic spectral analysis, nonlinear static analysis, linear or nonlinear time history analysis [9; 10; 11; 12; 13; 14; 15; 16; 17]), incremental dynamic analysis (IDA) is a parametric analysis method developed in 1998 and deeply discussed in 2002 [18; 19]. IDA became a worldwide method used by engineers and researchers, and it is still widespread. However, the introduction of uncertainties due to an excessively coarse description of seismic input with varying intensity is inevitable. Eventually, many authors have pointed out that a simple amplitude scaling of ground motion records is one of the main shortcomings in IDA, together with its high computational demand [20; 21; 22; 23]. In IDA, a certain number of inputs amplitude scaled to define \(IM=im\), then NL-THA is performed and \(DCR_{LS}\) so determined are used to define the distribution of \(DCR_{LS}\,IM=im\). This process is repeated by varying the scale factor to define the seismic response in a whole range of seismic intensities. According to previous research [24; 25; 26], the median response of a structure subjected to scaled ground motion records is comparable to that of a structure subjected to unscaled earthquake ground motion records. However, because a single \(IM\) is a highly simplified description of ground-motion severity, the value of \(DCR_{LS}\) from different ground-motion with \(IM=im\) may be different, and thus the relationship \(DCR_{LS}\sim\)(\(IM\)) is probabilistic. A common limitation in current databases is the lack of strong ground motion records covering high-intensity intervals at specific periods of the structure [27; 28]. Thus, an excessive scaling to fit within high-intensity intervals may occur, biasing the structural response [29], generating a false correlation between \(IM\)s and EDPs, and increasing uncertainties in the structural response.