# IMPA versus Cloud Analysis and IDA: Different Methods to Evaluate Structural Seismic Fragility
## 2 Methodology
### Performed Nonlinear Analysis  
#### 2.3.1 Cloud-Based Analysis  
The cloud-based Analysis (CA) is particularly suitable to assess structural fragility both for the simplicity of its formulation and for the low required computational effort. Conversely, it is extremely sensitive to the record selections and based on a few simplifying assumptions, such as fixed standard error of the regression [23; 31; 32; 39; 41; 49]. CA adopts a linear regression model in the logarithm scale to fit the pairs of demand to capacity ratio (\(DCR_{LS}\)) and \(IM\), where \(DCR_{LS}\) are calculated through nonlinear analysis. The regression-based probability model describes the \(DCR_{LS}\) for a given \(IM\) level and can be evaluated by Equations (3) and (4):  
\[\mathrm{E}[\ln DCR_{LS}|IM]=\ln\eta_{DCR_{LS}|IM}=\ln a+b\ln IM \tag{3}\]  
\[\sigma_{\ln\beta_{DCR_{LS}|IM}}\cong\beta_{DCR_{LS}|IM}=\sqrt{\sum_{i=1}^{N} \left(\ln DCR_{LS,i}-\ln\eta_{DCR_{LS}|IM_{i}}\right)^{2}/(N-2)} \tag{4}\]  
where \(\mathrm{E}[\ln DCR_{LS}|IM]\) is the expected value for the natural logarithm of \(DCR_{LS}\) given \(IM\), and \(\eta_{DCR_{LS}|IM}\) and \(\sigma_{\ln DCR_{LS}|IM}\) are the median and logarithmic standard deviation for \(DCR_{LS}\) given \(IM\), respectively. The constants \(\ln a\) and \(b\) are the linear least square regression coefficients. Finally, the structural fragility obtained based on the CA is (Equation (5)):  
\[\mathrm{P}(DCR_{LS}>1|IM)=\ \mathrm{P}(\ln DCR_{LS}>0|IM)=\Phi\left(\frac{\ln \eta_{DCR_{LS}|IM}}{\beta_{DCR_{LS}|IM}}\right) \tag{5}\]  
where \(\Phi\) is the standard Gaussian cumulative distribution function. In this work, two different methodologies are adopted to find the relationships of \(IM\) versus \(DCR_{LS}\) for the structure under investigation. Namely, in one case, time-history analyses are employed to evaluate the demand \(D_{\mathcal{H}}\) (demand of the \(j\)th element of \(l\)th mechanism) at each time step. Within this paper, this approach is named dynamic cloud analysis, or shortly D-CA. In the other case, the demand \(D_{\mathcal{H}}\) is computed by adopting the modal pushover analysis (MPA). This approach is referred to as MPA-CA. The modal pushover analysis [50; 51] is a nonlinear static procedure based on static analysis of the structure subjected to lateral forces distributed over the building height according to \(n\)th modal shape. [PERSON] and [PERSON] [51] showed that this procedure is accurate enough for practical application. The MPA procedure used in this work adopts the capacity spectrum method (CSM), a nonlinear static analysis procedure to assess the seismic vulnerability of buildings originally proposed by [PERSON] [52]. The procedure permits finding a correlation between earthquake ground motions and building performance [53] (ATC, 1982) comparing a response spectrum (representing structure demand) and a pushover curve (representing building capacity) by an iterative procedure. This latter aims to the definition of the performance point (PP), which represents the state of maximum inelastic displacement of a building for a given seismic event. To plot the two curves in the same chart, RS and pushover curves are transformed into an acceleration displacement response spectrum (ADRS). The whole procedure employed can be summarized in the following steps (see Figure 3):1. Determine demand: elastic response spectrum (\(T\), \(\xi\) = 5%);
2. Evaluate the capacity via pushover curves in terms of base shear \(V_{b,n}\) versus top displacement \(u_{r,n}\) for the MDOF structure subjected to lateral forces profile proportional to \(n\)th mode shape;
3. Determine maximum demand in terms of top displacement via the capacity spectrum method;  
Figure 3: Flowchart of MPA-CA and IMPA procedures. \(V_{b,n}\) and \(u_{r,n}\) are the base shear and the top displacement respectively, PP is the performance point and \(DCR_{LS}\) is the critical demand to the capacity ratio for the desired limit state (\(LS\)).