# Simplified Pushover Analysis for Rapid Assessment of Shear-Type Frames
## 5 Vulnerability Analysis through the Proposed Simplified Method
### Application of the Proposed Model in Vulnerability Assessment  
The potential of the proposed simplified mechanical model is demonstrated by developing, through Monte Carlo analysis, the fragility curves of a five-story building typology, whose geometry is shown in Figure 5. The proposed model allows conducting multiple capacity assessment checks with minimal computational effort, by randomly extracting the input data in each realization of a building sample. The variable data considered were concrete strength, steel strength, and flexural and shear reinforcement amounts. They were extracted from uniform distributions considering the ranges shown in Table 3. This results in a data cloud that can be statistically treated to construct fragility curves at predefined limit states.  
The cumulative probability of exceedance of a limit state \((xLS,\) with \(x=D,L,C)\) is expressed as function of the horizontal peak ground acceleration, \(a_{g}\). For each LS considered, for a given intensity measure \(a_{g,k}\), \(N_{it}\) Monte Carlo analyses are performed and a set of displacement-based capacity/demand ratios \(\rho=d_{xLS}/d_{i}\Big{(}a_{g,k}\Big{)}\) is obtained and subsequently statistically elaborated as:  
\[P_{xSL}\Big{(}a_{g,k}\Big{)}=P\Big{[}D\Big{(}a_{g,k}\Big{)}\geq d_{xLS}\Big{|} PGA=a_{g,k}\Big{]}=\frac{1}{N_{it}}\sum_{i=1}^{N_{it}}\frac{d_{xLS}}{d_{i} \Big{(}a_{g,k}\Big{)}} \tag{35}\]  
where \(P_{xLS}\) is the exceedance probability of the \(xLS\) (with \(x=D,L,C\)) for the \(k^{th}\) seismic intensity \(a_{g,k}\), with \(N_{it}=1000\) and \(k=1,\dots,52\), corresponding to \(a_{g}=0.01\)\(g,\dots,1.02\)\(g\).  
Once the probability for each \(a_{g,k}\) is obtained, LSE nonlinear regression [32; 33; 34] is carried out to determine the parameters \(\mu\) and \(V\) of the lognormal cumulative distribution function expressed in terms of \(a_{g}\):  
\[P_{xSL}\big{(}a_{g}\big{)}=\Phi\bigg{(}\frac{1}{V}ln\frac{a_{g}}{\mu}\bigg{)} \tag{36}\]  
where \(\Phi(\cdot)\) is the standard normal cumulative distribution function, \(\mu\) is the median of the capacity, and \(V\) is the coefficient of variation, assumed as equal to the log-standard deviation, which is generally accepted up to 0.6.  
The numerical simulations were carried out using the @_Risk_ software [35] on a simple spreadsheet. Figure 8 shows multiple capacity curves for the five-story building typology developed by randomly sampling the input data in Table 3 and performing 1000 iterations for each simulation. Table 4 shows the log-normal fragility curve parameters (mean \(\mu\) and coefficient of variation \(V\)) of the PGA-based capacity for three limit states (DLS, LLS, and CLS) of the five-story frame, with the resulting fragility curves at various limit states shown in Figure 9. It should be noted that these fragility curves do not include the record-to-record (R2R) variability, so their dispersion is only that pertaining to the capacity. This is a feature of this analytical approach: to show how to account for the uncertainties in the capacity by running simplified pushover analyses, as opposed to complex numerical pushover analyses, whose computational effort may be significant within a Monte Carlo approach. As for the additional uncertainty due to demand, according to FEMA 695 [36], the R2R variability can be accounted for by SRSS-adding a forfeit log-standard deviation of  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Concrete** & **Steel** & **Flexural** & **Stirrup** & **Stirrup** \\  & **Strength \(f_{ym}\)** & **Strength \(f_{ym}\)** & **Reinforcement \(\rho_{s}\)** & **Diameter \(\phi_{st}\)** & **Spacing \(S_{st}\)** \\  & **(MPa)** & **(MPa)** & **(\%)** & **(mm)** & **(mm)** \\ \hline  & 16–28 & 300–500 & 0.75–1.25 & 8–10 & 150–250 \\ \hline Uniform distribution type & Continuous & Continuous & Continuous & Discrete & Discrete \\ \hline \hline \end{tabular}
\end{table}
Table 3: Ranges adopted for material strengths, and for shear and flexural reinforcement.