# Vulnerability Analysis of Column-Supported Reinforced Concrete Silo Structures
## 4 Vulnerability Analysis of the Column-Supported Reinforced Concrete Silos  
### Fragility Analysis Methodology  
The seismic vulnerability of a structure refers to the damage probability of the structure under seismic action. In this section, the selection and treatment of the ground motion were both the most critical problems in the vulnerability analysis of the silo structure. With the initially selected 12 earthquake records in Section 3.1, the spectrum acceleration, in accordance with the same step of 0.10 g between 0.10 g and 1.20 g, was transformed into different IMs [37].  
Subsequently, the acceleration-time histories were inverted, yielding 144 records of the ground motion acceleration for a further dynamic IDA.  
In terms of its practical application, a silo can be divided into three working states, namely empty, half, and full, for the analysis considering the randomness of grain filling as well as the energy consumed for grain storage. With this, it can be expected that the dynamic elastic-plastic time-history analysis of the silo model will produce a total of 432 analytical cases.  
If the silo structure exceeds the assumed damage measure (_IM_) for a given ground motion intensity index, the assumed earthquake vulnerability equation of the structure can be written as follows:  
\[P_{\rm f}=P(D_{\rm e}>D_{\rm LSi}/IM=x),\ \text{where}\ i=1,\ 2,\ldots,\ n \tag{1}\]  
Here, \(P_{\rm f}\) denotes the probability of the structure exceeding the limit state, \(D_{\rm e}\) is the seismic demand of the structure, \(\rm LS_{i}\) expresses the i-th limit state, and IM represents the intensity measure of the seismic record.  
The maximum seismic response of a silo structure follows a log-normal distribution [38,39], which means that the relationship between the maximum displacement of the top column and IM is determined to follow a log-normal distribution, i.e., [40]. In (EDP)-N (\(\mu_{\ln x},\sigma_{\ln x}\)). Therefore, it is reasonable to assume that the probability of the structure exceeding the limit states can be defined using Equation (2).  
\[P(x>\rm LS_{i}|PGA)=1-\Phi\bigg{[}\frac{\ln(\rm LS_{i})-\mu_{\ln x}PGA}{ \sigma_{\ln x}|PGA}\bigg{]} \tag{2}\]  
\[\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp(-\frac{t^{2}}{2})dx \tag{3}\]  
In particular, the values of the deformation coefficient (\(\delta\)) for the four limit states \(\rm LS_{i}\) (\(i=1\)-\(4\)) were 1, 4, 6.5, and 14, respectively. Substituting each value of the LS into Equation (2) directly yields the probability of the structure exceeding the limit state.  
\[P_{f}=P(\text{LS}\geq\text{LS}_{1}|\text{IM}=x)=1-\Phi\bigg{(}\frac{\ln(1)-\mu_{ \text{ln},x}}{\sigma_{\text{ln},x}}\bigg{)} \tag{4}\]  
\[P_{f}=P(\text{LS}\geq\text{LS}_{2}|\text{IM}=x)=1-\Phi\bigg{(}\frac{\ln(4)-\mu_{ \text{ln},x}}{\sigma_{\text{ln},x}}\bigg{)} \tag{5}\]  
\[P_{f}=P(\text{LS}\geq\text{LS}_{3}|\text{IM}=x)=1-\Phi\bigg{(}\frac{\ln(6.5)- \mu_{\text{ln},x}}{\sigma_{\text{ln},x}}\bigg{)} \tag{6}\]  
\[P_{f}=P(\text{LS}\geq\text{LS}_{4}|\text{IM}=x)=1-\Phi\bigg{(}\frac{\ln(14)- \mu_{\text{ln},x}}{\sigma_{\text{ln},x}}\bigg{)} \tag{7}\]  
Here, \(\Phi(x)\) is the standard log-normal distribution function, \(\mu_{\text{ln},x}\) is the logarithm mean value of the top displacement of the supporting column \(\delta\), and \(\sigma_{\text{ln},x}\) is the logarithm standard deviation of \(\delta\).