# Vulnerability Analysis of Column-Supported Reinforced Concrete Silo Structures
## 4 Vulnerability Analysis of the Column-Supported Reinforced Concrete Silos
### Seismic Vulnerability Analysis of the Column-Supported Reinforced Concrete Silo  
From the IDA results described in Section 4.2, it can be found that the maximum displacement at the top of the support column was obtained each time, and the ground  
Figure 15: Quantile curves. motion demand parameter IM exhibited a logarithmic relationship. The least-squares polynomial regression was combined to make use of Equations (4)-(7), and the probabilities of the silo exceeding LS\({}_{1}\), LS\({}_{2}\), LS\({}_{3}\), and LS\({}_{4}\) were obtained. Figure 14 shows the vulnerability curves of the column-supported reinforced concrete silo in the different limit states under seismic action. As shown in Figure 16, the probability of the silo exceeding the limit states increases with the increase in the PGA. The damage states of the column-supported reinforced concrete silo could be divided into five categories based on the four exceedance probability curves: no damage, minor damage, moderate damage, severe damage, and collapse. When PGA = 0.035 g (frequent intensity), the overstepping probability for different LSi under the three working conditions was almost equal to 0, and the silo structure was completely in the normal state. With the increase in the amount of material stored in the silo, the minor damage vulnerability curve became increasingly steeper, indicating that the silo with the storage material would enter the subsequent damage level earlier. When PGA = 0.1 g (fortification intensity), the probability of minor damage increased significantly; the specific probability values in the empty-, half-, and full-silo states were 0.61%, 78.98%, and 96.60%, respectively; however, there was little chance of moderate or higher level of damage. The storage state of the structure has a great influence on the damage probability at different stages. When PGA = 0.22 g (rare intensity), the silo was slightly damaged, and the damage in the empty silo state was only limited to the LS\({}_{1}\) level. At this point, the probability values of moderate damage exceeding LS\({}_{2}\) in the half- and full-silo states were 15.61% and 54.23%, and the probability values of severe damage over LS\({}_{3}\) were 0.31% and 24.52%, whereas the collapse damage reaching LS\({}_{4}\) occurred only in the full-silo state, with a probability of 0.12%. The collapse resistance probability ability of the silo structure is relatively high under the empty-, half-, and full-silo states. When PGA = 0.40 g, the existence of a storage material significantly increased the probability of higher-level damage to the silo. The probability values of severe damage exceeding LS\({}_{3}\) in the half- and full-silo states were 36.60% and 93.11%, and the probability of collapse over LS4 under the full-silo condition was 19.30%. In summary, the results of the vulnerability analysis in the above three working states help further conclude that the full-silo state was the most dangerous working condition. Based on the IDA curve cluster (Figure 14c) and quantile curves (Figure 15c) in the full-silo state, the PGA median values of LS\({}_{4}\) and LS\({}_{3}\) near the collapse and severe limit states were 0.53 and 0.33 g, respectively, which were higher than the PGA value (0.22 g) of the rare earthquake in the area of seven-degree fortification, which meets the seismic defense requirement of \"no damage under small earthquake, repairable under moderate earthquake, and no collapse under large earthquakes\". Figure 16: Probability curves of the vulnerability. Based on the seismic vulnerability probability curve of the column-supported reinforced concrete silo, the annual exceedance probability along with the 50-year exceeding risk probability for each limit state of the silo can be calculated [41]. Under the full-silo condition, the exceeding probabilities of the silo reaching LS1, LS2, LS3, and LS4 in 50 years were 38.74%, 3.19%, 1.04%, and 0.38%, respectively.