# Vulnerability Analysis of Column-Supported Reinforced Concrete Silo Structures
## 3 Determination of Silo Damage Index and Seismic Demand Parameters
### Definition of Damage States
Moreover, the concrete at the bottom of the ring beam and supporting column of the silo structure began showing slight compression damage (Figure 10), and the IDA damage parameter results showed that the lateral stiffness of the reinforced concrete silo decreased further. Moderate damage: The third significant decrease in the lateral stiffness of the silo occurred at \(\delta\) = 6.5 cm, as shown in Figure 7b, where the yield zone of the structure was exhausted, and thus, the threshold value that can help distinguish between the moderate and severe damage states could be determined. In this stage, comp  
Figure 8: Contour plots of the tensile cracking failure-related damage variable \(Dt\). Figure 9: Cloud image of the damage variable Dt of the silo under tension. further on the basis of the previous stage but remained confined to the ring beam, the bin wall bottom, and the bottom of the supporting column. The dynamic response stage in Figure 11 evidently shows that the peak top displacement is approximately 6.6 cm and that the compressive strain of the concrete material reaches 0.4% [35], which corresponds to the initial spalling. Therefore, \(\delta\) = 6.5 cm was used to distinguish between the moderate and severe damage states. Severe damage: Figure 7b shows that the post-yield (plateau-like) zone of the lateral stiffness of the silo extends to \(\delta\) = 14.0 cm, which is considered to be the threshold value to distinguish between the severe and collapse damage states. Beyond this value, the lateral stiffness of the IDA curve decreases to zero when the growth rate of the median curve declines further. Based on the IDA results shown in Figure 11, there is serious concrete spalling at the bottom of the silo wall, the supporting column, and the ring beam, and there is further spreading. At this time, the most evident phenomena were the significant compression damage (indicated by the high value of the damage variable \(Dc\) = 0.968) and concrete spalling. The corresponding compressive strain of the concrete was 1.80%, which can be considered the strain limit for severe spalling of concrete silos [36]. Collapse: This state indicates that the silo structure is in a state of imminent collapse or has collapsed (Figure 12). The silo underwent a final small deterioration in its lateral  
Figure 11: Cloud images of the damage variable Dc of the silo under compression, along with its performance corresponding roughly to the limit between moderate and severe damage states. Figure 10: Cloud images of the damage variable Dc of the silo under compression. stiffness in the final stage of the structural response, gradually approaching a negative value (Figure 7b). Considering the calculation example, Wang et al. [29] conducted a shaking table test on a silo structure with a scale of 1:16. The maximum displacement response (the top displacement of the supporting column) of the structure was converted to that of the prototype on the basis of the test similarity relationship, and the displacement values of the structure under earthquake intensities of 0.1, 0.2, and 0.4 g were 3.7, 6.3, and 11.03 cm, respectively. Figure 13 shows the damage to the silo following an earthquake with an intensity of 0.40 g; serious cracks and concrete spalling can be observed at the end of the supporting column and the ring beam, resulting in the serious damage state of the silo. Figure 12: Cloud image of Dc for severe damage. Figure 13: Damage at different locations in the test site. Overall, the test results corresponded to the IDA damage parameters, and the damage degree of the silo structure following earthquakes with different intensities could be judged by the limit-state division standard listed in Table 3. Notably, the results listed in Table 3 are only the results of the damage state classification for a specific size action.